{"paper_meta":{"paper_id":"arxiv:0704.0229","title":"0704.0229","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"arXiv:0704.0229v4 [cs.CC] 22 Jan 2009\nGeometric Complexity Theory VI: the flip via\nsaturated and positive integer programming in\nrepresentation theory and algebraic geometry\nDedicated to Sri Ramakrishna\nKetan D. Mulmuley\nThe University of Chicago\nhttp://ramakrishnadas.cs.uchicago.edu\n(Technical report TR 2007-04, Comp. Sci. Dept.,\nThe University of Chicago, May, 2007)\nRevised version\nOctober 23, 2018\n\nAbstract\nThis article belongs to a series on geometric complexity theory (GCT), an\napproach to the P vs. NP and related problems through algebraic geometry\nand representation theory. The basic principle behind this approach is called\nthe flip. In essence, it reduces the negative hypothesis in complexity theory\n(the lower bound problems), such as the P vs. NP problem in characteristic\nzero, to the positive hypothesis in complexity theory (the upper bound prob-\nlems): specifically, to showing that the problems of deciding nonvanishing of\nthe fundamental structural constants in representation theory and algebraic\ngeometry, such as the well known plethysm constants [Mc, FH], belong to the\ncomplexity class P. In this article, we suggest a plan for implementing the\nflip, i.e., for showing that these decision problems belong to P. This is based\non the reduction of the preceding complexity-theoretic positive hypotheses to\nmathematical positivity hypotheses: specifically, to showing that there exist\npositive formulae–i.e. formulae with nonnegative coefficients–for the struc-\ntural constants under consideration and certain functions associated with\nthem. These turn out be intimately related to the similar positivity proper-\nties of the Kazhdan-Lusztig polynomials [KL1, KL2] and the multiplicative\nstructural constants of the canonical (global crystal) bases [Kas2, Lu2] in\nthe theory of Drinfeld-Jimbo quantum groups. The known proofs of these\npositivity properties depend on the Riemann hypothesis over finite fields\n(Weil conjectures proved in [Dl]) and the related results [BBD]. Thus the\nreduction here, in conjunction with the flip, in essence, says that the validity\nof the P ̸= NP conjecture in characteristic zero is intimately linked to the\nRiemann hypothesis over finite fields and related problems.\nThe main ingradients of this reduction are as follows.\nFirst, we formulate a general paradigm of saturated, and more strongly,\npositive integer programming, and show that it has a polynomial time al-\ngorithm, extending and building on the techniques in [DM2, GCT3, GCT5,\nGLS, KB, KTT, Ki, KT1].\nSecond, building on the work of Boutot [Bou] and Brion (cf. [Dh]), we\nshow that the stretching functions associated with the structural constants\nunder consideration are quasipolynomials, generalizing the known result that\nthe stretching function associated with the Littlewood-Richardson coefficient\nis a polynomial for type A [Der, Ki] and a quasi-polynomial for general types\n\n[BZ, Dh]. In particular, this proves Kirillov’s conjecture [Ki] for the plethysm\nconstants.\nThird, using these stretching quasi-polynomials, we formulate the math-\nematical saturation and positivity hypotheses for the plethysm and other\nstructural constants under consideration, which generalize the known sat-\nuration and conjectural positivity properties of the Littlewood-Richardson\ncoefficients [KT1, DM2, KTT]. Assuming these hypotheses, it follows that\nthe problem of deciding nonvanishing of any of these structural constants,\nmodulo a small relaxation, can be transformed in polynomial time into a\nsaturated, and more strongly, positive integer programming problem, and\nhence, can be solved in polynomial time.\nFourth, we give theoretical and experimental results in support of these\nhypotheses.\nFinally, we suggest an approach to prove these positivity hypotheses\nmotivated by the works on Kazhdan-Lusztig bases for Hecke algebras [KL1,\nKL2] and the canonical (global crystal) bases of Kashiwara and Lusztig [Lu2,\nLu4, Kas2] for representations of Drinfeld-Jimbo quantum groups [Dri, Ji].\nSteps in this direction are taken [GCT4, GCT7, GCT8].\nSpecifically, in [GCT4, GCT7] are constructed nonstandard quantum\ngroups, with compact real forms, which are generalizations of the Drinfeld-\nJimbo quantum group, and also associated nonstandard algebras, whose re-\nlationship with the nonstandard quantum groups is conjecturally similar\nto the relationship of the Hecke algebra with the Drinfeld-Jimbo quantum\ngroup. The article [GCT8] gives conjecturally correct algorithms to con-\nstruct canonical bases of the matrix coordinate rings of the nonstandard\nquantum groups and of nonstandard algebras that have conjectural posi-\ntivity properties analogous to those of the canonical (global crystal) bases,\nas per Kashiwara and Lusztig, of the coordinate ring of the Drinfeld-Jimbo\nquantum group, and the Kazhdan-Lusztig basis of the Hecke algebra. These\npositivity conjectures (hypotheses) lie at the heart of this approach. In view\nof [KL2, Lu2], their validity is intimately linked to the Riemann hypothesis\nover finite fields and the related works mentioned above.\n2\n\nContents\n1\nIntroduction\n4\n1.1\nThe decision problems . . . . . . . . . . . . . . . . . . . . . .\n7\n1.2\nDeciding nonvanishing of Littlewood-Richardson coefficients .\n12\n1.3\nBack to the general decision problems\n. . . . . . . . . . . . .\n16\n1.4\nSaturated and positive integer programming . . . . . . . . . .\n16\n1.5\nQuasi-polynomiality, positivity hypotheses, and the canonical\nmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n19\n1.6\nThe plethysm problem . . . . . . . . . . . . . . . . . . . . . .\n20\n1.7\nTowards PH1, SH, PH2,PH3 via canonial bases and canonical\nmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n27\n1.8\nBasic plan for implementing the flip\n. . . . . . . . . . . . . .\n29\n1.9\nOrganization of the paper . . . . . . . . . . . . . . . . . . . .\n30\n1.10 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n32\n2\nPreliminaries in complexity theory\n34\n2.1\nStandard complexity classes . . . . . . . . . . . . . . . . . . .\n34\n2.1.1\nExample: Littlewood-Richardson coefficients\n. . . . .\n35\n2.2\nConvex #P . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n36\n2.2.1\nLittlewood-Richardson coefficients\n. . . . . . . . . . .\n38\n2.2.2\nLittlewood-Richardson cone . . . . . . . . . . . . . . .\n38\n2.2.3\nEigenvalues of Hermitian matrices\n. . . . . . . . . . .\n39\n2.3\nSeparation oracle . . . . . . . . . . . . . . . . . . . . . . . . .\n39\n3\nSaturation and positivity\n41\n1\n\n3.1\nSaturated and positive integer programming . . . . . . . . . .\n41\n3.1.1\nA general estimate for the saturation index . . . . . .\n45\n3.1.2\nExtensions\n. . . . . . . . . . . . . . . . . . . . . . . .\n47\n3.1.3\nIs there a simpler algorithm? . . . . . . . . . . . . . .\n47\n3.2\nLittlewood-Richardson coefficients again . . . . . . . . . . . .\n47\n3.3\nThe saturation and positivity hypotheses\n. . . . . . . . . . .\n49\n3.4\nThe subgroup restriction problem . . . . . . . . . . . . . . . .\n52\n3.4.1\nExplicit polynomial homomorphism\n. . . . . . . . . .\n53\n3.4.2\nInput specification and bitlengths . . . . . . . . . . . .\n55\n3.4.3\nStretching function and quasipolynomiality . . . . . .\n57\n3.5\nThe decision problem in geometric invariant theory . . . . . .\n58\n3.5.1\nReduction from Problem 1.1.3 to Problem 1.1.4 . . . .\n59\n3.5.2\nInput specification . . . . . . . . . . . . . . . . . . . .\n59\n3.5.3\nStretching function and quasi-polynomiality . . . . . .\n60\n3.5.4\nPositivity hypotheses . . . . . . . . . . . . . . . . . . .\n61\n3.5.5\nG/P and Schubert varieties . . . . . . . . . . . . . . .\n62\n3.6\nPH3 and existence of a simpler algorithm\n. . . . . . . . . . .\n63\n3.7\nOther structural constants . . . . . . . . . . . . . . . . . . . .\n63\n4\nQuasi-polynomiality and canonical models\n65\n4.1\nQuasi-polynomiality\n. . . . . . . . . . . . . . . . . . . . . . .\n65\n4.1.1\nThe minimal positive form and modular index\n. . . .\n68\n4.1.2\nThe rings associated with a structural constant . . . .\n69\n4.2\nCanonical models . . . . . . . . . . . . . . . . . . . . . . . . .\n69\n4.2.1\nFrom PH0 to PH1,3 . . . . . . . . . . . . . . . . . . .\n70\n4.2.2\nOn PH0 in general . . . . . . . . . . . . . . . . . . . .\n72\n4.3\nNonstandard quantum group for the Kronecker and the plethysm\nproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n73\n4.4\nThe cone associated with the subgroup restriction problem\n.\n75\n4.5\nElementary proof of rationality . . . . . . . . . . . . . . . . .\n78\n5\nParallel and PSPACE algorithms\n84\n2\n\n5.1\nComplex semisimple Lie group\n. . . . . . . . . . . . . . . . .\n85\n5.2\nSymmetric group . . . . . . . . . . . . . . . . . . . . . . . . .\n89\n5.3\nGeneral linear group over a finite field . . . . . . . . . . . . .\n92\n5.3.1\nTensor product problem . . . . . . . . . . . . . . . . .\n93\n5.4\nFinite simple groups of Lie type . . . . . . . . . . . . . . . . .\n94\n6\nExperimental evidence for positivity\n95\n6.1\nLittlewood-Richardson problem . . . . . . . . . . . . . . . . .\n95\n6.2\nKronecker problem, n = 2 . . . . . . . . . . . . . . . . . . . .\n95\n6.3\nG/P and Schubert varieties . . . . . . . . . . . . . . . . . . .\n96\n6.4\nThe ring of symmetric functions\n. . . . . . . . . . . . . . . .\n97\n7\nOn verification and discovery of obstructions\n111\n7.1\nObstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111\n7.2\nDecision problems\n. . . . . . . . . . . . . . . . . . . . . . . . 113\n7.3\nVerification of obstructions\n. . . . . . . . . . . . . . . . . . . 113\n7.4\nRobust obstruction . . . . . . . . . . . . . . . . . . . . . . . . 115\n7.5\nVerification of robust obstructions\n. . . . . . . . . . . . . . . 116\n7.6\nArithemetic version of the P #P vs. NC problem in charac-\nteristric zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117\n7.6.1\nClass varieties . . . . . . . . . . . . . . . . . . . . . . . 117\n7.6.2\nObstructions\n. . . . . . . . . . . . . . . . . . . . . . . 118\n7.6.3\nRobust obstructions . . . . . . . . . . . . . . . . . . . 119\n7.6.4\nVerification of robust obstructions\n. . . . . . . . . . . 121\n7.6.5\nOn explicit construction of obstructions . . . . . . . . 122\n7.6.6\nWhy should robust obstructions exist? . . . . . . . . . 123\n7.6.7\nOn discovery of robust obstructions\n. . . . . . . . . . 124\n7.7\nArithmetic form of the P vs NP problem in characteristic zero126\n3\n\nChapter 1\nIntroduction\nThis article belongs to a series of papers, [GCT1] to [GCT11], on geomet-\nric complexity theory (GCT), which is an approach to the P vs. NP and\nrelated problems in complexity theory through algebraic geometry and rep-\nresentation theory. We assume here that the underlying field of computation\nis of characteristic zero. The usual P vs. NP problem is over a finite field.\nThe characteristic zero version is its weaker, formal implication, and philo-\nsophically, the crux.\nThe basic principle underlying GCT is called the flip [GCTflip]. The\nflip, in essence, reduces the negative hypotheses (lower bound problems) in\ncomplexity theory, such as the P ̸=?NP problem in characteristic zero, to\npositive hypotheses in complexity theory (upper bound problems): specifi-\ncally, to the problem of showing that a series of decision problems in rep-\nresentation theory and algebraic geometry belong to the complexity class\nP.\nEach of these decision problem is of the form: Given a nonnegative\nstructural constant in representation theory or geometric invariant theory,\nsuch as the well known plethysm constant, decide if it is nonzero (nonvan-\nishing), or rather, if is nonzero after a small relaxation. This flip from the\nnegative to the positive may be considered to be a nonrelativizable form of\nthe flip–from the undecidable to the decidable–that underlies the proof of\nG ̈odel’s incompleteness theorem. But the classical diagonalization technique\nin G ̈odel’s result is relativizable [BGS], and hence, not applicable to the P\nvs. NP problem. The flip, in contrast, is nonrelativizable. It is furthermore\nnonnaturalizable [GCT10]); i.e., it crosses the natural proof barrier [RR]\nthat any approach to the P vs. NP problem must cross.\nWe suggest here a plan for implementating the flip; i.e., for showing that\n4\n\nthe decision problems above belong to P. This is based on the reduction\nin this paper of the complexity-theoretic positivity hypotheses mentioned\nabove to mathematical positivity hypotheses: specifically, to showing that\nthere exist positive formulae for the structural constants under consideration\nand certain functions associated with them. We also give theoretical and\nexperimental evidence in support of the latter hypotheses.\nHere we say that a formula is positive if its coefficients are nonegative.\nThe problem finding the positive formulae as above turns out be intimately\nrelated to the analogous problem for the Kazhdan-Lusztig polynomials [KL1]\nand the multiplicative structural constants of the canonical (global crystal)\nbases [Kas2, Lu2] in the theory of Drinfeld-Jimbo quantum groups. The\nknown solution to the latter problem [KL2, Lu2] depends on the Riemann\nhypothesis over finite fields, proved in [Dl], and the related results in [BBD].\nThus the flip and the reduction here together roughly say that the valid-\nity of the P ̸= NP conjecture in characteristic zero is intimately linked\nto the Riemann hypothesis over finite fields and related problems. This is\nillustrated in Figure 1.1; the question marks there indicate unsolved prob-\nlems. It seems that substantial extension of the techniques related to the\nRiemann hypothesis over finite fields may be needed to prove the required\nmathematical positivity hypotheses here.\nWe do not have the necessary\nmathematical expertize for this task. But it is our hope that the experts in\nalgebraic geometry and representation theory will have something to say on\nthis matter.\nIt may be conjectured that the flip paradigm would also work in the\ncontext of the usual P vs. NP problem over F2 (the boolean field) or the\nfinite field Fp. But implementation of the flip over a finite field is expected\nto be much harder than in characteristic zero. That is why we focus on\ncharacteristic zero here, deferring discussion of the problems that arise over\nfinite field to [GCT11].\nNow we turn to a more detailed exposition of the main results in this\npaper and of Figure 1.1.\nAcknoledgements\nWe are grateful to the authors of [BOR] for pointing out an error in the\nsaturation hypothesis (SH) in the earlier version of this paper. It has been\ncorrected in this version with appropriate relaxation without affecting the\noverall approach of GCT (cf. Section 1.6 and also [GCT6erratum]). We\nare also grateful to Peter Littelmann for bringing the reference [Dh] to our\n5\n\nComplexity theoretic negative hypotheses (lower bound problems)\n|\n|\nThe flip\n|\n↓\nComplexity theoretic positive hypotheses (upper bound problems)\n|\n|\nThe reduction in this paper|\n|\n↓\nMathematical positivity hypotheses |\n|\n?\n|\n↓\n(?) The Riemann hypothesis over finite fields, related problems and their extensions\nFigure 1.1: Pictorial depiction of the basic plan for implementing the flip\n6\n\nattention, to H. Narayanan for suggesting the use of [KB] in the proof of\nTheorem 3.1.1 and bringing the positivity conjecture in [DM2] to our atten-\ntion, and to Madhav Nori for a helpful discussion. The experimental results\nin Chapter 6 were obtained using Latte [DHHH].\n1.1\nThe decision problems\nWe now describe the relevant decision problems in representation theory\nand algebraic geometry. The actual decision problems that arise in the flip\n(cf. the second box in Figure 1.1) are relaxed versions of these problems\ndescribed later (cf. Hypothesis 1.1.6).\nProblem 1.1.1 (Decision version of the Kronecker problem)\nGiven partitions λ, μ, π, decide nonvanishing of the Kronecker coefficient\nkπ\nλ,μ. This is the multiplicity of the irreducible representation (Specht mod-\nule) Sπ of the symmetric group Sn in the tensor product Sλ ⊗Sμ.\nEquivalently [FH], let H = GLn(C) × GLn(C) and ρ : H →G =\nGL(Cn ⊗Cn) = GLn2(C) the natural embedding. Then kπ\nλ,μ is the multi-\nplicity of the H-module Vλ(GLn(C)) ⊗Vμ(GLn(C)) in the G-module Vπ(G),\nconsidered as an H-module via the embedding ρ.\nHere Vλ(GLn(C)) denotes the irreducible representation (Weyl module)\nof GLn(C) corresponding to the partition λ; Vπ(G) is the Weyl module of\nG = GLn2(C).\nProblem 1.1.1 is a special case of the following generalized plethysm\nproblem.\nProblem 1.1.2 (Decision version of the plethysm problem)\nGiven partitions λ, μ, π, decide nonvanishing of the plethysm constant\naπ\nλ,μ. This is the multiplicity of the irreducible representation Vπ(H) of H =\nGLn(C) in the irreducible representation Vλ(G) of G = GL(Vμ), where Vμ =\nVμ(H) is an irreducible representation H. Here Vλ(G) is considered an H-\nmodule via the representation map ρ : H →G = GL(Vμ).\n(Decision version of the generalized plethysm problem)\nThe same as above, allowing H to be any connected reductive group.\nThis is a special case of the following fundamental problem of represen-\ntation theory (characteristic zero):\n7\n\nProblem 1.1.3 (Decision version of the subgroup restriction problem)\nLet G be connected reductive group, H a reductive group, possibly discon-\nnected, and ρ : H →G an explicit, polynomial homomorphism (as defined\nin Section 3.4). Here H will generally be a subgroup of G, and ρ its em-\nbedding. Let Vπ(H) be an irreducible representation of H, and Vλ(G) an\nirreducible representation of G. Here π and λ denote the classifying labels\nof the irreducible representations Vπ(H) and Vλ(G), respectively. Let mπ\nλ be\nthe multiplicity of Vπ(H) in Vλ(G), considered as an H-module via ρ.\nGiven specifications of the embedding ρ and the labels λ, π, as described\nin Section 3.4, decide nonvanishing of the multiplicity mπ\nλ.\nAll reductive groups in this paper are over C.\nThe reductive groups\nthat arise in GCT in characteristic zero are: the general and special linear\ngroups GLn(C) and SLn(C), algebraic tori, the symmetric group Sn, and\nthe groups formed from these by (semidirect) products. The reader may\nwish to focus on just these concrete cases, since all main ideas in this paper\nare illustrated therein.\nProblem 1.1.3 is, in turn, a special case of the following most general\nproblem.\nProblem 1.1.4 (Decision problem in geometric invariant theory)\nLet H be a reductive group, possibly disconnected, X a projective H-\nvariety (H-scheme), i.e., a variety with H-action.\nLet ρ denote this H-\naction. Let R = ⊕dRd be the homogeneous coordinate ring of X. Assume\nthat the singularities of spec(R) are rational.\nWe assume that X and ρ have special properties (as described in Sec-\ntion 3.5), so that, in particular, they have short specifications. Let Vπ(H)\nbe an irreducible representation of H. Let sπ\nd be the multiplicity of Vπ(H) in\nRd, considered as an H-module via the action ρ.\nGiven d, π, the specifications of X and ρ, decide nonvanishing of the\nmultiplicity sπ\nd.\nThis last problem is hopeless for general X. Indeed the usual specifi-\ncation of X, say in terms of the generators of the ideal of its appropriate\nembedding, is so large as to make this problem meaningless for a general\nX. But the instances of this decision problem that arise in GCT are for the\nfollowing very special kinds of projective H-varieties X, which, in particular,\nhave small specifications (Section 3.5):\n8\n\n1. G/P, where G is a connected, reductive group, P ⊆G its parabolic\nsubgroup, and H ⊆G a reductive subgroup with an explicit polyno-\nmial embedding. Problem 1.1.3 reduces to this special case of Prob-\nlem 1.1.4; cf. Section 3.5.\n2. Class varieties [GCT1, GCT2], which are associated with the funda-\nmental complexity classes such as P and NP. They are very special\nlike G/P, with conjecturally rational singularities [GCT10]. Each class\nvariety is specified by the complexity class and the parameters of the\nlower bound problem under consideration. Briefly, the P vs. NP prob-\nlem in characteristic zero is reduced in [GCT1, GCT2] to showing that\nthe class variety corresponding to the complexity class NP and the pa-\nrameters of the lower bound problem (such as the input size) cannot\nbe embedded in the class variety corresponding to the complexity class\nP and the same parameters. Efficient criteria for the decision prob-\nlems stated above are needed to construct explicit obstructions [GCT2]\nto such embeddings, thereby proving their nonexistence. Specifically,\nProblems 1.1.3 and 1.1.4 are the decision problems associated with\nProblems 2.5 and 2.6 in [GCT2], respectively. See Sections 7.6-7.7 for\na brief review of this story.\nFor these varieties Problem 1.1.4 turns out to be qualitatively similar to\nProblem 1.1.3 (cf. Section 3.5 and [GCT2, GCT10]). For this reason, the\nKronecker and the plethysm problems, which lie at the heart of the subgroup\nrestriction problem, can be taken as the main prototypes of the decision\nproblems that arise here.\nOne can now ask:\nQuestion 1.1.5 Do the decision problems above (Problems 1.1.1-1.1.3 and\nProblem1.1.4, when X therein is G/P or a class variety) belong to P? That\nis, can the nonvanishing of any of structural constants in these problems\nbe decided in poly(⟨x⟩) time, where x denotes the input-specification of the\nstructural constant and ⟨x⟩its bitlength?\nFor Problem 1.1.2, the input specification for the plethysm constant aπ\nλ,μ\nis given in the form of a triple x = (λ, μ, π). Here the partition λ is specified\nas a sequence of positive integers λ1 ≥λ2 ≥· · · λk > 0 (the zero parts of\nthe partition are suppressed); k is called the height or length of λ, and is\ndenoted by ht(λ). The bitlength ⟨λ⟩is defined to be the total bitlength of\nthe integers λr’s. The bitlength ⟨x⟩is defined to be ⟨λ⟩+ ⟨μ⟩+ ⟨π⟩. A\n9\n\ndetailed specification of the input specification x and its bitlength ⟨x⟩for\nthe other problems is given in Section 3.3.\nFor the reasons described in Section 1.6, Question 1.1.5 may not have\nan affirmative answer in general; i.e., these problems may not be in P in\ntheir strict form stated above. The following main conjectural complexity-\ntheoretic positivity hypothesis governing the flip says that the relaxed forms\nof these decision problems described in Section 3.3 belong to P. As we shall\nsee in Chapter 7, these relaxed forms suffice for the purposes of the flip.\nHypothesis 1.1.6 (PHflip) The relaxed forms (cf. Section 3.3) of Prob-\nlems 1.1.1, 1.1.2, 1.1.3, and the special cases of Problem 1.1.4, when X\ntherein is G/P or a class variety–which together include all decision prob-\nlems that arise in the flip–belong to the complexity class P.\nThis means nonvanishing of any of these structural constants, modulo a\nsmall relaxation (as described in Section 3.3), can be decided in poly(⟨x⟩)\ntime, where x denotes the input-specification of the structural constant and\n⟨x⟩its bitlength.\nThe phrase “modulo a small relaxation” in the relaxed form of the\nplethysm problem means the following:\n(a) Let h = dim G + htλ + htπ, where dim(G) is the dimension of the\ngroup G in Problem 1.1.2. Then there exist absolute nonnegative constants\nc and c′, independent of λ, μ and π, such that nonvanishing of the relaxed\n(stretched) plethysm constant abπ\nbλ,bμ, for any positive integral relaxation\nparameter b > chc′, can be decided in O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, ⟨b⟩)) time, where\n⟨b⟩denotes the bitlength b. The notation poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, ⟨b⟩) here means\nbounded by a polynomial of constant degree in ⟨λ⟩, ⟨μ⟩, ⟨π⟩and ⟨b⟩.\nIn\nparticular, the time is O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) if the relaxation parameter b is\nsmall; i.e. if its bitlength ⟨b⟩is O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)). (Observe that the bit\nlength of h is O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)).)\n(b) There exists a polynomial time algorithm for deciding nonvanishing of\naπ\nλ,μ, which works correctly on almost all λ, μ and π. Here polynomial time\nmeans O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) time. The meaning of “correctly on almost all”\nis specified in Hypothesis 1.6.5 below.\nA detailed specification of the relaxation, i.e., the meaning of the phrase\n“modulo a small relaxation” for the other problems is given in Section 3.3.\nThe structural constants in Problems 1.1.1-1.1.3 are of fundamental im-\nportance in representation theory. The kronecker and the plethysm con-\n10\n\nstants in Problems 1.1.1 and 1.1.2, in particular, have been studied inten-\nsively; see [FH, Mc, St4] for their significance.\nThere are many known\nformulae for these structural constants based on on the character formulae\nin representation theory. Several formulae for the characters of connected,\nreductive groups are known by now [FH], starting with the Weyl character\nformula. For the symmetric group, there is the Frobenius character formula\n[FH], for the general linear group over a finite field, Green’s formula [Mc],\nand for finite simple groups of Lie type, the character formula of Deligne-\nLusztig [DL], and Lusztig [Lu1]. (Finite simple groups of Lie type, other\nthan GLn(Fq), are not needed in GCT.)\nOne obvious method for deciding nonvanishing of the structural con-\nstants in Problems 1.1.1-1.1.4 is to compute them exactly. But all known al-\ngorithms for exact computation of the structural constants in Problems 1.1.1-\n1.1.3 take exponential time. This is expected, since this problem is #P-\ncomplete.\nIn fact, even the problem of exact computation of a Kostka\nnumber, which is a very special case of these structural constants, is #P-\ncomplete [N]. This means there is no polynomial time algorithm for com-\nputing any of them, assuming P ̸= NP.\nOf course, there are #P-complete quantities–e.g. the permanent of a\nnonnegative matrix [V]–whose nonvanishing can still be decided in polyno-\nmial time [Sc]. But the decision problems above are of a totally different\nkind and, at the surface, appear to have inherently exponential complexity.\nThis is because the dimensions of the irreducible representations that occur\nin their statements can be exponential in the ranks of the groups involved\nand the bit lengths of the classifying labels of these representations. For\nexample, the dimension of the Weyl module Vλ(GLn(C)) can be exponen-\ntial in n and the bit length of the partition λ. Furthermore, the number\nof terms in any of the preceding character formulae is also exponential. All\nthese decisions problems ask if one exponential dimensional representation\ncan occur within another exponential dimensional representation. To solve\nthem, it may seem necessary to take a detailed look into these representa-\ntions and/or the character formulae of exponential complexity. Hence, it\nseemed hard to believe that nonvanishing of these structural constants can,\nnevertheless, be decided in polynomial time (modulo a small relaxation).\nThis constituted the main philosophical obstacle in the course of GCT.\n11\n\n1.2\nDeciding nonvanishing of Littlewood-Richardson\ncoefficients\nThe first result, which indicated that this obstacle may be removable, came\nin the wake of the saturation theorem of Knutson and Tao [KT1].\nThis\nconcerns the following special case of Problem 1.1.3, with G = H × H, the\nembedding ρ : H →G being diagonal.\nProblem 1.2.1 (Littlewood-Richardson problem)\nGiven a complex semisimple, simply connected Lie group H, and its\ndominant weights α, β, λ, decide nonvanishing of a generalized Littelwood-\nRichardson coefficient cλ\nα,β. This is the multiplicity of the irreducible repre-\nsentation Vλ(H) of H in the tensor product Vα(H) ⊗Vβ(H).\nIt was shown in [GCT3, KT2, DM2] independently that nonvanishing of\nthe Littlewood-Richardson coefficient of type A can be decided in polyno-\nmial time; i.e., polynomial in the bit lengths of α, β, λ. Furthermore, the\nalgorithm in [GCT3] works in strongly polynomial time in the terminology\nof [GLS]; cf. Section 2.1. The three main ingradients in this result are:\n1. PH1: The Littlewood-Richarson rule, which goes back to 1940’s, and\nwhose most important feature is that it is positive–i.e., it involves no\nalternating signs as in character-based formulae–and its strengthening\nin [BZ], which gives a positive, polyhedral formula for the Littlewood-\nRichardson coefficient as the number of integer points in a polytope;\nthis can be the BZ-polytope [BZ] or the hive polytope [KT1].\nWe\nshall refer to this positivity property as the first positivity hypothesis\n(PH1).\n2. The polynomial and strongly polynomial time algorithms for linear\nprogramming [Kh, Ta], and\n3. SH: The saturation theorem of Knutson and Tao [KT1]. This says\nthat cλ\nα,β is nonzero if cnλ\nnα,nβ is nonzero for any n ≥1. We shall refer\nto this saturation property as the saturation hypothesis (SH).\nBrion [Z] observed that the verbatim translation of the saturation prop-\nerty in [KT1] fails to hold for the the generalized Littlewood-Richardson\ncoefficients of types B, C, D (it also fails for the Kronecker coefficients, as\nwell as the plethysm constants [Ki]). Hence, the algorithms in [GCT3, KT2,\n12\n\nDM2] do not work in types B, C and D. Fortunately, this situation can\nbe remedied. It is shown in [GCT5] that nonvanishing of the generalized\nLittewood-Richardson coefficient cλ\nα,β of arbitrary type can be decided in\n(strongly) polynomial time, assuming the positivity conjecture of De Loera\nand McAllister [DM2]. This conjectural hypothesis, based on considerable\nexperimental evidence, is as follows. Let\n ̃cλ\nα,β(n) = cnλ\nnα,nβ\n(1.1)\nbe the stretching function associated with the Littlewood-Richardson co-\nefficient cλ\nα,β. It is known to be a polynomial in type A [Der, Ki], and a\nquasi-polynomial, in general [BZ, Dh, DM2]. Recall that a fuction f(n) is\ncalled a quasi-polynomial if there exist l polynomials fj(n), 1 ≤j ≤l, such\nthat f(n) = fj(n) if n = j mod l. Here l is supposed to be the smallest such\ninteger, and is called the period of f(n). The period of ̃cλ\nα,β(n) for types\nB, C, D is either 1 or 2 [DM2]. In general, it is bounded by a fixed constant\ndepending on the types of the simple factors the Lie algebra.\nDefinition 1.2.2 We say that the quasi-polynomial f(n) is strictly posi-\ntive, if all coefficients of fj(n), for all j, are nonnegative; i.e., the nonzero\ncoefficients are positive. In general, we define the positivity index p(f) of\nf to be the smallest nonnegative integer such that f(n + p(f)) is strictly\npositive. We also say that f(n) is positive with index p(f).\nThus f(n) is strictly positive, iffits positivity index is zero.\nWith this terminology, the hypothesis mentioned above is the following.\nWe say a connected reductive group H is classical, if each simple factor of\nits Lie algebra H is of type A, B, C or D. We also say that the type of H\nor H is classical.\nHypothesis 1.2.3 (PH2): [KTT, DM2] Assume that H in Problem 1.2.1\nis classical.\nThen the Littlewood-Richardson stretching quasi-polynomial\n ̃cλ\nα,β(n) is strictly positive.\nWe shall refer to this as the second positivity hypothesis (PH2). This\nwas conjectured by King, Tollu and Toumazet [KTT] for type A, and De\nLoera and McAllister for types B, C, D. Since the stretching function above\nis a polynomial in type A, the positivity conjecture of King et al clearly\nimplies the saturation theorem of Knutson and Tao. That is, PH2 implies\nSH for type A.\n13\n\nWe can formulate an analogue of SH for a Lie algerbra of arbitrary clas-\nsical type so that PH2 implies SH for an arbitrary type. For this, we need\nto formulate the notion of a saturated quasi-polynomial, which is not con-\ntradicted by the counterexamples, mentioned above, to verbatim translation\nof the saturation property in [KT1, Ki] to the setting of quasi-polynomials.\nSpecifically, the notion of saturation in [KT1, Ki] works well if the stretching\nfunction is a polynomial, but not so if it is a quasipolynomial. Let f(n) be\na quasi-polynomial with period l. Let fj(n), 1 ≤j ≤l, be the polynomials\nsuch that f(n) = fj(n) if n = j mod l. The index of f, index(f), is defined\nto be the smallest j such that the polynomial fj(n) is not identically zero.\nIf f(n) is identically zero, we let index(f) = 0. If f(1) ̸= 0, then clearly\nindex(f) = 1.\nDefinition 1.2.4 We say that f(n) is strictly saturated if for any i: fi(n) >\n0 for every n ≥1 whenever fi(n) is not identically zero. The saturation in-\ndex s(f) of f is defined to be the smallest nonnegative integer such that\nf(n + s(f)) is strictly saturated. We also say that f(n) is saturated with\nindex s(f).\nThus f(n) is strictly saturated iffits saturation index is zero. Clearly\nthe saturation idex is bounded above by the positivity index. Thus if f(n)\nis strictly positive, it is strictly saturated. Hence, PH2 (Hypothesis 1.2.3)\nimplies:\nHypothesis 1.2.5 (SH): The Littlewood-Richardson stretching quasi-polynomial\ncλ\nα,β(n) of arbitary classical type is strictly saturated.\nThe polynomial time algorithm in [GCT5] works assuming SH as well.\nFor the Littlewood-Richardson coefficient of type A, the notion of strict\nsaturation here coincides with the notion of saturation in [KT1] since cλ\nα,β(n)\nis a polynomial in that case.\nKnutson and Tao [KT1] also conjectured\na generalized saturation property for arbitrary types. But that property,\nunlike the one defined above, is only conjectured to be sufficient, but not\nclaimed to be, or expected to be necessary. For this reason, it cannot be\nused in the complexity-theoretic applications in this paper.\nThere is another positivity conjecture for Littlewood-Richardson coeffi-\ncients that also implies the saturation theorem of Knutson and Tao. For\nthis consider the generating function\nCλ\nα,β(t) =\nX\nn≥0\n ̃cλ\nα,β(n)tn.\n(1.2)\n14\n\nIt is a rational function since ̃cλ\nα,β(n) is a quasi-polynomial [St1]. For type\nA, if ̃cλ\nα,β(n) is not identically zero, then Cλ\nα,β(t) is a rational function of\nform\nhdtd + · · · + h0\n(1 −t)d+1\n,\n(1.3)\nsince ̃cλ\nα,β(n) is a polynomial [St1]. It is conjectured in [KTT] that:\nHypothesis 1.2.6 (PH3:) The coefficients hi’s in eq.(1.3) are nonnegative\n(and h0 = 1).\nWe shall call this the third positivity hypothesis (PH3). It clearly implies SH\nfor Littlewood-Richardson coefficients of type A. To describe its analogue\nfor arbitrary classical type we need a definition.\nLet F(t) = P\nn f(n)tn be the generating function associated with the\nquasi-polynomial f(n). It is a rational function [St1].\nDefinition 1.2.7 We say that F(t) has a positive form, if, when f(n) is\nnot identically zero, it can be expressed in the form\nF(t) = hdtd + · · · + h0\nQk\ni=0(1 −tai)di ,\n(1.4)\nwhere (1) h0 = 1, and hi’s are nonnegative integers, (2) ai’s and di’s are\npositive integers, (3) P\ni di = d + 1, where d = max deg(fj(n)) is the degree\nof f(n).\nWe define the modular index of this positive form to be max{ai}.\nIf F(t) has a positive form with a0 = 1, then f(n) is strictly saturated\n(Definition 1.2.4); this easily follows from the power series expansion of the\nright hand side of eq.(1.4).\nThe analogue of Hypothesis 1.2.6 for arbitrary classical type is:\nHypothesis 1.2.8 (PH3:) The rational function Cλ\nα,β(t) has a positive\nform, with a0 = 1, of modular index bounded by a constant depending only\non the types of the simple factors of the Lie algebra of H.\nThis too implies SH for arbitrary classical type. For types B, C, D, the\nconstant above is 2. Experimental evidence for this hypothesis is given in\nSection 6.1.\n15\n\nThe analogue of the PH3, even in the more general q-setting, is known to\nhold for the generating function of the Kostant partition function of type A,\nand more generally, for a parabolic Kostant partition function; cf. Kirillov\n[Ki]. This also gives a support for the PH3 above, given a close relationship\nbetween Littlewood-Richardson coefficients and Kostant partition functions\n[FH].\n1.3\nBack to the general decision problems\nIt may be remarked that the Littlewood-Richardson problem actually never\narises in the flip. It is only used as a simplest proptotype of the actual (much\nharder) problems that arise–namely relaxed forms of Problems 1.1.1-1.1.4.\nNow we turn to these problems. The goal is to generalize the preced-\ning results and hypotheses for the Littlewood-Richardson coefficients to the\nstructural constants that arise in these problems. The problem of finding a\npositive, combinatorial formula for the plethysm constant (Problem 1.1.2),\nakin to the positive Littlewood-Richardson rule, has already been recog-\nnized as an outstanding, classical problem in representation theory [St4]–\nthe known formulae based on character theory mentioned in Section 1.1\nare not positive, because they involve alternating signs. Indeed, existence\nof such a formula is a part of the first positivity hypothesis (PH1) below\nfor the plethysm constant, and this problem is the main focus of the work\nin [GCT4, GCT7, GCT8, GCT9].\nIn view of the intensive work on the\nplethym constant in the literature, it has now become clear that the com-\nplexity of the plethysm problem (Problem 1.1.2) is far higher than that of\nthe Littlewood-Richardson problem (Problem 1.2.1). This gap in the com-\nplexity is the main source of difficulties that has to be addressed. We now\nstate the main ingradients in the plan in this paper to show that the relaxed\nforms of Problems 1.1.1, 1.1.2, 1.1.3, and 1.1.4, with X = G/P or a class\nvariety, belong to P.\n1.4\nSaturated and positive integer programming\nFirst, we formulate a general algorithmic paradigm of saturated and positive\ninteger programming that can be applied in the context of these problems.\nLet A be an m×n integer matrix, and b an integral m-vector. An integer\nprogramming problem asks if the polytope P : Ax ≤b contains an integer\n16\n\npoint. In general, it is NP-complete. We want to define its relaxed version,\nwhich will turn out to have a polynomial time algorithm.\nWe allow m, the number of constraints, to be exponential in n. Hence,\nwe cannot assume that A and b are explicitly specified. Rather, it is assumed\nthat the polytope P is specified in the form of a (polynomial-time) separation\noracle in the spirit of Gr ̈otschel, Lov ́asz and Schrijver [GLS]; cf. Section 2.3.\nGiven a point x ∈Rn, the separation oracle tells if x ∈P, and if not, gives\na hyperplane that separates x from P.\nLet fP(n) be the Ehrhart quasi-polynomial of P [St1]. By definition,\nfP (n) is the number of integer points in the dilated polytope nP.\nAn integer programming problem is called saturated, if\n1. The specification of P also contains a number sie(P), called the sat-\nuration index estimate, with the guarantee that the saturation in-\ndex s(fP) ≤sie(P); cf. Definition 1.2.4. In particular, this means\nfP(n + sie(P)) is strictly saturated.\n2. the goal of the problem is to give an efficient algorithm to decide if,\ngiven an integral relaxation parameter c > sie(P), if cP contains an\ninteger point.\nThe algorithm has to work only for relaxation parameters c > sie(P). In\nparticular, if sie(P) ≥1, the algorithm problem does not have to determine\nif P contains an integer point.\nAn integer programming problem is called positive, if\n1. the specification of P also contains a number pie(P), called the pos-\nitivity index estimate, with the guarantee that the positivity index\np(fP) ≤pie(P); cf. Definition 1.2.2. In particular, this means fP(n +\npie(P)) is strictly positive.\n2. the goal of the problem is to give an efficient algorithm to decide if,\ngiven an integral relaxation parameter c > pie(P), if cP contains an\ninteger point.\nAgain, the algorithm has to work only for relaxation parameters c > pie(P).\nSince s(fP) ≤p(fP), a positive integer programming problem is also satu-\nrated.\nThe following is the main complexity-theoretic result in this paper.\n17\n\nTheorem 1.4.1 (cf. Section 3.1)\n1. Index of the Ehrhart quasi-polynomial fP(n) of a polytope P presented\nby a separation oracle can be computed in oracle-polynomial time, and\nhence, in polynomial time, assuming that the oracle works in polyno-\nmial time.\n2. A saturated, and hence positive, integer programming problem has a\npolynomial time algorithm.\n3. Suppose the polytopes P’s that arise in a specific decision problem have\nthe following property: whenever P is nonempty, the Ehrhart quasi-\npolynomial fP(n) is “almost always” strictly saturated.\nThen there\nexists a polynomial time algorithm for deciding if P contains an integer\npoint that works correctly “almost always”.\nThe meaning of the phrase “almost always” in the context of the decision\nproblems in this paper will be specified later (cf. Theorem 3.1.1).\nIt may be remarked that the index as well as the period of the Ehrhart\nquasi-polynomial can be exponential in the bit length of the specification\nof P. In contrast to the polynomial time algorithm above to compute the\nindex, the known algorithms to compute the period (e.g. [W]) take time\nthat is exponential in the dimension of P. It may be conjectured that one\ncannot do much better: i.e., the period, unlike the index here, cannot be\ncomputed in polynomial time, in fact, even in 2o(dim(P )) time.\nThe algorithm in Theorem 1.4.1 is based on the separation-oracle-based\nlinear programming algorithm of Gr ̈otschel, Lov ́asz and Schrijver [GLS], and\na polynomial time algorithm for computing the Smith normal form [KB].\nThe paradigm of saturated integer programming is useful when one\nknows, a priori, a good estimate for the saturation index of the polytope\nunder consideration, or when the saturation index is almost always zero.\nFor example, if P is the hive polypolype for the Littlewood-Richardson co-\nefficient (type A), then sie(P) = 0, by the saturation theorem [KT1], and\npie(P) = 0, by PH2 (Hypothesis 1.2.3). For the polytopes P that would\narise in this paper, sie(P) and pie(P) would in general be nonzero, but con-\njecturally always small, and sie(P) would conjecturally be almost always\nzero.\n18\n\n1.5\nQuasi-polynomiality, positivity hypotheses, and\nthe canonical models\nThe basic goal now is to use Theorem 1.4.1 to get polynomial time algorithms\nto decide nonvanishing, modulo small relaxation, of the structural constants\nin Problems 1.1.1, 1.1.2, 1.1.3 and 1.1.4, with X = G/P or a class variety.\nThe main results in this paper which go towards this goal are as follows.\nQuasi-polynomiality\nWe associate stretching functions with the structural constants in Prob-\nlems 1.1.1-1.1.4, akin to the stretching function ̃cλ\nα,β(n) in eq.(1.1) associ-\nated with the Littlewwod-Richardson coefficient, and show that they are\nquasipolynomials; cf. Chapter 4. (But their periods need not be constants,\nas in the case of Littlewood-Richardson coefficients; in fact, they may be\nexponential in general.) In particular, this proves Kirillov’s conjecture [Ki]\nfor the plethysm constants. The proof is an extension of Brion’s remarkable\nproof (cf.\n[Dh]) of quasi-polynomiality of the stretching function associ-\nated with the Littlewood-Richardson coefficient.\nThe main ingradient in\nthe proof is Boutot’s result [Bou] that singularities of the quotient of an\naffine variety with rational singularities with respect to the action of a re-\nductive group are also rational. This is a generalization of an earlier result\nof Hochster and Roberts [Ho] in the theory of Cohen-Macauley rings.\nSaturation and positivity hypotheses\nUsing the stretching quasipolynomials above, we formulate (cf. Section 3.3)\nanalogues of the saturation and positivity hypotheses SH, PH1,PH2,PH3 in\nSection 1.2 for the structural constants in Problems 1.1.1-1.1.3 and Prob-\nlem 1.1.4, with X = G/P or a class variety. As for Littlewood-Richardson\ncoefficients, it turns out that PH2 implies SH. The hypotheses PH1 and SH\n(more strongly, PH2) together imply that the problem of deciding nonvan-\nishing of the structural constant in any of these problems, modulo a small\nrelaxation, can be transformed in polynomial time into a saturated (more\nstrongly, positive) integer programming problem, and hence, can be solved\nin polynomial time by Theorem 1.4.1.\nIn particular, this shows that all\nthe relaxed decision problems that arise in flip (cf. Hypothesis 1.1.6) have\npolynomial time algorithms, assuming these positivity hypotheses. Though\nthese algorithms are elementary, the positivity hypotheses on which their\n19\n\ncorrectness depends turn out to be nonelementary.\nThey are intimately\nlinked to the fundamental phenomena in algebraic geometry and the theory\nof quantum groups, as we shall see.\nWe also give theoretical and experimental results in support of these\nhypotheses; cf. Chapter 4-6.\nCanonical models\nThe proofs of quasi-polynomiality mentioned above also associate with each\nstructural constant under consideration a projective scheme, called the canon-\nical model, whose Hilbert function coicides with the stretching quasi-polynomial\nassociated with that structural constant, akin to the model associated by\nBrion [Dh] with the Littlewood-Richardson coefficient.\nThese canonical\nmodels play a crucial role in the approach to the posivity hypotheses sug-\ngested in Section 1.7.\n1.6\nThe plethysm problem\nWe now give precise statements of these results and hypotheses for the\nplethysm problem (Problem 1.1.2). It is the main prototype in this paper,\nwhich illustrates the basic ideas. Precise statements for the more general\nProblems 1.1.3 and 1.1.4 appear in Section 3.3.\nAs for the Littlewood-Richardson coefficients (cf.(1.1)), Kirillov [Ki] as-\nsociates with a plethysm constant aπ\nλ,μ a stretching function\n ̃aπ\nλ,μ(n) = anπ\nnλ,μ,\n(1.5)\nand a generating function\nAπ\nλ,μ(t) =\nX\nn≥0\nanπ\nnλ,μtn.\n(Note that μ is not stretched in these definitions.)\nHe conjectured that Aπ\nλ,μ(t) is a rational function. This is verified here\nin a stronger form:\nTheorem 1.6.1 (a) (Rationality) The generating function Aπ\nλ,μ(t) is ratio-\nnal.\n20\n\n(b) (Quasi-polynomiality) The stretching function ̃aπ\nλ,μ(n) is a quasi-polynomial\nfunction of n. This is equivalent to saying that all poles of Aπ\nλ,μ(t) are roots\nof unity, and the degree of the numerator of Aπ\nλ,μ(t) is strictly smaller than\nthat of the denominator.\n(c) There exist graded, normal C-algebras S = S(aπ\nλ,μ) = ⊕nSn, and T =\nT(aπ\nλ,μ) = ⊕nTn such that:\n1. The schemes spec(S) and spec(T) are normal and have rational singu-\nlarities.\n2. T = SH, the subring of H-invariants in S, where H = GLn(C) as in\nProblem 1.1.2,\n3. The quasi-polynomial ̃aπ\nλ,μ(n) is the Hilbert function of T. In other\nwords, it is the Hilbert function of the homogeneous coordinate ring of\nthe projective scheme Proj(T).\n(d) (Positivity) The rational function Aπ\nλ,μ(t) can be expressed in a positive\nform:\nAπ\nλ,μ(t) = h0 + h1t + · · · + hdtd\nQ\nj(1 −ta(j))d(j)\n,\n(1.6)\nwhere a(j)’s and d(j)’s are positive integers, P\nj d(j) = d + 1, where d is\nthe degree of the quasi-polynomial ̃aπ\nλ,μ(n), h0 = 1, and hi’s are nonnegative\nintegers.\nThe specific rings S(aπ\nλ,μ) and T(aπ\nλ,μ) constructed in the proof of The-\norem 1.6.1 are very special. We call them canonical rings associated with\nthe plethysm constant aπ\nλ,μ. We call Y (aπ\nλ,μ) = Proj(S(aπ\nλ,μ)), and Z(aπ\nλ,μ) =\nProj(T(aπ\nλ,μ)) the canonical models associated with aπ\nλ,μ. The canonical rings\nare their homogenous coordinate rings.\nIt may be remarked that the analogue of Theorem 1.6.1 (b) for Littlewood-\nRichardson coefficients has an elementary polyhedral proof. Specifically, the\nLittlewood-Richardson stretching function ̃cλ\nα,β(n) of any type is a quasi-\npolynomial since it coincides with the Ehrhart quasi-polynomial of the BZ-\npolytope [BZ]. Similarly, the analogue of Theorem 1.6.1 (d) for Littlewood-\nRichardson coefficients follows from Stanley’s positivity theorem for the\nEhrhart series of a rational polytope (which is implicit in [St3]).\nThese\npolyhderal proofs cannot be extended to the plethysm constant at this point,\n21\n\nsince no polyhedral expression for them is known so far–in fact, this is a part\nof the conjectural positivity hypothesis PH1 below.\nIn contrast, Brion’s\nproof in [Dh] of quasi-polynomiality of ̃cλ\nα,β(n) can be extended to prove\nTheorem 1.6.1 since it does not need a polyhedral interpretation for aπ\nλ,μ.\nBut Boutot’s result [Bou] that it relies on is nonelementary (because it needs\nresolution of singularities in characteristic zero [Hi], among other things).\nWe also give an elementary (nonpolyhedral proof) for Theorem 1.6.1 (a) (ra-\ntionality). But this does not extend to a proof of quasipolynomiality for all\nn, which turns out to be a far delicate problem. It is crucial in the context\nof saturated integer programming.\nTheorem 1.6.2 (Finitely generated cone)\nFor a fixed partition μ, let Tμ be the set of pairs (π, λ) such that the\nirreducible representation Vπ(H) of H = GLn(C) occurs in the irreducible\nrepresentation Vλ(G) of G = GL(Vμ(H)) with nonzero multiplicity. Then\nTμ is a finitely generated semigroup with respect to addition.\nThis is proved by an extension of Brion and Knop’s proof of the analogous\nresult for Littlewood-Richardson coefficients based on invariant theory. In\nthe case of Littlewood-Richardson coefficients, this again has an elementary\npolyhedral proof [Z].\nTheorem 1.6.3 (PSPACE)\nGiven partitions λ, μ, π, the plethysm constant aπ\nλ,μ can be computed in\npoly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) space.\nThe main observation in the proof of Theorem 1.6.3 is that the oldest\nalgorithm for computing the plethysm constant, based on the Weyl character\nformula, can be efficiently parallelized so as to work in polynomial parallel\ntime using exponentially many processors. After this, the result follows from\nthe relationship between parallel and space complexity classes. It may be\nremarked that the known algorithms for computing aπ\nλ,μ in the literature–\ne.g., the one based on Klimyk’s formula [FH]–take exponential time as well\nas space.\nTheorems 1.6.1, 1.6.2 and 1.6.3 lead to the following conjectural sat-\nuration and positivity hypotheses for the plethysm constant.\nThese are\nanalogues of PH1,PH2,PH3, SH in Section 1.2 for Littlewood-Richardson\ncoefficients.\n22\n\nHypothesis 1.6.4 (PH1)\nFor every (λ, μ, π) there exists a polytope P = P π\nλ,μ ⊆Rm such that:\n(1) The Ehrhart quasi-polynomial of P coincides with the stretching quasi-\npolynomial ̃aπ\nλ,μ(n) in Theorem 1.6.1. (This means P is given by a linear\nsystem of the form\nAx ≤b,\n(1.7)\nwhere A does not depend on λ and π and b depends only on λ and π in a\nhomogeneous, linear fashion.) In particular,\naπ\nλ,μ = φ(P),\n(1.8)\nwhere φ(P) is equal to the number of integer points in P.\n(2) The dimension m of the ambient space, and hence the dimension of P\nas well, and the bitlength of every entry in A are polynomial in the bitlength\nof μ and the heights of λ and π.\n(3) Whether a point x ∈Rm lies in P can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, ⟨x⟩)\ntime. That is, the membership problem belongs to the complexity class P.\nIf x does not lie in P, then this membership algorithm also outputs, in the\nspirit of [GLS], the specification of a hyperplane separating x from P.\nThe first statement here, in particular, would imply a positive, polyhedral\nformula for aμ\nλ,α, in the spirit of the known positive polyhedral formulae for\nthe Littlewood-Richardson coefficients in terms of the BZ- [BZ], hive [KT1]\nor other types of polytopes [Dh]. It would also imply polyhedral proofs for\nTheorem 1.6.1 (a), (b), (d), and Theorem 1.6.2. Conversely, Theorem 1.6.1\n(a), (b), (d), and Theorem 1.6.2 constitute a theoretical evidence for exis-\ntence of such a positive polyhedral formula.\nThe second statement in PH1 is justified by Theorem 1.6.3.\nSpecifi-\ncally, it should be possible to compute the number of integer points in P\nin PSPACE in view of Theorem 1.6.3. If dim(P) and m were exponential,\nthen the usual algorithms for this problem, e.g. Barvinok [Bar], cannot be\nmade to work in PSPACE. Indeed, it may be conjectured that the number\nof integer points in a general polytope P ⊆Rm can not be computed in\no(m) space.\nThe number of constraints in the hive [KT1] or the BZ-polytope [BZ]\nfor the Littlewood-Richardson coefficient cλ\nα,β is polynomial in the number\nof parts of α, β, λ. In contrast, the number of constraints defining P π\nλ,μ may\nbe exponential in the ⟨μ⟩and the number of parts of λ and π. But this is\n23\n\nnot a serious problem. As long as the faces of the polytope P have a nice\ndescription, the third statement in PH1 is a reasonable assumption. This\nhas been demonstrated in [GLS] for the well-behaved polytopes in combina-\ntorial optimization with exponentially many constraints. The situation in\nrepresentation theory should be similar, or even better. For example, the\nfacets of the hive polytope [KT1] are far nicer than the facets of a typical\npolytope in combinatorial optimization.\nIt is known that membership in a polytope is a “very easy” problem.\nFormally, if a polytope has polynomially many constraints, this problem\nbelongs to the complexity class NC ⊆P [KR], the subclass of problems\nwith efficient parallel algorithms, which is very low in the usual complexity\nhierarchy. Even if the number of constraints of P π\nλ,μ in PH1 is exponen-\ntial, the membership problem may still be conjectured to be in NC (cf.\nRemarkrnc)–which would be “very easy” compared to the decision problem\nwe began with (Problem 1.1.2). For this reason, PH1 is primarily a mathe-\nmatical positivity hypothesis as against PHflip (Hypothesis 1.1.6), and the\npositive, polyhedral formula for aπ\nλ,μ in (1.8) is its main content.\nThe remaining positivity hypotheses are purely mathematical.\nThey\ngeneralize SH,PH2 and PH3 for the Littlewood-Richardson coefficients to\nthe plethysm constants. We turn their specification next. We can begin\nby asking if the stretching quasipolynomial ̃aπ\nλ,μ(n) is strictly saturated or\npositive.\nThis need not be so.\nThe recent article [Ro] shows that strict\nsaturation need not hold for the Kronecker coefficients, as was conjectured\nin the earlier version of this paper. A similar phenomenon was also reported\nin [GCT7, GCT8], where it was observed that the structural constants of\nthe nonstandard quantum groups associated with the plethysm problem (of\nwhich the Kronecker problem is a special case) need not satisfy an analogue\nof PH2. But it was observed there that the positivity (and hence saturation)\nindices of these structural constants are small, though not always zero; eg.\nsee Figures 30,33,35 in [GCT8]. The same can be expected here. This is\nalso supported by the experimental evidence in [BOR] where too it may be\nobserved that the positivity index is small. Furthermore, in the special case\n(n = 2) of the Kronecker problem analysed in [BOR], the saturation index\nis zero for almost all Kronecker coefficients.\nThese considerations suggest:\nHypothesis 1.6.5 (SH)\n(a): The saturation index (Definition 1.2.4) of ̃aπ\nλ,μ(n) is bounded by a poly-\nnomial in the dimension of G in Problem 1.1.2 and the heights of λ and π.\n24\n\nThis means there exist absolute nonnegative constants c and c′, independent\nof n, λ, μ and π, such that the saturation index is bounded above by chc′,\nwhere h = dim G + htλ + htπ.\n(b): The quasi-polynomial ̃aπ\nλ,μ(n) is strictly saturated, i.e. the saturation\nindex is zero, for almost all λ, μ, π. Specifically, the density of the triples\n(λ, μ, π) of total bit length N with nonzero aπ\nλ,μ for which the saturation index\nis not zero is less than 1/N c′′, for any positive constant c′′, as N →∞.\nA stronger form of (a) is:\nHypothesis 1.6.6 (PH2) The positivity index (Definition 1.2.2) of the\nstretching quasi-polynomial ̃aπ\nλ,μ(n) is bounded by a polynomial in the di-\nmension of G and the heights of λ and π.\nThe following is another stronger form of SH (a). For this, we observe\nthat the positive rational form in Theorem 1.6.1 (d) is not unique. Indeed,\nthere is one such form for every h.s.o.p. (homogeneous sequence of param-\neters) of the homogenenous coordinate ring S; the a(j)’s in eq.(1.6) are the\ndegrees of these parameters.\nKirillov asked if the only possible pole of Aπ\nλ,μ is at t = 1–i.e. if aμ\nλ,α(n) is\na polynomial. This is not so (cf. Section 6.2). But it may be conjectured that\nthe structural constants a(j)’s are small. Specifcally, consider an h.s.o.p. of\nS with a (lexicographically) minimum degree sequence, and call the (unique)\npositive rational form in Theorem 1.6.1 (d) associated with such an h.s.o.p.\nminimal. The modular index χ(aπ\nλ,μ) of the plethysm constant is defined to\nbe the modular index (Definition 1.2.7) of this minimal positive form. Then:\nHypothesis 1.6.7 (PH3)\nThe function Aπ\nλ,μ(t) associated with aπ\nλ,μ has a positive rational form\nwith modular index bounded by a polynomial in the dimension of G and the\nheights of λ and π.\nMore specifically, this is so for the minimial positive rational form of\nAπ\nλ,μ(t) as above; i.e., the modular index χ(aπ\nλ,μ) is itself bounded by a poly-\nnomial in the dimension of G and the heights of λ and π.\nThis is a conjectural analogue of a stronger form of PH3 for Littlewood-\nRichardson coefficients (Hypothesis 1.2.6), which says that the modular in-\ndex of a Littlewood-Richardson coefficient, defined similarly, is one. PH3\n25\n\nhere would imply that the period of Aπ\nλ,μ(t) is smooth–i.e. has small prime\nfactors–though it may be exponential in the heights of λ, μ, π. It can be\nshown that PH3 implies SH (a) (Section 3.3).\nThe following result addresses the second arrow in Figure 1.1 in the\ncontext of the relaxed decision problem for the plethysm constant:\nTheorem 1.6.8 The complexity theoretic positivity hypothesis PHflip (Hy-\npothesis 1.1.6) for the plethysm constant is implied by the mathematical\npositivity hypotheses PH1 and SH above. Specifically, assuming PH1 and\nSH:\n(a) Nonvanishing of abπ\nbλ,bμ for any b > chc′, with c, c′, h as in SH, can be\ndecided in O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, ⟨b⟩)) time.\n(b) There is an O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)) time algorithm for deciding if aπ\nλ,μ is\nnonvanishing, which works correctly on almost all λ, μ and π; almost all\nmeans the same as in SH.\nHere (a) follows by applying Theorem 1.4.1 (2) to the polytope P π\nλ,μ in\nPH1, and letting the positivity index estimate for this polytope be chc′; (b)\nfollows from Theorem 1.4.1 (3).\nEvidence for the positivity hypotheses in special cases\nLittlewood-Richardson coefficients are special cases of (generalized) plethym\nconstants. We have already seen that PH1 holds in this case, and that there\nis considerable experimental evidence for PH2 and PH3 (Section 1.2). An-\nother crucial special case of the plethym problem is the Kronecker prob-\nlem (Problem 1.1.1)–in fact, this may be considered to be the crux of the\nplethysm problem. It follows from the results in [GCT9] that PH1 holds for\nthe Kronecker problem when n = 2; the earlier known formulae [RW, Ro]\nfor the Kronecker coefficient in this case are not positive. It can also be seen\nfrom the experimental evidence in [BOR] that the saturation and positivity\nindices of the Kronecker coefficient, for n = 2, are very small, and almost\nalways zero. We also give in Chapter 6 additional experimental evidence for\nPH2 for another basic special case of Problem 1.1.3, with H therein being\nthe symmetric group.\n26\n\n1.7\nTowards PH1, SH, PH2,PH3 via canonial bases\nand canonical models\nIn this section, we suggest an approach to prove PH1, SH, PH2 and PH3 for\nthe plethysm constant and the analogous hypotheses for the other structural\nconstants in Problems 1.1.3, and 1.1.4, with X = G/P or a class variety.\nIn the case of Littlewood-Richardson coefficients of type A, PH1 and SH\nhave purely combinatorial proofs. But it seems unrealistic to expect such\nproofs of the saturation and positivity hypotheses for the plethysm and\nother structural constants under consideration here given their substantially\nhigher complexity.\nThe approach that we suggest is motivated by the proof of PH1 for\nLittlewood-Richardson coefficients of arbitrary types based on the canonical\n(local/global crystal) bases of Kashiwara and Lusztig for representations of\nDrinfeld-Jimbo quantum groups [Dh, Kas2, Li, Lu2, Lu4]. By a Drinfeld-\nJimbo quantum group we shall mean in this paper quantization Gq of a\ncomplex, semisimple group G as in [RTF] that is dual to the Drinfeld-Jimbo\nquantized enveloping algebra [Dri]. Canonical bases for representions of a\nDrinfeld-Jimbo quantum group in type A are intimately linked [GrL] to the\nKazhdan-Lusztig basis for Hecke algebras [KL1, KL2]. A starting point for\nthe approach suggested here is:\nObservation 1.7.1 (PH0) The homogeneous coordinate rings of the canon-\nical models associated by Brion with the Littlewood-Richardson coefficients\nhave quantizations endowed with canonical bases as per Kashiwara and Lusztig.\nThis is a consequence of the work of Kashiwara [Kas3] and Lusztig [Lu3,\nLu4]; see Proposition 4.2.1 for its precise statment. This is why we call the\nmodels here canonical models.\nWe shall refer to the property above as the zeroeth positivity hypothesis\nPH0. Positivity here refers to the deep characteristic positivity property of\nthe canonical basis proved by Lusztig: namely its multiplicative and comul-\ntiplicative structure constants are nonnegative. For this reason, we say that\na canonical basis is positive. Similar positivity property is also known for\nthe Kazhdan-Lusztig basis [KL2]. The proofs of these positivity properties\nare based on the Riemann hypothesis over finite fields (Weil conjectures)\n[Dl] and the related work of Beilinson, Bernstein, Deligne [BBD].\nThe property above is called PH0 because it implies PH1 for Littlewood-\nRichardson coefficients of arbitrary types. Specifically, the latter is a formal\n27\n\nconsequence of the abstract properties of these canonical bases and is inti-\nmately related to their positivity; cf. Section 4.2.1, and [Dh, Kas2, Li, Lu4].\nThe saturation hypothesis SH in type A [KT1] is a refined property of the\npolyhedral formulae in PH1. In Section 4.2 we suggest an approach to prove\nSH, PH2 and PH3 for arbitrary types based on the properties of these canon-\nical bases. All this indicates that for the Littlewood-Richardson problem\nPH1, SH, PH2 and PH3 are intimately linked to PH0.\nThis suggests the following approach for proving PH1, SH, PH2 and PH3\nfor the plethysm and other structural constants under consideration in this\npaper (cf. Section 4.2.2):\n1. Construct quantizations of the homogeneous coordinate rings of the\ncanonical models associated with these structural constants,\n2. Show that they have canonical bases in some appropriate sense thereby\nextending PH0 to this general setting.\n3. Prove PH1, SH, PH2, and PH3 by a detailed analysis and study of\nthese canonical bases as per this extended PH0, just as in the case of\nLittlewood-Richardson coefficients.\nPictorially, this is depicted in Figure 1.2.\nQuantizations of the homogeneous coordinate rings of the canonical\nmodels associated with Littlewood-Richardson coefficients and their posi-\ntive canonical bases are constructed using the theory Drinfeld-Jimbo quan-\ntum group. In type A, it is intimately related to the theory of Hecke al-\ngebras. But, as expected, the theories of Drinfeld-Jimbo quantum groups\nand Hecke algebras do not work for the plethysm problem. What is needed\nis a quantum group and a quantized algebra that can play the same role\nin the plethysm problem that the Drinfeld-Jimbo quantum group and the\nHecke algebra play in the Littlewood-Richardson problem. These have been\nconstructed in [GCT4] for the Kronecker problem (Problem 1.1.1) and in\n[GCT7] for the generalized plethysm problem (Problem 1.1.2). We shall call\nthem nonstandard quantum groups and nonstandard quantized algebras; cf.\nSection 4.3 for their brief overview. In the special case of the Littlewood-\nRichardson problem, these specialize to the Drinfeld-Jimbo quantum group\nand the Hecke algebra, respectively. The article [GCT8] gives conjecturally\ncorrect algorithms to construct canonical bases of the matrix coordinate\nrings of the nonstandard quantum groups and of nonstandard algebras that\nhave conjectural positivity properties analogous to those of the canonical\n28\n\nConstruction of quantizations of the coordinate rings of canonical models\n|\n|\n|\n↓\nConstruction of canonical bases for these quantizations (PH0)\n|\n|\n|\n↓\nPositivity and saturation hypotheses PH1, SH |\n|\n|\n↓\nPolynomial-time algorithms for the relaxed decision problems\nFigure 1.2: Pictorial depiction of the approach\n(global crystal) bases, as per Kashiwara and Lusztig, of the coordinate ring\nof the Drinfeld-Jimbo quantum group, and the Kazhdan-Lusztig basis of the\nHecke algebra. These conjectures lie at the heart of the approach suggested\nhere, since they are crucial for the extension of PH0 (cf. Figure 1.2) to the\ngeneral setting here. Their verification seems to need substantial extension\nof the work surrounding the Riemann hypothesis over finite fields mentioned\nabove.\n1.8\nBasic plan for implementing the flip\nThe main application of the results and hypotheses in this paper in the\ncontext of the flip is the following result. As mentioned in Section 1.1, and\ndescribed in more detail in Sections 7.6-7.7, each lower bound problem, such\nas the P vs. NP problem over C, is reduced in [GCT1, GCT2] to the prob-\nlem of proving obstructions to embeddings among the class varieties that\narise in the problem. In Chapter 7 we define a robust obstruction, which is\nan obstruction that is well behaved with respect to relaxation, and whose\nvalidity (correctness) depends only on an appropriate PH1 but not SH. It is\n29\n\nconjectured that in each of the lower bound problems under consideration,\nrobust obstructions exist (Section 7.6.6). In the lower bound problems un-\nder consideration, ultimately one is only interested in proving existence of\nobstructions. So one may as well search for only robust obstructions.\nTheorem 1.8.1 (cf. Chapter 7) Consider the P vs. NP or the NC vs.\nP #P problem over C [GCT1].\nAssume that the homogeneous coordinate\nrings of the relevant class varieties [GCT1, GCT2] in this context have ra-\ntional singularities.\nAlso assume that the structural constants associated\nwith these class varieties satisfy analogous PH1 as specified in Chapter 7.\nThen:\n(a) The problem of verifying a robust obstruction in each of these problems\nbelongs to P, so also the relaxed form of the problem of verifying any ob-\nstruction (not necessarily robust).\n(b) There exists an explicit family of robust obstructions in each of these\nproblems assuming an additional hypothesis OH specified in Chapter 7; the\nmeaning of the term explicit is also given there.\n(b) The problem of deciding existence of a geometric obstruction also belongs\nto P, assuming a stronger form of PH1 specified in Chapter 7. Here geomet-\nric obstruction is a simpler type of robust obstruction, defined in Chapter 7,\nwhich is conjectured to exist in the lower bound problems under considera-\ntion.\nFor a precise statement of this theorem, see Chapter 7.\nThis theorem needs only PH1, but not SH, which is only needed to\nargue why robust obstructions should exist (Section 7.6.6), and furthermore,\nit is only needed for Problems 1.1.1-1.1.3 and not for the GIT Problem\n1.1.4. Thus PH1 is the main positivity hypothesis of GCT in the context\nproving existence of (robust) obstructions for the lower bound problems\nunder consideration.\nA basic plan for implementing the flip suggested by the considerations\nabove is summarized in Figure 1.3. It is an elaboration of Figure 1.1. Ques-\ntion marks in the figure indicate open problems.\n1.9\nOrganization of the paper\nThe rest of this paper is organized as follows.\n30\n\nNegative hypotheses in complexity theory (Lower bound problems)\n|\n|\nThe flip\n|\n↓\nPositive hypotheses in complexity theory (Upper bound problems)\n|\n|\nSaturated and positive integer programming, and\nthe quasi-polynomiality results in this paper\n|\n↓\nMathematical saturation and positivity hypotheses: PH1,SH (PH2,3)\n|\n|\nConstruction of the canonical models in this paper, and\nconstruction of the quantum groups in GCT4,7\n|\n??\n|\n↓\n(PH0): Construction of quantizations of the coordinate\nrings of the canonical models and their canonical bases\n|\n|\n|\n??\n|\n|\n↓\n(?): Problems related to the Riemann Hypothesis over finite\nfields, and their generalizations\nFigure 1.3: A basic plan for implementing the flip\n31\n\nIn Chapter 2 we describe the basic complexity theoretic notions that we\nneed in this paper and describe their significance in the context of represen-\ntation theory.\nIn Chapter 3, we give a polynomial time algorithm for saturated integer\nprogramming (Theorem 1.4.1), and give precise statements of the results\nand positivity hypotheses for Problems 1.1.3 and 1.1.4 (with X = G/P or\na class variety) mentioned in Section 1.5. These generalize the ones given\nin Section 1.6 for the plethysm constant. The framework of saturated in-\nteger programming in this paper may be applicable to many other struc-\ntural constants in representation theory and algebraic geometry, such as the\nKazhdan-Lusztig polynomials (cf. Sections 3.7).\nIn Chapter 4, we prove the basic quasi-polynomiality results–Theorem 1.6.1\nand its generalizations for Problems 1.1.3 and 1.1.4. We also define canonical\nmodels for the structural constants under consideration, and briefly describe\nthe relevance of the nonstandard quantum groups and the related results in\n[GCT4, GCT7, GCT8] in the context of quantizing the coordinate rings of\nthese canonical models and extending PH0 to them (Figure 1.2).\nIn Chapter 5, we prove the basic PSPACE results–Theorem 1.6.3 and\nits extensions for the various cases of Problem 1.1.3.\nIn Chapter 6, we give experimental evidence for the positivity hypotheses\nPH2 and PH3 in some special cases of the Problems 1.1.1-1.1.4.\nIn Chapter 7, we describe an application (Theorem 1.8.1) of the re-\nsults and positivity hypotheses in this paper to the problem of verifying or\ndiscovering a robust obstruction, i.e., a “proof of hardness” [GCT2] in the\ncontext of the P vs. NP and the permanent vs. determinant problems in\ncharacteristic zero.\n1.10\nNotation\nWe let ⟨X⟩denote the total bitlength of the specification of X. Here X can\nbe an integer, a partition, a classifying label of an irreducible representation\nof a reductive group, a polytope, and so on. The exact meaning of ⟨X⟩will\nbe clear from the context. The notation poly(n) means O(na), for some\nconstant a. The notation poly(n1, n2, . . .) similarly means bounded by a\npolynomial of a constant degree in n1, n2, . . .. Given a reductive group H,\nVλ(H) denotes the irreducible representation of H with the classifying label\nλ. The meaning depends on H. Thus if H = GLn(C), λ is a partition and\n32\n\nVλ(H) the Weyl module indexed by λ, if H = Sm, then λ is a partition of\nsize |λ| = m, and Vλ(H) the Specht module indexed by λ, and so on.\n33\n\nChapter 2\nPreliminaries in complexity\ntheory\nIn this chapter, we recall basic definitions in complexity theory, introduce\nadditional ones, and illustrate their significance in the context of represen-\ntation theory.\n2.1\nStandard complexity classes\nAs usual, P, NP and PSPACE are the classes of problems that can be\nsolved in polnomial time, nondeterministic polynomial time, and polyno-\nmial space, respectively. The class of functions that can be computed in\npolynomial time (space) is sometimes denoted by FP (resp. FPSPACE).\nBut, to keep the notation simple, we shall denote these classes by P and\nPSPACE again.\nLet SPACE(s(N)) denote the class of problems that can be solved in\nO(s(N)) space on inputs of bit length N; by convention s(N) counts only\nthe size of the work space. In other words, the size of the input, which is on\nthe read-only input tape, and the output, which is on the write-only output\ntape is not counted. Hence s(N) can be less than the size of the input or the\noutput, even logarithmic compared to these sizes. The class space(log(N))\nis denoted by LOGSPACE.\nAn algorithm is called strongly polynomial [GLS], if given an input x =\n(x1, . . . , xk),\n34\n\n1. the total number of arithmetic steps (+, ∗, −and comparisones) in the\nalgorithm is polynomial in k, the total number of input parameters,\nbut does not depend ⟨x⟩, where ⟨x⟩= P\ni⟨xi⟩denotes the bitlength of\nx.\n2. the bit length of every intermediate operand in the computation is\npolynomial in ⟨x⟩.\nClearly, a strongly polynomial algorithm is also polynomial. let strong P ⊆\nP denote the subclass of problems with strongly polynomial time algorithms.\nThe counting class associated with NP is denoted by #P. Specifically,\na function f : Nk →N, where N is the set of nonnegative integers, is in #P\nif it has a formula of the form:\nf(x) = f(x1, · · · , xk) =\nX\ny∈Nl\nχ(x, y),\n(2.1)\nwhere χ is a polynomial-time computable function that takes values 0 or 1,\nand y runs over all tuples such that ⟨y⟩= poly(⟨x⟩). The formula (2.1) is\ncalled a #P-formula. An important feature of a #P-formula in the context\nof representation theory is that it is positive; i.e., it does not contain any\nalternating signs.\nThe formula (2.1) is called a strong #P-formula, if, in addition, l is\npolynomial in k and χ is a strongly polynomial-time computable function.\nLet strong #P be the class of functions with strong #P-fomulae.\nIt is known and easy to see that\n#P ⊆PSPACE.\n(2.2)\n2.1.1\nExample: Littlewood-Richardson coefficients\nBy the Littlewood-Richardson rule [FH], the coefficient cλ\nα,β (cf.\nProb-\nlem 1.2.1) in type A is given by:\ncλ\nα,β =\nX\nT\nχ(T),\n(2.3)\nwhere T runs over all numbering of the skew shape λ/α, and χ(T) is 1 if\nT is a Littlewood-Richardson skew tableau of content β, and zero, other-\nwise. The total number of entries in T is quadratic in the total number of\n35\n\nnonzero parts in α, β, λ, and the number of arithmetic steps needed to com-\npute χ(T) is linear in this total number. Hence (2.3) is a strong #P-formula,\nand Littlewood-Richardson function c(α, β, λ) = cλ\nα,β belongs to strong #P.\nIt may be remarked that the character-based formulae for the Littlewood-\nRichardson coefficients are not #P-formulae, since they involve alternat-\ning signs. But the algorithms based on the these formulae for computing\nLittlewood-Richardson coefficients run in polynomial space. Thus, from the\nperspective of complexity theory, the main significance of the Littlewood-\nRichardson rule is that it puts the problem, which at the surface is only in\nPSPACE, in its smaller subclass (strong) #P.\nThough the Littlewood-Richardson rule is often called efficient in the\nrepresentation theory literature, it is not really so from the perspective of\ncomplexity theory. Because computation of cλ\nα,β using this formula takes\ntime that is exponential in both the total number of parts of α, β and λ, and\ntheir bit lengths. This is inevitable, since this problem is #P-complete [N].\nSpecifically, this means there is no polynomial time algorithm to compute\ncλ\nα,β, assuming P ̸= NP.\nAs remarked in earlier, nonzeroness (nonvanishing) of cλ\nα,β can be decided\nin poly(⟨α⟩, ⟨β⟩, ⟨λ⟩) time; [DM2, GCT3, KT1]. Furthermore, the algorithm\nin [GCT3] is strongly polynomial; i.e., the number of arithemtic steps in\nthis algorithm is a polynomial in the total number of parts of α, β, λ, and\ndoes not depend on the bit lengths of α, β, λ. Hence the problem of deciding\nnonvanishing of cλ\nα,β (type A) belongs to strong P.\nThe discussion above shows that the Littlewood-Richardson problem is\nakin to the problem of computing the permanent of an integer matrix with\nnonnegative coefficients. The latter is known to be #P-complete [V], but\nits nonvanishing can be decided in polynomial time, using the polynomial-\ntime algorithm for finding a perfect matching in bipartite graphs [Sc]. If\nthe positivity hypotheses in this paper hold, the situation would be similar\nfor many fundamental structural constants in representation theory and\nalgebraic geometry in a relaxed sense.\n2.2\nConvex #P\nNext we want to introduce a subclass of #P called convex #P.\nGiven a polytope P ⊆Rl, let χP denote the characteristic (membership)\nfunction of P: i.e., χP (y) = 1, if y ∈P, and zero otherwise. We say that\n36\n\nf = f(x) = f(x1, . . . , xk) has a convex #P-formula if, for every x ∈Zk,\nthere exists a convex polytope (or, more generally, a convex body) Px ⊆Rl,\nsuch that\n1. The membership function χPx(y) can be computed in poly(⟨x⟩, ⟨y⟩)\ntime, each integer point in Px has O(poly(⟨x⟩)) bitlength, and\n2.\nf(x) = φ(Px),\n(2.4)\nwhere φ(Px) denotes the number of integer points in Px. Equivalently,\nf(x) =\nX\ny∈Zl\nχPx(y),\n(2.5)\nwhere y runs over tuples in Zl of poly(⟨x⟩) bitlength, and χPx denotes\nthe membership function of the polytope Px.\nEquation (2.5) is similar to eq.(2.1). The main difference is that χ is now\nthe membership function of a convex polytope. Clearly, eq.(2.5), and hence,\neq.(2.4) is a #P-formula, when χPx can be computed in polynomial time.\nLet convex #P be the subclass of #P consisting of functions with convex\n#P-formulae.\nWe say that eq.(2.4) is a strongly convex #P-formula, if the character-\nistic function of Px is computable in strongly polynomial time. Let strongly\nconvex #P be the subclass of #P consisting of functions with strongly con-\nvex #P-formulae.\nWe do not assume in eq.(2.4) that the polytope Px is explicitly specified\nby its defining constraints. Rather, we only assume, following [GLS], that\nwe are given a computer program, called a membership oracle, which, given\ninput parameters x and y, tells whether y ∈Px in poly(⟨x⟩, ⟨y⟩) time.\nIf the number of constraints defining Px is polynomial in ⟨x⟩, then it\nis possible to specify Px by simply writing down these constraints. In this\ncase the membership question can be trivially decided in polynomial time–in\nfact, even in LOGSPACE–by verifying each constraint one at a time. This\nwould not work if Px has exponentially many constraints. In good cases,\nit is possible to answer the membership question in polynomial time even\nif Px has exponentially many facets. Many such examples in combinatorial\noptimization are given in [GLS]. One such illustrative example in repre-\nsentation theory is given in Section 2.2.2. The polytopes that would arise\n37\n\nin the plethysm and other problems of main interest in this paper are also\nexpected to be of this kind.\nWe now illustrate the notion of convex #P with a few examples in rep-\nresentation theory.\n2.2.1\nLittlewood-Richardson coefficients\nA geeneralized Littlewood-Richardson coefficient cλ\nα,β for arbitrary semisim-\nple Lie algebra (Problem 1.2.1) has a strong, convex #P-formula, because\ncλ\nα,β = φ(P λ\nα,β),\nwhere P λ\nα,β is the BZ-polytope [BZ] associated with the triple (α, β, λ).\nIt is easy to see from the description in [BZ] that the number of defin-\ning constraints of P λ\nα,β is polynomial in the total number of parts (coor-\ndinates) of α, β, λ.\nGiven α, β, λ, these constraints can be computed in\nstrongly polynomial time. Hence, the membership problem for P λ\nα,β belongs\nto LOGSPACE ⊆P. It follows that the Littlewood-Richardson function\nc(α, β, λ) = cλ\nα,β belongs to strongly convex #P.\n2.2.2\nLittlewood-Richardson cone\nWe now give a natural example of a polytope in representation theory, the\nnumber of whose defining constraints is exponential, but whose membership\nfunction can still be computed in polynomial time.\nGiven a complex, semisimple, simply connected group G, let the Littlewood-\nRichardson semigroup LR(G) be the set of all triples (α, β, λ) of dominant\nweights of G such that the irreducible module Vλ(G) appears in the tensor\nproduct Vα(G) ⊗Vβ(G) with nonzero multiplicity [Z]. Brion and Knop [El]\nhave shown that LR(G) is a finitely generated semigroup with respect to\naddition. This also follows from the polyhedral expression for Littlewood-\nRichardson coefficients in terms of BZ-polytopes [Z]. Let LRR(G) be the\npolyhedral cone generated by LR(G).\nWhen G = GLn(C), the facets of LRR(G) have an explicit description by\nthe affirmative solution to Horn’s conjecture in [Kl, KT1]. But their number\ncan be quite large (possibly exponential). Nevertheless, membership of any\nrational (α, β, λ) (not necessarily integral) in LRR(G) can be decided in\nstrongly polynomial time.\n38\n\nThis is because LRR(G) is the projection of a polytope P(G), the num-\nber of whose constraints is polynomial in the heights of α, β, λ [Z].\nIf\nφ : P(G) →LR(G) is this projection, we can choose P(G) so that for\nany integral (α, β, λ), φ−1(α, β, λ) is the BZ-polytope associated with the\ntriple (α, β, λ). To decide if (α, β, λ) ∈LR(G), we only have to decide if the\npolytope φ−1(α, β, λ) is nonempty. This can be done in strongly polynomial\ntime using Tardos’ linear programming algorithm [Ta].\n2.2.3\nEigenvalues of Hermitian matrices\nHere is another example of a polytope in representation theory with expo-\nnentially many facets, whose membership problem can still belong to P.\nFor a Hermitian matrix A, let λ(A) denote the sequence of eigenvalues\nof A arranged in a weakly decreasing order. Let HEr be the set of triple\n(α, β, λ) ∈Rr such that α = λ(A + B), β = λ(A), λ = λ(B) for some\nHermitian matrices A and B of dimension r. It is closely related to the\nLittlewood-Richardson semisgroup LRr = LR(GLr(C)): HEr ∩P 3\nr = LRr,\nwhere Pr is the semigroup of partitions of length ≤r. I. M. Gelfand asked\nfor an explicit description of HEr. Klyachko [Kl] showed that HEr is a\nconvex polyhedral cone. An explicit description of its facets is now known\nby the affirmative answer to Horn’s conjecture. But their number may be\nexponential. Hence, membership in HEr is still not easy to check using this\nexplicit description. This leads to the following complexity theoretic variant\nof Gelfand’s question:\nQuestion 2.2.1 Does the memembership problem for HEr belong to P?\nGiven that the answer is yes for the closely related LRr = LR(GLr(C))\n(Section 2.2.2), this may be so. If HEr were a projection of some polytope\nwith polynomially many facets, this would follow as in Section 2.2.2. But\nthis is not necessary. For example, Edmond’s perfect matching polytope for\nnon-bipartite graphs is not known to be a projection of any polytope with\npolynomially many constraints. Still the associated membership problem\nbelongs to P [Sc].\n2.3\nSeparation oracle\nSuppose P ⊆Rl is a convex polytope whose membership function χP is\npolynomial time computable. If χP(y) = 0 for some y ∈Rr, it is natural to\n39\n\nask, in the spirit of [GLS], for a “proof” of nonmembership in the form of a\nhyperplane that separates y from P.\nIn this paper, we assume that all polytopes are specified by the separation\noracle. This is a computer program, which given y, tells if y ∈P, and if\ny ̸∈P, returns such a separting hyperplane as a proof of nonmembership. We\nassume that the hyperplane is given in the form l = 0, where a linear function\nl such that P is contained in the half space l ≥0, but l(y) < 0. Furthermore.\nwe assume that P is a well-described polyhedron in the sense of [GLS]. This\nmeans P is specified in the form of a triple (χP , n, φ), where P ⊆Rn, χP\nis a program for computing the membership function given y ∈Rn, and\nthere exists a system of inequalities with rational coefficients having P as\nits solution set such that the encoding bit length of each inequality is at\nmost φ. We define the encoding length ⟨P⟩of P as n + φ. We also assume\nthat the separation oracle works in O(poly(⟨P⟩, ⟨y⟩) time.\nFor example, the polynomial time algorithm for the membership function\nof the Littlewood-Richardson cone (cf. Section 2.2.2) can be easily modified\nto return a separating hyperplane as a proof of nonmembership.\nIn what follows, we shall assume, as a part of the definition of a convex\n#P-formula, that Px in (2.4) is a well-described polyhedron specified by\na separation oracle that works in polynomial time with ⟨Px⟩= poly(⟨x⟩).\nThese additional requirements are needed for the saturated integer program-\nming algorithm in Chapter 3.\n40\n\nChapter 3\nSaturation and positivity\nIn this chapter we describe (Section 3.1) a polynomial time algorithm for\nsaturated and positive integer programming (Theorem 1.4.1). In Section 3.3\nwe state the main results and positivity hypotheses for the relaxed forms of\nProblem 1.1.3 and Problem 1.1.4, with X = G/P or a class variety therein.\nTogether they say that these relaxed decision problems can be efficiently\ntransformed into saturated (more strongly, positive) integer programming\nproblems, and hence can be solved in polynomial time.\n3.1\nSaturated and positive integer programming\nWe begin by proving Theorem 1.4.1.\nLet P ⊆Rn be a polytope given by a separation oracle (Section 2.3).\nLet ⟨P⟩be the encoding length of P as defined in Section 2.3. An oracle-\npolynomial time algorithm [GLS] is an algorithm whose running time is\nO(poly(⟨P⟩)), where each call to the separation oracle is computed as one\nstep.\nThus if the separation oracle works in polynomial time, then such\nan algorithm works in polynomial time in the usual sense.\nLet φ(P) be\nthe number of integer points in P.\nLet fP(n) = φ(nP) be the Ehrhart\nquasi-polynomial [St1] of P. Let l(P) be the least period of fP(n), if P\nis nonempty.\nLet fi,P(n), 1 ≤i ≤l(P), be the polynomials such that\nfP (n) = fi,P(n) if n = i modulo l(P). Let FP (t) = P\nn≥0 fP(n)tn denote\nthe Ehrhart series of P. It is a rational function.\nTheorem 3.1.1 (a) The index of fP (n), index(fP), can be computed in\noracle-polynomial time, and hence, in polynomial time, assuming that the\n41\n\noracle works in polynomial time. Furthermore, if index(fP) ̸= 0 (i.e. if P\nis nonempty), then fi,P(n) is not an identically zero polynomial for every i\ndivisible by index(fP).\n(b) The saturated, and hence, positive integer programming problem, as de-\nfined in Section 1.4, can be solved in oracle-polynomial time.\nHere it is\nassumed that the specification of P also contains the saturation index esti-\nmate sie(P), or the positivity index estimate pie(P), and that the bitlength\nof this estimate is O(poly(⟨P⟩)). Given a relaxation parameter c > sie(P)\n(or pie(P)), the problem is to determine if cP contains an integer point in\nO(poly(⟨P⟩, ⟨c⟩)) time.\n(c) Suppose {Px} is a family of polytopes, indexed by some parameter x,\nwith the following property: wherenver Px is nonempty, the Ehrhart quasi-\npolynomial fPx(n) is “almost always” strictly saturated.\nAlmost always\nmeans, the density of x’s of bitlength ≤N, with nonempty Px for which\nfPx(n) is not strictly saturated is less than 1/N c′′, for any positive c′′, as\nN →0. We also assume that Px is given by a separation oracle that works in\nO(poly(⟨x⟩)) time, where ⟨x⟩is the bitlength of x, and ⟨Px⟩= O(poly(⟨x⟩)).\nThen there exists a O(poly(⟨x⟩)) time algorithm for deciding if Px con-\ntains an integer point that works correctly “almost always”; i.e., on almost\nall x.\nProof:\n(a):\nNonemptyness of P can be decided in oracle-polynomial time using the\nalgorithm of Gr ̈otschel, Lov ́asz and Schrijver [GLS] (cf.\nTheorem 6.4.1\ntherein). An extension of this algorithm, furthermore, yields a specifica-\ntion of the affine space span(P) containing P if P is nonempty (cf. Theo-\nrems 6.4.9, and 6.5.5 in [GLS]). Specifically, it outputs an integral matrix\nC and an integral vector d such that span(P) is defined by Cx = d. This\nfinal specification is exact, even though the first part of the algorithm in\n[GLS] uses the ellipsoid method. Indeed, the use of simultneous diophan-\ntine approximation based on basis reduction in lattices is precisely to ensure\nthis exactness in the final answer. This is crucial for the next step of our\nalgorithm.\nIf P is empty, index(fP) = 0. So assume that it is nonempty. Let ̄C be\nthe Smith normal form of C; i.e., ̄C = ACB for some unimodular matrices\nA and B, where the leftmost principal submatrix of ̄C is a diagonal, integral\nmatrix, and all other columns are zero.\n42\n\nThe matrices ̄C, A and B can be computed in polynomial time using\nthe algorithm in [KB]. After a unimodular change of coordinates, by letting\nz = B−1x, span(P) is specified by the linear system ̄Cz = ̄d = Ad. The\nequations in this system are of the form:\n ̄cizi = ̄di,\n(3.1)\ni ≤codim(P), for some integers ̄ci and ̄di. By removing common factors if\nnecessary, we can assume that ̄ci and ̄di are relatively prime for each i. Let\n ̃c be the l.c.m. of ̄ci’s.\nThe statement (a) follows from:\nClaim 3.1.2 index(fP ) = ̃c and fi,P(n) is not an identically zero polyno-\nmial for every i divisible by ̃c.\nProof of the claim: Indeed, nP = {nz | z ∈P} contains no integer point\nunless ̃c divides n.\nHence, it is easy to see that FP (t) = F ̄P (t ̃c), where\nF ̄P (x) is the Ehrhart series of the dilated polytope ̄P = ̃cP. By eq.(3.1),\nthe equations defining ̄P are:\nzi = ̄di( ̃c/ ̄ci),\n(3.2)\nClearly, ̃c divides the least period l(P) of fP, and l( ̄P) = l(P)/ ̃c is the period\nof the Ehrhart quasipolynomial f ̄P(n). It suffices to show that the index of\nf ̄P (n) is one and that fj, ̄P(n) is not an identically zero polynomial for every\n1 ≤j ≤l( ̄P). This is equivalent to showing that ̄P contains a point z with\nwith zi = ai/b, for some integers ai’s and b such that b = j modulo l( ̄P).\nLet us call such a point j-admissible. Because of the form of the equations\n(3.2) defining span( ̄P), we can assume, without loss of generality, that ̄P is\nfull dimensional. This means the system (3.2) is empty. Then this follows\nfrom denseness of the set of j-admissible points. This proves the claim, and\nhence (a).\n(b): Let s = sie(P) be the given saturation index estimate. This means\nfP (n + s) is strictly saturated. This in conjunction with (a) implies that,\ngiven a relaxation parameter c > s, cP contains an integer point, iffc\nis divisible by index(fP ) (by letting n = c −s). This can be checked in\nO(poly(⟨P⟩, ⟨c⟩)) time since index(fP ) can be computed in polynomial time\nby (a).\n(c) The algorithm computes index(fPx) and says “Probably Yes”if the index\nis one, and “No” otherwise. Since the saturation index of fPx(n) is zero\n43\n\nalmost always, by the argument in (b) with s = 0 and c = 1, “Probably\nYes” really means “Yes” almost always. Q.E.D.\nThe algorithm in (c) has one drawback.\nIf the answer is “Probably\nYes”, we have no easy way of checking if Px really contains an integer point.\nIdeally, we would like an algorithm that says “Yes”, with an integer point\nin Px as a proof certificate, or “No”, or “Unsure”, and the density of x’s on\nwhich it says “Unsure” should be very small. This problem can be overcome\nif the family {Px} has the following stronger property, akin to the family of\nhive polytopes [KT1]: there is a linear function lx such that, for almost all x,\nif {Px} is nonempty, then the lx-optimum of Px is integral (this is stronger\nthan saying that fPx(n) is strictly saturated). In this case, the algorithm in\n(c) can be extended to yield the integral lx-optimum as a proof certificate. If\nthe lx-optimum is not integeral, the algorithm says “Unsure”. PH1 and SH\n(Section 1.6) for the plethysm (and more generally, the subgroup restriction)\nproblem may be strengthened by stipulating that the polytopes therein have\nthis property. But this is not needed in this paper.\nWe note down one corollary of the proof of Theorem 3.1.1 (this should\nbe well known):\nProposition 3.1.3 The rational function FP (t) = F ̄P (t ̃c), where F ̄P (x) is\nthe Ehrhart series of the dilated polytope ̄P = ̃cP, and ̃c is the index of\nfP (n).\nIf P is explicitly specified in the form a linear system\nAx ≤b,\n(3.3)\nwhere A is an m × n matrix, b an m vector and m = poly(n), then the\nfollowing stronger version of Theorem 3.1.1 holds. Let ⟨A⟩and ⟨A, b⟩denote\nthe bitlength of the specification of A and of the linear system (3.3).\nTheorem 3.1.4 Suppose P is specified in terms of an explicit linear system\n(3.3). Then the index of the Erhart quasi-polynomial fP(n) can be computed\nin poly(⟨A, b⟩) time, using poly(⟨A⟩) arithmetic operations.\nThus, saturated, and hence, positive integer programming problem spec-\nified in the form (3.3) can be solved in in poly(⟨A, b, c⟩) time, where c is the\nrelaxation parameter, using poly(⟨A⟩) arithmetic operations.\nProof: This is proved exactly as Theorem 3.1.1, but with Tardos’ strongly\npolynomial time algorithm for combinatorial linear programming [Ta] used\nin place of the algorithm in [GLS]. Q.E.D.\n44\n\n3.1.1\nA general estimate for the saturation index\nNow we give a general estimate for the saturation index of any polytope P\nwith a specification of the form\nAx ≤b,\n(3.4)\nwhere A is an m × n matrix, m possibly exponential. Let ∥P∥= n + ψ,\nwhere ψ is the maximum bitlength of any entry of A. Trivially, ∥P∥≤⟨P⟩.\nWe do not assume that we know the specification (3.4) of P explicitly. We\nonly assume that it exists, and that we are told ∥P∥. Then:\nTheorem 3.1.5 The saturation index of P is O(2poly(∥P ∥)).\nThus the\nbitlength of the saturation index is O(poly(∥P∥)).\nConjecturally, this also holds for the positivity index. This estimate is\nvery conservative, but useful when no better estimate is available.\nProof: There exists a triangulation of P into simplices such that every vertex\nof any simplex is also a vertex of P. Then\nfP(n) =\nX\n∆\nf∆(n),\nwhere ∆ranges over all open simplices in this triangulation; a zero-dimensional\nopen simplex is a vertex. The saturation index of fP(n) is clearly bounded\nby the maximum of the saturation indices of f∆(n).\nHence, we can assume, without loss of generality, that P is an open sim-\nplex. Let v0, . . . , vn be its vertices. Then, by Ehrhart’s result (cf. Theorem\n1.3 in [st5]),\nFP (t) =\nP\ni hiti\nQn\nj=0(1 −taj),\n(3.5)\nwhere h0 = 1, hi’s are nonnegative, and aj is the least positive integer\nsuch that ajvj is integral. By Cramer’s rule, the bit length of each aj is\npoly(∥P∥).\nWithout loss of generality, we can also assume that aj’s are\nrelatively prime. Otherwise, the estimate on the saturation index below has\nto be multiplied by the g.c.d. of aj’s. Then the result follows by applying\nthe following lemma to FP (t), since ⟨aj⟩= O(poly(∥P∥)). Q.E.D.\n45\n\nLemma 3.1.6 Let f(n) be a quasipolynomial whose generating function\nF(t) has a positive form\nF(t) =\nP\ni hiti\nQn\nj=0(1 −taj),\n(3.6)\nwhere h0 = 1, hi’s are nonnegative, and aj’s are positive and relatively\nprime.\nLet a = max{aj}.\nThen the saturation index s(f) of f(n) is\nO(poly(a, n)).\nProof: Let g(n) be the quasi-polynomial whose generating function G(t) =\nP g(n)tn is 1/Qn\nj=0(1 −taj). It is known that this is the Ehrhart quasipoly-\nnomial of the polytope N(a0, . . . , an) defined by the linear system\nX\najxj = 1, xj > 0.\nThe saturation index s(g) of g(n) is bounded by the Frobenius number\nassociated with the set of integers {aj}–this is the largest positive integer m\nsuch that the diophantine equation\nX\nj\najxj = m\nhas no positive integeral solution (x0, . . . , xn). It is known (e.g. [BDR]) that\nthe Frobenius number is bounded by\nX\nj\naj +\np\na0a1a2(a0 + a1 + a2) = O(poly(a)),\nassumming that a0 ≤a1 . . .. Hence, s(g) = O(poly(a)).\nSince f(n) is a quasi-polynomial, the degree of the numerator of F(t) is\nless than the degree of the denominator. Thus the maximum value of i that\noccurs in (3.6) is an.\nLet gi(n), i ≤an, be the quasi-polynomial whose generating function is\nti/Qn\nj=0(1 −taj). Then\ns(gi) ≤i + s(g) = O(poly(a, n)).\nSince, hi’s in (3.6) are nonnegative, s(f) = max s(gi). The result follows.\nQ.E.D.\n46\n\n3.1.2\nExtensions\nWe now mention a few straightforard extensions of Theorem 3.1.1.\nFirst, it is not necessary that P be a closed polytope. We can allow\nit to be half-closed.\nSpecifically, it can be a solution set of a system of\ninequalitites of the form:\nA1x ≤b1\nand\nA2x < b2,\n(3.7)\nwhere we have allowed strict inequalities. The function FP (n) = φ(nP), the\nnumber of integer points in nP, is again a quasi-polynomial. Hence, the\nnotions of saturation and positivity can be generalized to this setting in a\nnatural way.\nSecond, the algorithm in Theorem 3.1.1 (b) only needs a nonnegative\nnumber s(P) such that, for any positive integer c > s(P):\nSaturation guarantee: If the affine span of cP, contains an integer point,\nthen cP is guaranteed to contain an integer point.\nIf s(P) = sie(P), then this guarantee holds, as can be seen from the\nproof of Theorem 3.1.1.\n3.1.3\nIs there a simpler algorithm?\nThough the algorithm for saturated integer programming in Theorem 3.1.1\nis conceptually very simple, in reality it is quite intricate, because the work\nof Gr ̈otschel, Lov ́asz and Schrijver [GLS] needs a delicate extension of the el-\nlipsoid algorithm [Kh] and the polynomial-time algorithm for basis reduction\nin lattices due to Lenstra, Lenstra and Lov ́asz [LLL]. As has been empha-\nsized in [GLS], such a polynomial-time algorithm should only be taken as a\nproof of existence of an efficient algorithm for the problem under consider-\nation. It may be conjectured that for the problems under consideration in\nthis paper such simple, combinatorial algorithms exist. But for the design\nof such algorithms, saturation alone does not suffice. The stronger property\n(PH3), and more, is necessary. We shall address this issue in Section 3.6.\n3.2\nLittlewood-Richardson coefficients again\nTheorem 3.1.4 applied to the BZ-polytope [BZ], with saturation index esti-\nmate equal to zero, specializes to the following in the setting of the Littlewood-\n47\n\nRichardson problem (Problem 1.2.1):\nTheorem 3.2.1 [GCT5] Assuming SH (Hypothesis 1.2.5), nonvanishing of\ncλ\nα,β, given α, β, λ, can be decided in strongly polynomial time (Section 2.1)\nfor any semisimple classical Lie algebra G.\nIt is assumed here that α, β, λ are specified by their coordinates in the\nbasis of fundamental weights.\nFor type A, this reduces to the result in\n[GCT3], which holds unconditionly.\nThe saturation conjecture for type A arose [Z] in the context of Horn’s\nconjecture and the related result of Klyachko [Kl]. We now turn to implica-\ntions of Theorem 3.2.1 in this context.\nGiven a complex, semisimple, simply connected, classical group G, let\nLR(G) be the Littlewood-Richardson semigroup as in Section 2.2.2. The\nfollowing is a natural generalization of the problem raised by Zelevinsky [Z]\nto this general setting:\nProblem 3.2.2 Give an efficient description of LR(G).\nZelevinsky asks for a mathematically explicit description. This is a com-\nputer scientist’s variant of his problem.\nLet LRR(G) be the polyhedral convex cone generated by LR(G). For\nG = GLn(C), by the saturation theorem, a triple (α, β, λ) of dominant\nweights belongs to LR(G) iffit belongs to LRR(G).\nAssuming SH (Hy-\npothesis 1.2.5), Theorem 3.2.1 provides the following efficient description\nfor LR(G) in general. Recall that the period of the Littlewood-Richardson\nstretching polynomial ̃cλ\nα,β(n) divides a fixed constant d(G), which only de-\npends on the types of simple factors of G [DM2, GCT5]. Let αi’s denote\nthe coordinates of α in the basis of fundamental weights.\nCorollary 3.2.3 (a) Assuming SH, whether a given (α, β, λ) belongs to\nLR(G) can be determined in strongly polynomial time.\n(b) There exists a decomposition of LRR(G) into a set of polyhedral cones,\nwhich form a cell complex C(G), and, for each chamber C in this complex,\na set M(C) of O(rank(G)2) modular equations, each of the form\nX\ni\naiαi +\nX\ni\nbiβi +\nX\ni\nciλi = 0\n(mod d),\nfor some d dividing d(G), such that\n48\n\n1. SH (Hypothesis 1.2.5) is equivalent to saying that: (α, β, λ) ∈LR(G)\niff(α, β, λ) ∈LRR(G) and (α, β, λ) satisfies the modular equations in\nthe set M(Cα,β,λ) associated with the cone Cα,β,λ containing α, β, λ.\n2. Given (α, β, λ), whether (α, β, λ) ∈LRR(G) can be determined in\nstrongly polynomial time (cf. Section 1.2.5).\n3. If so, the cone Cα,β,λ and the associated set M(Cλ\nα,β) of modular equa-\ntions can also be determined in strongly polynomial time. After this,\nwhether (α, β, λ) satisfies the equations in M(Cλ\nα,β) can be trivially\ndetermined in strongly polynomial time.\nProof: (a) is a consequence of Theorem 3.2.1. (b) follows from a careful\nanalysis of the algorithm therein; see the proof of a more general result\n(Theorem 4.4.2) later. Q.E.D.\nWe call the labelled cell complex C(G), in which each cell C ∈C(G)\nis labelled with the set of modular equations M(C), the modular complex,\nassociated with LRR(G).\nWhen G = SLn(C), the modular complex is\ntrivial: it just consists of the whole cone LRR(G) with only one obvious\nmodular equation attached to it. But, for general G, the modular complex\nand the map C →M(C) are nontrivial.\nWe do not know their explicit\ndescription.\nCorollary 3.2.3 says that, given x = (α, β, λ), whether x ∈\nLRR(G), and whether the relevant modular equations are satisfied can be\nquickely verified on a computer, though the modular equations cannot be\neasily determined and verified by hand, as in type A.\nThis is the main\ndifference between type A and general types.\nThis naturally leads to:\nQuestion 3.2.4 Is there a mathematically explicit description of the mod-\nular complex C(G) for a general G?\n3.3\nThe saturation and positivity hypotheses\nNow let f(x), x ∈Nk, be a counting function associated with a structural\nconstant in representation theory or algebraic geometry.\nHere x denotes\nthe sequence of parameters associated with the constant. Let ⟨x⟩denote\nthe bitlength of x. Let ∥x∥and rank(x) denote its combinatorial size and\ncombinatorial rank–these measure complexity of the nonstretchable part in\n49\n\nthe specification of x and will be specified later for the f’s of interest in this\npaper.\nFor example, in the Littlewood-Richardson problem, x is the triple (α, β, λ),\nf(x) = f(α, β, λ) = cλ\nα,β, ⟨x⟩is the total bitlength of the coordinates of\nα, β, λ, ∥x∥is the total number of coordinates of α, β and λ, and rank(x) =\n∥x∥. The number of coordinates does not change during stretching, and\nhence, constitute the nonstretchable part of the input specification here.\nAssume that f(x) is nonnegative for all x ∈Nk, Then we can successively\nask the following questions:\n1. Does f ∈PSPACE? That is, can f(x) be computed in poly(⟨x⟩)\nspace?\n2. Does f ∈#P? (cf. Section 2.1)\n3. Does f ∈convex#P? (cf. Section 2.2)\n4. Can a stretching function ̃f(x, n) be associated with f(x) intrinsically\nso that ̃f(x, n) is quasi-polynomial?\n5. (PH1?): Is there a polytope Px, for every x, with ⟨Px⟩= O(poly(⟨x⟩))\nand ∥Px∥= O(poly(∥x∥)), such that ̃f(x, n) = fPx(n)?\n6. Are there good analogues of SH and/or PH2, PH3 for ̃f(x, n)?\nIf\nso, nonvanishing of f(x), modulo small relaxation, can be decided in\nO(poly(⟨Px⟩)) time by Theorem 3.1.1.\nIn the rest of this paper, we study these questions when f = f(x) is a\nnonnegative function associated with a structural constant in any of the deci-\nsion problems in Section 1.1. Exact specifications of x, ⟨x⟩, ∥x∥, rank(x), f(x),\nand ̃f(x, n) for these decision problems are given in Sections 3.4-3.5. It is\nshown in Chapter 5 that f(x) ∈PSPACE for Problem 1.1.2 and the special\ncases of Problem 1.1.3 that arise in the flip. This may be conjectured to be\nso for the f’s in Problem 1.1.4, with X therein a class variety; cf [GCT10]\nfor its justification. Quasipolynomiality of ̃f(x, n) is addressed in Chapter 4.\nThe hypotheses PH1, SH, PH2, and PH3 in these cases have the following\nunified form.\nHypothesis 3.3.1 (PH1) Let f = f(x) be the function associated with a\nstructural constant in\n50\n\n1. Problem 1.1.1, or\n2. 1.1.2, or\n3. Problem 1.1.3, or\n4. Problem 1.1.4, with X being a class variety therein.\nThen the function f(x) has a convex #P-formula (cf. (2.4))\nf(x) = φ(Px),\nsuch that:\n1. for every fixed x, the Ehrhart quasi-polynomial fPx(n) of Px coincides\nwith ̃f(x, n).\n2. ⟨P⟩= O(poly(P)) and ∥P∥= O(poly(∥x∥)).\nHypothesis 3.3.2 (SH)\n(a) Suppose f(x) is a structural constant as in PH1 above. Then for every x,\nthe saturation index s( ̃f) of ̃f(x, n) is O(poly(rank(x))). This means there\nexist absolute nonnegative constants c, c′ such that s( ̃f) ≤c(rank(x))c′.\n(b) For f(x) in Problems 1.1.1-1.1.3, the saturation index of ̃f(x, n) is zero–\ni.e., ̃f(x, n) is strictly saturated–for almost all x. This means the density of\nx, with ⟨x⟩≤N and f(x) nonzero, for which the saturation index s( ̃f) is\nnonzero is ≤1/N c′′, for any positive costant c′′, as N →∞.\nMore strongly than (a),\nHypothesis 3.3.3 (PH2) For f(x) as in PH1, the positivity index of ̃f(x, n)\nis O(poly(rank(x))).\nHypothesis 3.3.4 (PH3) For f(x) as in PH1, the generating function\nF(x, t) = P\nn ̃f(x, n)tn has a positive rational form of modular index O(poly(rank(x))).\nMore specifically, the modular index of ̃f(x, n), as defined in Section 4.1.1\nfor f’s that arise in this paper, is O(poly(rank(x))).\nPH3 implies SH (a); this follows from Lemma 3.1.6.\nThe following conservative bound follows from Theorem 3.1.5.\n51\n\nTheorem 3.3.5 (Weak SH)\nAssuming PH1 (Hypothesis 3.3.1), the saturation index of ̃f(x, n) is\nbounded by 2O(poly(∥x∥)); hence its bitlength is bounded by O(poly(∥x∥)).\nThe following result addresses the relaxed forms of the decision problems\nfor the structural constants under consideration (cf. Section 1.1).\nTheorem 3.3.6 Suppose f(x) is a structural constant as in PH1 above.\nThen PH1 (Hypothesis 3.3.1) and SH (Hypothesis 3.3.2) imply Hypothe-\nsis 1.1.6 (PHflip) in this case. Specifically:\n(a) For f(x) in Problems 1.1.1-1.1.4, nonvanishing of ̃f(x, a), for a given x\nand a relaxation parameter a > c(rank(x))c′, with c, c′ as in Hypothesis 3.3.2,\ncan be decided in poly(⟨x⟩, ⟨a⟩) time.\n(b) For f(x) as in Problems 1.1.1-1.1.3, there is a poly(⟨x⟩) time algorithm\nfor deciding nonvanishing of f(x) that works correctly on almost all x.\nThis follows from Theorem 3.1.1.\nThe following sections give precise descriptions of x, ⟨x⟩, ∥x∥, rank(x)\nand ̃f(x, n) for the structural constants under consideration.\n3.4\nThe subgroup restriction problem\nIn this section we consider the subgroup restriction problem (Problem 1.1.3).\nThe Kronecker and the plethysm problems (Problems 1.1.1, 1.1.2) are its\nspecial cases.\nLet G, H, ρ, λ, π, mπ\nλ be as in Problem 1.1.3. We shall define below an ex-\nplicit polynomial homomorphism ρ : H →G, as needed in the statement of\nProblem 1.1.3, and also the precise specifications [H], [ρ], [λ], [π] of H, ρ, λ, π,\nrespectively. We shall also define the bitlengths ⟨H⟩, ⟨ρ⟩, ⟨λ⟩, ⟨π⟩and the\ncombinatorial bit lengths ∥λ∥, ∥π∥. We let ∥H∥= ⟨H⟩and ∥ρ∥= ⟨ρ⟩, since\nH and ρ belong to the nonstretchable part of the input. On the other hand,\nλ and π will be stretched in the definition of ̃f(x, n), and hence their com-\nbinatorial bit lengths will differ from the usual bit lengths. The input x in\nthe subgroup restriction problem is the tuple ([H], [ρ], [λ], [π]). Its bitlength\n⟨x⟩is defined to be the sum of the bitlengths ⟨H⟩, ⟨ρ⟩, ⟨λ⟩, ⟨π⟩, and ∥x∥is\ndefined to be the sum of ∥H∥, ∥ρ∥, ∥λ∥and ∥π∥. Finally rank(x) is defined\nto the sum of the ranks of H and G and ∥λ∥and ∥π∥. Here that rank of\n52\n\na (reductive) group is defined in a standard way. For example, the rank of\nthe symmentric group Sn is n, that of GLn(C) is n. The rank of a general\nfinite or connected simple group can be defined similarly, and the rank of a\nmore complex reductive group is defined to be the sum of the ranks of its\nsimple components. With this terminology, we let f(x) = mπ\nλ, with x as\ndefined here in Hypotheses 3.3.1-3.3.4 and Theorem 3.3.6 for the subgroup\nrestriction problem. Here H and ρ are implicit in the definition of mπ\nλ.\nFor example, in the plethym problem (Problem 1.1.2), these specifi-\ncations are as follows.\nThe specification [H] is just the root system for\nH = GLn(C). Its bitlength ⟨H⟩is n. The specification [ρ] of the repre-\nsentation map ρ : H →G = GL(Vμ(H)) consists of just the partition μ\nspecified in terms of its nonzero parts. Its bitlength ⟨ρ⟩= ⟨μ⟩. The ranks\nof H and G are as usual. The partitions λ and μ are specified in terms of\ntheir nonzero parts. Their bitlength is the total bitlength of the parts, and\nthe combinatorial bit length is the total number of parts (the height). It\nis crucial here that only nonzero parts of λ are specified, because the rank\nof G can be exponential in the rank of H and the bitlength of μ. Hence,\nthe bitlength of this compact representation of λ can be polynomial in the\nrank of H and the bitlength of μ, even if the dimension of G is exponential.\nThe main difference between ⟨x⟩and ∥x∥is that the stretchable data λ and\nπ contribute their bitlengths to the former, and their heights to the latter.\nThe plethysm problem is the main prototype of the subgroup restriction\nproblem. If the reader wishes, (s)he can skip the rest of this subsection and\njump to Section 3.4.3 in the first reading.\nIn general, we assume that H in Problem 1.1.3 is a finite simple group, or\na complex simple, simply connected Lie group, or an algebraic torus (C∗)k,\nor a direct product of such groups. The results and hypotheses in this paper\nare also applicable if we allow simple types of semidirect products, such as\nwreath products, which is all that we need for the sake of the flip. But these\nextensions are routine, and hence, for the sake of simplicity, we shall confine\nourselves to direct products.\n3.4.1\nExplicit polynomial homomorphism\nNow let us define an explicit polynomial homomorphism. This will be done\nby defining basic explicit homomorphisms, and composing them functorially.\nBasic explicit homomorphisms:\nLet V be an irreducible polynomial representation of H (character-\n53\n\nistic zero), or more generally, an explicit polynomial representation that\nis constructed functorially from the irreducible polynomial representations\nusing the operations ⊕and ⊗.\nThen the corresponding homomorphism\nρ : H →G = GL(V ) is an explicit polynomial homomorphism. The iden-\ntity map H →H is also an explicit polynomial homomorphism.\nThe polynomiality restriction here only applies to the torus component\nof H. If H is a finite simple group, or a complex semisimple group, then\nany irreducible representation of H is, by definition, polynomial. In general,\na representation is polynomial if its restriction to the torus component is\npolynomial; i.e., a sum of polynomial (one dimensional) characters.\nTo see why the polynomiality restriction is essential, let H be a torus,\nV its rational representation, and G = GL(V ).\nLet Vλ(G) = Symd(V ),\nthe symmetric representation of G, and let π be the label of the trivial\ncharacter of H. Then the multiplicity mπ\nλ is the number of H-invariants in\nSymd(V ). This is easily seen to be the number of nonnegative solutions of a\nsystem of linear diophontine equations. But the problem of deciding whether\na given system of linear diophontine equations has a nonnegative solution\nis, in general, NP-complete. Though the system that arises above is of a\nspecial form, it is not expected to be in P if V is allowed to be any rational\nrepresentation; the associated decision problem may be NP-complete even\nin this special case. If V is a polynomial representation of a torus H, then\nall coefficients of the system are nonnegative, and the decision problem is\ntrivially in P.\nComposition:\nWe can now compose the basic explicit (polynomial) homomorphisms\nabove functorially:\n1. If ρi : H →Gi are explicit, the product map ρ : H →Q\ni Gi is also\nexplicit.\n2. If ρi : Hi →Gi are explicit, the product map ρ : Q Hi →Q Gi is also\nexplicit.\nInstead of products, we can also allow simple semi-direct products such\nas wreath products here. We may also allow other functorial constructions\nsuch as induced representations and restrictions. For example, if ρ : H →G\nis an explicit polynomial homomorphism, and G′ ⊆G is an explicit subgroup\nof G such that ρ(H) ⊆G′, then the restricted homomorphism ρ′ : H →G′\ncan also be considered to be an explicit polynomial homomorphism. But\n54\n\nfor the sake of simplicity, we shall confine ourselves to the simple functorial\nconstructions above.\n3.4.2\nInput specification and bitlengths\nNow we describe the specifications [H], [ρ], [λ], [μ], their bitlengths. These\nare very similar to the ones in the plethysm problem.\nThe specification [H]:\nWe assume that H is specified as follows.\n(1) If H is a complex, simple, simply connected Lie group, then the specifica-\ntion [H] consists of the root system of H or the Dynkin diagram. Let ⟨H⟩be\nthe bitlength of this specification. Thus, if H = SLn(C), then ⟨H⟩= O(n).\n(2) If H is a simple group of Lie type (Chevalley group) then it has a similar\nspecification [Ca]. The only finite groups of Lie type that arise in GCT are\nSLn(Fpk) and GLn(Fpk). In this case the specification [H] is easy: we only\nhave to specify n, p, k. We define ⟨H⟩in this case to be n + k + log2 p; not\nlog2 n + log2 k + log2 n. As a rule, ⟨H⟩is defined to be the sum of the rank\nparameters (such as n and k here) and bit lengths of the weight parameters\n(such as p here) in the specification. This is equivalent to assuming that the\nrank parameters are specified in unary.\n(3) If H is the alternating group An, we only specify n. Let ⟨H⟩= n.\n(4) The torus is specified by its dimension. We define ⟨H⟩to be the dimen-\nsion.\n(5) If H is a product of such groups, its specification is composed from the\nspecifications of its factors, and the bitlength ⟨H⟩is defined to be the sum\nof the bitlengths of the constituent specifcations.\nThe specification [ρ]:\nLet us first assume that ρ is a basic explicit polynomial homomorphism.\nIn this case the specification of ρ : H →G = GL(V ) is a pair [ρ] = ([H], [V ])\nconsisting of the specification [H] of H as above, and the combinatorial\nspecification [V ] of the representation V as defined below:\n(1) If H is a semisimple, simply connected Lie group, and V = Vμ(H) its\nirrreducible representation for a dominant weight μ of H, then V is specified\nby simply giving the coordinates of μ in terms of the fundamental weights\nof H.\nThus [V ] = μ, and its bitlength ⟨V ⟩is the total bitlength of all\ncoordinates of μ, and the combinatorial bit length ∥V ∥is the total number\nof coordinates of μ.\n55\n\n(2) If H = Sn, and V = Sγ its irreducible representation (Specht module),\nthen [V ] is the partition γ labelling this Specht module. We define ⟨V ⟩to\nbe the bitlength of this partition, and ∥V ∥= ⟨V ⟩.\n(3) If H is a finite general linear group GLn(Fpk), and V its irreducible rep-\nresentation, as classified by Green [Mc], then [V ] is the combinatorial clas-\nsifying label of V as given in [Mc]. It is a certain partition-valued function,\nwhich can be specified by listing the places where the function is nonzero\nand the nonzero partition values at these places. Let ⟨V ⟩be the bitlength\nof this specification; it is O(poly(n, k, ⟨p⟩)). We let ∥V ∥= ⟨V ⟩. More gener-\nally, if H is a finite group of Lie type, and V its irreducible representation,\nthen [V ] is the combinatorial classifying label of V as given by Lusztig [Lu1].\n(4) If H is a torus and V is a polynomial character, then [V ] is the speci-\nfication of the character. Its bitlength is the bitlength of the specification,\nand combinatorial bit length is the dimension of H.\n(5) If V is composed from irreducible representations, then [V ] is composed\nfrom the specifications of the irreducible representations in an obvious way.\nBitlengths and combinatorial bitlengths are defined additively.\nThe bitlength ⟨ρ⟩is defined to be ⟨H⟩+ ⟨V ⟩, where ⟨V ⟩is the bitlength\nof [V ].\nIf ρ is a composite homomorphism, its specification [ρ] is composed from\nthe specifications of its basic constituents in an obvious way. The bitlength\n⟨ρ⟩is defined to be the sum of the bitlengths of these basic specifications.\nThe specifications [λ] and [π]:\nVπ(H) is the tensor product of the irreducible representations of the\nfactors of H. We let [π] be the tuple of the combinatorial classifying labels\nof each of these irreducible representations, as specified above. Let ⟨π⟩be\ntheir total bit length, and ∥π∥the total combinatorial bit length. Similarly,\nVλ(G) is the tensor product of the irreducible representations of the factors\nof G. When G = GLm(C), λ is a partition, which we specify by only giving\nits nonzero parts, whose number is equal to the height of λ. This is crucial\nsince the height of λ can be much less than than the rank m of G, as in\nthe plethysm problem (Problem 1.1.2). We shall leave a similar compact\nspecification [λ] for a general connected, reductive G to the reader. Let ⟨λ⟩\nbe its bitlength and ∥λ∥its combinatorial bit length.\n56\n\n3.4.3\nStretching function and quasipolynomiality\nLet f(x) = mπ\nλ as above, with x = ([H], [ρ], [λ], [π]). Here λ is the dominant\nweight of G. First, assume that H is connected, reductive. Then π is the\ndominant weight of H. For a given x, let us define the stretching function\nas\n ̃f(x, n) = ̃mπ\nλ(n) = mnπ\nnλ,\n(3.8)\nwhich is the multiplicity of Vnπ(H) in Vnλ(G), considered as an H-module\nvia ρ : H →G. Let Mπ\nλ (t) = P\nn≥0 ̃mπ\nλ(n)tn be the generating function of\nthis stretching quasi-polynomial.\nThe following is the generalization of Theorem 1.6.1 in this setting.\nTheorem 3.4.1 (a) (Rationality) The generating function Mπ\nλ (t) is ratio-\nnal.\n(b) (Quasi-polynomiality) The stretching function ̃mπ\nλ(n) is a quasi-polynomial\nfunction of n.\n(c) There exist graded, normal C-algebras S = S(mπ\nλ) = ⊕nSn and T =\nT(mπ\nλ) = ⊕nTn such that:\n1. The schemes spec(S) and spec(T) are normal and have rational singu-\nlarities.\n2. T = SH, the subring of H-invariants in S.\n3. The quasi-polynomial ̃mπ\nλ(n) is the Hilbert function of T.\n(d) (Positivity) The rational function Mπ\nλ (t) can be expressed in a positive\nform:\nMπ\nλ (t) = h0 + h1t + · · · + hdtd\nQ\nj(1 −ta(j))d(j)\n,\n(3.9)\nwhere a(j)’s and d(j)’s are positive integers, P\nj d(j) = d+1, where d is the\ndegree of the quasi-polynomial, h0 = 1, and hi’s are nonnegative integers.\nThe specific rings S(mπ\nλ) and T(mπ\nλ) constructed in the proof of this\nresult are called the canonical rings associated with the structrural con-\nstant mπ\nλ. The projective schemes Y (mπ\nλ) = Proj(S(mπ\nλ)), and Z(mπ\nλ) =\nProj(T(mπ\nλ)) are called the canonical models associated with mπ\nλ.\nTheorem 3.4.1 and its generalization, when H can be disconnected, is\nproved in Chapter 4; cf. Theorem 4.1.1.\n57\n\nFinitely generated semigroup\nThe following is an analogue of Theorem 1.6.2.\nTheorem 3.4.2 Assume that H is connected. For a fixed ρ : H →G, let\nT(H, G) be the set of pairs (μ, λ) of dominant weights of H and G such that\nthe irreducible representation Vπ(H) of H occurs in the irreducible repre-\nsentation Vλ(G) of G with nonzero multiplicity. Then T(H, G) is a finitely\ngenerated semigroup with respect to addition.\nThis is proved in Section 4.4.\nPSPACE\nThe following is a generalization of Theorem 1.6.3.\nTheorem 3.4.3 Assume that H in Problem 1.1.3 is a direct product, whose\neach factor is a complex simple, simply connected Lie group, or an alternat-\ning (or symmetric) group, or SLn(Fpk) (or GLn(Fpk)), or a torus. Then\nf(x) = mπ\nλ can be computed in poly(⟨x⟩) space, with x as specified above.\nThis is proved in Chapter 5. It may be conjectured that Theorem 3.4.3\nholds even when the composition factors of H are allowed to be general\nfinite simple groups of Lie type. This will be so if Lusztig’s algorithm [Lu5]\nfor computing the characters of finite simple groups of Lie type can be\nparallelized; cf. Section 5.4.\nPositivity hypotheses\nTheorem 3.4.1-3.4.3, along with the experimental results in special cases\n(cf. Chapter 6), constitute the main evidence in support of the positivity\nHypotheses 3.3.1-3.3.4 for the subgroup restrition problem.\n3.5\nThe decision problem in geometric invariant\ntheory\nFinally, let us turn to the most general Problem 1.1.4.\n58\n\n3.5.1\nReduction from Problem 1.1.3 to Problem 1.1.4\nFirst, let us note that the subgroup restriction problem (Problem 1.1.3)\nis a special case of Problem 1.1.4.\nTo see this, let H, ρ and G be as in\nProblem 1.1.3, and let X be the closed G-orbit of the point vλ corresponding\nto the highest weight vector of Vλ(G) in the projective space P(Vλ(G)). Then\nX = Gvλ ∼= G/Pλ,\n(3.10)\nwhere the P = Pλ = Gvλ is the parabolic stabilizer of vλ. We have a natural\naction of H on X via ρ. Let R be the homogeneous coordinate ring of X. By\n[Ha, MR, Rm, Sm], the singularities of spec(R) are rational. By Borel-Weil\n[FH], the degree one component R1 of the homogeneous coordinate ring R\nof X is Vλ(G). Hence, sπ\n1 in this special case of Problem 1.1.4 is precisely mπ\nλ\nin Problem 1.1.3. The results in Section 3.4 for sπ\n1 generalize in a natural\nway for sπ\nd.\n3.5.2\nInput specification\nThe variety X in the above example is completely specified by H, ρ and λ.\nHence its specification [X] can be given in the form a tuple ([H], [ρ], [λ]),\nwhere [H], [ρ] and [λ] are the specifications of H, ρ and λ as in Section 3.4,\nThe input specification x for Problem 1.1.4 in the special case above is the\ntuple ([X], d, [π]) = ([H], [ρ], [λ], d, [π]), where [π] is the specification of π as\nin Section 3.4.\nWe now describe a class of varieties X which have similar compact spec-\nifications.\nLet G be a connected, reductive group, H a reductive, possibly discon-\nnected, reductive group, and ρ : H →G an explicit polynomial homomor-\nphism as in Section 3.4. Let V = Vλ(G) be an irreducible representation of G\nfor a dominant weight λ. Let P(V ) be the projective space associated with\nV . It has a natural action of H via ρ. Let v ∈P(V ) be a point that is char-\nacterized by its stabilizer Gv ⊆G. This means it is the only point in P(V )\nthat is stabilized by Gv. For example, the point vλ above is characterized by\nits parabolic stabilier. We assume that we know the Levi decompositioon of\nGv explicity, and its compact specification [Gv], like that of H, and also an\nexplicit compact specification of the embedding ρ′ : Gv →G, aking to that\nof the explicit homomorphism ρ : H →G. Let X ⊆P(V ) be the projective\nclosure of the G-orbit of v in P(V ). Then X as well as the action of H on\nX are completely specified by λ, H, ρ, Gv and ρ′. Hence, we can let [X] be\n59\n\nthe tuple (λ, [H], [ρ], [Gv], [ρ′]). The input specification x for Problem 1.1.4\nwith the X of this form is the tuple ([X], d, [π]). The bitlengths ⟨x⟩and ∥x∥\nare defined additively. The rank(x) is defined to be the sum of the ranks of\nH and G, dim(V ) and ∥π∥. Since the point vλ above is characterized by its\nstabilizer, G/P is a variety of this form.\nThe class varieties [GCT1, GCT2] are either of this form, or a slight ex-\ntension of this form, and admit such compact specifications. The algebraic\ngeometry of an X of the above form is completely determined by the repre-\nsentation theories of the two homomorphisms ρ : H →G and ρ′ : Gv →G.\nFurthermore, the results in [GCT2] say that Problem 1.1.4 for a class variety\nis intimately linked with the subgroup restriction problem and its variants\nfor the homomomorphisms ρ and ρ′. Hence it is qualitatively similar to the\nsubgroup restriction problem in this case; cf. [GCT10] for further elabora-\ntion of the connection between these two problems.\n3.5.3\nStretching function and quasi-polynomiality\nNow let H, X, R and sπ\nd be as in Problem 1.1.4, with H therein assumed to\nbe connected. We associate with f(x) = sπ\nd the following stretching fucntion:\n ̃f(x, n) = ̃sπ\nd(n) = snπ\nnd,\n(3.11)\nwhere snπ\nnd is the multiplicity of the irrreducible representation Vnπ(H) of H\nin Rnd, the componenent of the homogeneous coordinate ring R of X with\ndegree nd. Let S(t) = P\nn≥0 ̃sπ\nd(n)tn.\nTheorem 3.5.1 Assume that the singularities of spec(R) are rational.\n(a) (Rationality) The generating function Sπ\nd (t) is rational.\n(b) (Quasi-polynomiality) The stretching function ̃sπ\nd(n) is a quasi-polynomial\nfunction of n.\n(c) There exist graded, normal C-algebras S = S(sπ\nd) = ⊕nSn and T =\nT(sπ\nd) = ⊕nTn such that:\n1. The schemes spec(S) and spec(T) are normal and have rational singu-\nlarities.\n2. T = SH, the subring of H-invariants in S.\n3. The quasi-polynomial ̃sπ\nd(n) is the Hilbert function of T.\n60\n\n(d) (Positivity) The rational function Sπ\nd (t) can be expressed in a positive\nform:\nSπ\nd (t) = h0 + h1t + · · · + hktk\nQ\nj(1 −ta(j))k(j)\n,\n(3.12)\nwhere a(j)’s and k(j)’s are positive integers, P\nj k(j) = k + 1, where k is\nthe degree of the quasi-polynomial ̃sπ\nd(n), h0 = 1, and hi’s are nonnegative\nintegers.\nThis is proved in Chapter 4. Theorem 3.4.1 is a special case of this theorem,\nin view of the reduction in Section 3.5.1. Theorem 3.5.1 is applicable when\nX is a class variety, assuming that its singularities are rational.\n3.5.4\nPositivity hypotheses\nEven though Theorem 3.5.1 holds for any X, with spec(R) having ratio-\nnal singularities, the positivity hypotheses PH1, SH, PH2, and PH3 can\nbe expected to hold for only very special X’s. In general, characterizing\nthe X’s with compact specification for which these hypotheses hold is a\ndelicate problem.\nHypotheses 3.3.1-3.3.4 say that these hold when X in\nProblem 1.1.4 is G/P (as in Section 3.5.1) or a class variety, with the input\nspecification x as described above. For future reference, we shall reformulate\nthese hypotheses purely in geometric terms.\nFor this we need a definition.\nLet T = P\nn Tn be a graded complex C-algebra so that the singularities\nof spec(T) rational.\nLet Z = Proj(T).\nAssume that Z has a compact\nspecification [Z]; we shall specify it below for the Z’s of interest to us.\nWe let [T], the specification of T, to be [Z].\nThis will play the role of\nthe input in the definition below. Let ⟨T⟩denote its bitlength, and ∥T∥\ncombinatorial bit length.\nLet hT (n) = dim(Tn) be its Hilbert function,\nwhich is a quasipolynomial, since the singularities of spec(T) are rational;\ncf. Lemma 4.1.3.\nDefinition 3.5.2 We say that PH1 holds for T (or Z) if the Hilbert quasi-\npolynomial hT (n) is convex. This means there exists a polytope P = PT\ndepending on the input [T], whose Ehrhart quasipolynomial fP(n) coincides\nwith the Hilbert function hT (n), and whose membership function χP(y) can\nbe computed in poly(⟨T⟩, y) time. We assume that a separating hyperplane\ncan also be computed in polynomial time if y ̸∈P (Section 2.3).\n61\n\nIf PH1 holds we can also ask if analogues of SH, PH2, and PH3–whose\nformulation is similar and hence omitted–hold.\n3.5.5\nG/P and Schubert varieties\nLet us illustrate this definition with an example. Let X ∼= G/Pλ be as in\nSection 3.5.1 and R its homogeneous coordinate ring. We have already seen\nthat it has a compact specification: namely [X] = λ. Since singularities\nof spec(R) are rational, PH1 makes sense.\nFor G/P it follows from the\nBorel-Weil theorem. The Hilbert series of R is of the form\nh0 + · · · + hdtd\n(1 −t)d+1\n,\nwith h0 = 1 and hi’s nonnegative. This is so because R is Cohen-Macauley\n[Rm] and is generated by its degree one component. Hence, the modular\nindex of the Hilbert function is one (PH3). PH2 turns out to be nontriv-\nial. Experimental evidence in its support for the classical G/P is given in\nSection 6.3. Considerations for the Schubert subvarieties are similar. Ex-\nperimental evidence for PH2 for the classical Schubert varieties is also given\nin Section 6.3.\nNow let s = sπ\nd be the multiplicity as Problem 1.1.4, with X having a\ncompact specification [X] as above. Let T = T(s) be the ring associated\nwith s as in Theorem 3.5.1 (c).\nLet Z = Z(s) = Proj(T).\nWe let the\nspecification [Z] = ([X], d, π). Let ⟨Z⟩be its bitlength.\nSo Theorem 3.1.1 in this context implies:\nTheorem 3.5.3 If PH1 and SH holds for Z(s) then nonvanishing of s,\nmodulo small relaxation, can be decided in poly(⟨Z⟩) time.\nWe also have the following reformulation:\nProposition 3.5.4 Hypotheses 3.3.1-3.3.4 are equivalent to PH1,SH,PH2,PH3\nfor Z(s), where s is a stucture constant that corresponds the structure con-\nstant f(x) in Hypotheses 3.3.1. Thus, in the case of the subgroup restriction\nproblem, s = sπ\n1 = mπ\nλ as in Section 3.5.1.\nThis is just a consequence of definitions.\n62\n\n3.6\nPH3 and existence of a simpler algorithm\nAs we remarked in Section 3.1.3, the use of the ellipsoid method and basis\nreduction in lattices makes the the algorithm for saturated integer program-\nming (cf. Theorem 3.1.1) fairly intricate. For the flip (cf. [GCTflip] and\nChapter 7), it is desirable to have simpler algorithms for the relaxed forms\nof the decision problems under consideration, akin to the the polynomial\ntime combinatorial algorithms in combinatorial optimization [Sc] that do\nnot rely on the elliposoid method or basis reduction. We briefly examine in\nthis section the role of PH3 in this context.\nThe simple combinatorial algorithms in combinatorial optimization work\nonly when the problem under consideration is unimodular–in which case the\nvertices of the underlying polytope P are integral–or almost unimodular–\ne.g. when the vertices of P are half integral. Edmond’s algorithm for finding\nminimum weight perfect matching in nonbipartite graphs [Sc] is a classic\nexample of the second case.\nIn the unimodular case, Stanley’s positivity result [St1] implies that the\nrational function FP (t) has a positive form\nFP (t) = h(d)td + · · · + h(0)\n(1 −t)d+1\n.\nIf PH3 (Hypothesis 3.3.4) holds for a structural function f(x) under con-\nsideration then the Ehrhart series FPx(t) of the polytope Px associated with\nx in PH1 (Hypothesis 3.3.1) has a minimal positive form in which each root\nof the denominator has O(poly(∥x∥)) order. Roughly, this says that the\nsituation is “close” to the unimodular case. Hence, in such a case we can\nexpect a purely combinatorial polynomial-time algorithm for deciding non-\nvanishing of f(x), modulo small relaxation, that does not need the ellipsoid\nmethod or basis reduction.\n3.7\nOther structural constants\nThe paradigm of saturated and positive integer programming in this paper,\nalong with appropriate analogues of PH1,SH,PH2,PH3, may be applicable\nseveral other fundamental structural constants in representation theory and\nalgebraic geometry, in addition to the ones in Problems 1.1.1-1.1.4 treated\nabove, such as\n63\n\n1. the value of a Kazhdan-Lusztig polynomial at q = 1, [KL1];\n2. the values at q = 1 of the well behaved special cases of the parabolic\nKostka polynomials and their q-analogues [Ki];\n3. the structural coefficients of the multiplication of Schubert polynomi-\nals, and so on.\n64\n\nChapter 4\nQuasi-polynomiality and\ncanonical models\nIn this chapter we prove quasipolynomiality of the stretching functions as-\nsociated with the various structural constants under consideration (Sec-\ntion 4.1), describe the associated canonical models (Section 4.2), describe\nthe role of nonstandard quantum groups in [GCT4, GCT7, GCT8] in the\ndeeper study of these models (Section 4.3), prove finite generation of the\nsemigroup of weights (Theorem 3.4.2) (Section 4.4), and give an elementary\nproof of rationality in Theorem 3.4.1 (a) (Section 4.5).\n4.1\nQuasi-polynomiality\nHere we prove Theorem 3.5.1; Theorems 1.6.1 and 3.4.1 are its special cases\nin view of the reduction in Section 3.5.1. This, in turn, follows from the\nfollowing more general result.\nLet R = ⊕kRd be a normal graded C-algebra with an action of a reduc-\ntive group H. Assume that spec(R) has rational singularities. Let H0 be\nthe connected component of H containing the identity. Let HD = H/H0 be\nits discrete component. Given a dominant weight π of H0, we consider the\nmodule Vπ = Vπ(H0), an H-module with trivial action of HD. Let sπ\nd denote\nthe multiplicity of the H-module Vπ in Rd. Let ̃sπ\nd(n) be the multiplicity of\nthe H-module Vnπ in Rnd. This is a stretching function associated with the\nmulitplicity sπ\nd. Let Sπ\nd (t) = P\nn≥0 ̃sπ\nd(n)tn.\n65\n\nTheorem 4.1.1 (a) (Rationality) The generating function Sπ\nd (t) is ratio-\nnal.\n(b) (Quasi-polynomiality) The stretching function ̃sπ\nd(n) is a quasi-polynomial\nfunction of n.\n(c) There exist graded, normal C-algebras S = S(sπ\nd) = ⊕nSn and T =\nT(sπ\nd) = ⊕nTn such that:\n1. The schemes spec(S) and spec(T) are normal and have rational singu-\nlarities.\n2. T = SH, the subring of H-invariants in S.\n3. The quasi-polynomial ̃sπ\nd(n) is the Hilbert function of T.\n(d) (Positivity) The rational function Sπ\nd (t) can be expressed in a positive\nform:\nSπ\nd (t) = h0 + h1t + · · · + hktk\nQ\nj(1 −ta(j))k(j)\n,\n(4.1)\nwhere a(j)’s and k(j)’s are positive integers, P\nj k(j) = k + 1, where k is\nthe degree of the quasi-polynomial ̃sπ\nd(n), h0 = 1, and hi’s are nonnegative\nintegers.\nTheorem 3.5.1 follows from this by letting R be the homogeneous coordinate\nring of X.\nMore generally, if W is an irreducible representation of HD, we can\nconsider the H-module Vπ ⊗W. Let sπ,W\nd\nbe its multiplicity in Rd. Let\n ̃sπ,W\nd\n(n) be the multiplicity of the trivial H-representation in the H-module\nRnd ⊗V ∗\nnπ ⊗Symn(W ∗). Then\nTheorem 4.1.2 Analogue of Theorem 4.1.1 holds for ̃sπ,W\nd\n(n).\nFor the purposes of the flip, Theorem 4.1.1 suffices.\nProof: We shall only prove Theorem 4.1.1, the proof of Theorem 4.1.2 be-\ning similar.\nThe proof is an extension of M. Brion’s proof (cf.\n[Dh]) of\nquasi-polynomiality of the stretching function associated with a Littlewood-\nRichardson coefficient of any semisimple Lie algebra.\nClearly (a) follows from (b); cf. [St1].\n66\n\n(b) and (c):\nLet Cd be the cyclic group generated by the primitive root ζ of unity of\norder d. It has a natural action on R: x ∈Cd maps z ∈Rk to xkz. Let\nB = RCd = P\nn≥0 Rnd ⊆R be the subring of Cd-invariants. By Boutot\n[Bou], B is a normal C-algebra and spec(B) has rational singularities.\nAssume that H0 is semisimple; extension to the reductive case being easy.\nLet π∗be the dominant weight of H0 such that V ∗\nπ = Vπ∗. By Borel-Weil\n[FH],\nCπ∗= ⊕n≥0V ∗\nnπ = ⊕n≥0Vnπ∗,\nis the homogeneous coordinate ring of the H0-orbit of the point vπ∗∈P(Vπ∗)\ncorresponding to the highest weight vector. This H0-orbit is isomorphic to\nH0/Pπ∗, where Pπ∗⊆H0 is the parabolic stabilizer of vπ∗. Hence Cπ∗is\nnormal and spec(Cπ∗) has rational signularities; cf.\n[Ha, MR, Rm, Sm].\nIt follows that B ⊗Cπ∗is also normal, and spec(B ⊗Cπ∗) has rational\nsingularities. Consider the action of C∗on B ⊗Cπ∗given by:\nx(b ⊗c) = (x · b) ⊗(x−1 · c),\nwhere x ∈C∗maps b ∈Bn to xnb, the action on Cπ∗being similar. Consider\nthe invariant ring\nS = (B ⊗Cπ∗)C∗= ⊕nSn = ⊗n≥0Rnd ⊗V ∗\nnπ.\n(4.2)\nBy Boutot [Bou], it is a normal, and spec(D) has rational singularities.\nSince Vnπ is an H-module, the algebra S has an action of H. Let\nT = T(sπ\nd) = SH = ⊕n≥0Tn\n(4.3)\nbe its subring of H-invariants. By Boutot [Bou], it is normal, and spec(T)\nhas rational singularities–this is the crux of the proof. By Schur’s lemma, the\nmultiplicity of the trivial H-representation in Sn = Rnd⊗V ∗\nnπ is precisely the\nmultiplicity ̃sπ\nd(n) of the H-module Vnπ in Rnd. Hence, the Hilbert function\nof T, i.e., dim(Tn), is precisely ̃sπ\nd(n), and the Hilbert series P\nn≥0 dim(Tn)tn\nis Sπ\nd (t).\nQuasipolynomiality of ̃sπ\nd(n) follows by applying the following\nlemma:\nLemma 4.1.3 (cf. [Dh]) If T = ⊕∞\nn=0Tn is a graded C-algebra, such that\nspec(T) is normal and has rational simgularites, then dim(Tn), the Hilbert\nfunction of T, is a quasi-polynomial function of n.\n67\n\n(d) Since spec(T) has rational singularities, T is Cohen-Macaualey.\nLet\nt1, . . . , tu be its homogeneous sequence of parameters (h.s.o.p.), where u =\nk + 1 is the Krull dimension of T. By the theory of Cohen-Macauley rings\n[St2], it follows that its Hilbert series Sπ\nd (t) is of the form\nh0 + h1t + · · · + hktk\nQk+1\ni=1 (1 −tdi)\n,\n(4.4)\nwhere (1) h0 = 1, (2) di is the degree of ti, and (3) hi’s are nonnegative\nintegers. This proves (d). Q.E.D.\nRemark 4.1.4 A careful examination of the proof above shows that ratio-\nnality of Sπ\nd (t), and more strongly, asymptotic quasi-polynomiality of ̃sπ\nd(n)\nas n →∞, can be proved using just Hilbert’s result on finite generation of\nthe algebra of invariants of a reductive-group action. Boutot’s result is nec-\nessary to prove quasi-polynomiality for all n. This is crucial for saturated\nand positive integer programming (Chapter 3).\n4.1.1\nThe minimal positive form and modular index\nThe form (4.4) of Sπ\nd (t) is not unique because it depends on the degrees di’s\nof the paramters ti’s. For future use, let us record the following consequences\nof the proof. Let T be the ring constructed in the proof above.\nCorollary 4.1.5 Suppose T has an h.s.o.p.\nt = (t1, . . . , tu) with di =\ndeg(ti). Then Sπ\nd (T) has a positive rational form (4.4) with di = deg(ti)\ntherein.\nThe proof above is lets us define a minimal positive form of the rational\nfunction Sπ\nd (t) associated with a structural constant s. For this, let us or-\nder h.s.o.p.’s of T lexicographically as per their degree sequences. Here the\ndegree seqeunce of an h.s.o.p. t = (t1, . . . , tu) is defined to be (d1, . . . , du),\nwhere di = deg(ti). The form (4.4) is the same for any h.s.o.p. of lexi-\ncographically minimum degree sequence. We call it the minimal positive\nform of Sπ\nd (t). The modular index of sπ\nd is defined to be max{di}, where\n(d1, . . . , du) is the degree sequence of a lexicographically minimal h.s.o.p.\nSince Problems 1.1.1, 1.1.2,1.1.3, 1.2.1 are special cases of Problem 1.1.4,\nthis defines minimal positive forms of the rational generating functions of the\nstretching quasi-polynomials (cf. Theorem 3.4.1) associated with the struc-\ntural constants in these problems, and also the modular indices of these\nstructural constants.\n68\n\n4.1.2\nThe rings associated with a structural constant\nThe preceding proof also associates with the structural constant s a few\nrings which will be important later. Specifically, let S = S(s) and T = T(s)\nbe the rings as in Theorem 4.1.1 (c) associated with the structural constant\ns = sπ\nd.\nLet R = R(s) be the homogeneous coordinate ring of X as in\nTheorem 4.1.1. We call R(s), S(s) and T(s) the rings associated with the\nstructure constant s.\nWhen s = mπ\nλ, as in the subgroup restriction problem (Problem 1.1.3),\nX ∼= G/P as given in eq.(3.10. Then these rings are explicitly as follows:\nR(mπ\nλ)\n=\n⊕n≥0Vnλ(G),\nS(mπ\nλ)\n=\n⊕n≥0Vnλ(G) ⊗Vnπ(H)∗,\nT(mπ\nλ)\n=\n⊕n≥0(Vnλ(G) ⊗Vnπ(H)∗)H.\n(4.5)\nBy specializing the subgroup restriction problem further to the Littlewood-\nRichardson problem (Problem 1.2.1), we get the following rings associated\nby Brion (cf. [Dh]) with the Littlewood-Richardson coefficient cλ\nα,β:\nR(cλ\nα,β)\n=\n⊕n≥0Vnα(H) ⊗Vnβ(H),\nS(cλ\nα,β)\n=\n⊕n≥0Vnα(H) ⊗Vnβ(H) ⊗Vnλ(H)∗,\nT(cλ\nα,β)\n=\n⊕n≥0(Vnα(H) ⊗Vnβ(H) ⊗Vnλ(H)∗)H.\n(4.6)\n4.2\nCanonical models\nThere are several rings other than T(cλ\nα,β) whose Hilbert function coincides\nwith the Littlewood-Richardson stretching quasi-polynomial ̃cλ\nα,β(n).\nFor\nexample, let P = P λ\nα,β be the BZ-polytope [BZ] whose Ehrhart quasi-\npolynomial coincides with ̃cλ\nα,β(n).\nWe can associate with P a ring TP\nas in Stanley [St3] whose Hilbert function coincides with ̃cλ\nα,β(n).\nThere\nare many other choices for P. For example, in type A, we can consider a\nhive polytope or a honeycomb polytope [KT1] instead of the BZ-polytope.\nThe rings TP ’s associated with different P’s will, in general, be different,\nand there is nothing canonical about them. In contrast, the ring T(cλ\nα,β) is\nspecial because:\nProposition 4.2.1 (PH0) The rings R(cλ\nα,β), S(cλ\nα,β), T(cλ\nα,β) have quan-\ntizations Rq(cλ\nα,β), Sq(cλ\nα,β), Tq(cλ\nα,β) endowed with canonical bases in the ter-\nminology of Lusztig [Lu4]. Furthermore, the canonical bases of Rq(cλ\nα,β), Sq(cλ\nα,β)\n69\n\nare compatible with the action of the Drinfeld-Jimbo quantum group associ-\nated with H = GLn(C), and the canonical basis of Sq(cλ\nα,β) is an extension\nof the canonical basis of Tq(cλ\nα,β) in a natural way.\nThis follows from the work of Lusztig (cf. [Lu3], Chapter 27 in [Lu4]) and\nKashiwara (cf.\nTheorem2 in [Kas3]).\nSpecializations of these canonical\nbases at q = 1 will be called canonical bases of R(cλ\nα,β), S(cλ\nα,β), T(cλ\nα,β).\nLusztig [Lu4] has conjectured that the structural constants associated with\nthe canonical bases in Proposition 4.2.1 are polynomials in q with nonnega-\ntive integral coefficients as in the case of the canonical basis of the (negative\npart of the) Drinfeld-Jimbo enveloping algebra. We refer to Proposition 4.2.1\nas PH0 in view of this (conjectural) positivity property.\nIn view of this proposition, we call the rings R(cλ\nα,β), S(cλ\nα,β) and T(cλ\nα,β)\nthe canonical rings associated with the Littlewood-Richardson coefficient\ncλ\nα,β, and X = Proj(R(cλ\nα,β)), Y = Proj(S(cλ\nα,β)) and Z = Proj(T(cλ\nα,β)) the\ncanonical models associated with cλ\nα,β.\n4.2.1\nFrom PH0 to PH1,3\nNow we study the relevance of PH0 above in the context of PH1,SH,PH2,\nand PH3 for Littlewood-Richardson coefficients (Section 1.2).\nPH1\nAs already remarked in Section 1.7, PH1 for Littlewood-Richardson coeffi-\ncients is a formal consequence of the properties of Kashiwara’s crystal oper-\nators on the canonical bases in PH0 (Proposition 4.2.1); [Dh, Kas2, Li, Lu4].\nSpecifically, the canonical basis of the ring Rq(cλ\nα,β) also yields a canon-\nical basis for the tensor product Vq,α ⊗Vq,β of the irreducible Hq modules\nwith highest weights α and β. The Littlwood-Richardson rule for arbitrary\ntypes follows from the study of Kashiwara’s crystal operators on this canon-\nical basis for the tensor product; [Lu4]. This rule is equivalent to the one\nin [Li] based on combinatorial interpretation of the crystal operators in the\npath model therein. The article [Dh] derives a convex polyhedral formula\nfor Littlewood-Richardson coefficients (of arbitrary type) using this com-\nbinatorial interpretation. Though the complexity-theoretic issues are not\naddressed in [Dh], it can be verified that the polyhedral formula therein is a\nconvex #P-formula. This yields PH1 for Littlewood-Richardson coefficients\nof arbitrary types using PH0.\n70\n\nSH\nNow let us see the relevance of PH0 in the context of SH for Littlewood-\nRichardson coefficients of arbitrary type.\nThe polytope in [Dh], mentioned above, for type A is equivalent to the\nhive polytope in [KT1] in the sense that the number integer points in both\nthe polytopes is the same. Knutson and Tao prove SH for type A by show-\ning that the hive polytope always has in integral vertex. To extend this\nproof to an arbitrary type, one has to convert the polytope in [Dh] into a\npolytope that is guaranteed to contain an integral vertex if the index of the\nstretching quasipolynomial ̃cλ\nα,β(n) is one. The main difficulty here is that\nwe do not have a nice mathematical interpretation for the index. Algorithm\nin Theorem 3.1.1 applied to the polytope in [Dh] computes this index in\npolynomial time. But it does not give a nice interpretation that can be used\nin a proof as above.\nThis index is simply the largest integer dividing the degrees of all ele-\nments in any basis of the canonical ring T(cλ\nα,β)–in particular, the canon-\nical basis. This follows by applying Proposition 3.1.3 to the polytope in\n[Dh]. This leads us to ask: is there an interpretation for the index based on\nLusztig’s topological construction of the canonical basis in Proposition 4.2.1?\nIf so, this may be used to extend the known polyhedral proof for SH in type\nA to arbitrary types. Alternatively, it may be possible to prove SH using\ntopological properties of the canonical basis in the spirit of the topological\n(intersection-theoretic) proof [Bl] of SH in type A.\nPH3\nNow let us see the relevance of PH0 in the context of PH3 for Littlewood-\nRichardson coefficients.\nFirst, let us consider the minimal positive form (Section 4.1.1) associated\nwith a Littlewood-Richardson coefficient cλ\nα,β of type A. Let T = T(cλ\nα,β)\ndenote the ring that arises in this case; cf. eq.(4.6). Now we can ask:\nQuestion 4.2.2 Are all di’s occuring in the minimal positive form (cf.\n(4.4)) one in this special case?\nThis is equivalent to asking if the ring\nT = T(cλ\nα,β) in this case is integral over T1, the degree one component of T.\nIf so, this would provide an explanation for the conjecture of King at al\n[KTT] (cf. eq.(1.3)) in the theory of Cohen-Macauley rings:\n71\n\nProposition 4.2.3 Assuming yes, the conjecture of King et al [KTT] (Hy-\npothesis 1.2.6) holds.\nRemark 4.2.4 In contrast, the ring TP associated with the hive polytope\n(cf. beginning of Section 4.2) need not be integral over its degree one compo-\nnent, in view of the fact that the hive polytope can have nonintegral vertices\n[DM1].\nRemark 4.2.5 T = T(cλ\nα,β) need not be generated by its degree one compo-\nnent T1. If this were always so, the h-vector (hd, · · · , h0) in eq.(1.3) would\nbe an M-vector (Macauley-vector) [St2]. But one can construct α, β and λ\nfor which this does not hold.\nProof: (of the proposition) Since T is integral over T1, it has an h.s.o.p., all of\nwhose elements have degree 1. By Theorem 3.4.1, the singularities of spec(T)\nare rational.\nHence T is Cohen-Macaulay.\nNow the result immediately\nfollows from the theory of Cohen-Macauley rings [St2]. Q.E.D.\nIn view of this Proposition, the conjecture of King et al will follow if all\ncanonical basis elements of T(cλ\nα,β) can be shown to be integral over the basis\nelements of degree one. This requires a further study of the multiplicative\nstructure of this canonical basis. Considerations for PH3 (Hypothesis 1.2.8)\nfor Littlewood-Richardson coefficients of arbitrary type are similar.\nPH2\nSimilarly, the positivity property (PH2) of the stretching quasipolynomial\nassociated with Littlewood-Richardson coefficients may possibly follow from\na deep study of the multiplicative structure of the canonical basis as per\nPH0 (Proposition 4.2.1), just as positivity of the multiplicative structural\ncoefficients of the canonical basis for the (negative part of the) Drinfeld-\nJimbo enveloping algebra follows from a deep study of the multiplicative\nstructure of this basis [Lu4].\n4.2.2\nOn PH0 in general\nThe discussion above indicates that for Littlewood-Richardson coefficients\nPH1,SH,PH3, and plausibly PH2 as well are intimately related to PH0\n(Proposition 4.2.1).\nThis leads us to ask if the rings associated in Sec-\ntion 4.1.2 with other structural constants under consideration in this paper\n72\n\nhave quantizations which satisfy appropriate forms of PH0. If so, this PH0\nmay be used to derive PH1, SH, PH3, and PH2 (Hypotheses 3.3.1-3.3.4)\nfor these structural constants. Note that SH (a) follows from PH3 (see the\nremark after Hypothesis 3.3.4); PH2 may also follow from PH3. Thus PH1\nand PH3 are the ones to focus on.\nTo formalize this, let s be a structural constant which is either the Kro-\nnecker coefficient as in Problem 1.1.1, or the plethysm constant as in Prob-\nlem 1.1.2, or the multiplicity mπ\nλ in Problem 1.1.3, or the multiplicity sπ\nd, as\nin Problem 1.1.4, when X therein is a class variety. Let R(s), S(s), T(s) be\nthe rings associated with s (Section 4.1.2). Let X(s) = Proj(R(s)), Y (s) =\nProj(S(s)) and Z(s) = Proj(R(s)). We call R = R(s), S = S(s), T = T(s)\nthe canonical rings associated with s, and X(s), Y (s), Z(s) the canonical\nmodels associated with s, because we expect these rings and models to be\nspecial as in the case of the Littlewood-Richardson coefficients.\nLet H be as in Problem 1.1.3 or Problem 1.1.4.\nAssume that H is\nconnected. Let Hq denote the Drifeld-Jimbo quantization of H. Now we\nask:\nQuestion 4.2.6 (PH0??) Are there quatizations Rq, Sq of R, S, with Hq-\naction, and a quantization Tq of T with “canonical” bases (in some appro-\npriate sense) B(Rq), B(Sq), B(Tq), where B(Rq) and B(Sq) are compatible\nwith the Hq-action and B(Sq) is an extension of B(Tq)? Furthermore, do\nthese canonical bases have appropriate positivity properties?\nIn other words, are there quantizations of R, S and T for which PH0\n(Proposition 4.2.1) can be extended in a natural way?\nIf so, this extended PH0 may be used to prove PH1 and SH for s just as\nin the case of Littlewood-Richardson coefficients (of type A).\n4.3\nNonstandard quantum group for the Kronecker\nand the plethysm problems\nWe now consider this question when s is the kronecker or the plethysm\nconstant (cf. Problems 1.1.1 and 1.1.2). PH0 for Littlewood-Richardson\ncoefficients (Proposition 4.2.1) depends critically on the theory of Drinfeld-\nJimbo quantum groups. This is intimately related (in type A) [GrL] to the\nrepresentation theory of Hecke algebras. To extend PH0 in the context of the\nkronecker and the plethysm constants, one needs extensions of these theories\n73\n\nin the context of Problems 1.1.1-1.1.2. In this section, we briefly review the\nresults in [GCT4, GCT8, GCT7] in this direction and the theoretical and\nexperimental evidence it provides in support of PH0–that is, affirmative\nanswer to Question 4.2.6–in this context.\nSo let us consider the generalized plethysm problem (Problem 1.1.2).\nAs expected, the representation theory of Drinfeld-Jimbo quantum groups\nand Hecke algebras does not work in the context of this general problem.\nBriefly, the problem is that if H is a connected, reductive group and V its\nrepresentation, then the homomorphism H →G = GL(V ) does not quan-\ntize in the setting of Drinfeld-Jimbo quantum groups. That is, there is no\nquantum group homomorphism from Hq, the Drinfeld-Jimbo quantization\nof H, to Gq, the Drinfeld-Jimbo quantization of G. In [GCT4, GCT7], a new\nnonstandard quantization GH\nq of G– called a nonstandard quantum group–is\nconstructed so that there is a quantum group homomorphism Hq →GH\nq .\nWhen H = G, GH\nq coincides with the Drinfeld-Jimbo quantum group. The\narticle [GCT8] gives a conjectural scheme for constructing a nonstandard\ncanonical basis for the matrix coordinate ring of GH\nq\nthat is akin to the\ncanonical basis for the matrix coordinate ring of the Drinfeld-Jimbo quan-\ntum group [Lu4, Kas3].\nIt is known that the Drinfeld-Jimbo quantum group Gq = GLq(V ) and\nthe Hecke algebra Hn(q) are dually paired: i.e., they have commuting ac-\ntions on V ⊗n\nq\nfrom the left and the right that determine each other, where Vq\ndenotes the standard quantization of V . Furthermore, the Kazhdan-Lusztig\nbasis for Hn(q) is intimately related to the canonical basis for Gq [GrL]. Sim-\nilarly, [GCT7] constructs a nonstandard generalization BH\nn (q) of the Hecke\nalgebra which is (conjecturally) dually paired to GH\nq . The article [GCT8]\ngives a conjectural scheme for constructing a nonstandard canonical basis of\nBH\nn (q) akin to the Kazhdan-Lusztig basis of the Hecke algebra Hn(q).\nThe nonstandard quantum group GH\nq and the nonstandard algebra BH\nn (q)\nturn out to be fundamentally different from the standard Drinfeld-Jimbo\nquantum group Gq and the Hecke algebra Hn(q). For example, the non-\nstandard quantum group GH\nq is a nonflat deformation of G in general. This\nmeans the Poincare series of the matrix coordinate ring of GH\nq is different\nfrom the Poincare series of the matrix coordinate ring of G. Specifically,\nthe terms of the first series can be smaller than the respective terms of the\nsecond series. Similarly, BH\nn (q) is a nonflat deformation of the group algebra\nC[Sn] of the symmetric group Sn; i.e., its dimension can be bigger than that\nof C[Sn].\n74\n\nNonflatness of GH\nq intuitively means that it is “smaller” than G in gen-\neral. Hence, it may seem that there is a loss of information when one goes\nfrom G to GH\nq . Fortunately, there is none, as per the reciprocity conjecture\nin [GCT7]. This roughly says that the information which is lost in the tran-\nsition from G to GH\nq simply gets transfered to BH\nn (q), which is bigger than\nHn(q). In other words, there is no information loss overall. Hence analogues\nof the properties in the standard setting should also hold in the nonstandard\nsetting, though in a far more complex way.\nThat is what seems to happen to positivity. Specifically, experimental ev-\nidence suggests that the conjectural nonstandard canonical bases in [GCT8]\nhave nonstandard positivity properties which are complex versions of the\npositivity properties in the standard setting. See [GCT7, GCT8, GCT10]\nfor a detailed story.\n4.4\nThe cone associated with the subgroup restric-\ntion problem\nIn this section, we prove Theorem 3.4.2, by extending the proof of Brion\nand Knop (cf. [El]) for the Littlewood-Richardson problem. The proof is in\nthe spirit of the proof of quasipolynomiality in Section 4.1.\nLet G be a connected, reductive group, H a connected, reductive sub-\ngroup, and ρ : H →G a homomorphism. Theorem 3.4.2 has the following\nequivalent formulation.\nLet S(H, G) be the set of pairs (μ, λ) such that\nVμ(H) ⊗Vλ(G) has a nonzero H-invariant. Then,\nTheorem 4.4.1 The set S(H, G) is a finitely generated semigroup with re-\nspect to addition.\nWhen G = H × H and the embedding H ⊆G is diagonal, this special-\nizes to the Brion-Knop result mentioned above. The proof follows by an\nextension the technique therein.\nProof: Let B be a Borel subgroup of G, U the unipotent radical of B and\nT the maximal torus in B. Similarly, let B′ be a Borel subgroup of H, U ′\nthe unipotent radical of B′ and T ′ the maximal torus in B′. Without loss\nof generality, we can assume that B′ ⊆B, U ′ ⊆U, T ′ ⊆T. Let A = C[G]U\nbe the algebra of regular functions on G that are invariant with respect to\nthe right multiplication by U. It is known to be finitely generated [El]. The\ngroups G and T act on A via left and right multiplication, respectively. As\n75\n\na G × T-module,\nA = ⊕λVλ(G),\n(4.7)\nwhere the torus T acts on Vλ(G) via multiplication by the highest weight\nλ∗of the dual module. Similarly,\nA′ = C[H]U′ = ⊕λVμ(H),\n(4.8)\nwhere the torus T ′ acts on Vμ(H) via multiplication by the highest weight\nμ∗of the dual module.\nNow A ⊗A′ is finitely generated since A and A′ are. Let X = (A ⊗A′)H\nbe the ring of invariants of H acting diagonally on A⊗A′. The torus T ×T ′\nacts on X from the right. Since H is reductive, X is finitely generated [PV].\nHence, the semigroup of the weights of the right action of T × T ′ on X is\nfinitely generated. We have\nX = (A ⊗A′)H = ((⊕Vλ(G)) ⊗(⊕Vμ(H)))H = ⊕(Vλ(G) ⊗Vμ(H))H,\nand the weights of the algebra X are of the form (λ∗, μ∗) such that Vλ(G) ⊗\nVμ(H) contains a nontrivial H-invariant.\nTherefore these pairs form a\nfinitely generated semigroup. Q.E.D.\nFor the sake of simplicity, assume that G and H are semisimple in what\nfollows.\nLet TR(H, G) denote the polyhedral convex cone in the weight\nspace of H × G generated by T(H, G), as defined in Theorem 3.4.2. This is\na generalization of the Littlewood-Richardson cone (Section 2.2.2).\nThe following generalization of Corollary 3.2.3 is a consequence of The-\norem 3.1.1 and its proof.\nTheorem 4.4.2 Assume that the positivity hypothesis PH1 (Section 3.3)\nholds for the subgroup restriction problem for the pair (H, G), where both H\nand G are classical. Given dominant weights μ, λ of H and G, the polytope\nPμ,λ as in PH1 has a specification of the form\nAx ≤b\n(4.9)\nwhere A depends only on H and G, but not on μ or λ, and b depends\nhomogeneously and linearly on μ, λ. Let n be the total number of columns\nin A.\nThen, there exists a decomposition of TR(H, G) into a set of polyhedral\ncones, which form a cell complex C(H, G), and, for each chamber C in this\n76\n\ncomplex, a set M(C) of O(n) modular equations, each of the form\nX\ni\naiμi +\nX\ni\nbiλi = 0\n(mod d),\nsuch that\n1. Saturation hypothesis SH is equivalent to saying that: (μ, λ) ∈T(H, G)\niff(μ, λ) ∈TR(H, G) and (μ, λ) satisfies the modular equations in the\nset M(Cμ,λ) associated with the smallest cone Cμ,λ ∈C(H, G) contain-\ning (μ, λ).\n2. Given (μ, λ), whether (μ, λ) ∈TR(H, G) can be determined in polyno-\nmial time.\n3. If so, whether (μ, λ) satisfies the modular equations associated with\nthe smallest cone in C(H, G) containing it can also be determined in\npolynomial time.\nProof: Given a point p = (μ′, λ′) in the weight space of H ×G, where μ′ and\nλ′ are arbitrary rational points, let S(p) denote the constraints (half-spaces)\nin the sytem (4.9) whose bounding hyperplanes contain the polytope Pμ′,λ′.\nWe can decompose TR(H, G) into a conical, polyhedral cell complex, so that\ngiven a cone C in this complex, and a point p in its interior, the set S(p)\ndoes not depend on p. We shall denote this set by S(C). Thus the affine\nspan of Pμ,λ, for any (μ, λ) ∈C, is determined by the linear system\nA′x = b′,\nwhere [A′, b′] consists of the rows of [A, b] in (4.9) corresponding to the set\nS(C). By finding the Smith normal form of A′, we can associate with C a set\nof modular equations that the entries of b′ must satisfy for this affine span to\ncontain an integer point; see the proof of Theorem 3.1.1. Since the entries of\nA′ depend only on H and G, these equations depend only on C. If (μ, λ) ∈\nT(H, G), then (μ, λ) is integral, and hence these equations are satisfied.\nConversely, if (μ, λ) ∈TR(H, G) and these equations are satisfied, then the\nsaturation property implies that (μ, λ) ∈T(H, G), as seen by examining\nthe proof of Theorem 3.1.1. Furthermore, given (μ, λ), the algorithm in the\nproof of Theorem 3.1.1 implicitly determines if (μ, λ) ∈TR(H, G) and if\nthese modular equations are satisfied in polynomial time. Q.E.D.\n77\n\n4.5\nElementary proof of rationality\nIn this section we give an elementary proof of rationality in Theorem 3.4.1\n(a), when H therein is connected–actually of a slightly stronger statement:\nnamely, the stretching function ̃mπ\nλ(n) is asymptotically a quasipolynomial,\nas n →∞; cf. Remark 4.1.4. But this proof cannot be extended to prove\nquasipolynomiality for all n. The proof here is motivated by the work of\nRassart [Rs], De Loera and McAllister on the stretching function associated\nwith a Littlewood-Richardson coefficient.\nFirst, we recall some standard results that we will need.\nVector partition functions\nGiven an integral s×n matrix B and integral n-vector c, consider the vector\nparitition function φB(c), which is the number of integer solutions to the\ninteger programming problem\nBy = c,\ny ≥0.\n(4.10)\nFor a fixed c, b, let\nφB,c(n) = φB(nc)\nφB,c,b(n) = φB(nc + b).\n(4.11)\nBy Sturmfels [Stm] and Szenes-Vergne residue formula [SV], φB(c) is a\npiecewise quasipolynomial function of c. That is, Rn can be decomposed into\npolyhedral cones, called chambers, so that the restriction of φB(c) to each\nchamber R is a multivariate quasipolynomial function of the coordinates of c.\nThis implies that φB,c(n) is a quasipolynomial function of n. It also implies\nthat the function φB,c,b(n) is asymptotically a quasipolynomial function of\nn, as n →∞, because the points nc + b, as n →∞, lie in just one chamber.\nThe Szenes-Verne residue formula [SV] for vector partition functions also\nimplies that there is a constant d(B), depending only on B, such that the\nperiod of φB,c(n), for any c, divides d(B).\nKlimyk’s formula\nLet H ⊆G and mπ\nλ be as in Theorem 3.4.1 (a), with H connected. Let us\nassume that H is semisimple, the general case being similar. Let H and G\nbe the Lie algebras of H and G respectively. We recall Klimyk’s formula for\nmπ\nλ. Without loss of generality, we can assume that the Cartan subalgebra\n78\n\nC ⊆H is a subalgebra of the Cartan subalgebra D ⊆G.\nSo we have a\nrestriction from D∗to C∗, and we assume that the half-spaces determining\npositive roots are compatible. We denote weights of H by symbols such as μ\nand of G by symbols such as ̄μ. To be consistent, we shall use the notation\nmπ ̄λ instead of mπ\nλ in this proof. We write ̄μ ↓μ if the weight ̄μ of G restricts\nto the weight μ of H. We denote a typical element of the Weyl group of\nH by W, and a typical element of the Weyl group of G by ̄W. Given a\ndominant weight π of G and a weight ̄μ of G, let n ̄μ( ̄λ) denote the dimension\nof the weight space for ̄μ in B ̄λ = V ̄λ(G).\nWe assume that:\n(A): For any weight μ of H, the number of ̄μ’s such that ̄μ ↓μ is finite.\nFor example, this is so in the plethysm problem (Problem 1.1.2). We\nshall see later how this assumption can be removed.\nBy Klimyk’s formula (cf. page 428, [FH]),\nmπ ̄λ =\nX\nW\n(−1)W\nX\n ̄μ↓π−ρ−W (ρ)\nn ̄μ(V ̄λ),\n(4.12)\nwhere ρ is half the sum of positive roots of H. We allow ̄μ in the inner sum\nto range over all weights ̄μ of G such that ̄μ ↓π −ρ −W(ρ) by defining\nn ̄μ(V ̄λ) to be zero if ̄μ does not occur in V ̄λ.\nProof of Theorem 3.4.1 (a)\nThe goal is to express ̃mπ ̄λ(n) as a linear combination of vector partition\nfunctions φB,c,b(n)’s, for suitable B, c, b’s, using Klimyk’s formula for mπ ̄λ.\nAfter this, we can deduce asymptotic quasipolynomiality of ̃mπ ̄λ(n) from\nasymptotic quasipolynomiality of φB,c,b(n)’s.\nBy Kostant’s multiplicity formula (cf. page 421 [FH]),\nn ̄μ(V ̄λ) =\nX\n ̄\nW\n(−1)\n ̄\nW P( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ)),\n(4.13)\nwhere P( ̄λ), for a weight ̄λ of G, denotes the Kostant partition function;\ni.e., the number of ways to write ̄λ as a sum of positive roots of G. It is\nimportant for the proof that Kostant’s formula (4.13) holds even if ̄μ is not\na weight that occurs in the representation V ̄λ–in this case, n ̄μ(V ̄λ) = 0, and\nthe right hand side of (4.13) vanishes.\nBy eq.(4.12) and (4.13),\n79\n\nmπ ̄λ =\nX\nW\nX\n ̄\nW\n(−1)W (−1)\n ̄\nW\nX\n ̄μ↓π−ρ−W (ρ)\nP( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ)).\n(4.14)\nLet D denote the dominant Weyl chamber in the weight space of G. Let\nC denote the Weyl chamber complex associated with the weight space of G.\nThe cells in this complex are closed polyhedral cones. Each cone is either\nthe chamber ̄W(D), for some Weyl group element ̄W, or a closed face of\n ̄W(D) of any dimension.\nUsing M ̈obius inversion, the inner sum\nX\n ̄μ↓π−ρ−W (ρ)\nP( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ))\nin eq.(4.14) can be written as a linear combination\nX\nC\na(C)\nX\n ̄μ∈C: ̄μ↓π−ρ−W (ρ)\nP( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ)),\nwhere C ranges over chambers in the Weyl chamber complex C, a(C) is an\nappropriate constant for each C.\nHence,\nmπ ̄λ =\nX\nW\nX\n ̄\nW\n(−1)W (−1)\n ̄\nW X\nC\na(C)\nX\n ̄μ∈C: ̄μ↓π−ρ−W (ρ)\nP( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ)).\n(4.15)\nNow think of π and ̄λ as variables. But H and G are fixed, and hence\nalso the quantities such as ρ and ̄ρ.\nClaim 4.5.1 For fixed Weyl group elements W, ̄\nW and a fixed C, the sum\nX\n ̄μ∈C: ̄μ↓π−ρ−W (ρ)\nP( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ))\n(4.16)\ncan be expressed as a vector partition function associated with an appropriate\nlinear system\nBy = c,\ny ≥0,\n(4.17)\nwhere the matrix\nB = BH,G,C,\n80\n\ndepends only on C and the root systems of H and G, but not on π and ̄λ,\nand the coordinates of the vector\nc = mW, ̄\nW,C( ̄λ, π, ρ, ̄ρ),\ndepend on W, ̄W, C, ρ, ̄ρ, π, π, and furthermore, their dependence on π, ̄λ, ρ, ̄ρ\nis linear.\nHere assumption (A) is crucial. Without it, the sum (4.16) can diverge. Of\ncourse, without assumption (A), we can still make the sum finite, by requir-\ning that ̄μ lie within the convex hull H ̄λ generated by the points { ̄W( ̄λ)},\nwhere ̄W ranges over all Weyl group elements. This means we have to add\nconstraints to the system (4.17) corresponding to the facets of H ̄λ.\nBut\nthe entries of the resulting B would depend on ̄λ, and the theory of vector\npartition functions will no longer apply.\nProof of the claim: Let ̄μi’s denote the integer coordinates of ̄μ in the basis\nof fundamental weights.\nWe denote the integer vector ( ̄μ1, ̄μ2, · · · ) by ̄μ\nagain. The Kostant partition function P(ν) is a vector partition function\nassociated with an integer programming problem:\nBPv = ν,\nv ≥0,\nwhere the columns of BP correspond to positive roots of G. The sum in\n(4.16) is equal to the number of integral pairs ( ̄μ, v) such that\n1. ̄μ ∈C,\n2. ̄μ ↓π −ρ −W(ρ),\n3. BPv = ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ), v ≥0.\nThe first two condititions here can be expressed in terms of linear con-\nstraints (equalities and inequalities) on the coordinates ̄μi’s. Thus the three\nconditions together can be expressed in terms of linear constraints on ( ̄μ, v).\nBy the finiteness assumption (A), the polytope determined by these con-\nstraints is a bounded polytope.\nThe number of integer points in such a\npolytope can be expressed as a vector partition function (cf. [BBCV]). This\nproves the claim.\nLet us denote the vector partition associated with the integer program-\nming problem (4.17) in the claim by φW, ̄\nW,C(c( ̄λ, π, ρ, ̄ρ)). Then\nmπ ̄λ =\nX\nW\nX\n ̄\nW\n(−1)W (−1)\n ̄\nW X\nC\na(C)φW, ̄\nW,C(c( ̄λ, π, ρ, ̄ρ)).\n(4.18)\n81\n\nHence,\n ̃mπ ̄λ(n) = mn ̄λ\nnπ =\nX\nW\nX\n ̄\nW\n(−1)W (−1)\n ̄\nW X\nC\na(C)φW, ̄\nW,C(c(n ̄λ, nπ, ρ, ̄ρ)).\n(4.19)\nIt follows from Claim 4.5.1 and the standard results on vector partition\nfunctions mentioned in the begining of this section that\ngW, ̄\nW,C(n) = φW, ̄\nW,C(c(n ̄λ, nπ, ρ, ̄ρ)),\nis asymptitically a quasipolynomial function of n.\nHence, ̃mπ ̄λ(n) is also\nasymptotically a quasipolynomial function of n.\nThis implies (cf.\n[St1])\nthat\nMπ ̄λ (t) =\nX\nn≥0\n ̃mπ ̄λ(n)tn\n(4.20)\nis rational function of t.\nThis proves Theorem 3.4.1 (a) under the finiteness assumption (A).\nIt remains to remove the assumption (A). Let G′ ⊇H be the smallest\nLevi subalgebra of G containing H. Then\nmπ ̄λ =\nX\nπ′\nmπ′\n ̄λ mπ\nπ′,\n(4.21)\nwhere π′ ranges over dominant weights of G′, mπ′\n ̄λ denotes the multiplicity of\nVπ′(G′) in V ̄λ(G), and mπ\nπ′ the multiplicity of Vπ(H) in Vπ′(G′). Furthermore,\n1. the finiteness asssumption (A) is now satisfied for the pair (G′, H): i.e.,\nfor any weight μ of H, the number of weights μ′’s of G′ such that μ′ ↓μ\nis finite.\n2. There is a polyhedral expression for mπ′\n ̄λ ; this follows from [Li, Dh].\nBy the first condition and the argument above, we get an expression for\nmπ\nπ′ akin to (4.18). Substituting this expression and the polyhedral expres-\nsion for mπ′\n ̄λ in (4.21), leads to a formula for ̃mπ ̄λ(n) as a linear combination\nof φB,c,b(n)’s for appropriate B, c, b’s. After this, we proceed as before.\nThis proves Theorem 3.4.1 (a). Q.E.D.\nWe also note down the following consequence of the proof.\nProposition 4.5.2 There is a constant D depending only G and H, such\nthat for any ̄λ, π, orders of the poles of Mπ ̄λ (t) (cf. (4.20), as roots of unity,\ndivide D.\n82\n\nA bound on D provided by the proof below is very weak: D = O(2O(rank(G))).\nProof: It suffices to to bound the period of the quasipolynomial ̃mπ ̄λ(n). For\nthis, it suffices to let n →∞. For a fixed W, ̄\nW , C, the chamber containing\nc(n ̄λ, nπ, ρ, ̄ρ)) is completely determined by ̄λ and π as n →∞. Under these\nconditions, the degree of φW, ̄\nW,C(c(n ̄λ, nπ, ρ, ̄ρ)) is equal to the dimension of\nthe polytope associated with this vector partition function. This dimension\nis clearly O(rank(G)2).\nBy Szenes-Vergne residue formula [SV], there is a constant D depending\non only G, H, W, ̄\nW , C, such that the period of the quasipolynomial h(n) =\nφW, ̄\nW,C(c(n ̄λ, nπ, 0, 0)) divides D for every ̄λ, π; here we are putting ρ and\n ̄ρ equal to zero, since we are interested in what happens as n →∞. Q.E.D.\n83\n\nChapter 5\nParallel and PSPACE\nalgorithms\nIn this chapter we give PSPACE algorithms (cf. Theorem 3.4.3) for com-\nputing the various structural constants under consideration . We shall only\nprove Theorem 3.4.3, when H is therein is either a complex, semisimple\ngroup, or a symmetric group, or a general linear group over a finite field,\nthe extension to the general case being routine.\nWe recall two standard results in parallel complexity theory [KR], which\nwill be used repeatedly.\nLet NC(t(N), p(N) denote the class of problems that can be solved\nin O(t(N)) parallel time using O(p(N)) processors, where N denotes the\nbitlength of the input. Let\nNC = ∪iNC(logi(N), poly(N)).\nThis is the class of problems having efficient parallel algorithms.\nProposition 5.0.3 [Cs, KR] Let A be an n × n-matrix with entries in a\nring R of characteristic zero. Then the determinant of A, and A−1, if A\nis nonsingular, can be computed in O(log2 n) parallel steps using poly(n)\nprocessors; here each operation in the ring is considered one step. Hence, if\nR = Q, the problems of computing the determinant, the inverse and solving\nlinear systems belong to NC.\nProposition 5.0.4 The class NC(t(N), 2t(N)) ⊆SPACE(O(t(N))).\nIn\nparticular, NC(poly(N), 2O(poly(N))) ⊆PSPACE.\n84\n\n5.1\nComplex semisimple Lie group\nIn this section we prove a special case of Theorem 3.4.3 for the general-\nized plethym problem (Problem 1.1.2). Accordingly, let H be a complex,\nsemisimple, simply connected Lie group, G = GL(V ), where V = Vμ(H) is\nan irreducible representation of H with dominant weight μ, ρ : H →G the\nhomomorphism corresponding to the representation, and mπ\nλ the multiplic-\nity of Vπ(H) in Vλ(G), considered as an H-module via ρ; cf. Problem 1.1.3.\nThen:\nTheorem 5.1.1 The multiplicity mπ\nλ can be computed in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, dim(H))\nspace.\nHere it is assumed that the partition λ = λ1 ≥λ2 ≥· · · λr > 0 is rep-\nresented in a compact form by specifying only its nonzero parts λ1, . . . , λr.\nThis is important since dim(G) can be exponential in dim(H) and ⟨μ⟩. A\ncompact representation allows ⟨λ⟩to be small, say poly(dim(H), ⟨μ⟩), in this\ncase.\nWe begin with a simpler special case.\nProposition 5.1.2 If dim(V ) = poly(dim(H)), then mπ\nλ can be computed\nin PSPACE; i.e., in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, dim(H)) space.\nThis implies that the Kronecker coefficient (Problem 1.1.1) can be computed\nin PSPACE.\nProof: Let us use the notation ̄λ instead of λ to be consistent with the\nnotation used in Klimyk’s formula (4.12). By the latter, mπ ̄λ can be computed\nin PSPACE if n ̄μ(V ̄λ) in that formula can be computed in PSPACE for every\n ̄μ and ̄λ. In type A, this is just the number of Gelfand-Tsetlin tableau with\nthe shape ̄λ and weight ̄μ.\nIf dim(V ) = poly(dim(H)), the size of such\na tableau is O(dim(V )2) = poly(dim(H)).\nSo we can count the number\nof such tableu in PSPACE as follows: Begin with a zero count, and cycle\nthrough all tableaux of shape ̄λ in polynomial space one by one, increasing\nthe count by one everytime the tableau satisfies all constraints for Gelfand-\nTsetlin tableau and has weight ̄μ. In general, the role of Gelfand-Tsetlin\ntableaux is played by Lakshmibai-Seshadri (LS) paths [Li, Dh]. Q.E.D.\nThe argument above does not work if dim(V ) is not poly(dim(H)), as\nin the plethym problem (Problem 1.1.2), where dim(V ) = dim(Vμ) can\nbe exponential in n = dim(H) and the bitlength of μ. In this case, the\n85\n\nalgorithm cannot even afford to write down a tableau since its size need not\nbe polynomial.\nNext we turn to Theorem 5.1.1.\nFor the sake of simplicity, we shall\nprove it only for H = SLn(C), or rather GLn(C)–i.e., the usual plethysm\nproblem. This illustrates all the basic ideas. The general case is similar. We\nshall prove a slightly stronger result in this case:\nTheorem 5.1.3 The plethysm constant aπ\nλ,μ can be can be computed in\npoly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) space.\nHere the dependence on n = dim(H) is not there. This makes a difference\nif the heights of μ and π are less than n = dim(H)–remember that we are\nusing a compact representation of a partition in which only nonzero parts\nare specified. This is really not a big issue. Because aπ\nλ,μ depends only on\nthe partitions λ, μ, π and not n. Hence, without loss of generality, we can\nassume that n is the maximum of the heights of μ and π. It is possible to\nstrengthen Theorem 5.1.1 similarly.\nTo prove Theorem 5.1.3, we shall give an efficient parallel algorithm to\ncompute ̃aπ\nλ,μ that works in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) parallel time using O(2poly(⟨λ⟩,⟨μ⟩,⟨π⟩))\nprocessors. This will show that the problem of computing ̃aπ\nλ,μ is in the com-\nplexity class NC(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩), 2poly(⟨λ⟩,⟨μ⟩,⟨π⟩)), which is contained in\nPSPACE by Proposition 5.0.4. The basic idea is to parallelize the classical\ncharacter-based algorithm for computing aπ\nλ,μ by using efficient parallel algo-\nrithm for inverting a matrix and solving a linear system (Proposition 5.0.3).\nWe begin by recalling the standard facts concerning the characters of\nthe general linear group.\nGiven a representation W of GLm(C), let ρ :\nGLm(C) →GL(W) be the representation map.\nLet χρ(x1, . . . , xm) de-\nnote the formal character of this representation W. This is the trace of\nρ(diag(x1, . . . , xm)), where diag(x1, . . . , xn) denotes the generic diagonal ma-\ntrix with variable entries x1, . . . , xm on its diagonal. If W is an irreducible\nrepresentation Vλ(GLm(C)), then χρ(x1, . . . , xm) is the Schur polynomial\nSλ(x1, . . . , xm). By the Weyl character formula,\nSλ =\n|xλi+m−i\nj\n|\n|xm−i\nj\n| ,\n(5.1)\nwhere |ai\nj| denotes the determinant of an m×m-matrix a. The Schur polyno-\nmials form a basis of the ring of symmetric polynomials in x1, . . . , xm. The\n86\n\nsimplest basis of this ring consists of the complete symmetric polynomials\nMβ(x1, . . . , xm) defined by\nMβ(x1, . . . , xm) =\nX\nγ\ntγ,\nwhere γ ranges over all permutations of β and tγ = Q\ni xγi\ni . Schur polyno-\nmials are related to Mβ by:\nSλ =\nX\nβ\nkβ\nλMβ,\n(5.2)\nwhere kβ\nλ is the Kostka number. This is the number of semistandard tableau\nof shape λ and weight β.\nIf the representation W is reducible, its decomposition into irreducibles\nis given by:\nW =\nX\nπ\nm(π)Vπ(GLn(C)),\n(5.3)\nwhere m(π)’s are the coefficients of the formal character χρ(x1, . . . , xm) in\nthe Schur basis:\nχρ =\nX\nπ\nm(π)Sπ.\nProof of Theorem 5.1.3\nLet λ, μ, π be as in Theorem 5.1.3. Let H = GLn(C), V = Vμ(H), G =\nGL(V ). Let sλ(x1, . . . , xm) be the formal character of the representation\nVλ(G) of G. Here m = dim(Vμ) can be exponential in n and ⟨μ⟩. The basis\nof Vμ(H) is indexed by semistandard tableau of shape μ with entries in [1, n].\nLet us order these tableau, say lexicographically, and let Ti, 1 ≤i ≤m,\ndenote the i-th tableau in this order. With each tableau T, we associate a\nmonomial\nt(T) =\nn\nY\ni=1\ntwi(T)\ni\n,\nwhere wi(T) denotes the number of i’s in T. Given a polynomial f(x1, . . . , xm),\nlet us define fμ = fμ(t1, . . . , tn) to be the polynomial obtained by substi-\ntuting xi = t(Ti) in f(x1, . . . , xm). Then the formal character of Vλ(G),\nconsidered as an H-representation of via the homomorphism H →G =\n87\n\nGL(Vμ(H)), is the symmetric polynomial Sλ,μ(t1, . . . , tn) = (Sλ)μ.\nThe\nplethysm constant aπ\nλ,μ is defined by:\nSλ,μ(t1, . . . , tn) =\nX\nπ\naπ\nλ,μSπ(t1, . . . , tn).\n(5.4)\nAn efficient parallel algorithm to compute aπ\nλ,μ is as follows. Here by an\nefficient parallel algorithm, we mean an algorithm that works in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)\ntime using 2poly(⟨λ⟩,⟨μ⟩,⟨π⟩) processors.\nWe will repeatedly use Proposi-\ntion 5.0.3.\nAlgorithm\n(1) Compute Sλ,μ(t1, . . . , tn). By the Weyl character formula (5.1),\nSλ,μ(t1, . . . , tn) = Aλ,μ(t1, . . . , tn)\nBλ,μ(t1, . . . , tn),\nwhere Aλ(x1, . . . , xm) and Bλ(x1, . . . , xm) denote the numerator and denom-\ninator in (5.1), and Aλ,μ = (Aλ)μ, and Bλ,μ = (Bλ)μ. Let R = C[t1, . . . , tn].\nThen\nAλ,μ(t1, . . . , tn) = |t(Tj)λi+m−i|.\nThis is the determinant of an m × m matrix with entries in R, where m =\ndim(V ) can be exponential in n and ⟨μ⟩. It can be evaluated in O(log2 m)\nparallel ring operations using poly(m) processors. Each ring element that\narises in the course of this algorithm is a polynomial in t1, . . . , tn of total\ndegree O(|λ|m), where |λ| denotes the size of λ. The total number of its\ncoefficients is r = O((|λ|m)n). Hence each ring operation can be carried\nout efficiently in O(log2(r)) parallel time using poly(r) processors. Since\nlog m = poly(n, ⟨μ⟩) and log r = poly(n, ⟨λ⟩, ⟨μ⟩), it follows that Aλ,μ can\nbe evaluated in poly(n, ⟨μ⟩, ⟨λ⟩) parallel time using 2poly(n,⟨μ,λ⟩) processors.\nThe determinant Bλ,μ can also be computed efficiently in parallel in a similar\nfashion. To compute Sλ,μ, we have to divide Aλ,μ by Bλ,μ. This can be done\nby solving an r × r linear system, which, again, can be done efficiently in\nparallel. This computation yields representation of Sλ,μ in the monomial\nbasis {Mβ} of the ring of symmetric polynomials in t1, . . . , tn.\n(2) To get the coefficients aπ\nλ,μ, we have to get the representation of Sλ,μ(t)\nin the Schur basis. This change of basis requires inversion of the matrix\nin the linear system (5.2). The entries of the matrix K occuring in this\n88\n\nlinear system are Kostka numbers. Each Kostka number can be computed\nefficiently in parallel.\nHence, all entries of this matrix can be computed\nefficiently in parallel. After this, the matrix can be inverted efficiently in\nparallel, and the coefficients aπ\nλ,μ’s of Sλ,μ in the Schur basis can be computed\nefficiently in parallel. Finally, we use Proposition 5.0.4 to conclude that aπ\nλ,μ\ncan be computed in PSPACE. Q.E.D.\n5.2\nSymmetric group\nNext we prove Theorem 3.4.3 when H = Sm.\nLet X = Vμ(Sm) be an\nirreducible representation (the Specht module) of Sm corresponding to a\npartition μ of size m.\nLet ρ : H →G = GL(X) be the corresponding\nhomomorphism.\nTheorem 5.2.1 Given partitions λ, μ, π, where μ and π have size m, the\nmultiplicity mπ\nλ,μ of the Specht module Vπ(Sm) in Vλ(G) can be computed in\npoly(m, ⟨λ⟩) space.\nRemark 5.2.2 The bitlengths ⟨μ⟩and ⟨π⟩are not mentioned in the com-\nplexity bound because they are bounded by m.\nFor this, we need three lemmas.\nLemma 5.2.3 The character of a symmetric group can be computed in\nPSPACE.\nSpecifically, given a partition π of size m, and a sequence\ni = (i1, i2, . . .) of nonnegative integers such that P jij = m, the value of\nthe character χπ of Sm on the conjugacy class Ci of permutations indexed\nby i can be computed in poly(m) parallel time using 2poly(m) processors.\nHence it can be computed in poly(m) space (cf. Proposition 5.0.4).\nHere the conjugacy class Ci consists of those permutations that have i1\n1-cycles, i2 2-cycles, and so on.\nProof: Let k be the height of the partition π. Let x = (x1, . . . , xk) be the\ntuple of variables xi’s. Given a formal series f(x) and a tuple (l1, . . . , lk) of\nnonnegative integers, let [f(x)](l1,...,lk) denote the coefficient of xl1\n1 · · · xlk\nk in\nf.\nBy the Frobenius character formula [FH],\nχλ(Ci) = [f(x)](l1,...,lk),\n(5.5)\n89\n\nwhere\nl1 = π1 + k −1, l2 = π2 + k −2, . . . , lk = πk,\nand\nf(x) = ∆(x)\nm\nY\nj=1\nPj(x)ij,\nwith\n∆(x)\n=\nQ\ni<j(xi −xj),\nPj(x)\n=\nxj\n1 + · · · + xj\nk.\n(5.6)\nSince deg(f) = poly(m) and k ≤m, the total number of coefficients of\nf(x) is 2poly(m). Hence, we can evaluate f(x) in PSPACE by setting up\nappropriate recurrence relations.\nAlternatively, we can easily evaluate f(x) in poly(m) parallel time using\n2poly(m) processors, and then extract its required coefficient. After this, the\nresult follows from Proposition 5.0.4. Q.E.D.\nLemma 5.2.4 Suppose φ is a character of Sm whose value on any conju-\ngacy class Ci can be computed in O(s) space, for some parameter s. Then,\nthe multiplicity of the representation Vπ(Sm) in the representation Vφ(Sm)\ncorresponding φ can be computed in O(poly(m) + s) space.\nProof: The multiplicity is given by the inner product\n⟨φ, χπ⟩= 1\nm!\nX\nσ∈Sm\n ̄\nφ(σ)χπ(σ).\n(5.7)\nBy assumption, φ(σ) can be computed in O(s) space, and by Lemma 5.2.3,\nχπ(σ) can be computed in poly(m) space. Hence, the result follows from\nthe preceding formula. Q.E.D.\nGiven an irreducible representation X = Vμ(Sm) and an irreducible rep-\nresentation W = Vλ(G) of G = GL(X)), let ρμ denote the representation\nmap Sm →G, ρλ the representation map G →GL(W), and\nρ : Sm →G →GL(W)\ntheir composition. This is a representation of Sm. Let χρ be the character\nof ρ.\nLemma 5.2.5 For any σ ∈Sm, χρ(σ) can be computed in poly(m, ⟨λ⟩) in\npoly(m, ⟨λ⟩) space.\n90\n\nThe bitlength ⟨μ⟩is not mentioned in the complexity bound because it\nis bounded by m.\nProof: Let r = dim(X). The formal character of the representation Vλ(G)\nof G = GL(X) is the Schur polynomial Sλ(x1, . . . , xr), r = dim(X). Hence,\nχρ(σ) = Sλ(α)\nwhere α = (α1, . . . , αr) is the tuple of eigenvalues of ρμ(σ). We shall compute\nthe right hand side fast in parallel–i.e., in poly(m, ⟨λ⟩) parallel time using\n2poly(m,⟨λ⟩) processors–and then use Proposition 5.0.4 to conclude the proof.\nThis is done as follows.\n(1) Let χμ denote the character of the representation ρμ. Let pi(α) = αi\n1 +\n· · · + αi\nr denote the i-th power sum of the eigenvalues. For any i,\npi(α) = χμ(σi).\nWe can compute σi, for i ≤|λ|, where |λ| denotes the size of λ, in poly(log i, m) =\npoly(m, ⟨λ⟩) time using repeated squaring. After this χμ(σi) can be com-\nputed fast in parallel in poly(m) time using Lemma 5.2.3. Thus each pi(α)\ncan be computed in poly(m, ⟨λ⟩) time in parallel using 2poly(m,⟨λ⟩) proces-\nsors. We calculate pi(α) in parallel for all i ≤|λ|, and all pγ(α) = Q\nj pγj(α)\nin parallel for all partitions γ of size at most m.\n(2) After this, we calculate the complete symmetric function hi(α), for\neach i ≤|λ|, fast in parallel, by using the relation [Mc]:\nhi =\nX\n|γ|=i\nz−1\nγ pγ,\nwhere zγ = Q\ni≥1 imimi!, and mi = mi(γ) denotes the number of parts of γ\nequal to i. Thus we can calculate hγ(α) = Q\nj hγj(α), for all partitions γ of\nsize m, fast in parallel.\n(3) To compute Sλ(α), we recall that the transition matrix between\nthe Schur basis {Sλ} and the complete symmetric basis {hγ} of the ring\nof symmetric functions is K∗, the transpose inverse of the Kostka matrix\nK = [Kλ,γ], where Kλ,γ denote the Kostka number; cf. [Mc]. As we noted in\nthe proof of Theorem 5.1.3, each Kostka number can be computed in fast in\nparallel. Hence, K can be computed fast in parallel. After this, its inverse\nK−1 can be computed fast in parallel by Proposition 5.0.3–this is the crux\nof the proof–and finally K∗as well. Thus Sλ(α) can be computed fast in\nparallel, since each hγ(α) can be computed fast in parallel. Q.E.D.\nTheorem 5.2.1 follows from Lemma 5.2.3,5.2.4 and 5.2.5. Q.E.D.\n91\n\n5.3\nGeneral linear group over a finite field\nIn this section we prove Theorem 3.4.3, when H therein is the general lin-\near group GLn(Fpk) over a finite field Fpk. Irreducible representations of\nH = GLn(Fpk) have been classified by Green [Mc]. They are labelled by\ncertain partition-valued functions. See [Mc] for a precise description of these\nlabelling functions. It is clear from the description therein that each labelling\nfunction has a compact representation of bitlength O(n + k + ⟨p⟩), where\n⟨p⟩= log2 p; we specify a function by giving its partition values at the places\nwhere it is nonzero. Let μ denote any such label. Let X = Vμ(H) be the\ncorresponding irreducible representation of H, and ρ : H →G = GL(X)\nthe corresponding homomorphism.\nTheorem 5.3.1 Given a partition λ and labelling functions μ and π as\nabove, the multiplicity mπ\nλ,μ of the irreducible representation Vπ(H) in Vλ(G)\ncan be computed in poly(n, k, ⟨p⟩, ⟨λ⟩) space.\nThe proof is similar to that of Theorem 5.2.1 for the symmetric group\nwith the following result playing the role of Lemma 5.2.3:\nLemma 5.3.2 Given a label γ of an irreducible character χγ of H = GLn(Fpk)\nand a label δ of a conjugacy class in H, the value χγ(δ) can be computed\nin poly(n, k, ⟨p⟩) parallel time using 2poly(n,k,⟨p⟩) processors, and hence by\nProposition 5.0.4, in poly(n, k, ⟨p⟩) space.\nThe label δ of a conjugacy class in H is also a partition valued function\n[Mc], which admits a compact representation of bitlength poly(n, k, ⟨p⟩).\nProof: We shall parallelize Green’s algorithm [Mc] for computing the charac-\nter values, and then conclude by Proposition 5.0.3. Green shows that χγ(δ)’s\nare entries of a transition matrix between a two polynomial bases: the first\nconstructed using Hall-Littlewood polynomials, and the second using Schur\npolynomials. We have construct this transition matrix fast in parallel. We\nshall only indicate here how the transition matrix between the basis of Hall-\nLittlewood polynomials and the Schur basis for the ring symmetric functions\nover Z[t] can be constructed fast in parallel. This idea can then be easily\nextended to complete the proof.\nFirst, we recall the definition of the Hall-Littlewood polynomial Pπ(x; t) =\nPπ(x1, . . . , xk; t) [Mc].\nThis is a symmetric polynomial in xi’s with co-\nefficients in Z[t].\nIt interpolates between the Schur function sπ(x) and\n92\n\nthe monomial symmetric function mπ(x) because Pπ(x; 0) = sπ(x) and\nPπ(x; 1) = mπ(x). The formal definition is as follows:\nFor a given partition π, let vπ(t) = Q\ni≥0 vmi(t), where mi is the number\nof parts of π equal to i, and\nvm(t) =\nm\nY\ni=1\n1 −ti\n1 −t .\nThen\nPπ(x; t) = Aπ(x, t)\nBπ(x, t),\n(5.8)\nwhere\nAπ(x, t) = P\nσ∈Sk sgn(σ)σ(xπ1\n1 · · · xπk\nk\nQ\ni<j xi −txj,\nBπ(x, t) = vπ(t) Q\ni<j(xi −xj).\n(5.9)\nHere sgn(σ) denotes the sign of σ.\nLet wπ,α(t)’s be the coeffcients of Pπ(x, t) in the Schur basis:\nPπ(x; t) =\nX\nα\nwπ,α(t)sα(x).\n(5.10)\nWe want to calculate the matrix W = [wπ,α] fast in parallel.\nUsing\nformula (5.9), we calculate Aπ(x; t) fast in parallel; i.e., we calculate the\ncoefficients of Aπ(x; t) in the basis of monomials in x and t. We calculate\nBπ(x; t) similarly.\nAfter this the division in (5.8) can be carried out by\nsolving a an appropriate linear system. This can be done fast in parallel\nby Proposition 5.0.3.\nSince, Pπ(x; t) is symmetric in xi’s, this yields its\ncoefficients in the monomial symmetric basis {mα(x)} with the coefficients\nbeing in Z[t]. The transition matrix [Mc] from the monomial symmetric\nbasis to the Schur basis is given by the inverse of the Kostka matrix. This\ninverse can be computed fast in parallel by Proposition 5.0.3. After this,\nthe coefficients wπ,α’s of Pπ(x; t) in the Schur basis can be computed fast in\nparallel.\nFurthermore, the inverse of W = [Wπ,α] can also be computed fast in\nparallel by Proposition 5.0.3. Q.E.D.\n5.3.1\nTensor product problem\nAnalogue of the Kronecker problem (Problem 1.1.1) for H = GLn(Fpk) is:\n93\n\nProblem 5.3.3 Given partition valued functions λ, μ, π, decide if the mul-\ntiplicity bπ\nλ,μ of Vπ(H) in the tensor product Vλ(H) ⊗Vμ(H) is nonzero.\nIn this context:\nTheorem 5.3.4 The multiplicity mπ\nλ,μ can be computed in PSPACE; i.e.,\nin poly(n, k, ⟨p⟩) space.\nProof: This follows from Lemma 5.3.2 and analogues of Lemmas 5.2.4 and\n5.2.5 in this setting. Q.E.D.\nA possible canditate for a stretching function assoociated with bπ\nλ,μ is:\n ̃bπ\nλ,μ(n) = bnπ\nnλ,nμ,\nwhere nλ denotes the stretched partition-valed function obtained by stretch-\ning each partition value of λ by a factor of n. In other words ̃bπ\nλ,μ(n) is\nthe multiplicity of Vnπ(H(n)) in Vnλ(H(n)) ⊗Vnμ(H(n)), where H(n) =\nGLnm(Fpk) is the stretched group. Is it a quasi-polynomial? If so, we can\nalso ask for a good bound on its saturation and positivity indices.\n5.4\nFinite simple groups of Lie type\nThe work of Deligne-Lusztig [DL] and Lusztig[Lu5] yield an algorithm for\ncomputing the character values for finite simple groups of Lie type.\nQuestion 5.4.1 Can this algorithm be parallelized?\nIf so, Lemma 5.3.2, and hence Theorem 5.3.1, can be extended to finite\nsimple groups of Lie type.\n94\n\nChapter 6\nExperimental evidence for\npositivity\nIn this chapter we give experimental evidence for positivity (PH2,3).\n6.1\nLittlewood-Richardson problem\nExperimental evidence for PH2 in the context of the Littlewood-Richardson\nproblem (Problem 1.2.1) has been given in [DM2], and for PH3 in type A\nin [KTT]. We give experimental evidence for PH3 in types B, C, D here.\nLet Cλ\nα,β(t) be as in eq.(1.2). Its reduced positive form for various values\nof α, β, λ is shown in Figure 6.1 for type B, in Figure 6.2 for type C, and\nFigure 6.3 for type D. The rank of the Lie algebra is three in all cases.\nIn these types, the period of the stretching quasipolynomial ̃cλ\nα,β(n) is at\nmost two. Accordingly, the period of every pole of Cλ\nα,β(t) is at most two.\nThe tables were computed from the tables in [DM2] for the stretching quasi-\npolynomial ̃cλ\nα,β(n) in these cases.\n6.2\nKronecker problem, n = 2\nLet kπ\nλ,μ be the Kronecker coefficient in Problem 1.1.1. Let ̃kπ\nλ,μ(n) = ̃knπ\nnλ,nμ\nbe the associated stretching quasi-polynomial, and\nKπ\nλ,μ(t) =\nX\nn≥0\n ̃kπ\nλ,μ(n)tn,\n95\n\nthe associated rational function. An explicit formula (with alternating signs)\nfor the Kronecker coefficient, when n = 2, has given by Remmel and White-\nhead [RW] and Rosas [Ro], and a positive formula in [GCT9]. This case\nturns out to be nontrivial. For example, the number of chambers (domains)\nof quasi-polynomiality in this case turns out to be more than a million. Their\nexplicit description can be found out using the formula for the Kronecker\ncoefficient in [RW].\nWe implemented Rosas’ formula to check PH2 for the quasipolynomial\n ̃kπ\nλ,μ(n) for a few thousand values of μ, ν and λ with the help of a computer.\nA large number of samples was chosen to ensure that a significant fraction\nof the chambers were sampled. The quasi-polynomial ̃kπ\nλ,μ(n) and a positive\nform of the rational function Cπ\nλ,μ(t) are shown Figures 6.4 and 6.5 for few\nsample values of λ = (λ1, λ2), μ = (μ1, μ2), and π = (π1, π2, π3, π4).\nIt\nmay be noted that ̃kπ\nλ,μ(n) need not be a polynomial; this answers Kirillov’s\nquestion [Ki] in the negative. But its period is at most two for n = 2. This\nfollows from the formula in [RW].\nFor the λ, μ and π that we sampled,\npositivity index of ̃kπ\nλ,μ(n) is always zero.\nBut it turns out [BOR] that\nthere are some λ, μ and π for which the saturation and positivity indices of\n ̃kπ\nλ,μ(n) are nonzero (one), but very small and thus consistent with SH and\nPH2 (Hypothesis1.6.6) in this paper; in the earlier version of this paper, SH\nand PH2 stipulated that the saturation and positivity indices are always\nzero. These (λ, μ, π) escaped our random sampling, because their density is\nextremely small [BOR].\n6.3\nG/P and Schubert varieties\nLet V = Vλ(G) be an irreducible representation of G = SLk(C) correspond-\ning to a partition λ.\nLet vλ be the point in P(V ) corresponding to the\nhighest weight vector, and X = Gvλ ∼= G/Pλ its closed orbit. Let hk,λ(n)\nbe the Hilbert function of the homogeneous coordinate ring R of X. It is\na quasipolynomial since spec(R) has rational singularities. In fact, it is a\npolynomial, since t = 1 is the only pole of the Hilbert series\nHk,λ(t) =\nX\nn≥0\nhk,λ(n)tn.\nFigure 6.6 gives experimental evidence for strict positivity (PH2) of hk,λ(n)\n(as discussed in Section 3.5.5) for a few sample values of k and λ. Figure 6.7\ngives experimental evidence for strict positivity of the Hilbert polynomial\n96\n\nof the Schubert subvarieties of the Grassmanian; there Gn,k denotes the\nGrassmannian of k-planes in V = Cn, and Ωa, a = (a(1), . . . , a(d)) its\nSchubert subvariety consisting of the k-subspaces W such that dim(W ∩\nVn−k+i−a(i)) ≥i for all i, where V = Vn ⊃· · · V1 ⊃0 is a complete flag of\nsubspaces in V . The Hilbert polynomials were computed using the explicit\npolyhedral interpretation for them deduced from the theory of algebras with\nstraightening laws (Hodge algebras) [DEP2].\n6.4\nThe ring of symmetric functions\nLet V = Ck, G = GL(V ), H = Sk, with the natural embedding H →\nG.\nLet us consider the spacial case of the subgroup restriction problem\n(Problem 1.1.3), with Vλ(G) = V , and Vπ(H) the trivial representation of\nH.\nThen s = mπ\nλ, the multiplicity of the trivial representation in V , is\none. Though the decision problem (Problem 1.1.3) is trivial in this case, the\ncanonical model associated with s is nontrivial.\nThe canonical rings R = R(s) and S = S(s) associated with s in this\ncase coincide with C[V ] = C[x1, . . . , xk].\nThe ring T = T(s) = SH =\nC[x1, . . . , xk]Sk is the subring of symmetric functions. Its Hilbert function\nh(n) is a quasipolynomial. PH1 and PH3 for Z = Proj(T), as per Defini-\ntion 3.5.2, follow easily, the latter from the well known rational generating\nfunction for the partition function [St1]. But PH2 turns out to be nontriv-\nial. Figures 6.8-6.13 give experimental evidence for strict positivity of h(n)\n(PH2). In these figures, the i-th row of the table for a given k shows hi(n),\nwhere hi(n), 1 ≤i ≤l, are such that h(n) = hi(n), when n = i modulo the\nperiod l of h(n).\n97\n\nα\nβ\nλ\nCλ\nα,β(t)\n(0, 15, 5)\n(12, 15, 3)\n(6, 15, 6)\n350 t8+19121 t7+123576 t6+297561 t5+342064 t4+192779 t3+46992 t2+2641 t+1\n(1−t)3(1−t2)3\n(4, 8, 11)\n(3, 15, 10)\n(10, 1, 3)\n1+5 t+6 t2+t3\n(1−t2)3\n(8, 1, 3)\n(11, 13, 3)\n(8, 6, 14)\n2 t8+45 t7+259 t6+591 t5+773 t4+522 t3+198 t2+29 t+1\n(1−t)3(1−t2)4\n(8, 9, 14)\n(8, 4, 5)\n(1, 5, 15)\n136 t9+3422 t8+20204 t7+53608 t6+76076 t5+60986 t4+26674 t3+5568 t2+345 t+1\n(1−t)3(1−t2)4\n(10, 5, 6)\n(5, 4, 10)\n(0, 7, 12)\n219 t8+12135 t7+79231 t6+193003 t5+223919 t4+127907 t3+31704 t2+1870 t+1\n(1−t)6(1+t)3\n \n \nFigure 6.1: Cλ\nα,β(t) for B3\n98\n\nα\nβ\nλ\nCλ\nα,β(t)\n(1, 13, 6)\n(14, 15, 5)\n(5, 11, 7)\n18145 t8+267151 t7+1070716 t6+1917716 t5+1735692 t4+778184 t3+144596 t2+5538 t+1\n(1−t)4(1−t2)3\n(4, 15, 14)\n(12, 12, 10)\n(4, 9, 8)\n2280 t9+267658 t8+2746131 t7+9276935 t6+14682332 t5+11903923 t4+4746803 t3+751126 t2+21249 t+1\n(1−t)4(1−t2)3\n(9, 0, 8)\n(8, 12, 9)\n(7, 7, 3)\n3 t2+4 t+1\n(1−t)6\n(10, 2, 7)\n(8, 10, 1)\n(7, 5, 5)\n8984 t8+132826 t7+534183 t6+960491 t5+873227 t4+394045 t3+74067 t2+2941 t+1\n(1−t)4(1−t2)3\n(10, 10, 15)\n(11, 3, 15)\n(10, 7, 15)\n7162 t9+736327 t8+7178960 t7+23540366 t6+36359642 t5+28788904 t4+11166361 t3+1693696 t2+43515 t+1\n(1−t)7(1+t)3\n \n \nFigure 6.2: Cλ\nα,β(t) for C3\n99\n\nα\nβ\nλ\nCλ\nα,β(t)\n(0, 15, 5)\n(12, 15, 3)\n(6, 15, 6)\n633 t7+24259 t6+142236 t5+252113 t4+168220 t3+36131 t2+1414 t+1\n(1−t)7(1−t2)\n(4, 8, 11)\n(3, 15, 10)\n(10, 1, 3)\n7962 t8+503679 t7+4525372 t6+11944350 t5+12218255 t4+4879052 t3+586370 t2+10862 t+1\n(1−t)8(1−t2)\n(8, 1, 3)\n(11, 13, 3)\n(8, 6, 14)\n81 t7+19407 t6+211964 t5+513585 t4+426652 t3+110317 t2+4609 t+1\n(1−t)6(1−t2)\n(8, 9, 14)\n(8, 4, 5)\n(1, 5, 15)\n9 t2+8 t+1+2 t3\n(1−t)3\n(10, 5, 6)\n(5, 4, 10)\n(0, 7, 12)\n3647 t7+111208 t6+570739 t5+920201 t4+560336 t3+106748 t2+3435 t+1\n(1−t)8(1+t)\n \n \nFigure 6.3: Cλ\nα,β(t) for D3\n100\n\nλ1\nλ2\nμ1\nμ2\nπ1\nπ2\nπ3\nπ4\n ̃kπ\nλ,μ(n); n odd\n ̃kπ\nλ,μ(n); n even\nKπ\nλ,μ(t)\n87\n62\n97\n52\n64\n39\n24\n22\n1/2 + 4 n + 11/2 n2\n1 + 4 n + 11/2 n2\n1+8 t+11 t2+2 t3\n(1−t)2(1−t2)\n104\n95\n149\n50\n95\n78\n15\n11\n1/2 + 13/2 n + 18 n2\n1 + 13/2 n + 18 n2\n1+23 t+36 t2+12 t3\n(1−t)2(1−t2)\n101\n85\n102\n84\n78\n72\n24\n12\n17/2 n + 71\n2 n2\n1 + 17/2 n + 71\n2 n2\n1+42 t+72 t2+27 t3\n(1−t)2(1−t2)\n79\n63\n93\n49\n88\n37\n14\n3\n3/4 + 27\n2 n + 303\n4 n2\n1 + 27\n2 n + 303\n4 n2\n1+88 t+151 t2+63 t3\n(1−t)2(1−t2)\n97\n93\n114\n76\n77\n66\n47\n0\n1/2 + 15/2 n + 21 n2\n1 + 15/2 n + 21 n2\n1+27 t+42 t2+14 t3\n(1−t)2(1−t2)\n88\n56\n113\n31\n99\n35\n7\n3\n1/2 + 11/2 n + 10 n2\n1 + 11/2 n + 10 n2\n1+14 t+20 t2+5 t3\n(1−t)2(1−t2)\n134\n82\n140\n76\n91\n72\n49\n4\n3/4 + 21 n + 669\n4 n2\n1 + 21 n + 669\n4 n2\n1+187 t+334 t2+147 t3\n(1−t)2(1−t2)\n133\n69\n149\n53\n98\n55\n43\n6\n1 + 6 n + 8 n2\n1 + 6 n + 8 n2\n15 t2+13 t+1+3 t3\n(1−t)3\n80\n63\n111\n32\n88\n38\n10\n7\n1\n1\n1+t\n1−t\n118\n69\n151\n36\n95\n63\n20\n9\n1 + 4 n + 4 n2\n1 + 4 n + 4 n2\n7 t2+7 t+1+t3\n(1−t)3\n96\n51\n103\n44\n90\n53\n3\n1\n1/2 + 39\n2 n + 36 n2\n1 + 39\n2 n + 36 n2\n1+54 t+72 t2+17 t3\n(1−t)2(1−t2)\n117\n72\n133\n56\n82\n57\n41\n9\n1 + 9 n + 18 n2\n1 + 9 n + 18 n2\n35 t2+26 t+1+10 t3\n(1−t)3\n72\n63\n77\n58\n49\n38\n28\n20\n1/2 + 7 n + 55\n2 n2\n1 + 7 n + 55\n2 n2\n1+33 t+55 t2+21 t3\n(1−t)2(1−t2)\n48\n37\n49\n36\n34\n24\n16\n11\n1/2 + 6 n + 37\n2 n2\n1 + 6 n + 37\n2 n2\n1+23 t+37 t2+13 t3\n(1−t)2(1−t2)\n108\n56\n113\n51\n73\n50\n29\n12\n1 + 4 n + 4 n2\n1 + 4 n + 4 n2\n7 t2+7 t+1+t3\n(1−t)3\n \n \nFigure 6.4: The quasipolynomial ̃kπ\nλ,μ and the rational function Kπ\nλ,μ(t) for the Kronecker problem, n = 2.\n101\n\nλ1\nλ2\nμ1\nμ2\nπ1\nπ2\nπ3\nπ4\n ̃kπ\nλ,μ(n); n odd\n ̃kπ\nλ,μ(n); n even\nKπ\nλ,μ(t)\n77\n40\n78\n39\n58\n29\n24\n6\n1 + 19/2 n + 57\n2 n2\n1 + 19/2 n + 57\n2 n2\n56 t2+37 t+1+20 t3\n(1−t)3\n153\n81\n157\n77\n96\n63\n61\n14\n1 + 3 n + 2 n2\n1 + 3 n + 2 n2\n3 t2+4 t+1\n(1−t)3\n90\n89\n102\n77\n90\n42\n30\n17\n1/2 + 13/2 n + 6 n2\n1 + 13/2 n + 6 n2\n1+11 t+12 t2\n(1−t)2(1−t2)\n145\n102\n160\n87\n96\n84\n39\n28\n1 + 10 n + 25 n2\n1 + 10 n + 25 n2\n49 t2+34 t+1+16 t3\n(1−t)3\n109\n95\n136\n68\n78\n60\n46\n20\n1 + 3 n + 2 n2\n1 + 3 n + 2 n2\n3 t2+4 t+1\n(1−t)3\n100\n42\n104\n38\n85\n27\n27\n3\n1 + 8 n\n1 + 8 n\n8 t+1+7 t2\n(1−t)2\n74\n51\n86\n39\n52\n34\n26\n13\n1\n1\n1+t\n1−t\n98\n90\n124\n64\n92\n67\n22\n7\n1/2 + 23/2 n + 60 n2\n1 + 23/2 n + 60 n2\n1+70 t+120 t2+49 t3\n(1−t)2(1−t2)\n57\n38\n75\n20\n52\n25\n17\n1\n1 + 3 n + 2 n2\n1 + 3 n + 2 n2\n3 t2+4 t+1\n(1−t)3\n159\n140\n170\n129\n89\n82\n73\n55\n1 + 3/2 n + 1/2 n2\n1 + 3/2 n + 1/2 n2\n1+t\n(1−t)3\n144\n122\n157\n109\n88\n86\n74\n18\n3/4 + n + 1/4 n2\n1 + n + 1/4 n2\n1\n(1−t)2(1−t2)\n90\n68\n92\n66\n88\n37\n23\n10\n1/4 + 12 n + 351\n4 n2\n1 + 12 n + 351\n4 n2\n1+98 t+176 t2+76 t3\n(1−t)2(1−t2)\n89\n42\n100\n31\n76\n28\n19\n8\n1 + 6 n + 8 n2\n1 + 6 n + 8 n2\n15 t2+13 t+1+3 t3\n(1−t)3\n88\n56\n107\n37\n71\n39\n20\n14\n1 + 9/2 n + 9/2 n2\n1 + 9/2 n + 9/2 n2\n8 t2+8 t+1+t3\n(1−t)3\n124\n111\n133\n102\n98\n89\n27\n21\n1/2 + 7 n + 53\n2 n2\n1 + 7 n + 53\n2 n2\n1+32 t+53 t2+20 t3\n(1−t)2(1−t2)\n \n \nFigure 6.5: Continuation of Figure 6.4\n102\n\nk\nλ\nhk,λ(n)\n3\n(21, 19)\n399 n3 + 35527969472513\n137438953472 n2 + 4329327034365\n137438953472 n + 1\n5\n(21, 19)\n3700378042361\n4194304\nn7 + 575575719967\n524288\nn6 + 2157156441\n4096\nn5 + 266554253\n2048\nn4 + 4643843\n256\nn3 + 1468423\n1024\nn2 + 7619\n128 n + 1\n3\n(21, 9, 6)\n270 n3 + 40819369181185\n274877906944 n2 + 3092376453119\n137438953472 n + 1\n3\n(12, 9, 5)\n42 n3 + 40132174413825\n1099511627776 n2 + 11544872091645\n1099511627776 n + 1\n3\n(21, 9, 6)\n27396522639355\n536870912\nn6 + 463063744509\n8388608\nn5 + 6265700353\n262144\nn4 + 5577375771\n1048576\nn3 + 84246529\n131072 n2 + 20971505\n524288 n + 1048573\n1048576\n3\n(21, 19, 16)\n15 n3 + 81363860455425\n4398046511104 n2 + 8246337208319\n1099511627776 n + 1\n4\n(9, 7, 5)\n7215545057279\n17179869184 n6 + 4183298146289\n4294967296\nn5 + 247765925897\n268435456\nn4 + 1914699777\n4194304\nn3 + 4160749567\n33554432 n2 + 587202553\n33554432 n + 67108863\n67108864\n4\n(21, 12, 9)\n16437913583613\n268435456\nn6 + 132498063359\n2097152\nn5 + 109509083155\n4194304\nn4 + 1462763527\n262144\nn3 + 171442179\n262144\nn2 + 10485755\n262144 n + 524287\n524288\n4\n(21, 9, 5)\n32469952757755\n536870912\nn6 + 129805320191\n2097152\nn5 + 108129157137\n4194304\nn4 + 2926313487\n524288\nn3 + 86638593\n131072 n2 + 10616825\n262144 n + 262143\n262144\n4\n(21, 9, 6)\n27396522639355\n536870912\nn6 + 463063744509\n8388608\nn5 + 6265700353\n262144\nn4 + 5577375771\n1048576\nn3 + 84246529\n131072 n2 + 20971505\n524288 n + 1048573\n1048576\n4\n(31, 19, 5)\n35969680015355\n33554432\nn6 + 1424674346311\n2097152\nn5 + 22705493343\n131072\nn4 + 46973953\n2048\nn3 + 3423915\n2048\nn2 + 65365\n1024 n + 16383\n16384\nFigure 6.6: Hilbert polynomial for G/Pλ, G = SLk(C). There is a slight rounding error caused by interpolation–\ne.g., the constant term of each polynomial should be one.\n103\n\nn\nk\na\n7\n3\n(1, 3, 5)\n1/3 n3 + 59373627899905\n39582418599936 n2 + 28587302322173\n13194139533312 n + 1\n7\n3\n(1, 2, 4)\nn + 1\n7\n3\n(1, 4, 6)\n22265110462465\n534362651099136 n5 + 4638564679679\n11132555231232 n4 + 13\n8 n3 + 105942526633\n34359738368 n2 + 146028888073\n51539607552 n + 34359738361\n34359738368\n6\n2\n(1, 4, 5)\n15637498706143\n2251799813685248 n6 + 3665038759245\n35184372088832 n5 + 1389660529559\n2199023255552 n4 + 272014595421\n137438953472 n3\n+ 230973796809\n68719476736 n2 + 100215903571\n34359738368 n + 1\n6\n2\n(1, 4, 6)\n69578470195\n25048249270272 n7 + 1217623228439\n25048249270272 n6 + 372534725887\n1043677052928 n5 + 30953963537\n21743271936 n4 + 12044363351\n3623878656 n3\n+ 683671553\n150994944 n2 + 1335466297\n402653184 n + 268435457\n268435456\n7\n3\n(1, 4, 6)\n23456248059223\n562949953421312 n5 + 7330077518505\n17592186044416 n4 + 1786706395137\n1099511627776 n3 + 423770106525\n137438953472 n2 + 24338148015\n8589934592 n + 17179869169\n17179869184\n6\n3\n(1, 3, 6)\n1/8 n4 + 16126170540715\n17592186044416 n3 + 19\n8 n2 + 710101259605\n274877906944 n + 1\n8\n3\n(1, 3, 6)\n171798691840001\n1374389534720000 n4 + 31496426837333\n34359738368000 n3 + 4080218931199\n1717986918400 n2 + 443813287253\n171798691840 n + 1\nFigure 6.7: Hilbert polynomial of the Schubert subvariety Ωa, a = (a(1), . . . , a(k)), of the Grassmannian Gn,k.\n104\n\nk = 2\n\"\n1/2 n + 1/2\n1/2 n + 1\n#\nk = 3\n \n \n1/12 n2 + 1/2 n + 5\n12\n1/12 n2 + 1/2 n + 2/3\n1/12 n2 + 1/2 n + 3/4\n1/12 n2 + 1/2 n + 46912496118443\n70368744177664\n1/12 n2 + 1/2 n + 58640620148053\n140737488355328\n1/12 n2 + 1/2 n + 1\n \n \nk = 4\n \n \n1\n144 n3 + 5\n48 n2 + 61572651155457\n140737488355328 n + 15881834623431\n35184372088832\n1\n144 n3 + 117281240296107\n1125899906842624 n2 + 140737488355325\n281474976710656 n + 19\n36\n1\n144 n3 + 234562480592215\n2251799813685248 n2 + 123145302310909\n281474976710656 n + 19791209299969\n35184372088832\n1\n144 n3 + 234562480592215\n2251799813685248 n2 + 70368744177667\n140737488355328 n + 62549994824587\n70368744177664\n1\n144 n3 + 5\n48 n2 + 61572651155453\n140737488355328 n + 748278746681\n2199023255552\n1\n144 n3 + 117281240296107\n1125899906842624 n2 + 70368744177665\n140737488355328 n + 26388279066621\n35184372088832\n1\n144 n3 + 117281240296107\n1125899906842624 n2 + 7\n16 n + 7940917311717\n17592186044416\n1\n144 n3 + 117281240296107\n1125899906842624 n2 + 35184372088831\n70368744177664 n + 6841405683939\n8796093022208\n1\n144 n3 + 29320310074027\n281474976710656 n2 + 30786325577729\n70368744177664 n + 9\n16\n1\n144 n3 + 58640620148053\n562949953421312 n2 + 35184372088831\n70368744177664 n + 5619726097523\n8796093022208\n1\n144 n3 + 117281240296105\n1125899906842624 n2 + 7\n16 n + 2993114986727\n8796093022208\n1\n144 n3 + 58640620148055\n562949953421312 n2 + 1/2 n + 1\n \n \nFigure 6.8: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk; k = 2, 3, 4.\n105\n\n1\n2880 n4 +\n46912496118441\n4503599627370496 n3 + 3787206717893\n35184372088832 n2 + 469583091025\n1099511627776 n + 499743305817\n1099511627776\n1\n2880 n4 +\n46912496118445\n4503599627370496 n3 + 3787206717891\n35184372088832 n2 + 503942829399\n1099511627776 n + 310001195055\n549755813888\n1\n2880 n4 +\n46912496118441\n4503599627370496 n3 + 3787206717897\n35184372088832 n2 + 469583091031\n1099511627776 n + 30279519437\n68719476736\n400319966877379\n1152921504606846976 n4 +\n23456248059221\n2251799813685248 n3 + 3787206717893\n35184372088832 n2 + 503942829403\n1099511627776 n + 47340083975\n68719476736\n400319966877379\n1152921504606846976 n4 +\n5864062014805\n562949953421312 n3 + 3787206717895\n35184372088832 n2 + 117395772755\n274877906944 n + 89955703917\n137438953472\n400319966877379\n1152921504606846976 n4 +\n5864062014805\n562949953421312 n3 + 3787206717893\n35184372088832 n2 + 31496426837\n68719476736 n + 92771293595\n137438953472\n400319966877379\n1152921504606846976 n4 +\n46912496118447\n4503599627370496 n3 + 1893603358947\n17592186044416 n2 + 234791545515\n549755813888 n + 5661005505\n17179869184\n1\n2880 n4 +\n23456248059223\n2251799813685248 n3 + 1893603358949\n17592186044416 n2 + 62992853675\n137438953472 n + 94680167945\n137438953472\n400319966877379\n1152921504606846976 n4 +\n46912496118441\n4503599627370496 n3 + 3787206717891\n35184372088832 n2 + 117395772757\n274877906944 n + 38869454029\n68719476736\n1\n2880 n4 +\n46912496118441\n4503599627370496 n3 + 3787206717897\n35184372088832 n2 + 62992853675\n137438953472 n + 52494044729\n68719476736\n1\n2880 n4 +\n5864062014805\n562949953421312 n3 + 946801679473\n8796093022208 n2 + 234791545515\n549755813888 n + 2830502753\n8589934592\n400319966877379\n1152921504606846976 n4 +\n23456248059219\n2251799813685248 n3 + 1893603358949\n17592186044416 n2 + 251971414695\n549755813888 n + 27487790695\n34359738368\n400319966877379\n1152921504606846976 n4 +\n23456248059223\n2251799813685248 n3 + 1893603358945\n17592186044416 n2 + 234791545509\n549755813888 n + 31233956605\n68719476736\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 251971414699\n549755813888 n + 19375074691\n34359738368\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 29348943189\n68719476736 n + 11005853695\n17179869184\n400319966877379\n1152921504606846976 n4 +\n23456248059219\n2251799813685248 n3 + 473400839737\n4398046511104 n2 + 251971414699\n549755813888 n + 5917510497\n8589934592\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 234791545511\n549755813888 n + 15616978311\n34359738368\n1\n2880 n4 +\n23456248059223\n2251799813685248 n3 + 1893603358947\n17592186044416 n2 + 62992853673\n137438953472 n + 23192823403\n34359738368\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 946801679475\n8796093022208 n2 + 117395772757\n274877906944 n + 2830502755\n8589934592\n400319966877379\n1152921504606846976 n4 +\n11728124029609\n1125899906842624 n3 + 1893603358949\n17592186044416 n2 + 31496426837\n68719476736 n + 30541989663\n34359738368\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 117395772759\n274877906944 n + 9717363505\n17179869184\n1\n2880 n4 +\n2932031007403\n281474976710656 n3 + 1893603358945\n17592186044416 n2 + 125985707353\n274877906944 n + 4843768673\n8589934592\n400319966877379\n1152921504606846976 n4 +\n23456248059223\n2251799813685248 n3 + 59175104967\n549755813888 n2 + 117395772755\n274877906944 n + 2830502755\n8589934592\n400319966877379\n1152921504606846976 n4 +\n23456248059221\n2251799813685248 n3 + 946801679475\n8796093022208 n2 + 62992853675\n137438953472 n + 3435973837\n4294967296\n400319966877379\n1152921504606846976 n4 +\n23456248059223\n2251799813685248 n3 + 1893603358947\n17592186044416 n2 + 58697886379\n137438953472 n + 11244462985\n17179869184\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 946801679475\n8796093022208 n2 + 15748213419\n34359738368 n + 9687537337\n17179869184\n1\n2880 n4 +\n11728124029609\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 117395772753\n274877906944 n + 1892469965\n4294967296\n1\n2880 n4 +\n23456248059221\n2251799813685248 n3 + 946801679473\n8796093022208 n2 + 7874106709\n17179869184 n + 11835020991\n17179869184\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 117395772753\n274877906944 n + 3904244579\n8589934592\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 125985707349\n274877906944 n + 1879048193\n2147483648\n \n \nFigure 6.9: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 5; the\nfirst 30 rows.\n106\n\n1\n2880 n4 +\n11728124029609\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 58697886375\n137438953472 n + 88453211\n268435456\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 15748213419\n34359738368 n + 2958755247\n4294967296\n400319966877379\n1152921504606846976 n4 +\n23456248059223\n2251799813685248 n3 + 946801679475\n8796093022208 n2 + 58697886375\n137438953472 n + 4858681755\n8589934592\n400319966877379\n1152921504606846976 n4 +\n11728124029609\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 62992853673\n137438953472 n + 151367771\n268435456\n1\n2880 n4 +\n23456248059221\n2251799813685248 n3 + 946801679473\n8796093022208 n2 + 58697886375\n137438953472 n + 2274244835\n4294967296\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 236700419869\n2199023255552 n2 + 62992853671\n137438953472 n + 858993459\n1073741824\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 14674471595\n34359738368 n + 3904244571\n8589934592\n400319966877379\n1152921504606846976 n4 +\n23456248059219\n2251799813685248 n3 + 59175104967\n549755813888 n2 + 7874106709\n17179869184 n + 2421884337\n4294967296\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 59175104967\n549755813888 n2 + 29348943189\n68719476736 n + 3784939927\n8589934592\n400319966877379\n1152921504606846976 n4 +\n23456248059223\n2251799813685248 n3 + 473400839737\n4398046511104 n2 + 62992853673\n137438953472 n + 1908874355\n2147483648\n400319966877379\n1152921504606846976 n4 +\n5864062014805\n562949953421312 n3 + 946801679473\n8796093022208 n2 + 29348943189\n68719476736 n + 1952122291\n4294967296\n400319966877379\n1152921504606846976 n4 +\n11728124029609\n1125899906842624 n3 + 946801679471\n8796093022208 n2 + 62992853675\n137438953472 n + 2899102929\n4294967296\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 473400839735\n4398046511104 n2 + 58697886381\n137438953472 n + 707625689\n2147483648\n400319966877379\n1152921504606846976 n4 +\n11728124029613\n1125899906842624 n3 + 59175104967\n549755813888 n2 + 15748213419\n34359738368 n + 2958755253\n4294967296\n1\n2880 n4 +\n11728124029609\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 58697886377\n137438953472 n + 3288334339\n4294967296\n100079991719345\n288230376151711744 n4 +\n5864062014805\n562949953421312 n3 + 59175104967\n549755813888 n2 + 62992853675\n137438953472 n + 1210942169\n2147483648\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 946801679477\n8796093022208 n2 + 29348943193\n68719476736 n + 1415251375\n4294967296\n1\n2880 n4 +\n5864062014805\n562949953421312 n3 + 59175104967\n549755813888 n2 + 31496426839\n68719476736 n + 3435973835\n4294967296\n100079991719345\n288230376151711744 n4 +\n1466015503701\n140737488355328 n3 + 473400839737\n4398046511104 n2 + 29348943195\n68719476736 n + 976061147\n2147483648\n100079991719345\n288230376151711744 n4 +\n11728124029611\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 31496426839\n68719476736 n + 3280877793\n4294967296\n1\n2880 n4 +\n5864062014805\n562949953421312 n3 + 946801679473\n8796093022208 n2 + 29348943191\n68719476736 n + 946234983\n2147483648\n100079991719345\n288230376151711744 n4 +\n2932031007403\n281474976710656 n3 + 946801679475\n8796093022208 n2 + 15748213419\n34359738368 n + 1479377623\n2147483648\n1\n2880 n4 +\n5864062014805\n562949953421312 n3 + 59175104967\n549755813888 n2 + 29348943195\n68719476736 n + 122007643\n268435456\n100079991719345\n288230376151711744 n4 +\n2932031007403\n281474976710656 n3 + 473400839737\n4398046511104 n2 + 3937053355\n8589934592 n + 1449551461\n2147483648\n100079991719345\n288230376151711744 n4 +\n2932031007403\n281474976710656 n3 + 236700419869\n2199023255552 n2 + 29348943193\n68719476736 n + 71070151\n134217728\n1\n2880 n4 +\n1466015503701\n140737488355328 n3 + 59175104967\n549755813888 n2 + 15748213419\n34359738368 n + 739688813\n1073741824\n100079991719345\n288230376151711744 n4 +\n11728124029611\n1125899906842624 n3 + 473400839739\n4398046511104 n2 + 14674471597\n34359738368 n + 607335219\n1073741824\n1\n2880 n4 +\n2932031007403\n281474976710656 n3 + 236700419869\n2199023255552 n2 + 3937053355\n8589934592 n + 605471085\n1073741824\n100079991719345\n288230376151711744 n4 +\n11728124029611\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 29348943193\n68719476736 n + 88453211\n268435456\n100079991719345\n288230376151711744 n4 +\n1466015503701\n140737488355328 n3 + 473400839737\n4398046511104 n2 + 3937053355\n8589934592 n + 2147483647\n2147483648\n \n \nFigure 6.10: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 5;\nthe last 30 rows.\n107\n\n53375995583651\n4611686018427387904 n5 +\n21892498188609\n36028797018963968 n4 +\n418085902907\n35184372088832 n3 + 115486898397\n1099511627776 n2 + 26847522421\n68719476736 n + 8448724291\n17179869184\n53375995583651\n4611686018427387904 n5 +\n10946249094305\n18014398509481984 n4 +\n836171805815\n70368744177664 n3 + 7396888121\n68719476736 n2 + 15302809397\n34359738368 n + 9853503363\n17179869184\n106751991167299\n9223372036854775808 n5 +\n21892498188599\n36028797018963968 n4 +\n1672343611625\n140737488355328 n3 + 57743449207\n549755813888 n2 + 14060052663\n34359738368 n + 975427339\n2147483648\n13343998895913\n1152921504606846976 n5 +\n10946249094301\n18014398509481984 n4 +\n1672343611641\n140737488355328 n3 + 59175104961\n549755813888 n2 + 15302809423\n34359738368 n + 610309549\n1073741824\n106751991167305\n9223372036854775808 n5 +\n21892498188611\n36028797018963968 n4 +\n1672343611623\n140737488355328 n3 + 115486898411\n1099511627776 n2 + 13423761203\n34359738368 n + 4461910959\n8589934592\n13343998895913\n1152921504606846976 n5 +\n21892498188605\n36028797018963968 n4 +\n1672343611631\n140737488355328 n3 + 29587552483\n274877906944 n2 + 15939100865\n34359738368 n + 1927366573\n2147483648\n213503982334599\n18446744073709551616 n5 +\n10946249094305\n18014398509481984 n4 +\n418085902909\n35184372088832 n3 + 57743449199\n549755813888 n2 + 13423761217\n34359738368 n + 835973461\n2147483648\n106751991167299\n9223372036854775808 n5 +\n10946249094305\n18014398509481984 n4 +\n836171805821\n70368744177664 n3 + 59175104963\n549755813888 n2 + 7651404713\n17179869184 n + 320001575\n536870912\n106751991167299\n9223372036854775808 n5 +\n10946249094299\n18014398509481984 n4 +\n1672343611637\n140737488355328 n3 + 115486898395\n1099511627776 n2 + 14060052667\n34359738368 n + 255936429\n536870912\n213503982334597\n18446744073709551616 n5 +\n2736562273577\n4503599627370496 n4 +\n1672343611629\n140737488355328 n3 + 7396888121\n68719476736 n2 + 7651404703\n17179869184 n + 2859997491\n4294967296\n106751991167299\n9223372036854775808 n5 +\n342070284197\n562949953421312 n4 + 104521475727\n8796093022208 n3 + 57743449199\n549755813888 n2 + 1677970153\n4294967296 n + 1790721319\n4294967296\n1\n86400 n5 +\n10946249094299\n18014398509481984 n4 +\n836171805815\n70368744177664 n3 + 59175104959\n549755813888 n2 + 3984775219\n8589934592 n + 987842475\n1073741824\n1\n86400 n5 +\n5473124547153\n9007199254740992 n4 +\n209042951453\n17592186044416 n3 + 57743449193\n549755813888 n2 + 3355940307\n8589934592 n + 884291849\n2147483648\n106751991167303\n9223372036854775808 n5 +\n10946249094301\n18014398509481984 n4 +\n836171805817\n70368744177664 n3 + 118350209919\n1099511627776 n2 + 1912851173\n4294967296 n + 264972307\n536870912\n213503982334605\n18446744073709551616 n5 +\n2736562273577\n4503599627370496 n4 +\n836171805827\n70368744177664 n3 + 57743449201\n549755813888 n2 + 7030026327\n17179869184 n + 1233125369\n2147483648\n53375995583651\n4611686018427387904 n5 +\n10946249094299\n18014398509481984 n4 +\n836171805819\n70368744177664 n3 + 118350209933\n1099511627776 n2 + 1912851177\n4294967296 n + 1478317113\n2147483648\n106751991167297\n9223372036854775808 n5 +\n1368281136787\n2251799813685248 n4 +\n836171805817\n70368744177664 n3 + 57743449195\n549755813888 n2 + 1677970151\n4294967296 n + 117959881\n268435456\n1667999861989\n144115188075855872 n5 +\n10946249094303\n18014398509481984 n4 + 104521475727\n8796093022208 n3 + 59175104961\n549755813888 n2 + 3984775215\n8589934592 n + 109722991\n134217728\n106751991167297\n9223372036854775808 n5 +\n342070284197\n562949953421312 n4 +\n836171805815\n70368744177664 n3 + 28871724597\n274877906944 n2 + 1677970153\n4294967296 n + 332087389\n1073741824\n213503982334607\n18446744073709551616 n5 +\n10946249094307\n18014398509481984 n4 +\n836171805821\n70368744177664 n3 + 59175104963\n549755813888 n2 + 7651404709\n17179869184 n + 384426083\n536870912\n \n \nFigure 6.11: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 6; the first 20 rows.\n108\n\n213503982334615\n18446744073709551616 n5 +\n10946249094307\n18014398509481984 n4 +\n836171805813\n70368744177664 n3 + 28871724597\n274877906944 n2 + 7030026337\n17179869184 n + 640721865\n1073741824\n106751991167309\n9223372036854775808 n5 +\n10946249094297\n18014398509481984 n4 +\n209042951455\n17592186044416 n3 + 29587552479\n274877906944 n2 + 1912851179\n4294967296 n + 157275009\n268435456\n26687997791827\n2305843009213693952 n5 +\n342070284197\n562949953421312 n4 +\n836171805825\n70368744177664 n3 + 57743449199\n549755813888 n2 + 3355940301\n8589934592 n + 90445245\n268435456\n13343998895913\n1152921504606846976 n5 +\n5473124547153\n9007199254740992 n4 + 104521475729\n8796093022208 n3 + 29587552481\n274877906944 n2 + 1992387605\n4294967296 n + 450971557\n536870912\n106751991167303\n9223372036854775808 n5 +\n10946249094307\n18014398509481984 n4 +\n836171805827\n70368744177664 n3 + 28871724597\n274877906944 n2 + 209746269\n536870912 n + 17843591\n33554432\n106751991167303\n9223372036854775808 n5 +\n10946249094297\n18014398509481984 n4 +\n836171805819\n70368744177664 n3 + 924611015\n8589934592 n2 + 239106397\n536870912 n + 5146825\n8388608\n1667999861989\n144115188075855872 n5 +\n5473124547153\n9007199254740992 n4 +\n418085902911\n35184372088832 n3 + 7217931151\n68719476736 n2 + 54922081\n134217728 n + 265331665\n536870912\n1667999861989\n144115188075855872 n5 +\n10946249094291\n18014398509481984 n4 +\n836171805825\n70368744177664 n3 + 59175104963\n549755813888 n2 + 478212795\n1073741824 n + 326629601\n536870912\n1667999861989\n144115188075855872 n5 +\n10946249094297\n18014398509481984 n4 +\n418085902909\n35184372088832 n3 + 14435862303\n137438953472 n2 + 3355940305\n8589934592 n + 24121261\n67108864\n53375995583655\n4611686018427387904 n5 +\n10946249094291\n18014398509481984 n4 +\n209042951453\n17592186044416 n3 + 59175104973\n549755813888 n2 + 3984775217\n8589934592 n + 503316475\n536870912\n26687997791827\n2305843009213693952 n5 +\n10946249094285\n18014398509481984 n4 +\n836171805825\n70368744177664 n3 + 57743449207\n549755813888 n2 + 3355940305\n8589934592 n + 115234099\n268435456\n106751991167303\n9223372036854775808 n5 +\n2736562273573\n4503599627370496 n4 +\n209042951459\n17592186044416 n3 + 7396888121\n68719476736 n2 + 3825702363\n8589934592 n + 42684549\n67108864\n106751991167301\n9223372036854775808 n5 +\n10946249094305\n18014398509481984 n4 +\n209042951453\n17592186044416 n3 + 28871724605\n274877906944 n2 + 1757506587\n4294967296 n + 277411267\n536870912\n106751991167303\n9223372036854775808 n5 +\n10946249094299\n18014398509481984 n4 +\n418085902905\n35184372088832 n3 + 924611015\n8589934592 n2 + 3825702353\n8589934592 n + 33950041\n67108864\n106751991167307\n9223372036854775808 n5 +\n1368281136787\n2251799813685248 n4 +\n52260737865\n4398046511104 n3 + 28871724601\n274877906944 n2 + 209746269\n536870912 n + 61328751\n134217728\n53375995583651\n4611686018427387904 n5 +\n2736562273575\n4503599627370496 n4 + 104521475727\n8796093022208 n3 + 29587552487\n274877906944 n2 + 249048451\n536870912 n + 128849017\n134217728\n106751991167301\n9223372036854775808 n5 +\n5473124547145\n9007199254740992 n4 +\n209042951459\n17592186044416 n3 + 7217931151\n68719476736 n2 + 1677970157\n4294967296 n + 30318473\n67108864\n106751991167303\n9223372036854775808 n5 +\n342070284197\n562949953421312 n4 +\n418085902919\n35184372088832 n3 + 14793776243\n137438953472 n2 + 1912851181\n4294967296 n + 143223581\n268435456\n53375995583651\n4611686018427387904 n5 +\n5473124547153\n9007199254740992 n4 +\n209042951459\n17592186044416 n3 + 1804482787\n17179869184 n2 + 878753295\n2147483648 n + 13898875\n33554432\n106751991167303\n9223372036854775808 n5 +\n5473124547147\n9007199254740992 n4 +\n418085902907\n35184372088832 n3 + 29587552479\n274877906944 n2 + 956425585\n2147483648 n + 195527051\n268435456\n \n \nFigure 6.12: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 6; the middle 20 rows.\n109\n\n13343998895913\n1152921504606846976 n5 +\n5473124547145\n9007199254740992 n4 +\n418085902919\n35184372088832 n3 + 7217931149\n68719476736 n2 + 209746269\n536870912 n + 64348647\n134217728\n53375995583651\n4611686018427387904 n5 +\n5473124547143\n9007199254740992 n4 + 104521475729\n8796093022208 n3 + 14793776237\n137438953472 n2 + 498096901\n1073741824 n + 28772927\n33554432\n106751991167301\n9223372036854775808 n5 +\n5473124547141\n9007199254740992 n4 +\n209042951457\n17592186044416 n3 + 3608965575\n34359738368 n2 + 1677970153\n4294967296 n + 11719905\n33554432\n13343998895913\n1152921504606846976 n5 +\n5473124547151\n9007199254740992 n4 +\n418085902905\n35184372088832 n3 + 3698444059\n34359738368 n2 + 478212797\n1073741824 n + 74631683\n134217728\n106751991167311\n9223372036854775808 n5 +\n5473124547147\n9007199254740992 n4 +\n418085902907\n35184372088832 n3 + 28871724591\n274877906944 n2 + 878753299\n2147483648 n + 10682367\n16777216\n106751991167313\n9223372036854775808 n5 +\n5473124547151\n9007199254740992 n4 + 104521475729\n8796093022208 n3 + 14793776237\n137438953472 n2 + 478212797\n1073741824 n + 84006215\n134217728\n106751991167307\n9223372036854775808 n5 +\n2736562273573\n4503599627370496 n4 +\n209042951459\n17592186044416 n3 + 28871724589\n274877906944 n2 + 104873135\n268435456 n + 3161957\n8388608\n53375995583655\n4611686018427387904 n5 +\n5473124547149\n9007199254740992 n4 +\n209042951451\n17592186044416 n3 + 29587552485\n274877906944 n2 + 996193803\n2147483648 n + 118111589\n134217728\n106751991167309\n9223372036854775808 n5 +\n1368281136787\n2251799813685248 n4 +\n418085902907\n35184372088832 n3 + 28871724601\n274877906944 n2 + 419492539\n1073741824 n + 49899533\n134217728\n106751991167305\n9223372036854775808 n5 +\n2736562273573\n4503599627370496 n4 +\n52260737863\n4398046511104 n3 + 29587552479\n274877906944 n2 + 1912851173\n4294967296 n + 87717907\n134217728\n106751991167309\n9223372036854775808 n5 +\n1368281136787\n2251799813685248 n4 + 104521475729\n8796093022208 n3 + 28871724595\n274877906944 n2 + 878753303\n2147483648 n + 8962701\n16777216\n106751991167305\n9223372036854775808 n5 +\n2736562273571\n4503599627370496 n4 +\n209042951453\n17592186044416 n3 + 29587552499\n274877906944 n2 + 956425585\n2147483648 n + 10878263\n16777216\n53375995583651\n4611686018427387904 n5 +\n5473124547157\n9007199254740992 n4 + 104521475727\n8796093022208 n3 + 3608965575\n34359738368 n2 + 838985083\n2147483648 n + 53611225\n134217728\n13343998895913\n1152921504606846976 n5 +\n5473124547145\n9007199254740992 n4 +\n418085902907\n35184372088832 n3 + 14793776249\n137438953472 n2 + 498096903\n1073741824 n + 104354293\n134217728\n106751991167299\n9223372036854775808 n5 +\n2736562273575\n4503599627370496 n4 +\n209042951447\n17592186044416 n3 + 3608965575\n34359738368 n2 + 838985087\n2147483648 n + 7873221\n16777216\n106751991167303\n9223372036854775808 n5 +\n5473124547151\n9007199254740992 n4 +\n418085902899\n35184372088832 n3 + 14793776243\n137438953472 n2 + 956425599\n2147483648 n + 90737805\n134217728\n53375995583655\n4611686018427387904 n5 +\n5473124547155\n9007199254740992 n4 + 104521475727\n8796093022208 n3 + 14435862303\n137438953472 n2 + 878753295\n2147483648 n + 18680385\n33554432\n106751991167305\n9223372036854775808 n5 +\n2736562273573\n4503599627370496 n4 + 104521475725\n8796093022208 n3 + 3698444063\n34359738368 n2 + 478212799\n1073741824 n + 36634403\n67108864\n106751991167311\n9223372036854775808 n5 +\n684140568395\n1125899906842624 n4 +\n209042951463\n17592186044416 n3 + 7217931153\n68719476736 n2 + 52436567\n134217728 n + 19926951\n67108864\n26687997791827\n2305843009213693952 n5 +\n1368281136789\n2251799813685248 n4 +\n6532592233\n549755813888 n3 + 14793776243\n137438953472 n2 + 498096903\n1073741824 n + 67108871\n67108864\n \n \nFigure 6.13: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 6; the last 20 rows.\n110\n\nChapter 7\nOn verification and discovery\nof obstructions\nIn this chapter we give applications of the results and positivity hypotheses\nin this paper to the problem of verifying or discovering an obstruction, i.e.,\na “proof of hardness” [GCT2] in the context of the P vs.\nNP and the\npermanent vs. determinant problems in characteristic zero.\n7.1\nObstruction\nAn obstruction in an abstract setting of Problem 1.1.4 is defined as follows.\nLet X and Y be H-varieties with compact specifications (Section 3.5),\nH a connected reductive group. Let ⟨X⟩and ⟨Y ⟩denote the bit lengths of\ntheir specifications (Section 3.5). Suppose we wish to show that X cannot\nbe embedded as an H-subvariety of Y . Pictorially:\nX ̸֒→Y.\n(7.1)\nFor example, in the context of the P vs. NP problem in characteristic\nzero [GCT1, GCT2], X is a class variety XNP (n, l) associated with the\ncomplexity class NP for the given input size parameter n and the circuit\nsize parameter l. The variety Y is the class variety XP (l) associated with\nthe class P for given l. And H is SLl(C). If NP ⊆P (over C) to the\ncontrary, then it would turn out that\nXNP (n, l) ֒→XP (l)\n111\n\nas an H-subvariety, for every l = poly(n). The goal is to show that this\nembedding cannot exist when l = poly(n) and n →∞.\nLet R(X) and R(Y ) be the homogeneous coordinate rings of X and\nY , respectively. Let R(X)d and R(Y )d denote their degree d-compoenents.\nSuppose to the contrary that an H-embedding as in (7.1) exists. Then there\nexists a degre preserving H-equivariant surjection from R(Y )d to R(X)d for\nevery d, and hence, a degree-preserving H-equivariant injection from R(X)∗\nd\nto R(Y )∗\nd. Hence, every irreducible H-module Vλ(G) that occurs in R(X)∗\nd\nalso occurs in R(Y )∗\nd. This leads to:\nDefinition 7.1.1 An irreducible representation Vλ(H) is called an obstruc-\ntion for the pair (X,Y) if it occurs (as an H-submodule) in R(X)∗\nd but not\nin R(Y )∗\nd, for some d. We say that Vλ(H) is an obstruction of degree d.\nRemark 7.1.2 The obstruction as defined here is dual to the obstrution as\ndefined in [GCT2].\nExistence of such an obstruction implies that X cannot be embedded in\nY as an H-subvariety.\nLet us assume that X and Y are H-subvarieties of P(V ), where V is an\nH-module, and that we are given a point y ∈Y ⊆P(V ) that is distiguished\nin the following sense. Let Hy ⊆H be the stabilizer of y. Then Cy, the line\nin V corresponding to y, is invariant under Hy. Let [y] be the set of points\nin P(V ) stabilized by Hy. We say that y is characterized by its stabilizer Hy\nif y = [y]; i.e., y is the only point in P(V ) stabilized by Hy. Let\nH[y] = ∪z∈[y]Hz\nbe the union of the H-orbits of all points in [y]. We say that y is a distin-\nguished point of Y if Y equals the projective closure of H[y] in P(V ). If y\nis characterized by its stabilizer, this means Y is the projective closure of\nthe orbit Hy of y.\nIf Vλ(H) occurs in R(Y )∗\nd, then it can be shown (cf. Proposition 4.2 in\n[GCT2]) that Vλ(H) contains an Hy-submodule isomorphic to (Cy)d, the\nd-th tensor power of Cy.\nThis leads to the following stronger notion of\nobstruction:\nDefinition 7.1.3 [GCT2] We say that Vλ(H) is a strong obstruction for\nthe pair (X, Y ) if, for some d, it occurs in R(X)∗\nd, but it does not contain\nan Hy-module isomorphic to (Cy)d.\n112\n\nExistence of a strong obstruction also implies that X cannot be embedded\nin Y as an H-subvariety. The results in [GCT2] suggest that strong obstruc-\ntions exist in the context of the lower bound problems under cosideration.\nThe goal then is to show their existence.\n7.2\nDecision problems\nThe decisions problems that arise in this context are the following. Let sλ\nd\nbe the multiplicity of Vλ(H) in R(X)∗\nd, and md\nλ the multiplicity of the Hy-\nmodule (Cy)d in Vλ(H), considered an Hy-module via the the embedding\nρ : Hy ֒→H. Thus λ is a strong obstruction of degree d iffsλ\nd is nonzero\nand md\nλ is zero.\nProblem 7.2.1 (Decision Problems)\n(a) Given d, λ and the specification of X, decide if sλ\nd is nonzero.\n(b) Given d, λ and the specifications of H, Hy and ρ, decide if md\nλ is nonzero.\n(c) Given d, λ and the specifications of X, H, Hy and ρ, decide if λ is a\nstrong obstruction of degree d.\nThe first is an instance of the decision Problem 1.1.4 in geometric in-\nvariant theory, and the second of the subgroup restriction Problem 1.1.3.\nBy the results in Chapter 3-4, relaxed forms of the decision problems in (a)\nand (b) belong to P assuming appropriate PH1 and SH; this implies that\na relaxed form of the decision problem in (c) also belongs to P assumming\nPH1 and SH. We will only need a weak relaxed form of (c), for which the\nweak form of SH that is implied by PH1 (cf. Theorem 3.3.5) will suffice.\n7.3\nVerification of obstructions\nThe relevant PH1 are as follows.\nAssume that that singularities of spec(R(X)) are rational.\nBy Theo-\nrem 3.5.1, the stretching function ̃sλ\nd(k) = skλ\nkd is a quasipolynomial. Hence\nPH1 for sλ\nd (Hypothesis 3.3.1, or rather its slight variant obtained by replac-\ning R(X)d with R(X)∗\nd) is well defined. It is:\nHypothesis 7.3.1 (PH1):\nThere exists a polytope P λ\nd such that:\n113\n\n1. The number of integer points in P λ\nd is equal to sλ\nd.\n2. The Ehrhart quasi-polynomial of P λ\nd coincides with the stretching quasi-\npolynomial ̃sλ\nd(n) (cf. Theorem 3.5.1).\n3. The polytope P λ\nd is given by a separating oracle, as in Section 2.3. Its\nencoding bitlength ⟨P λ\nd ⟩is poly(⟨d⟩, ⟨λ⟩, ⟨X⟩), and the combinatorial\nsize ∥P λ\nd ∥is poly(ht(λ), ∥X∥), where ∥X∥is the combinatorial size of\nX (Section 3.5), and ht(λ) is the height of λ.\nSimilarly by Theorem 3.4.1 (or rather its slight variant which can be\nproved similarly), the stretching function mkd\nkλ is a quasipolynomial. Hence\nPH1 for md\nλ (cf. Hypothesis 3.3.1 and Section 3.4) is also well defined. It is:\nHypothesis 7.3.2 (PH1:)\nThere exists a polytope Qd\nλ such that:\n1. The number of integer points in Qd\nλ is equal to md\nλ.\n2. The Ehrhart quasi-polynomial of Qd\nλ coincides with the stretching quasi-\npolynomial ̃md\nλ(n) (Theorem 3.5.1).\n3. The polytope Qd\nλ is given by a separating oracle. Its encoding bitlength\n⟨Qd\nλ⟩is poly(⟨d⟩, ⟨λ⟩, ⟨ρ⟩, ⟨Hy⟩, ⟨H⟩), and the combinatorial size ∥Qd\nλ∥\nis O(poly(ht(λ), ⟨H⟩, ⟨Hy⟩, ⟨ρ⟩)). Here ⟨H⟩, ⟨Hy⟩and ⟨ρ⟩denote the\nbitlengths of H, Hy and ρ (Section 3.4).\nTheorem 7.3.3 (Weak SH:)\n(a) Assuming PH1 (Hypothesis 7.3.1), the saturation index of ̃sλ\nd(n) is at\nmost apoly(∥P λ\nd ∥), for some explicit constant a > 0.\n(b) Assuming PH1 (Hypothesis 7.3.2), the saturation index of ̃md\nλ(n) is at\nmost bpoly(∥Qd\nλ∥), for some explicit constant b > 0.\nThis follows from Theorem 3.3.5.\nTheorem 7.3.4 Assume PH1 (Hypotheses 7.3.1-7.3.2). Then, given d, λ,\nthe specifications of X, H, Hy and ρ, and a relaxation parameter c greater\nthan the explicit bounds on the saturation indices in Theorem 7.3.3, whether\ncλ is an obstruction of degree d can be decided in\npoly(⟨d⟩, ⟨λ⟩, ⟨X⟩, ⟨H⟩, ⟨Hy⟩, ⟨ρ⟩, ⟨c⟩)\ntime.\n114\n\nThis follows by applying Theorem 3.1.1 to the polytopes P λ\nd and Qd\nλ with\nthe saturation index estimates in Theorem 7.3.3.\n7.4\nRobust obstruction\nWe now define a notion of obstruction that is well behaved with respect to\nrelaxation.\nDefinition 7.4.1 Assume PH1 for both sλ\nd and md\nλ (Hypotheses 7.3.1-7.3.2).\nWe say that Vλ(H) is a robust obstruction for the pair (X, Y ) if one of the\nfollowing hold:\n1. Qd\nλ is empty, and P λ\nd is nonempty.\n2. Both Qdλ and P λ\nd are nonempty, the affine span of Qd\nλ does not contain\nan integer point and the affine span of P λ\nd contains an integer point.\nIf Vλ(H) is a robust obstruction, so is Vlλ(H), for all or most positive\nintegral l, hence the name robust.\nProposition 7.4.2 Assume PH1 for both sλ\nd and md\nλ as above. If Vλ(H)\nis a robust obstruction for the pair (X, Y ), then for some positive integer\nk–called a relaxation parameter–Vkλ(H) is a strong obstruction for (X, Y ).\nIn fact, this is so for most large enough k.\nProof:\n(1) Suppose Qd\nλ is empty, and P λ\nd is nonempty. Let k be a large enough\npositive integer k such that kP λ\nd = P kλ\ndk contains an integer point. Then skλ\nkd\nis nonzero. But mkd\nkλ is zero since Qkd\nkλ = kQd\nλ is empty. Thus kλ is a strong\nobstruction.\n(2) Suppose both Qdλ and P λ\nd are nonempty, the affine span of Qd\nλ does not\ncontain an integer point and the affine span of P λ\nd contains an integer point.\nWe can choose a positive integer k such that the affine span of kQd\nλ = Qdk\ndλ\ndoes not contain an integer point, but kP λ\nd = P kλ\ndk contains an integer point;\nmost large enough k have this property. This means skλ\nkd is nonzero, but mkd\nkλ\nis zero. Thus kλ is a strong obstruction. Q.E.D.\n115\n\n7.5\nVerification of robust obstructions\nTheorem 7.5.1 Assume that the singularities of spec(R(X)) are rational.\nAssume PH1 for both sλ\nd and md\nλ as above. Then, given λ, d and the speci-\nfications of ρ : Hy ֒→H and X, whether Vλ(H) is a robust obstruction can\nbe verified in poly(⟨ρ⟩, ⟨Hy⟩, ⟨H⟩, ⟨X⟩, ⟨d⟩, ⟨λ⟩) time. Furthermore, a positive\nintegral relaxation parameter k such that Vkλ(G) is a strong obstruction can\nalso be found in the same time.\nThe crucial result used implicitly here is the quasipolynomiality theorem\n(Theorem 4.1.1) because of which PH1 for both sλ\nd and md\nλ are well defined.\nProof: By linear programming [GLS], whether Qd\nλ is nonempty or not can\nbe determined in poly(⟨Qd\nλ⟩) = poly(⟨ρ⟩, ⟨Hy⟩, ⟨H⟩, ⟨d⟩, ⟨λ⟩) time.\nIf it\nis nonempty, the linear programming algorithm also gives its affine span.\nWhether this contains an integer point can be determined in polynomial\ntime, using the polynomial time algorithm for computing the Smith normal\nform, as in the proof of Theorem 3.1.1.\nSimilarly, whether P λ\nd is nonempty or not can be determined in poly(⟨P λ\nd ⟩) =\npoly(⟨X⟩, ⟨d⟩, ⟨λ⟩) time. If it is nonempty, whether its affine span contains\nan integer point can be determined in polynomial time similarly. Further-\nmore, the algorithm can also be made to return a vertex v of the polytope\nP λ\nd if it is nonempty.\nUsing these observations, whether Vλ(G) is a robust obstruction can be\ndetermined in polynomial time.\nAs far as the computation of the relaxation parameter k is concerned,\nlet us consider the second case in Definition 7.4.1–when both Qd\nλ and P λ\nd are\nnonempty, the affine span of Qd\nλ does not contain an integer point and the\naffine span of P λ\nd contains an integer point–the first case being simpler. In\nthis case, by examining the Smith normal forms of the defining equations\nof the affine spans of P λ\nd and Qd\nλ and the rational coordinates of a vertex\nv ∈P λ\nd , we can find a large enough k so that the affine span of Qkd\nkλ does not\ncontain an integer point, the affine span of P kλ\nkd contains an integer point,\nand P kλ\nkd contains an integer point that is some multiple of v. Q.E.D.\nThe value of the relaxation parameter k computed above is rather con-\nservative.\nOne may wish to compute as small value of k as possible for\nwhich Vkλ(G) is a strong obstruction (though in our application this is not\nnecessary).\nIf SH for holds for the structural constant sλ\nd (cf.\nHypothe-\nsis 3.3.2 and Section 3.5), then we can let k be the smallest integer larger\n116\n\nthan the saturation index (estimate) for P λ\nd such that affine span of Qkd\nkλ (if\nnonempty) does not contain an integer point (as can be ensured by looking\nat the Smith normal of the defining equations of the affine span).\n7.6\nArithemetic version of the P #P vs. NC prob-\nlem in characteristric zero\nWe now specialize the discussion in the preceding sections in the context\nof the arithmetic form of the P #P vs. NC problem in characteristric zero\n[V]. In concrete terms, the problem is to show that the permanent of an\nn × n complex matrix X cannot be expressed as a determinant of an m × m\ncomplex matrix, whose entries are (possibly nonhomogeneous) linear com-\nbinations of the entries of X.\n7.6.1\nClass varieties\nThe class varieties in this context are as follows [GCT1]. Let Y be an m×m\nvariable matrix, which can also be thought of as a variable l-vector, l = m2.\nLet X be its, say, principal bottom-right n × n submatrix, n < m, which\ncan be thought of as a variable k-vector, k = n2. Let V = Symm(Y ) be the\nspace of homogeneous forms of degree m in the variable entries of Y . The\nspace V , and hence P(V ), has a natural action of G = GL(Y ) = GLl(C)\ngiven by\n(σf)(Y ) = f(σ−1Y ),\nfor any f ∈V , σ ∈G, and thinking of Y as an l-vector. Let W = Symn(X)\nbe the space of homogeneous forms of degree n in the variable entries of\nX. The space W, and also P(W), has a similar action of K = GL(X) =\nGLk(C). We use any entry y of Y not in X as the homogenizing variable\nfor embedding W in V via the map φ : W →V defined by:\nφ(h)(Y ) = ym−nh(X),\n(7.2)\nfor any h(X) ∈W. We also think of φ as a map from P(W) to P(V ).\nLet g = det(Y ) ∈P(V ) be the determinant form, and f = φ(h), where\nh = perm(X) ∈P(W). Let ∆V [g], ∆V [f] ⊆P(V ) be the projective closures\nof the orbits Gg and Gf, respectively, in P(V ). Let ∆W [h] ⊆P(W) be the\nprojective closure of the K-orbit Kh of h in P(W). Then ∆V [g] is called the\nclass variety associated with NC and ∆V [f] the class variety associated with\n117\n\nP #P ; ∆W[h] is called the base class variety associated with P #P . (The base\nclass variety is not used in what follows. Rather its variant, called a reduced\nclass variety defined below, will be used.) These class varieties depend on\nthe lower bound parameters n and m. If we wish to make these explicit, we\nwould write ∆V [f, n, m] and ∆V [g, m] instead of ∆V [f] and ∆V [g].\nThe class varieties ∆V [g] = ∆V [g, m] and ∆V [f] = ∆V [f, n, m] are\nG-subvarieties of P(V ), and their homogeneous coordinate rings RV [g] =\nRV [g, m] and RV [f] = RV [f, n, m] have natural degree-preserving G-action.\nIt is conjectured in [GCT1] that, if m = poly(n) and n →∞, then\nf ̸∈∆V [g]; this is equivalent to saying that the class variety ∆V [f, n, m]\ncannot be embedded in the class variety ∆V [g, m] (as a subvariety). This\nimplies the arithmetic form of the P #P ̸= NC conjecture in characteristic\nzero.\n7.6.2\nObstructions\nThe obstruction in this context is defined as follows. A G-module Vλ(G) is\ncalled an obstruction for the pair (f, g) if it occurs in RV [f, n, m]∗\nd but not\nRV [g, m]∗\nd for some d. It is called a strong obstruction if, for some d, it occurs\nin RV [f, n, m]∗\nd but it does not contain (Cg)d as a Gg-submodule, where\n(Cg) ⊆V denotes the one dimensional line corresponding to g, and Gg ⊆G\nis the stabilizer of g = det(Y ) ∈P(V ). If Vλ(G) is a (strong) obstruction\nof degree d, then the size |λ| = dm; hence d is completely determined by λ\nand m.\nExistence of an obstruction or a strong obstruction implies that the\nclass variety ∆V [f, n, m] cannot be embedded in the class variety ∆V [g, m],\nas sought. The main algebro-geometric results of [GCT1, GCT2] suggest\nthat strong obstructions should indeed exist for all n →∞, assuming m =\npoly(n); cf. Section 4, Conjecture 2.10 and Theorem 2.11 in [GCT2]. The\ngoal then is to prove existence of strong obstructions for all n.\nThe definition of a strong obstruction can be simplified further as follows.\nLet X′ denote the set of variables, which consists of the variable entries in\nX and the homogenizing variable y above. Let W ′ = Symm(X′) ⊆V =\nSymm(Y ) be the space of homogeneous forms of degree m in the variables\nof X′.\nWe have a natural action of H = GL(X′) = GLn2+1(C) on W ′\nand hence on P(W ′).\nWe have a natural map φ′ : W →W ′ given by\nφ′(h)(X′) = ym−nh(X). The map φ in (7.2) is φ′ followed by the inclusion\nfrom W ′ to V . We also think of φ′ as a map from P(W) to P(W ′).\n118\n\nLet f ′ = φ′(h), for h = perm(X) ∈P(W). Let ∆W ′[f ′] ⊆P(W ′) be\nthe orbit closure of Hf ′.\nIt is an H-subvariety of P(W ′), and hence its\nhomogeneous coordinate ring RW ′[f ′] has the natural degree preserving H-\naction. We call ∆W ′[f ′] the reduced class variety for P #P . It is known (cf.\nTheorem 8.2 in [GCT2]) that Vλ(G) occurs in RV [f]∗\nd iffVλ(H) occurs in\nRW ′[f ′]∗\nd. Here the dominant weight λ of G is considered a dominant weight\nof H by restriction from G to H.\nHence Vλ(G) is a strong obstruction for the pair (f, g), ifffor some d,\nVλ(H) occurs in RW ′[f ′]∗\nd as an H-submodule and Vλ(G) does not contain\n(Cg)d as a Gg-submodule.\nIn particular, we can assume without loss of\ngenerality that the height of the Young diagram for λ is at most n2 + 1;\notherwise Vλ(H) would be zero.\n7.6.3\nRobust obstructions\nIt is known that the stabilizer Gg of g = det(Y ) ∈P(V ) consists of lin-\near transformations in G of the form Y →AY ∗B−1, thinking of Y as an\nm × m matrix, where Y ∗is either Y or Y T , A, B ∈GLm(C). Thus the con-\nnected component of Gg is essentially GLm(C) × GLm(C) ⊆G = GLl(C) =\nGLm2(C). This means the subgroup restriction problem for the embedding\nρ : Gg ֒→G is essentially the Kronecker problem (Problem 1.1.1).\nAssume PH1 (Hypothesis 7.3.2) for the subgroup restriction ρ : Gg ֒→G;\nwhich is essentially PH1 for the Kronecker problem. It now assumes the\nfollowing concrete form. Let md\nλ denote the multiplicity of the Gg-module\n(Cg)d in Vλ(G). Assume that the height of λ is at most n2+1 for the reasons\ngive above.\nHypothesis 7.6.1 (PH1:)\nThere exists a polytope Qd\nλ such that:\n1. The number of integer points in Qd\nλ is equal to md\nλ.\n2. The Ehrhart quasi-polynomial of Qd\nλ coincides with the stretching quasi-\npolynomial ̃md\nλ(n) (cf. Theorem 3.5.1).\n3. The polytope Qd\nλ is given by a separating oracle, and its encoding\nbitlength ⟨Qd\nλ⟩is poly(n, ⟨m⟩, ⟨d⟩, ⟨λ⟩) time.\nWe have to explain why ⟨Qd\nλ⟩is stipulated to depend polynomially on\nn and ⟨m⟩, rather than m. After all, the bitlengths ⟨G⟩, ⟨Gg⟩and ⟨ρ⟩are\n119\n\nO(poly(m2)) as per the definitions in Section 3.4. So, as per PH1 for sub-\ngroup restriction in Section 3.4.3, ⟨Qd\nλ⟩should depend polynomially on m.\nWe are stipulating a stronger condition for the following reason. First, as we\nalready mentioned, the above hypothesis is essentially PH1 for the Kronecker\nproblem, which is obtained by specializing PH1 for the plethysm problem\n(Hypothesis 1.6.4). In Hypothesis 1.6.4, the encoding bitlength of the poly-\ntope depends polynomially on the bitlengths of the various partition param-\neters λ, π, μ of the plethysm constant aπ\nλ,μ, but is independent of the rank of\nthe group G therein. (As explained in the remarks after Hypothesis 1.6.4,\nthis is justified because the bound in Theorem 1.6.3 is also independent of\nthe rank of G). For the same reason, the encoding bitlength of the polytope\nhere should be independent of the rank of G (which is m2), but should de-\npend polynomiallly on the total bit length of the partitions parametrizing\nthe representations Vλ(G) and (Cy)d. This is O(n + ⟨m⟩+ ⟨d⟩+ ⟨λ⟩). (Note\nthat the one dimensional representation (Cy)d of Gg is essentially the d-th\npower of the determinant representation of Gg, since the connected compo-\nnent of Gg is isomorphic to GLm(C)×GLm(C). The Young diagram for the\npartition corresponding to the d-th power of the determinant representation\nof GLm(C) is a rectangle of height m and width d. It can be specified by\nsimply giving m and d–this specification has bit length ⟨m⟩+ ⟨d⟩.)\nNext let us specialize PH1 as per Hypothesis 7.3.1. The class variety\n∆V [f] = ∆[f, n, m] will now play the role of X in Hypothesis 7.3.1. But,\nfor the reasons explained in the proof of Proposition 7.6.4 below, we shall\ninstead specialize Hypothesis 7.3.1 to the (simpler) reduced class variety\nZ = ∆W ′[f ′]. It now assumes that following concrete form. Let sλ\nd denote\nthe multiplicity of Vλ(H) in RW ′[f ′]∗\nd. Putting Z in place of X in Hypothe-\nsis 7.3.1, we get:\nHypothesis 7.6.2 (PH1):\nThere exists a polytope P λ\nd such that:\n1. The number of integer points in P λ\nd is equal to sλ\nd.\n2. The Ehrhart quasi-polynomial of P λ\nd coincides with the stretching quasi-\npolynomial ̃sλ\nd(n) (cf. Theorem 3.5.1).\n3. The polytope P λ\nd is given by a separating oracle, and its encoding\nbitlength ⟨P λ\nd ⟩is\npoly(⟨d⟩, ⟨λ⟩, ⟨Z⟩) = poly(⟨d⟩, ⟨λ⟩, n, ⟨m⟩).\n(7.3)\n120\n\nHere (7.3) follows because ⟨Z⟩= n+⟨m⟩. To see why, let us observe that\nZ = ∆W ′[f ′] is completely specified once the point f ′ = ym−nh ∈P(W ′) is\nspecified. To specify f ′, it sufficies to specify m, n and the point h ∈P(W).\nIt is known [GCT2] that the point h = perm(X) ∈P(W) is completely\ncharacterized by its stabilizer Kh ⊆K = GL(X) = GLk(C). Furthermore,\nKh is explicitly known [Mc]. It is generated by the linear transformation in\nK of the form X →λXμ−1, thinking of X as an n × n matrix, where λ and\nμ are either diagonal or permutation matrices. So to specifiy h, it suffices\nto specify Kh, K and the embedding ρ′ : Kh ֒→K. The bit length of this\nspecification is O(n) (cf. Section 3.4). To specify f ′, and hence Z, it suffices\nto specify m, n, K, Kh and ρ′. The total bit length of this specification is\nO(n + ⟨m⟩).\nAssume PH1 for both md\nλ and sλ\nd, i.e., Hypotheses 7.6.1 and 7.6.2.\nDefinition 7.6.3 We say that Vλ(G) is a robust obstruction for the pair\n(f, , g) if one of the following hold:\n1. Qd\nλ is empty, and P λ\nd is nonempty.\n2. Both Qd\nλ and P λ\nd are nonempty, the affine span of Qd\nλ does not contain\nan integer point and the affine span of P λ\nd contains an integer point.\nIf the first condition holds, we say that Vλ(G) is a geometric obstruction.\nIf the second condition holds, it is called a modular obstruction.\nProposition 7.6.4 Assume PH1 for both md\nλ and sλ\nd (Hypotheses 7.6.1 and\n7.6.2). If Vλ(G) is a robust obstruction for the pair (f, g), then for some\npositive integral relaxation parameter k, Vkλ(G) is a strong obstruction for\n(f, g). In fact, this is so for most large enough k.\nProof: This essentially follows from Proposition 7.4.2. It only remains to\nclarify why we can use PH1 for the reduced class variety ∆W ′[f ′]–as we are\ndoing here– in place of PH1 for the class variety ∆V [f]. This is because,\nas already mentioned, Vλ(G) occurs in RV [f]∗\nd iffVλ(H) occurs in RW ′[f ′]∗\nd.\nQ.E.D.\n7.6.4\nVerification of robust obstructions\nTheorem 7.6.5 Assume that the singularities of spec(RW ′[f ′]) are ratio-\nnal. Assume PH1 for both md\nλ and sλ\nd as above (Hypotheses 7.6.1 and 7.6.2).\n121\n\nThen, given n, m, λ and d, whether Vλ(H) is a robust obstruction can be\nverified in poly(n, ⟨m⟩, ⟨d⟩, ⟨λ⟩) time. Furthermore, a positive integral relax-\nation parameter k such that kλ is a strong obstruction can also be computed\nin this much time.\nOnce n and m are specified, the various class varieties and K, Kh, ρ′, G, Gg, ρ\nabove are automatically specified implicitly.\nProof: This follows from Theorem 7.5.1; cf. also the remark following its\nproof. Q.E.D.\nTheorem 7.3.4 can be similarly specialized in this context; we leave that\nto the reader.\n7.6.5\nOn explicit construction of obstructions\nTheorem 7.6.6 Assume that m = poly(n) or even 2polylog(n), and:\n1. (RH) [Rationality Hypothesis]: The singularities of spec(RW ′[f ′]) are\nrational.\n2. PH1 for both md\nλ and sλ\nd (Hypotheses 7.6.1 and 7.6.2).\n3. OH [Obstruction Hypothesis]: For every (large enough) n, there exists\nλ of poly(n) bit length such that |λ| is divisible by m and one of the\nfollowing holds (with d = |λ|/m):\n(a) Qd\nλ is empty, and P λ\nd is nonempty.\n(b) Both Qdλ and P λ\nd are nonempty, the affine span of Qd\nλ does not\ncontain an integer point and the affine span of P λ\nd,x contains an\ninteger point.\nThen there exists an explicit family {λn} of robust obstructions.\nHere we say that {λn} is an explicit family of robust obstructions if each\nλn is short and easy to verify. Short means ⟨λn⟩is O(poly(n)). Easy to verify\nmeans whether λn is a robust obstruction can be verified in O(poly(n)) time.\nThe poly(n) bound here and in OH is meant to be independent of m,\nas long as m << 2n; i.e., it should hold even when m = 2polylog(n). In\nother words, the family {λn} should continue to remain an explicit robust\nobstruction family, as we vary m over all values ≤2polylog(n), and perhaps\n122\n\neven values ≤2o(n), but will cease to be an obstruction family for some large\nenough m = 2Ω(n). This is an important uniformity condition.\nProof: OH basically says that there exists a short robust obstruction λn for\nevery n. By Theorem 7.6.5, it is easy to verify. Q.E.D.\n7.6.6\nWhy should robust obstructions exist?\nThe main question now is: why should OH hold?\nThat is, why should\n(short) robust obstructions exist?\nAs we already mentioned, the results in [GCT1, GCT2] indicate that\nstrong obstructions should exist for every n, assuming m = poly(n). We\nshall give a heuristic argument for existence of robust obstructions assuming\nthat strong obstructions exist. This will crucially depend on the following SH\nfor md\nλ, which is essentially SH for the Kronecker problem (i.e. specialization\nof Hypothesis 1.6.5 to the Kronecker problem), good experimental evidence\nfor which is provided in [BOR].\nHypothesis 7.6.7 (SH:) (a): The saturation index of ̃md\nλ(k) is bounded by\na polynomial in m. (Observe that the rank of G is poly(m) and the height of\nλ is at most n2 + 1). (b): The quasi-polynomial ̃md\nλ(n) is strictly saturated,\ni.e. the saturation index is zero, for almost all λ (and d).\nIf Vλ(G) is a strong obstruction, sλ\nd is nonzero but md\nλ is zero. Thus,\nassumming PH1, there are three possibilities:\n1. Qd\nλ is empty, and P λ\nd is nonempty and contains an integer point.\n2. Both Qdλ and P λ\nd are nonempty, the affine span of Qd\nλ does not contain\nan integer point and P λ\nd contains an integer point.\n3. Both Qd\nλ and P λ\nd are nonempty. The affine span of Qd\nλ contains an\ninteger point, but Qd\nλ does not. And P λ\nd contains an integer point.\nIn the first two cases, λ is a robust obstruction. As per SH (Hypothe-\nsis 7.6.7), for almost all λ, the Ehrhart quasipolynomial of Qd\nλ is saturated:\nthis means (cf. the proof of Theorem 3.1.1), if the affine span of Qd\nλ contains\nan integer point then Qd\nλ also contains an integer point. And hence, with\na high probability, the third case should not occur. In other words, strong\nobstructions can be expected to be robust with a high probability.\n123\n\nLet us call a strong obstruction λ fragile if it is not robust; this means\nthe affine span of Qd\nλ contains an integer point, but Qd\nλ does not. By SH\n(Hypothesis 7.6.7), if λ is fragile, then for some k = poly(m), Qkd\nkλ contains\nan intger point, and hence, kλ is not obstruction. Thus fragile obstructions\nare close to not being obstructions, and furthermore, are expected to be\nrare, as argued above. This is why we are focussing on robust obstructions.\nIt may be remarked that the only SH needed in the argument above\nis the one (Hypothesis 7.6.7) for the structural constant md\nλ.\nThis is a\nspecial case of the SH for the subgroup restriction problem (cf. Section 3.4)\nspecialized to the embedding Gg ֒→G. In particular, we do not need SH\nfor the structural constant sλ\nd; i.e., for the more difficult decision problem in\ngeometric invariant theory (cf. Problem 1.1.4 and Section 3.5).\n7.6.7\nOn discovery of robust obstructions\nIt may be conjectured that not just the verification (cf.\nTheorem 7.6.5)\nbut also the discovery of robust obstructions is easy for the problem under\nconsideration. In this section we shall give an argument in support of this\nconjecture for geometric (robust) obstructions (which may be conjectured to\nexist in the problem under consideration). For this we need to reformulate\nthe notions of strong and robust obstructions (Definition 7.6.3) as follows.\nLet TZ be the set of pairs (d, λ) such that sλ\nd is nonzero and SZ the set\nof pairs (d, λ) such that md\nλ is nonzero.\nProposition 7.6.8 Assuming PH1 above (Hypotheses 7.6.1 and 7.6.2), TZ\nand SZ are finitely generated semigroups with respect to addition.\nThese semi-groups are analogues of the Littlewood-Richardson semigroup\n(Section 2.2.2) in this setting.\nProof: The proof is similar to that for the Littlewood-Richardson semigroup\n[Z].\nFor given d and λ, the polytope P λ\nd in PH1 for sλ\nd (Hypothesis 7.6.2) has\na specification of the form\nAx ≤b\n(7.4)\nwhere A depends only the variety Z = ∆W ′[f ′], but not on d or λ, and\nb depends homogeneously and linearly on d and λ. Let P be the polytope\ndefined by the inequalities (7.4) where both d and λ are treated as variables.\nThen P is a polyhedral cone (through the origin) in the ambient space\n124\n\ncontaining P with the coordinates x, d and λ. Let PZ be the set of integer\npoints in P. It is a finitely generated semigroup since P is a polyhedral cone.\nLet TR be the orthogonal projection of P on the hyperplane corresponding\nto the coordinates d and λ. Now TZ is simply the projection of PZ. Hence\nit is a finitely generated semigroup as well.\nThe proof for SZ is similar, with SR defined similarly. Q.E.D.\nThe polyhedral cones TR and SR here are analogues of the Littlewood-\nRichardson cone (Section 2.2.2) in this setting. Note that (d, λ) ∈TR iffP λ\nd\nis nonempty; similarly for SR.\nA Weyl module Vλ(G) is a strong obstruction for the pair (f, g) of degree\nd iff(d, λ) occurs in TZ but not in SZ. It is a robust obstruction iffit occurs\nin TR but not in SZ. It is a geometric obstruction iffit occurs in TR but not\nin SR. It is a modular obstruction iffit occurs in TR and also in SR but not\nin SZ.\nAssuming PH1 (Hypothesis 7.6.2), whether (d, λ) belongs to TR can be\ndetermined in polynomial time by linear programming, since (d, λ) ∈TR\niffP λ\nd is nonempty. Similarly, assuming PH1 (Hypothesis 7.6.1), whether\n(d, λ) ∈SR can be determined in polynomial time.\nThe following is a stronger complement to PH1.\nHypothesis 7.6.9 (PH1*)\nWhether TR\\SR is nonempty can be determined in polynomial time; i.e.,\npoly(n, ⟨m⟩) time. If so, the algorithm can also output (d, λ) ∈TR \\ SR of\npolynomial bit length.\nProposition 7.6.10 Assuming PH1*, given n and m, the problem of decid-\ning if a geometric obstruction exists for the pair (f, g), and finding one if one\nexists, belongs to the complexity class P; i.e., it can be done in poly(n, ⟨m⟩)\ntime.\nThis immediately follows from Hypothesis 7.6.9 since (d, λ) is a geometric\nobstruction iff(d, λ) ∈TR \\ SR.\nHypothesis 7.6.9 is supported by the following:\nProposition 7.6.11 Assuming PH1 (Hypotheses 7.6.1 and 7.6.2), Hypoth-\nesis 7.6.9 holds if TR and SR have polynomially many explicitly given con-\nstraints with the specification of polynomial bit length; here polynomial means\npoly(n, ⟨m⟩).\n125\n\nThe proposition holds even if the polytope SR has exponentially many\nconstraints, as long as it is given by a separation oracle that works in poly-\nnomial time.\nProof: It suffices to check if SR satisfies each constraint of TR. This can be\ndone in polynomial time using the linear programming algorithm in [GLS].\nSpecifically, let l(y) ≥0 be a constraint of TR. Then we just need to minimize\nl(y) on SR and check if the minimum exceeds zero. Q.E.D.\nBut this method does not work when the number of constraints of TR is\nexponential, as expected in the context of the lower bound problems under\nconsideration. In fact, no generic black-box-type algorithm, like the one in\n[GLS] based on just a membership or separation oracle for TR, can be used\nto prove (4) when the number of constraints of TR is exponential.\nFortunately, this is not a serious problem. A basic principle in combi-\nnatorial optimization, as illustrated in [GLS], is that a complexity theoretic\nproperty that holds for polytopes with polynomially many constraints will\nalso hold for polytopes with exponentially many constraints, provided these\nconstraints are sufficiently well-behaved.\nFor example, Edmond’s perfect\nmatching polytope for nonbipartite graphs has complexity-theoretic proper-\nties similar to the perfect matching polytope for bipartite graphs, though it\ncan have exponentially many constraints. We have already remarked that\nTR and SR are analogues of the Littlewood-Richardson cones. The facets of\nthe Littlewood-Richardson cone have a very nice explicit description [Kl, Z].\nThe cones TR, SR here are expected to have similar nice explicit descrip-\ntion.\nThis is why Hypothesis 7.6.9 can be expected to hold even if the\nnumber of constraints of TR is exponential, just as it holds even when SR\nhas exponentially many constraints. But a polynomial-time algorithm as in\nHypothesis 7.6.9 would have to depend crucially on the specific nature of\nthe facets (constraints) of TR in the spirit of the linear-programming-based\nalgorithm for the construction of a maximum-weight perfect matching in\nnonbipartite graphs [Ed], where too the number of constraints is exponen-\ntial but the algorithm still works because of the structure theorems based\non the specific nature of the constraints.\n7.7\nArithmetic form of the P vs NP problem in\ncharacteristic zero\nWe turn now to the arithemetic form of the P vs. NP problem in character-\nistic zero. The arguments are essentially verbatim translations of those for\n126\n\nthe arithmetic form of the P #P vs. NC problem in the preceding section.\nHence we shall be brief.\nIn the preceding section h(X) was perm(X) and g(Y ) was det(Y ). Now\nh(X) and g(Y ) would be explicit (co)-NP-complete and P-complete func-\ntions E(X) and H(Y ) constructed in [GCT1]. They can be thought of as\npoints in suitable W = Symk(X) and V = Syml[Y ], k = O(n2), l = O(m2),\nwith the natural action of GL(X) and G = GL(Y ), where n denotes the\nnumber of input parameters and m denotes the circuit size parameter in\nthe lower bound problem. These functions are extremely special like the\ndeterminant and the permanent in the sense that they are “almost” char-\nacterized by their stabilizers as explained in [GCT1]–and this is enough for\nour purposes.\nWe again have a natural embedding φ : P(W) →P(V ), which lets us\ndefine f = φ(h). The class variety for NP is defined to be ∆V [f] ⊆P(V ),\nthe projective closure of the orbit Gf. The class variety for P is ̃∆V [g] ⊆\nP(V ), which is defined to be the projective closure of G[g], where [g] denotes\nthe set of points in P(V ) that are stabilized by Gg ⊆G, the stabilizer of g.\nAn explicit description of Gg is given in [GCT1]; cf. Section 7 therein. To\nshow P ̸= NP in characteristic zero, it suffices to show that ∆V [f] is not a\nsubvariety of ̃∆V [g] for all large enough n, if m = poly(n) (cf. Conjecture\n7.4. in [GCT1]). For this, in turn, it suffices to show existence of strong\nobstructions, defined very much as in Section 7.6, for all n, assumming\nm = poly(n).\nWe can then formulate PH1 for the new h(X) and g(Y ) just as in Hy-\npotheses 7.6.1 and 7.6.2, and the notion of a robust obstruction as in Defi-\nnition 7.6.3. We then have:\nTheorem 7.7.1 (Verification of obstructions)\nAnalogues of Theorems 7.6.5 and 7.6.6 holds for h(X) = E(X) and\ng(Y ) = H(Y ).\nFurthermore, even discovery of robust obstructions can be conjectured\nto be easy (poly-time)–this would follow from the obvious analogue of Hy-\npothesis 7.6.9 here.\nHeuristic argument for existence of robust obstructions is very similar\nto the one in Section 7.6.6. It needs SH for the special case of the subgroup\nrestriction problem for the embedding Gg ֒→G. 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Lusztig, Canonical bases arising from quantized enveloping al-\ngebras, J. Amer. Math. Soc. 3, (1990), 447-498.\n[Lu3]\nG. Lusztig, Canonical bases in tensor products, Proc. Nat. Acad.\nSci. USA, vo. 89, pp 8177-8179, 1992.\n[Lu4]\nG. Lusztig, Introduction to quantum groups, Birkh ̈auser, 1993.\n[Lu5]\nG. Lusztig, Character sheaves, (1985/1986), Advances in Math.\n56, 193-237; II, 57, 226-265; III, 57, 266-315; IV 59, 1-63; V, 61,\n103-155.\n134\n\n[Mc]\nI. Macdonald, Symmetric functions and Hall polynomials, Oxford\nscience publications, Clarendon press, 1995.\n[MR]\nV. Mehta, A. Ramanathan, Frobeniuns splitting and cohomology\nvanishing for Schubert varieties, Ann. Math. 122, 1985, 27-40.\n[Mc]\nH. Minc, Permanents, Addison-Wesley, 1978.\n[N]\nH. Narayanan, On the complexity of computing Kostka numbers\nand Littlewood-Richardson coefficients, J. of Algebraic combina-\ntorics, vol. 24, issue 3, Nov. 2006.\n[PV]\nV. Popov, E. Vinberg, Invariant theory, in Encyclopaedia of Math-\nematical Sciences, Algebraic Geometry IV, Eds. A. Parshin, I. Sha-\nfarevich, Springer-Verlag, 1989.\n[Rm]\nA. Ramanathan, Schubert varieties are arithmetically Cohen-\nMacaley, Invent. Math 80, No. 2, 283-294 (1985).\n[Rs]\nE. Rassart, A polynomiality property for Littlewood-Richardson\ncoefficients, arXiv:math.CO/0308101, 16 Aug. 2003.\n[RW]\nJ. Remmel, T. Whitehead, On the Kronecker product of Schur\nfunctions of two row shapes, Bull. Belg. Math. Soc. 1 (1994), 649-\n683.\n[Ro]\nM. Rosas, The Kronecker Product of Schur Functions Indexed by\nTwo-Row Shapes or Hook Shapes, Journal of Algebraic Combi-\nnatorics, An international journal, Volume 14, issue 2, September\n2001.\n[RR]\nA. Razborov, S. Rudich, Natural proofs, J. Comput. System Sci.,\n55 (1997), pp. 24-35.\n[RTF]\nN. Reshetikhin, L. Takhtajan, L. Faddeev, Quantization of Lie\ngroups and Lie algebras, Leningrad Math. J., 1 (1990), 193-225.\n[Sc]\nA. Schrijver, Combinatorial optimization, Vol. A-C, Springer, 2004.\n[Sm]\nK. Smith, F-rational rings have rational singularities, Amer. J.\nMath. 119 (1997).\n[Sp]\nT. Springer, Linear algebraic groups, in Algebraic Geometry IV,\nEncyclopaedia of Mathematical Sciences, Springer-Verlag, 1989.\n135\n\n[St1]\nR. Stanley, Enumerative combinatorics, vol. 1, Wadsworth and\nBrooks/Cole, Advanced Books and Software, 1986.\n[St2]\nR. Stanley, Combinatorics and commutative algebra, Birkh ̈auser,\n1996.\n[St3]\nR. Stanley, Generalized h-vectors, intersection cohomology of toric\nvarieties, and related results, Advanced studies in pure mathemat-\nics 11, 1987, commutative algebra and combinatorics, pp. 187-213.\n[St4]\nR. Stanley, Positivity problems and conjectures in algebraic com-\nbinatorics, manuscript, to appear in Mathematics: Frontiers and\nPerpsectives, 1999.\n[st5]\nR. Stanley, Decompositions of rational polytopes, Annals of dis-\ncrete mathematics 6 (1980) 333-342.\n[Stm]\nB. Sturmfels, On vector partition functions, J. Combinatorial The-\nory, Seris A 72 (1995), 302-309.\n[SV]\nA. Szenes, M. Vergne, Residue formulae for vector partitions and\nEuler-Maclarin sums, Advances in Apllied Mathematics, vol. 30,\nissue 1/2, January 2003.\n[Ta]\nE. Tardos, A strongly polynomial algorithm to solve combinatorial\nlinear programs, Operations Research 34 (1986), 250-256.\n[W]\nK. Woods, Computing the period of an Ehrhart quasipolynomial.\nThe Electron. J. Combin. 12 (2005), Research paper 34.\n[V]\nL. Valiant, The complexity of computing the permanent, Theoret-\nical Computer Science 8, pp 189-201, 1979.\n[Z]\nA.\nZelevinsky,\nLittlewood-Richardson\nsemigroups,\narXiv:math.CO/9704228 v1 30 Apr 1997.\n136","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0704.0229v4 [cs.CC] 22 Jan 2009\nGeometric Complexity Theory VI: the flip via\nsaturated and positive integer programming in\nrepresentation theory and algebraic geometry\nDedicated to Sri Ramakrishna\nKetan D. Mulmuley\nThe University of Chicago\nhttp://ramakrishnadas.cs.uchicago.edu\n(Technical report TR 2007-04, Comp. Sci. Dept.,\nThe University of Chicago, May, 2007)\nRevised version\nOctober 23, 2018"},{"paragraph_id":"p2","order":2,"text":"Abstract\nThis article belongs to a series on geometric complexity theory (GCT), an\napproach to the P vs. NP and related problems through algebraic geometry\nand representation theory. The basic principle behind this approach is called\nthe flip. In essence, it reduces the negative hypothesis in complexity theory\n(the lower bound problems), such as the P vs. NP problem in characteristic\nzero, to the positive hypothesis in complexity theory (the upper bound prob-\nlems): specifically, to showing that the problems of deciding nonvanishing of\nthe fundamental structural constants in representation theory and algebraic\ngeometry, such as the well known plethysm constants [Mc, FH], belong to the\ncomplexity class P. In this article, we suggest a plan for implementing the\nflip, i.e., for showing that these decision problems belong to P. This is based\non the reduction of the preceding complexity-theoretic positive hypotheses to\nmathematical positivity hypotheses: specifically, to showing that there exist\npositive formulae–i.e. formulae with nonnegative coefficients–for the struc-\ntural constants under consideration and certain functions associated with\nthem. These turn out be intimately related to the similar positivity proper-\nties of the Kazhdan-Lusztig polynomials [KL1, KL2] and the multiplicative\nstructural constants of the canonical (global crystal) bases [Kas2, Lu2] in\nthe theory of Drinfeld-Jimbo quantum groups. The known proofs of these\npositivity properties depend on the Riemann hypothesis over finite fields\n(Weil conjectures proved in [Dl]) and the related results [BBD]. Thus the\nreduction here, in conjunction with the flip, in essence, says that the validity\nof the P ̸= NP conjecture in characteristic zero is intimately linked to the\nRiemann hypothesis over finite fields and related problems.\nThe main ingradients of this reduction are as follows.\nFirst, we formulate a general paradigm of saturated, and more strongly,\npositive integer programming, and show that it has a polynomial time al-\ngorithm, extending and building on the techniques in [DM2, GCT3, GCT5,\nGLS, KB, KTT, Ki, KT1].\nSecond, building on the work of Boutot [Bou] and Brion (cf. [Dh]), we\nshow that the stretching functions associated with the structural constants\nunder consideration are quasipolynomials, generalizing the known result that\nthe stretching function associated with the Littlewood-Richardson coefficient\nis a polynomial for type A [Der, Ki] and a quasi-polynomial for general types"},{"paragraph_id":"p3","order":3,"text":"[BZ, Dh]. In particular, this proves Kirillov’s conjecture [Ki] for the plethysm\nconstants.\nThird, using these stretching quasi-polynomials, we formulate the math-\nematical saturation and positivity hypotheses for the plethysm and other\nstructural constants under consideration, which generalize the known sat-\nuration and conjectural positivity properties of the Littlewood-Richardson\ncoefficients [KT1, DM2, KTT]. Assuming these hypotheses, it follows that\nthe problem of deciding nonvanishing of any of these structural constants,\nmodulo a small relaxation, can be transformed in polynomial time into a\nsaturated, and more strongly, positive integer programming problem, and\nhence, can be solved in polynomial time.\nFourth, we give theoretical and experimental results in support of these\nhypotheses.\nFinally, we suggest an approach to prove these positivity hypotheses\nmotivated by the works on Kazhdan-Lusztig bases for Hecke algebras [KL1,\nKL2] and the canonical (global crystal) bases of Kashiwara and Lusztig [Lu2,\nLu4, Kas2] for representations of Drinfeld-Jimbo quantum groups [Dri, Ji].\nSteps in this direction are taken [GCT4, GCT7, GCT8].\nSpecifically, in [GCT4, GCT7] are constructed nonstandard quantum\ngroups, with compact real forms, which are generalizations of the Drinfeld-\nJimbo quantum group, and also associated nonstandard algebras, whose re-\nlationship with the nonstandard quantum groups is conjecturally similar\nto the relationship of the Hecke algebra with the Drinfeld-Jimbo quantum\ngroup. The article [GCT8] gives conjecturally correct algorithms to con-\nstruct canonical bases of the matrix coordinate rings of the nonstandard\nquantum groups and of nonstandard algebras that have conjectural posi-\ntivity properties analogous to those of the canonical (global crystal) bases,\nas per Kashiwara and Lusztig, of the coordinate ring of the Drinfeld-Jimbo\nquantum group, and the Kazhdan-Lusztig basis of the Hecke algebra. These\npositivity conjectures (hypotheses) lie at the heart of this approach. In view\nof [KL2, Lu2], their validity is intimately linked to the Riemann hypothesis\nover finite fields and the related works mentioned above.\n2"},{"paragraph_id":"p4","order":4,"text":"Contents\n1\nIntroduction\n4\n1.1\nThe decision problems . . . . . . . . . . . . . . . . . . . . . .\n7\n1.2\nDeciding nonvanishing of Littlewood-Richardson coefficients .\n12\n1.3\nBack to the general decision problems\n. . . . . . . . . . . . .\n16\n1.4\nSaturated and positive integer programming . . . . . . . . . .\n16\n1.5\nQuasi-polynomiality, positivity hypotheses, and the canonical\nmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n19\n1.6\nThe plethysm problem . . . . . . . . . . . . . . . . . . . . . .\n20\n1.7\nTowards PH1, SH, PH2,PH3 via canonial bases and canonical\nmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n27\n1.8\nBasic plan for implementing the flip\n. . . . . . . . . . . . . .\n29\n1.9\nOrganization of the paper . . . . . . . . . . . . . . . . . . . .\n30\n1.10 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n32\n2\nPreliminaries in complexity theory\n34\n2.1\nStandard complexity classes . . . . . . . . . . . . . . . . . . .\n34\n2.1.1\nExample: Littlewood-Richardson coefficients\n. . . . .\n35\n2.2\nConvex #P . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n36\n2.2.1\nLittlewood-Richardson coefficients\n. . . . . . . . . . .\n38\n2.2.2\nLittlewood-Richardson cone . . . . . . . . . . . . . . .\n38\n2.2.3\nEigenvalues of Hermitian matrices\n. . . . . . . . . . .\n39\n2.3\nSeparation oracle . . . . . . . . . . . . . . . . . . . . . . . . .\n39\n3\nSaturation and positivity\n41\n1"},{"paragraph_id":"p5","order":5,"text":"3.1\nSaturated and positive integer programming . . . . . . . . . .\n41\n3.1.1\nA general estimate for the saturation index . . . . . .\n45\n3.1.2\nExtensions\n. . . . . . . . . . . . . . . . . . . . . . . .\n47\n3.1.3\nIs there a simpler algorithm? . . . . . . . . . . . . . .\n47\n3.2\nLittlewood-Richardson coefficients again . . . . . . . . . . . .\n47\n3.3\nThe saturation and positivity hypotheses\n. . . . . . . . . . .\n49\n3.4\nThe subgroup restriction problem . . . . . . . . . . . . . . . .\n52\n3.4.1\nExplicit polynomial homomorphism\n. . . . . . . . . .\n53\n3.4.2\nInput specification and bitlengths . . . . . . . . . . . .\n55\n3.4.3\nStretching function and quasipolynomiality . . . . . .\n57\n3.5\nThe decision problem in geometric invariant theory . . . . . .\n58\n3.5.1\nReduction from Problem 1.1.3 to Problem 1.1.4 . . . .\n59\n3.5.2\nInput specification . . . . . . . . . . . . . . . . . . . .\n59\n3.5.3\nStretching function and quasi-polynomiality . . . . . .\n60\n3.5.4\nPositivity hypotheses . . . . . . . . . . . . . . . . . . .\n61\n3.5.5\nG/P and Schubert varieties . . . . . . . . . . . . . . .\n62\n3.6\nPH3 and existence of a simpler algorithm\n. . . . . . . . . . .\n63\n3.7\nOther structural constants . . . . . . . . . . . . . . . . . . . .\n63\n4\nQuasi-polynomiality and canonical models\n65\n4.1\nQuasi-polynomiality\n. . . . . . . . . . . . . . . . . . . . . . .\n65\n4.1.1\nThe minimal positive form and modular index\n. . . .\n68\n4.1.2\nThe rings associated with a structural constant . . . .\n69\n4.2\nCanonical models . . . . . . . . . . . . . . . . . . . . . . . . .\n69\n4.2.1\nFrom PH0 to PH1,3 . . . . . . . . . . . . . . . . . . .\n70\n4.2.2\nOn PH0 in general . . . . . . . . . . . . . . . . . . . .\n72\n4.3\nNonstandard quantum group for the Kronecker and the plethysm\nproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n73\n4.4\nThe cone associated with the subgroup restriction problem\n.\n75\n4.5\nElementary proof of rationality . . . . . . . . . . . . . . . . .\n78\n5\nParallel and PSPACE algorithms\n84\n2"},{"paragraph_id":"p6","order":6,"text":"5.1\nComplex semisimple Lie group\n. . . . . . . . . . . . . . . . .\n85\n5.2\nSymmetric group . . . . . . . . . . . . . . . . . . . . . . . . .\n89\n5.3\nGeneral linear group over a finite field . . . . . . . . . . . . .\n92\n5.3.1\nTensor product problem . . . . . . . . . . . . . . . . .\n93\n5.4\nFinite simple groups of Lie type . . . . . . . . . . . . . . . . .\n94\n6\nExperimental evidence for positivity\n95\n6.1\nLittlewood-Richardson problem . . . . . . . . . . . . . . . . .\n95\n6.2\nKronecker problem, n = 2 . . . . . . . . . . . . . . . . . . . .\n95\n6.3\nG/P and Schubert varieties . . . . . . . . . . . . . . . . . . .\n96\n6.4\nThe ring of symmetric functions\n. . . . . . . . . . . . . . . .\n97\n7\nOn verification and discovery of obstructions\n111\n7.1\nObstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111\n7.2\nDecision problems\n. . . . . . . . . . . . . . . . . . . . . . . . 113\n7.3\nVerification of obstructions\n. . . . . . . . . . . . . . . . . . . 113\n7.4\nRobust obstruction . . . . . . . . . . . . . . . . . . . . . . . . 115\n7.5\nVerification of robust obstructions\n. . . . . . . . . . . . . . . 116\n7.6\nArithemetic version of the P #P vs. NC problem in charac-\nteristric zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117\n7.6.1\nClass varieties . . . . . . . . . . . . . . . . . . . . . . . 117\n7.6.2\nObstructions\n. . . . . . . . . . . . . . . . . . . . . . . 118\n7.6.3\nRobust obstructions . . . . . . . . . . . . . . . . . . . 119\n7.6.4\nVerification of robust obstructions\n. . . . . . . . . . . 121\n7.6.5\nOn explicit construction of obstructions . . . . . . . . 122\n7.6.6\nWhy should robust obstructions exist? . . . . . . . . . 123\n7.6.7\nOn discovery of robust obstructions\n. . . . . . . . . . 124\n7.7\nArithmetic form of the P vs NP problem in characteristic zero126\n3"},{"paragraph_id":"p7","order":7,"text":"Chapter 1\nIntroduction\nThis article belongs to a series of papers, [GCT1] to [GCT11], on geomet-\nric complexity theory (GCT), which is an approach to the P vs. NP and\nrelated problems in complexity theory through algebraic geometry and rep-\nresentation theory. We assume here that the underlying field of computation\nis of characteristic zero. The usual P vs. NP problem is over a finite field.\nThe characteristic zero version is its weaker, formal implication, and philo-\nsophically, the crux.\nThe basic principle underlying GCT is called the flip [GCTflip]. The\nflip, in essence, reduces the negative hypotheses (lower bound problems) in\ncomplexity theory, such as the P ̸=?NP problem in characteristic zero, to\npositive hypotheses in complexity theory (upper bound problems): specifi-\ncally, to the problem of showing that a series of decision problems in rep-\nresentation theory and algebraic geometry belong to the complexity class\nP.\nEach of these decision problem is of the form: Given a nonnegative\nstructural constant in representation theory or geometric invariant theory,\nsuch as the well known plethysm constant, decide if it is nonzero (nonvan-\nishing), or rather, if is nonzero after a small relaxation. This flip from the\nnegative to the positive may be considered to be a nonrelativizable form of\nthe flip–from the undecidable to the decidable–that underlies the proof of\nG ̈odel’s incompleteness theorem. But the classical diagonalization technique\nin G ̈odel’s result is relativizable [BGS], and hence, not applicable to the P\nvs. NP problem. The flip, in contrast, is nonrelativizable. It is furthermore\nnonnaturalizable [GCT10]); i.e., it crosses the natural proof barrier [RR]\nthat any approach to the P vs. NP problem must cross.\nWe suggest here a plan for implementating the flip; i.e., for showing that\n4"},{"paragraph_id":"p8","order":8,"text":"the decision problems above belong to P. This is based on the reduction\nin this paper of the complexity-theoretic positivity hypotheses mentioned\nabove to mathematical positivity hypotheses: specifically, to showing that\nthere exist positive formulae for the structural constants under consideration\nand certain functions associated with them. We also give theoretical and\nexperimental evidence in support of the latter hypotheses.\nHere we say that a formula is positive if its coefficients are nonegative.\nThe problem finding the positive formulae as above turns out be intimately\nrelated to the analogous problem for the Kazhdan-Lusztig polynomials [KL1]\nand the multiplicative structural constants of the canonical (global crystal)\nbases [Kas2, Lu2] in the theory of Drinfeld-Jimbo quantum groups. The\nknown solution to the latter problem [KL2, Lu2] depends on the Riemann\nhypothesis over finite fields, proved in [Dl], and the related results in [BBD].\nThus the flip and the reduction here together roughly say that the valid-\nity of the P ̸= NP conjecture in characteristic zero is intimately linked\nto the Riemann hypothesis over finite fields and related problems. This is\nillustrated in Figure 1.1; the question marks there indicate unsolved prob-\nlems. It seems that substantial extension of the techniques related to the\nRiemann hypothesis over finite fields may be needed to prove the required\nmathematical positivity hypotheses here.\nWe do not have the necessary\nmathematical expertize for this task. But it is our hope that the experts in\nalgebraic geometry and representation theory will have something to say on\nthis matter.\nIt may be conjectured that the flip paradigm would also work in the\ncontext of the usual P vs. NP problem over F2 (the boolean field) or the\nfinite field Fp. But implementation of the flip over a finite field is expected\nto be much harder than in characteristic zero. That is why we focus on\ncharacteristic zero here, deferring discussion of the problems that arise over\nfinite field to [GCT11].\nNow we turn to a more detailed exposition of the main results in this\npaper and of Figure 1.1.\nAcknoledgements\nWe are grateful to the authors of [BOR] for pointing out an error in the\nsaturation hypothesis (SH) in the earlier version of this paper. It has been\ncorrected in this version with appropriate relaxation without affecting the\noverall approach of GCT (cf. Section 1.6 and also [GCT6erratum]). We\nare also grateful to Peter Littelmann for bringing the reference [Dh] to our\n5"},{"paragraph_id":"p9","order":9,"text":"Complexity theoretic negative hypotheses (lower bound problems)\n|\n|\nThe flip\n|\n↓\nComplexity theoretic positive hypotheses (upper bound problems)\n|\n|\nThe reduction in this paper|\n|\n↓\nMathematical positivity hypotheses |\n|\n?\n|\n↓\n(?) The Riemann hypothesis over finite fields, related problems and their extensions\nFigure 1.1: Pictorial depiction of the basic plan for implementing the flip\n6"},{"paragraph_id":"p10","order":10,"text":"attention, to H. Narayanan for suggesting the use of [KB] in the proof of\nTheorem 3.1.1 and bringing the positivity conjecture in [DM2] to our atten-\ntion, and to Madhav Nori for a helpful discussion. The experimental results\nin Chapter 6 were obtained using Latte [DHHH].\n1.1\nThe decision problems\nWe now describe the relevant decision problems in representation theory\nand algebraic geometry. The actual decision problems that arise in the flip\n(cf. the second box in Figure 1.1) are relaxed versions of these problems\ndescribed later (cf. Hypothesis 1.1.6).\nProblem 1.1.1 (Decision version of the Kronecker problem)\nGiven partitions λ, μ, π, decide nonvanishing of the Kronecker coefficient\nkπ\nλ,μ. This is the multiplicity of the irreducible representation (Specht mod-\nule) Sπ of the symmetric group Sn in the tensor product Sλ ⊗Sμ.\nEquivalently [FH], let H = GLn(C) × GLn(C) and ρ : H →G =\nGL(Cn ⊗Cn) = GLn2(C) the natural embedding. Then kπ\nλ,μ is the multi-\nplicity of the H-module Vλ(GLn(C)) ⊗Vμ(GLn(C)) in the G-module Vπ(G),\nconsidered as an H-module via the embedding ρ.\nHere Vλ(GLn(C)) denotes the irreducible representation (Weyl module)\nof GLn(C) corresponding to the partition λ; Vπ(G) is the Weyl module of\nG = GLn2(C).\nProblem 1.1.1 is a special case of the following generalized plethysm\nproblem.\nProblem 1.1.2 (Decision version of the plethysm problem)\nGiven partitions λ, μ, π, decide nonvanishing of the plethysm constant\naπ\nλ,μ. This is the multiplicity of the irreducible representation Vπ(H) of H =\nGLn(C) in the irreducible representation Vλ(G) of G = GL(Vμ), where Vμ =\nVμ(H) is an irreducible representation H. Here Vλ(G) is considered an H-\nmodule via the representation map ρ : H →G = GL(Vμ).\n(Decision version of the generalized plethysm problem)\nThe same as above, allowing H to be any connected reductive group.\nThis is a special case of the following fundamental problem of represen-\ntation theory (characteristic zero):\n7"},{"paragraph_id":"p11","order":11,"text":"Problem 1.1.3 (Decision version of the subgroup restriction problem)\nLet G be connected reductive group, H a reductive group, possibly discon-\nnected, and ρ : H →G an explicit, polynomial homomorphism (as defined\nin Section 3.4). Here H will generally be a subgroup of G, and ρ its em-\nbedding. Let Vπ(H) be an irreducible representation of H, and Vλ(G) an\nirreducible representation of G. Here π and λ denote the classifying labels\nof the irreducible representations Vπ(H) and Vλ(G), respectively. Let mπ\nλ be\nthe multiplicity of Vπ(H) in Vλ(G), considered as an H-module via ρ.\nGiven specifications of the embedding ρ and the labels λ, π, as described\nin Section 3.4, decide nonvanishing of the multiplicity mπ\nλ.\nAll reductive groups in this paper are over C.\nThe reductive groups\nthat arise in GCT in characteristic zero are: the general and special linear\ngroups GLn(C) and SLn(C), algebraic tori, the symmetric group Sn, and\nthe groups formed from these by (semidirect) products. The reader may\nwish to focus on just these concrete cases, since all main ideas in this paper\nare illustrated therein.\nProblem 1.1.3 is, in turn, a special case of the following most general\nproblem.\nProblem 1.1.4 (Decision problem in geometric invariant theory)\nLet H be a reductive group, possibly disconnected, X a projective H-\nvariety (H-scheme), i.e., a variety with H-action.\nLet ρ denote this H-\naction. Let R = ⊕dRd be the homogeneous coordinate ring of X. Assume\nthat the singularities of spec(R) are rational.\nWe assume that X and ρ have special properties (as described in Sec-\ntion 3.5), so that, in particular, they have short specifications. Let Vπ(H)\nbe an irreducible representation of H. Let sπ\nd be the multiplicity of Vπ(H) in\nRd, considered as an H-module via the action ρ.\nGiven d, π, the specifications of X and ρ, decide nonvanishing of the\nmultiplicity sπ\nd.\nThis last problem is hopeless for general X. Indeed the usual specifi-\ncation of X, say in terms of the generators of the ideal of its appropriate\nembedding, is so large as to make this problem meaningless for a general\nX. But the instances of this decision problem that arise in GCT are for the\nfollowing very special kinds of projective H-varieties X, which, in particular,\nhave small specifications (Section 3.5):\n8"},{"paragraph_id":"p12","order":12,"text":"1. G/P, where G is a connected, reductive group, P ⊆G its parabolic\nsubgroup, and H ⊆G a reductive subgroup with an explicit polyno-\nmial embedding. Problem 1.1.3 reduces to this special case of Prob-\nlem 1.1.4; cf. Section 3.5.\n2. Class varieties [GCT1, GCT2], which are associated with the funda-\nmental complexity classes such as P and NP. They are very special\nlike G/P, with conjecturally rational singularities [GCT10]. Each class\nvariety is specified by the complexity class and the parameters of the\nlower bound problem under consideration. Briefly, the P vs. NP prob-\nlem in characteristic zero is reduced in [GCT1, GCT2] to showing that\nthe class variety corresponding to the complexity class NP and the pa-\nrameters of the lower bound problem (such as the input size) cannot\nbe embedded in the class variety corresponding to the complexity class\nP and the same parameters. Efficient criteria for the decision prob-\nlems stated above are needed to construct explicit obstructions [GCT2]\nto such embeddings, thereby proving their nonexistence. Specifically,\nProblems 1.1.3 and 1.1.4 are the decision problems associated with\nProblems 2.5 and 2.6 in [GCT2], respectively. See Sections 7.6-7.7 for\na brief review of this story.\nFor these varieties Problem 1.1.4 turns out to be qualitatively similar to\nProblem 1.1.3 (cf. Section 3.5 and [GCT2, GCT10]). For this reason, the\nKronecker and the plethysm problems, which lie at the heart of the subgroup\nrestriction problem, can be taken as the main prototypes of the decision\nproblems that arise here.\nOne can now ask:\nQuestion 1.1.5 Do the decision problems above (Problems 1.1.1-1.1.3 and\nProblem1.1.4, when X therein is G/P or a class variety) belong to P? That\nis, can the nonvanishing of any of structural constants in these problems\nbe decided in poly(⟨x⟩) time, where x denotes the input-specification of the\nstructural constant and ⟨x⟩its bitlength?\nFor Problem 1.1.2, the input specification for the plethysm constant aπ\nλ,μ\nis given in the form of a triple x = (λ, μ, π). Here the partition λ is specified\nas a sequence of positive integers λ1 ≥λ2 ≥· · · λk > 0 (the zero parts of\nthe partition are suppressed); k is called the height or length of λ, and is\ndenoted by ht(λ). The bitlength ⟨λ⟩is defined to be the total bitlength of\nthe integers λr’s. The bitlength ⟨x⟩is defined to be ⟨λ⟩+ ⟨μ⟩+ ⟨π⟩. A\n9"},{"paragraph_id":"p13","order":13,"text":"detailed specification of the input specification x and its bitlength ⟨x⟩for\nthe other problems is given in Section 3.3.\nFor the reasons described in Section 1.6, Question 1.1.5 may not have\nan affirmative answer in general; i.e., these problems may not be in P in\ntheir strict form stated above. The following main conjectural complexity-\ntheoretic positivity hypothesis governing the flip says that the relaxed forms\nof these decision problems described in Section 3.3 belong to P. As we shall\nsee in Chapter 7, these relaxed forms suffice for the purposes of the flip.\nHypothesis 1.1.6 (PHflip) The relaxed forms (cf. Section 3.3) of Prob-\nlems 1.1.1, 1.1.2, 1.1.3, and the special cases of Problem 1.1.4, when X\ntherein is G/P or a class variety–which together include all decision prob-\nlems that arise in the flip–belong to the complexity class P.\nThis means nonvanishing of any of these structural constants, modulo a\nsmall relaxation (as described in Section 3.3), can be decided in poly(⟨x⟩)\ntime, where x denotes the input-specification of the structural constant and\n⟨x⟩its bitlength.\nThe phrase “modulo a small relaxation” in the relaxed form of the\nplethysm problem means the following:\n(a) Let h = dim G + htλ + htπ, where dim(G) is the dimension of the\ngroup G in Problem 1.1.2. Then there exist absolute nonnegative constants\nc and c′, independent of λ, μ and π, such that nonvanishing of the relaxed\n(stretched) plethysm constant abπ\nbλ,bμ, for any positive integral relaxation\nparameter b > chc′, can be decided in O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, ⟨b⟩)) time, where\n⟨b⟩denotes the bitlength b. The notation poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, ⟨b⟩) here means\nbounded by a polynomial of constant degree in ⟨λ⟩, ⟨μ⟩, ⟨π⟩and ⟨b⟩.\nIn\nparticular, the time is O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) if the relaxation parameter b is\nsmall; i.e. if its bitlength ⟨b⟩is O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)). (Observe that the bit\nlength of h is O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)).)\n(b) There exists a polynomial time algorithm for deciding nonvanishing of\naπ\nλ,μ, which works correctly on almost all λ, μ and π. Here polynomial time\nmeans O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) time. The meaning of “correctly on almost all”\nis specified in Hypothesis 1.6.5 below.\nA detailed specification of the relaxation, i.e., the meaning of the phrase\n“modulo a small relaxation” for the other problems is given in Section 3.3.\nThe structural constants in Problems 1.1.1-1.1.3 are of fundamental im-\nportance in representation theory. The kronecker and the plethysm con-\n10"},{"paragraph_id":"p14","order":14,"text":"stants in Problems 1.1.1 and 1.1.2, in particular, have been studied inten-\nsively; see [FH, Mc, St4] for their significance.\nThere are many known\nformulae for these structural constants based on on the character formulae\nin representation theory. Several formulae for the characters of connected,\nreductive groups are known by now [FH], starting with the Weyl character\nformula. For the symmetric group, there is the Frobenius character formula\n[FH], for the general linear group over a finite field, Green’s formula [Mc],\nand for finite simple groups of Lie type, the character formula of Deligne-\nLusztig [DL], and Lusztig [Lu1]. (Finite simple groups of Lie type, other\nthan GLn(Fq), are not needed in GCT.)\nOne obvious method for deciding nonvanishing of the structural con-\nstants in Problems 1.1.1-1.1.4 is to compute them exactly. But all known al-\ngorithms for exact computation of the structural constants in Problems 1.1.1-\n1.1.3 take exponential time. This is expected, since this problem is #P-\ncomplete.\nIn fact, even the problem of exact computation of a Kostka\nnumber, which is a very special case of these structural constants, is #P-\ncomplete [N]. This means there is no polynomial time algorithm for com-\nputing any of them, assuming P ̸= NP.\nOf course, there are #P-complete quantities–e.g. the permanent of a\nnonnegative matrix [V]–whose nonvanishing can still be decided in polyno-\nmial time [Sc]. But the decision problems above are of a totally different\nkind and, at the surface, appear to have inherently exponential complexity.\nThis is because the dimensions of the irreducible representations that occur\nin their statements can be exponential in the ranks of the groups involved\nand the bit lengths of the classifying labels of these representations. For\nexample, the dimension of the Weyl module Vλ(GLn(C)) can be exponen-\ntial in n and the bit length of the partition λ. Furthermore, the number\nof terms in any of the preceding character formulae is also exponential. All\nthese decisions problems ask if one exponential dimensional representation\ncan occur within another exponential dimensional representation. To solve\nthem, it may seem necessary to take a detailed look into these representa-\ntions and/or the character formulae of exponential complexity. Hence, it\nseemed hard to believe that nonvanishing of these structural constants can,\nnevertheless, be decided in polynomial time (modulo a small relaxation).\nThis constituted the main philosophical obstacle in the course of GCT.\n11"},{"paragraph_id":"p15","order":15,"text":"1.2\nDeciding nonvanishing of Littlewood-Richardson\ncoefficients\nThe first result, which indicated that this obstacle may be removable, came\nin the wake of the saturation theorem of Knutson and Tao [KT1].\nThis\nconcerns the following special case of Problem 1.1.3, with G = H × H, the\nembedding ρ : H →G being diagonal.\nProblem 1.2.1 (Littlewood-Richardson problem)\nGiven a complex semisimple, simply connected Lie group H, and its\ndominant weights α, β, λ, decide nonvanishing of a generalized Littelwood-\nRichardson coefficient cλ\nα,β. This is the multiplicity of the irreducible repre-\nsentation Vλ(H) of H in the tensor product Vα(H) ⊗Vβ(H).\nIt was shown in [GCT3, KT2, DM2] independently that nonvanishing of\nthe Littlewood-Richardson coefficient of type A can be decided in polyno-\nmial time; i.e., polynomial in the bit lengths of α, β, λ. Furthermore, the\nalgorithm in [GCT3] works in strongly polynomial time in the terminology\nof [GLS]; cf. Section 2.1. The three main ingradients in this result are:\n1. PH1: The Littlewood-Richarson rule, which goes back to 1940’s, and\nwhose most important feature is that it is positive–i.e., it involves no\nalternating signs as in character-based formulae–and its strengthening\nin [BZ], which gives a positive, polyhedral formula for the Littlewood-\nRichardson coefficient as the number of integer points in a polytope;\nthis can be the BZ-polytope [BZ] or the hive polytope [KT1].\nWe\nshall refer to this positivity property as the first positivity hypothesis\n(PH1).\n2. The polynomial and strongly polynomial time algorithms for linear\nprogramming [Kh, Ta], and\n3. SH: The saturation theorem of Knutson and Tao [KT1]. This says\nthat cλ\nα,β is nonzero if cnλ\nnα,nβ is nonzero for any n ≥1. We shall refer\nto this saturation property as the saturation hypothesis (SH).\nBrion [Z] observed that the verbatim translation of the saturation prop-\nerty in [KT1] fails to hold for the the generalized Littlewood-Richardson\ncoefficients of types B, C, D (it also fails for the Kronecker coefficients, as\nwell as the plethysm constants [Ki]). Hence, the algorithms in [GCT3, KT2,\n12"},{"paragraph_id":"p16","order":16,"text":"DM2] do not work in types B, C and D. Fortunately, this situation can\nbe remedied. It is shown in [GCT5] that nonvanishing of the generalized\nLittewood-Richardson coefficient cλ\nα,β of arbitrary type can be decided in\n(strongly) polynomial time, assuming the positivity conjecture of De Loera\nand McAllister [DM2]. This conjectural hypothesis, based on considerable\nexperimental evidence, is as follows. Let\n ̃cλ\nα,β(n) = cnλ\nnα,nβ\n(1.1)\nbe the stretching function associated with the Littlewood-Richardson co-\nefficient cλ\nα,β. It is known to be a polynomial in type A [Der, Ki], and a\nquasi-polynomial, in general [BZ, Dh, DM2]. Recall that a fuction f(n) is\ncalled a quasi-polynomial if there exist l polynomials fj(n), 1 ≤j ≤l, such\nthat f(n) = fj(n) if n = j mod l. Here l is supposed to be the smallest such\ninteger, and is called the period of f(n). The period of ̃cλ\nα,β(n) for types\nB, C, D is either 1 or 2 [DM2]. In general, it is bounded by a fixed constant\ndepending on the types of the simple factors the Lie algebra.\nDefinition 1.2.2 We say that the quasi-polynomial f(n) is strictly posi-\ntive, if all coefficients of fj(n), for all j, are nonnegative; i.e., the nonzero\ncoefficients are positive. In general, we define the positivity index p(f) of\nf to be the smallest nonnegative integer such that f(n + p(f)) is strictly\npositive. We also say that f(n) is positive with index p(f).\nThus f(n) is strictly positive, iffits positivity index is zero.\nWith this terminology, the hypothesis mentioned above is the following.\nWe say a connected reductive group H is classical, if each simple factor of\nits Lie algebra H is of type A, B, C or D. We also say that the type of H\nor H is classical.\nHypothesis 1.2.3 (PH2): [KTT, DM2] Assume that H in Problem 1.2.1\nis classical.\nThen the Littlewood-Richardson stretching quasi-polynomial\n ̃cλ\nα,β(n) is strictly positive.\nWe shall refer to this as the second positivity hypothesis (PH2). This\nwas conjectured by King, Tollu and Toumazet [KTT] for type A, and De\nLoera and McAllister for types B, C, D. Since the stretching function above\nis a polynomial in type A, the positivity conjecture of King et al clearly\nimplies the saturation theorem of Knutson and Tao. That is, PH2 implies\nSH for type A.\n13"},{"paragraph_id":"p17","order":17,"text":"We can formulate an analogue of SH for a Lie algerbra of arbitrary clas-\nsical type so that PH2 implies SH for an arbitrary type. For this, we need\nto formulate the notion of a saturated quasi-polynomial, which is not con-\ntradicted by the counterexamples, mentioned above, to verbatim translation\nof the saturation property in [KT1, Ki] to the setting of quasi-polynomials.\nSpecifically, the notion of saturation in [KT1, Ki] works well if the stretching\nfunction is a polynomial, but not so if it is a quasipolynomial. Let f(n) be\na quasi-polynomial with period l. Let fj(n), 1 ≤j ≤l, be the polynomials\nsuch that f(n) = fj(n) if n = j mod l. The index of f, index(f), is defined\nto be the smallest j such that the polynomial fj(n) is not identically zero.\nIf f(n) is identically zero, we let index(f) = 0. If f(1) ̸= 0, then clearly\nindex(f) = 1.\nDefinition 1.2.4 We say that f(n) is strictly saturated if for any i: fi(n) >\n0 for every n ≥1 whenever fi(n) is not identically zero. The saturation in-\ndex s(f) of f is defined to be the smallest nonnegative integer such that\nf(n + s(f)) is strictly saturated. We also say that f(n) is saturated with\nindex s(f).\nThus f(n) is strictly saturated iffits saturation index is zero. Clearly\nthe saturation idex is bounded above by the positivity index. Thus if f(n)\nis strictly positive, it is strictly saturated. Hence, PH2 (Hypothesis 1.2.3)\nimplies:\nHypothesis 1.2.5 (SH): The Littlewood-Richardson stretching quasi-polynomial\ncλ\nα,β(n) of arbitary classical type is strictly saturated.\nThe polynomial time algorithm in [GCT5] works assuming SH as well.\nFor the Littlewood-Richardson coefficient of type A, the notion of strict\nsaturation here coincides with the notion of saturation in [KT1] since cλ\nα,β(n)\nis a polynomial in that case.\nKnutson and Tao [KT1] also conjectured\na generalized saturation property for arbitrary types. But that property,\nunlike the one defined above, is only conjectured to be sufficient, but not\nclaimed to be, or expected to be necessary. For this reason, it cannot be\nused in the complexity-theoretic applications in this paper.\nThere is another positivity conjecture for Littlewood-Richardson coeffi-\ncients that also implies the saturation theorem of Knutson and Tao. For\nthis consider the generating function\nCλ\nα,β(t) =\nX\nn≥0\n ̃cλ\nα,β(n)tn.\n(1.2)\n14"},{"paragraph_id":"p18","order":18,"text":"It is a rational function since ̃cλ\nα,β(n) is a quasi-polynomial [St1]. For type\nA, if ̃cλ\nα,β(n) is not identically zero, then Cλ\nα,β(t) is a rational function of\nform\nhdtd + · · · + h0\n(1 −t)d+1\n,\n(1.3)\nsince ̃cλ\nα,β(n) is a polynomial [St1]. It is conjectured in [KTT] that:\nHypothesis 1.2.6 (PH3:) The coefficients hi’s in eq.(1.3) are nonnegative\n(and h0 = 1).\nWe shall call this the third positivity hypothesis (PH3). It clearly implies SH\nfor Littlewood-Richardson coefficients of type A. To describe its analogue\nfor arbitrary classical type we need a definition.\nLet F(t) = P\nn f(n)tn be the generating function associated with the\nquasi-polynomial f(n). It is a rational function [St1].\nDefinition 1.2.7 We say that F(t) has a positive form, if, when f(n) is\nnot identically zero, it can be expressed in the form\nF(t) = hdtd + · · · + h0\nQk\ni=0(1 −tai)di ,\n(1.4)\nwhere (1) h0 = 1, and hi’s are nonnegative integers, (2) ai’s and di’s are\npositive integers, (3) P\ni di = d + 1, where d = max deg(fj(n)) is the degree\nof f(n).\nWe define the modular index of this positive form to be max{ai}.\nIf F(t) has a positive form with a0 = 1, then f(n) is strictly saturated\n(Definition 1.2.4); this easily follows from the power series expansion of the\nright hand side of eq.(1.4).\nThe analogue of Hypothesis 1.2.6 for arbitrary classical type is:\nHypothesis 1.2.8 (PH3:) The rational function Cλ\nα,β(t) has a positive\nform, with a0 = 1, of modular index bounded by a constant depending only\non the types of the simple factors of the Lie algebra of H.\nThis too implies SH for arbitrary classical type. For types B, C, D, the\nconstant above is 2. Experimental evidence for this hypothesis is given in\nSection 6.1.\n15"},{"paragraph_id":"p19","order":19,"text":"The analogue of the PH3, even in the more general q-setting, is known to\nhold for the generating function of the Kostant partition function of type A,\nand more generally, for a parabolic Kostant partition function; cf. Kirillov\n[Ki]. This also gives a support for the PH3 above, given a close relationship\nbetween Littlewood-Richardson coefficients and Kostant partition functions\n[FH].\n1.3\nBack to the general decision problems\nIt may be remarked that the Littlewood-Richardson problem actually never\narises in the flip. It is only used as a simplest proptotype of the actual (much\nharder) problems that arise–namely relaxed forms of Problems 1.1.1-1.1.4.\nNow we turn to these problems. The goal is to generalize the preced-\ning results and hypotheses for the Littlewood-Richardson coefficients to the\nstructural constants that arise in these problems. The problem of finding a\npositive, combinatorial formula for the plethysm constant (Problem 1.1.2),\nakin to the positive Littlewood-Richardson rule, has already been recog-\nnized as an outstanding, classical problem in representation theory [St4]–\nthe known formulae based on character theory mentioned in Section 1.1\nare not positive, because they involve alternating signs. Indeed, existence\nof such a formula is a part of the first positivity hypothesis (PH1) below\nfor the plethysm constant, and this problem is the main focus of the work\nin [GCT4, GCT7, GCT8, GCT9].\nIn view of the intensive work on the\nplethym constant in the literature, it has now become clear that the com-\nplexity of the plethysm problem (Problem 1.1.2) is far higher than that of\nthe Littlewood-Richardson problem (Problem 1.2.1). This gap in the com-\nplexity is the main source of difficulties that has to be addressed. We now\nstate the main ingradients in the plan in this paper to show that the relaxed\nforms of Problems 1.1.1, 1.1.2, 1.1.3, and 1.1.4, with X = G/P or a class\nvariety, belong to P.\n1.4\nSaturated and positive integer programming\nFirst, we formulate a general algorithmic paradigm of saturated and positive\ninteger programming that can be applied in the context of these problems.\nLet A be an m×n integer matrix, and b an integral m-vector. An integer\nprogramming problem asks if the polytope P : Ax ≤b contains an integer\n16"},{"paragraph_id":"p20","order":20,"text":"point. In general, it is NP-complete. We want to define its relaxed version,\nwhich will turn out to have a polynomial time algorithm.\nWe allow m, the number of constraints, to be exponential in n. Hence,\nwe cannot assume that A and b are explicitly specified. Rather, it is assumed\nthat the polytope P is specified in the form of a (polynomial-time) separation\noracle in the spirit of Gr ̈otschel, Lov ́asz and Schrijver [GLS]; cf. Section 2.3.\nGiven a point x ∈Rn, the separation oracle tells if x ∈P, and if not, gives\na hyperplane that separates x from P.\nLet fP(n) be the Ehrhart quasi-polynomial of P [St1]. By definition,\nfP (n) is the number of integer points in the dilated polytope nP.\nAn integer programming problem is called saturated, if\n1. The specification of P also contains a number sie(P), called the sat-\nuration index estimate, with the guarantee that the saturation in-\ndex s(fP) ≤sie(P); cf. Definition 1.2.4. In particular, this means\nfP(n + sie(P)) is strictly saturated.\n2. the goal of the problem is to give an efficient algorithm to decide if,\ngiven an integral relaxation parameter c > sie(P), if cP contains an\ninteger point.\nThe algorithm has to work only for relaxation parameters c > sie(P). In\nparticular, if sie(P) ≥1, the algorithm problem does not have to determine\nif P contains an integer point.\nAn integer programming problem is called positive, if\n1. the specification of P also contains a number pie(P), called the pos-\nitivity index estimate, with the guarantee that the positivity index\np(fP) ≤pie(P); cf. Definition 1.2.2. In particular, this means fP(n +\npie(P)) is strictly positive.\n2. the goal of the problem is to give an efficient algorithm to decide if,\ngiven an integral relaxation parameter c > pie(P), if cP contains an\ninteger point.\nAgain, the algorithm has to work only for relaxation parameters c > pie(P).\nSince s(fP) ≤p(fP), a positive integer programming problem is also satu-\nrated.\nThe following is the main complexity-theoretic result in this paper.\n17"},{"paragraph_id":"p21","order":21,"text":"Theorem 1.4.1 (cf. Section 3.1)\n1. Index of the Ehrhart quasi-polynomial fP(n) of a polytope P presented\nby a separation oracle can be computed in oracle-polynomial time, and\nhence, in polynomial time, assuming that the oracle works in polyno-\nmial time.\n2. A saturated, and hence positive, integer programming problem has a\npolynomial time algorithm.\n3. Suppose the polytopes P’s that arise in a specific decision problem have\nthe following property: whenever P is nonempty, the Ehrhart quasi-\npolynomial fP(n) is “almost always” strictly saturated.\nThen there\nexists a polynomial time algorithm for deciding if P contains an integer\npoint that works correctly “almost always”.\nThe meaning of the phrase “almost always” in the context of the decision\nproblems in this paper will be specified later (cf. Theorem 3.1.1).\nIt may be remarked that the index as well as the period of the Ehrhart\nquasi-polynomial can be exponential in the bit length of the specification\nof P. In contrast to the polynomial time algorithm above to compute the\nindex, the known algorithms to compute the period (e.g. [W]) take time\nthat is exponential in the dimension of P. It may be conjectured that one\ncannot do much better: i.e., the period, unlike the index here, cannot be\ncomputed in polynomial time, in fact, even in 2o(dim(P )) time.\nThe algorithm in Theorem 1.4.1 is based on the separation-oracle-based\nlinear programming algorithm of Gr ̈otschel, Lov ́asz and Schrijver [GLS], and\na polynomial time algorithm for computing the Smith normal form [KB].\nThe paradigm of saturated integer programming is useful when one\nknows, a priori, a good estimate for the saturation index of the polytope\nunder consideration, or when the saturation index is almost always zero.\nFor example, if P is the hive polypolype for the Littlewood-Richardson co-\nefficient (type A), then sie(P) = 0, by the saturation theorem [KT1], and\npie(P) = 0, by PH2 (Hypothesis 1.2.3). For the polytopes P that would\narise in this paper, sie(P) and pie(P) would in general be nonzero, but con-\njecturally always small, and sie(P) would conjecturally be almost always\nzero.\n18"},{"paragraph_id":"p22","order":22,"text":"1.5\nQuasi-polynomiality, positivity hypotheses, and\nthe canonical models\nThe basic goal now is to use Theorem 1.4.1 to get polynomial time algorithms\nto decide nonvanishing, modulo small relaxation, of the structural constants\nin Problems 1.1.1, 1.1.2, 1.1.3 and 1.1.4, with X = G/P or a class variety.\nThe main results in this paper which go towards this goal are as follows.\nQuasi-polynomiality\nWe associate stretching functions with the structural constants in Prob-\nlems 1.1.1-1.1.4, akin to the stretching function ̃cλ\nα,β(n) in eq.(1.1) associ-\nated with the Littlewwod-Richardson coefficient, and show that they are\nquasipolynomials; cf. Chapter 4. (But their periods need not be constants,\nas in the case of Littlewood-Richardson coefficients; in fact, they may be\nexponential in general.) In particular, this proves Kirillov’s conjecture [Ki]\nfor the plethysm constants. The proof is an extension of Brion’s remarkable\nproof (cf.\n[Dh]) of quasi-polynomiality of the stretching function associ-\nated with the Littlewood-Richardson coefficient.\nThe main ingradient in\nthe proof is Boutot’s result [Bou] that singularities of the quotient of an\naffine variety with rational singularities with respect to the action of a re-\nductive group are also rational. This is a generalization of an earlier result\nof Hochster and Roberts [Ho] in the theory of Cohen-Macauley rings.\nSaturation and positivity hypotheses\nUsing the stretching quasipolynomials above, we formulate (cf. Section 3.3)\nanalogues of the saturation and positivity hypotheses SH, PH1,PH2,PH3 in\nSection 1.2 for the structural constants in Problems 1.1.1-1.1.3 and Prob-\nlem 1.1.4, with X = G/P or a class variety. As for Littlewood-Richardson\ncoefficients, it turns out that PH2 implies SH. The hypotheses PH1 and SH\n(more strongly, PH2) together imply that the problem of deciding nonvan-\nishing of the structural constant in any of these problems, modulo a small\nrelaxation, can be transformed in polynomial time into a saturated (more\nstrongly, positive) integer programming problem, and hence, can be solved\nin polynomial time by Theorem 1.4.1.\nIn particular, this shows that all\nthe relaxed decision problems that arise in flip (cf. Hypothesis 1.1.6) have\npolynomial time algorithms, assuming these positivity hypotheses. Though\nthese algorithms are elementary, the positivity hypotheses on which their\n19"},{"paragraph_id":"p23","order":23,"text":"correctness depends turn out to be nonelementary.\nThey are intimately\nlinked to the fundamental phenomena in algebraic geometry and the theory\nof quantum groups, as we shall see.\nWe also give theoretical and experimental results in support of these\nhypotheses; cf. Chapter 4-6.\nCanonical models\nThe proofs of quasi-polynomiality mentioned above also associate with each\nstructural constant under consideration a projective scheme, called the canon-\nical model, whose Hilbert function coicides with the stretching quasi-polynomial\nassociated with that structural constant, akin to the model associated by\nBrion [Dh] with the Littlewood-Richardson coefficient.\nThese canonical\nmodels play a crucial role in the approach to the posivity hypotheses sug-\ngested in Section 1.7.\n1.6\nThe plethysm problem\nWe now give precise statements of these results and hypotheses for the\nplethysm problem (Problem 1.1.2). It is the main prototype in this paper,\nwhich illustrates the basic ideas. Precise statements for the more general\nProblems 1.1.3 and 1.1.4 appear in Section 3.3.\nAs for the Littlewood-Richardson coefficients (cf.(1.1)), Kirillov [Ki] as-\nsociates with a plethysm constant aπ\nλ,μ a stretching function\n ̃aπ\nλ,μ(n) = anπ\nnλ,μ,\n(1.5)\nand a generating function\nAπ\nλ,μ(t) =\nX\nn≥0\nanπ\nnλ,μtn.\n(Note that μ is not stretched in these definitions.)\nHe conjectured that Aπ\nλ,μ(t) is a rational function. This is verified here\nin a stronger form:\nTheorem 1.6.1 (a) (Rationality) The generating function Aπ\nλ,μ(t) is ratio-\nnal.\n20"},{"paragraph_id":"p24","order":24,"text":"(b) (Quasi-polynomiality) The stretching function ̃aπ\nλ,μ(n) is a quasi-polynomial\nfunction of n. This is equivalent to saying that all poles of Aπ\nλ,μ(t) are roots\nof unity, and the degree of the numerator of Aπ\nλ,μ(t) is strictly smaller than\nthat of the denominator.\n(c) There exist graded, normal C-algebras S = S(aπ\nλ,μ) = ⊕nSn, and T =\nT(aπ\nλ,μ) = ⊕nTn such that:\n1. The schemes spec(S) and spec(T) are normal and have rational singu-\nlarities.\n2. T = SH, the subring of H-invariants in S, where H = GLn(C) as in\nProblem 1.1.2,\n3. The quasi-polynomial ̃aπ\nλ,μ(n) is the Hilbert function of T. In other\nwords, it is the Hilbert function of the homogeneous coordinate ring of\nthe projective scheme Proj(T).\n(d) (Positivity) The rational function Aπ\nλ,μ(t) can be expressed in a positive\nform:\nAπ\nλ,μ(t) = h0 + h1t + · · · + hdtd\nQ\nj(1 −ta(j))d(j)\n,\n(1.6)\nwhere a(j)’s and d(j)’s are positive integers, P\nj d(j) = d + 1, where d is\nthe degree of the quasi-polynomial ̃aπ\nλ,μ(n), h0 = 1, and hi’s are nonnegative\nintegers.\nThe specific rings S(aπ\nλ,μ) and T(aπ\nλ,μ) constructed in the proof of The-\norem 1.6.1 are very special. We call them canonical rings associated with\nthe plethysm constant aπ\nλ,μ. We call Y (aπ\nλ,μ) = Proj(S(aπ\nλ,μ)), and Z(aπ\nλ,μ) =\nProj(T(aπ\nλ,μ)) the canonical models associated with aπ\nλ,μ. The canonical rings\nare their homogenous coordinate rings.\nIt may be remarked that the analogue of Theorem 1.6.1 (b) for Littlewood-\nRichardson coefficients has an elementary polyhedral proof. Specifically, the\nLittlewood-Richardson stretching function ̃cλ\nα,β(n) of any type is a quasi-\npolynomial since it coincides with the Ehrhart quasi-polynomial of the BZ-\npolytope [BZ]. Similarly, the analogue of Theorem 1.6.1 (d) for Littlewood-\nRichardson coefficients follows from Stanley’s positivity theorem for the\nEhrhart series of a rational polytope (which is implicit in [St3]).\nThese\npolyhderal proofs cannot be extended to the plethysm constant at this point,\n21"},{"paragraph_id":"p25","order":25,"text":"since no polyhedral expression for them is known so far–in fact, this is a part\nof the conjectural positivity hypothesis PH1 below.\nIn contrast, Brion’s\nproof in [Dh] of quasi-polynomiality of ̃cλ\nα,β(n) can be extended to prove\nTheorem 1.6.1 since it does not need a polyhedral interpretation for aπ\nλ,μ.\nBut Boutot’s result [Bou] that it relies on is nonelementary (because it needs\nresolution of singularities in characteristic zero [Hi], among other things).\nWe also give an elementary (nonpolyhedral proof) for Theorem 1.6.1 (a) (ra-\ntionality). But this does not extend to a proof of quasipolynomiality for all\nn, which turns out to be a far delicate problem. It is crucial in the context\nof saturated integer programming.\nTheorem 1.6.2 (Finitely generated cone)\nFor a fixed partition μ, let Tμ be the set of pairs (π, λ) such that the\nirreducible representation Vπ(H) of H = GLn(C) occurs in the irreducible\nrepresentation Vλ(G) of G = GL(Vμ(H)) with nonzero multiplicity. Then\nTμ is a finitely generated semigroup with respect to addition.\nThis is proved by an extension of Brion and Knop’s proof of the analogous\nresult for Littlewood-Richardson coefficients based on invariant theory. In\nthe case of Littlewood-Richardson coefficients, this again has an elementary\npolyhedral proof [Z].\nTheorem 1.6.3 (PSPACE)\nGiven partitions λ, μ, π, the plethysm constant aπ\nλ,μ can be computed in\npoly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) space.\nThe main observation in the proof of Theorem 1.6.3 is that the oldest\nalgorithm for computing the plethysm constant, based on the Weyl character\nformula, can be efficiently parallelized so as to work in polynomial parallel\ntime using exponentially many processors. After this, the result follows from\nthe relationship between parallel and space complexity classes. It may be\nremarked that the known algorithms for computing aπ\nλ,μ in the literature–\ne.g., the one based on Klimyk’s formula [FH]–take exponential time as well\nas space.\nTheorems 1.6.1, 1.6.2 and 1.6.3 lead to the following conjectural sat-\nuration and positivity hypotheses for the plethysm constant.\nThese are\nanalogues of PH1,PH2,PH3, SH in Section 1.2 for Littlewood-Richardson\ncoefficients.\n22"},{"paragraph_id":"p26","order":26,"text":"Hypothesis 1.6.4 (PH1)\nFor every (λ, μ, π) there exists a polytope P = P π\nλ,μ ⊆Rm such that:\n(1) The Ehrhart quasi-polynomial of P coincides with the stretching quasi-\npolynomial ̃aπ\nλ,μ(n) in Theorem 1.6.1. (This means P is given by a linear\nsystem of the form\nAx ≤b,\n(1.7)\nwhere A does not depend on λ and π and b depends only on λ and π in a\nhomogeneous, linear fashion.) In particular,\naπ\nλ,μ = φ(P),\n(1.8)\nwhere φ(P) is equal to the number of integer points in P.\n(2) The dimension m of the ambient space, and hence the dimension of P\nas well, and the bitlength of every entry in A are polynomial in the bitlength\nof μ and the heights of λ and π.\n(3) Whether a point x ∈Rm lies in P can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, ⟨x⟩)\ntime. That is, the membership problem belongs to the complexity class P.\nIf x does not lie in P, then this membership algorithm also outputs, in the\nspirit of [GLS], the specification of a hyperplane separating x from P.\nThe first statement here, in particular, would imply a positive, polyhedral\nformula for aμ\nλ,α, in the spirit of the known positive polyhedral formulae for\nthe Littlewood-Richardson coefficients in terms of the BZ- [BZ], hive [KT1]\nor other types of polytopes [Dh]. It would also imply polyhedral proofs for\nTheorem 1.6.1 (a), (b), (d), and Theorem 1.6.2. Conversely, Theorem 1.6.1\n(a), (b), (d), and Theorem 1.6.2 constitute a theoretical evidence for exis-\ntence of such a positive polyhedral formula.\nThe second statement in PH1 is justified by Theorem 1.6.3.\nSpecifi-\ncally, it should be possible to compute the number of integer points in P\nin PSPACE in view of Theorem 1.6.3. If dim(P) and m were exponential,\nthen the usual algorithms for this problem, e.g. Barvinok [Bar], cannot be\nmade to work in PSPACE. Indeed, it may be conjectured that the number\nof integer points in a general polytope P ⊆Rm can not be computed in\no(m) space.\nThe number of constraints in the hive [KT1] or the BZ-polytope [BZ]\nfor the Littlewood-Richardson coefficient cλ\nα,β is polynomial in the number\nof parts of α, β, λ. In contrast, the number of constraints defining P π\nλ,μ may\nbe exponential in the ⟨μ⟩and the number of parts of λ and π. But this is\n23"},{"paragraph_id":"p27","order":27,"text":"not a serious problem. As long as the faces of the polytope P have a nice\ndescription, the third statement in PH1 is a reasonable assumption. This\nhas been demonstrated in [GLS] for the well-behaved polytopes in combina-\ntorial optimization with exponentially many constraints. The situation in\nrepresentation theory should be similar, or even better. For example, the\nfacets of the hive polytope [KT1] are far nicer than the facets of a typical\npolytope in combinatorial optimization.\nIt is known that membership in a polytope is a “very easy” problem.\nFormally, if a polytope has polynomially many constraints, this problem\nbelongs to the complexity class NC ⊆P [KR], the subclass of problems\nwith efficient parallel algorithms, which is very low in the usual complexity\nhierarchy. Even if the number of constraints of P π\nλ,μ in PH1 is exponen-\ntial, the membership problem may still be conjectured to be in NC (cf.\nRemarkrnc)–which would be “very easy” compared to the decision problem\nwe began with (Problem 1.1.2). For this reason, PH1 is primarily a mathe-\nmatical positivity hypothesis as against PHflip (Hypothesis 1.1.6), and the\npositive, polyhedral formula for aπ\nλ,μ in (1.8) is its main content.\nThe remaining positivity hypotheses are purely mathematical.\nThey\ngeneralize SH,PH2 and PH3 for the Littlewood-Richardson coefficients to\nthe plethysm constants. We turn their specification next. We can begin\nby asking if the stretching quasipolynomial ̃aπ\nλ,μ(n) is strictly saturated or\npositive.\nThis need not be so.\nThe recent article [Ro] shows that strict\nsaturation need not hold for the Kronecker coefficients, as was conjectured\nin the earlier version of this paper. A similar phenomenon was also reported\nin [GCT7, GCT8], where it was observed that the structural constants of\nthe nonstandard quantum groups associated with the plethysm problem (of\nwhich the Kronecker problem is a special case) need not satisfy an analogue\nof PH2. But it was observed there that the positivity (and hence saturation)\nindices of these structural constants are small, though not always zero; eg.\nsee Figures 30,33,35 in [GCT8]. The same can be expected here. This is\nalso supported by the experimental evidence in [BOR] where too it may be\nobserved that the positivity index is small. Furthermore, in the special case\n(n = 2) of the Kronecker problem analysed in [BOR], the saturation index\nis zero for almost all Kronecker coefficients.\nThese considerations suggest:\nHypothesis 1.6.5 (SH)\n(a): The saturation index (Definition 1.2.4) of ̃aπ\nλ,μ(n) is bounded by a poly-\nnomial in the dimension of G in Problem 1.1.2 and the heights of λ and π.\n24"},{"paragraph_id":"p28","order":28,"text":"This means there exist absolute nonnegative constants c and c′, independent\nof n, λ, μ and π, such that the saturation index is bounded above by chc′,\nwhere h = dim G + htλ + htπ.\n(b): The quasi-polynomial ̃aπ\nλ,μ(n) is strictly saturated, i.e. the saturation\nindex is zero, for almost all λ, μ, π. Specifically, the density of the triples\n(λ, μ, π) of total bit length N with nonzero aπ\nλ,μ for which the saturation index\nis not zero is less than 1/N c′′, for any positive constant c′′, as N →∞.\nA stronger form of (a) is:\nHypothesis 1.6.6 (PH2) The positivity index (Definition 1.2.2) of the\nstretching quasi-polynomial ̃aπ\nλ,μ(n) is bounded by a polynomial in the di-\nmension of G and the heights of λ and π.\nThe following is another stronger form of SH (a). For this, we observe\nthat the positive rational form in Theorem 1.6.1 (d) is not unique. Indeed,\nthere is one such form for every h.s.o.p. (homogeneous sequence of param-\neters) of the homogenenous coordinate ring S; the a(j)’s in eq.(1.6) are the\ndegrees of these parameters.\nKirillov asked if the only possible pole of Aπ\nλ,μ is at t = 1–i.e. if aμ\nλ,α(n) is\na polynomial. This is not so (cf. Section 6.2). But it may be conjectured that\nthe structural constants a(j)’s are small. Specifcally, consider an h.s.o.p. of\nS with a (lexicographically) minimum degree sequence, and call the (unique)\npositive rational form in Theorem 1.6.1 (d) associated with such an h.s.o.p.\nminimal. The modular index χ(aπ\nλ,μ) of the plethysm constant is defined to\nbe the modular index (Definition 1.2.7) of this minimal positive form. Then:\nHypothesis 1.6.7 (PH3)\nThe function Aπ\nλ,μ(t) associated with aπ\nλ,μ has a positive rational form\nwith modular index bounded by a polynomial in the dimension of G and the\nheights of λ and π.\nMore specifically, this is so for the minimial positive rational form of\nAπ\nλ,μ(t) as above; i.e., the modular index χ(aπ\nλ,μ) is itself bounded by a poly-\nnomial in the dimension of G and the heights of λ and π.\nThis is a conjectural analogue of a stronger form of PH3 for Littlewood-\nRichardson coefficients (Hypothesis 1.2.6), which says that the modular in-\ndex of a Littlewood-Richardson coefficient, defined similarly, is one. PH3\n25"},{"paragraph_id":"p29","order":29,"text":"here would imply that the period of Aπ\nλ,μ(t) is smooth–i.e. has small prime\nfactors–though it may be exponential in the heights of λ, μ, π. It can be\nshown that PH3 implies SH (a) (Section 3.3).\nThe following result addresses the second arrow in Figure 1.1 in the\ncontext of the relaxed decision problem for the plethysm constant:\nTheorem 1.6.8 The complexity theoretic positivity hypothesis PHflip (Hy-\npothesis 1.1.6) for the plethysm constant is implied by the mathematical\npositivity hypotheses PH1 and SH above. Specifically, assuming PH1 and\nSH:\n(a) Nonvanishing of abπ\nbλ,bμ for any b > chc′, with c, c′, h as in SH, can be\ndecided in O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, ⟨b⟩)) time.\n(b) There is an O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)) time algorithm for deciding if aπ\nλ,μ is\nnonvanishing, which works correctly on almost all λ, μ and π; almost all\nmeans the same as in SH.\nHere (a) follows by applying Theorem 1.4.1 (2) to the polytope P π\nλ,μ in\nPH1, and letting the positivity index estimate for this polytope be chc′; (b)\nfollows from Theorem 1.4.1 (3).\nEvidence for the positivity hypotheses in special cases\nLittlewood-Richardson coefficients are special cases of (generalized) plethym\nconstants. We have already seen that PH1 holds in this case, and that there\nis considerable experimental evidence for PH2 and PH3 (Section 1.2). An-\nother crucial special case of the plethym problem is the Kronecker prob-\nlem (Problem 1.1.1)–in fact, this may be considered to be the crux of the\nplethysm problem. It follows from the results in [GCT9] that PH1 holds for\nthe Kronecker problem when n = 2; the earlier known formulae [RW, Ro]\nfor the Kronecker coefficient in this case are not positive. It can also be seen\nfrom the experimental evidence in [BOR] that the saturation and positivity\nindices of the Kronecker coefficient, for n = 2, are very small, and almost\nalways zero. We also give in Chapter 6 additional experimental evidence for\nPH2 for another basic special case of Problem 1.1.3, with H therein being\nthe symmetric group.\n26"},{"paragraph_id":"p30","order":30,"text":"1.7\nTowards PH1, SH, PH2,PH3 via canonial bases\nand canonical models\nIn this section, we suggest an approach to prove PH1, SH, PH2 and PH3 for\nthe plethysm constant and the analogous hypotheses for the other structural\nconstants in Problems 1.1.3, and 1.1.4, with X = G/P or a class variety.\nIn the case of Littlewood-Richardson coefficients of type A, PH1 and SH\nhave purely combinatorial proofs. But it seems unrealistic to expect such\nproofs of the saturation and positivity hypotheses for the plethysm and\nother structural constants under consideration here given their substantially\nhigher complexity.\nThe approach that we suggest is motivated by the proof of PH1 for\nLittlewood-Richardson coefficients of arbitrary types based on the canonical\n(local/global crystal) bases of Kashiwara and Lusztig for representations of\nDrinfeld-Jimbo quantum groups [Dh, Kas2, Li, Lu2, Lu4]. By a Drinfeld-\nJimbo quantum group we shall mean in this paper quantization Gq of a\ncomplex, semisimple group G as in [RTF] that is dual to the Drinfeld-Jimbo\nquantized enveloping algebra [Dri]. Canonical bases for representions of a\nDrinfeld-Jimbo quantum group in type A are intimately linked [GrL] to the\nKazhdan-Lusztig basis for Hecke algebras [KL1, KL2]. A starting point for\nthe approach suggested here is:\nObservation 1.7.1 (PH0) The homogeneous coordinate rings of the canon-\nical models associated by Brion with the Littlewood-Richardson coefficients\nhave quantizations endowed with canonical bases as per Kashiwara and Lusztig.\nThis is a consequence of the work of Kashiwara [Kas3] and Lusztig [Lu3,\nLu4]; see Proposition 4.2.1 for its precise statment. This is why we call the\nmodels here canonical models.\nWe shall refer to the property above as the zeroeth positivity hypothesis\nPH0. Positivity here refers to the deep characteristic positivity property of\nthe canonical basis proved by Lusztig: namely its multiplicative and comul-\ntiplicative structure constants are nonnegative. For this reason, we say that\na canonical basis is positive. Similar positivity property is also known for\nthe Kazhdan-Lusztig basis [KL2]. The proofs of these positivity properties\nare based on the Riemann hypothesis over finite fields (Weil conjectures)\n[Dl] and the related work of Beilinson, Bernstein, Deligne [BBD].\nThe property above is called PH0 because it implies PH1 for Littlewood-\nRichardson coefficients of arbitrary types. Specifically, the latter is a formal\n27"},{"paragraph_id":"p31","order":31,"text":"consequence of the abstract properties of these canonical bases and is inti-\nmately related to their positivity; cf. Section 4.2.1, and [Dh, Kas2, Li, Lu4].\nThe saturation hypothesis SH in type A [KT1] is a refined property of the\npolyhedral formulae in PH1. In Section 4.2 we suggest an approach to prove\nSH, PH2 and PH3 for arbitrary types based on the properties of these canon-\nical bases. All this indicates that for the Littlewood-Richardson problem\nPH1, SH, PH2 and PH3 are intimately linked to PH0.\nThis suggests the following approach for proving PH1, SH, PH2 and PH3\nfor the plethysm and other structural constants under consideration in this\npaper (cf. Section 4.2.2):\n1. Construct quantizations of the homogeneous coordinate rings of the\ncanonical models associated with these structural constants,\n2. Show that they have canonical bases in some appropriate sense thereby\nextending PH0 to this general setting.\n3. Prove PH1, SH, PH2, and PH3 by a detailed analysis and study of\nthese canonical bases as per this extended PH0, just as in the case of\nLittlewood-Richardson coefficients.\nPictorially, this is depicted in Figure 1.2.\nQuantizations of the homogeneous coordinate rings of the canonical\nmodels associated with Littlewood-Richardson coefficients and their posi-\ntive canonical bases are constructed using the theory Drinfeld-Jimbo quan-\ntum group. In type A, it is intimately related to the theory of Hecke al-\ngebras. But, as expected, the theories of Drinfeld-Jimbo quantum groups\nand Hecke algebras do not work for the plethysm problem. What is needed\nis a quantum group and a quantized algebra that can play the same role\nin the plethysm problem that the Drinfeld-Jimbo quantum group and the\nHecke algebra play in the Littlewood-Richardson problem. These have been\nconstructed in [GCT4] for the Kronecker problem (Problem 1.1.1) and in\n[GCT7] for the generalized plethysm problem (Problem 1.1.2). We shall call\nthem nonstandard quantum groups and nonstandard quantized algebras; cf.\nSection 4.3 for their brief overview. In the special case of the Littlewood-\nRichardson problem, these specialize to the Drinfeld-Jimbo quantum group\nand the Hecke algebra, respectively. The article [GCT8] gives conjecturally\ncorrect algorithms to construct canonical bases of the matrix coordinate\nrings of the nonstandard quantum groups and of nonstandard algebras that\nhave conjectural positivity properties analogous to those of the canonical\n28"},{"paragraph_id":"p32","order":32,"text":"Construction of quantizations of the coordinate rings of canonical models\n|\n|\n|\n↓\nConstruction of canonical bases for these quantizations (PH0)\n|\n|\n|\n↓\nPositivity and saturation hypotheses PH1, SH |\n|\n|\n↓\nPolynomial-time algorithms for the relaxed decision problems\nFigure 1.2: Pictorial depiction of the approach\n(global crystal) bases, as per Kashiwara and Lusztig, of the coordinate ring\nof the Drinfeld-Jimbo quantum group, and the Kazhdan-Lusztig basis of the\nHecke algebra. These conjectures lie at the heart of the approach suggested\nhere, since they are crucial for the extension of PH0 (cf. Figure 1.2) to the\ngeneral setting here. Their verification seems to need substantial extension\nof the work surrounding the Riemann hypothesis over finite fields mentioned\nabove.\n1.8\nBasic plan for implementing the flip\nThe main application of the results and hypotheses in this paper in the\ncontext of the flip is the following result. As mentioned in Section 1.1, and\ndescribed in more detail in Sections 7.6-7.7, each lower bound problem, such\nas the P vs. NP problem over C, is reduced in [GCT1, GCT2] to the prob-\nlem of proving obstructions to embeddings among the class varieties that\narise in the problem. In Chapter 7 we define a robust obstruction, which is\nan obstruction that is well behaved with respect to relaxation, and whose\nvalidity (correctness) depends only on an appropriate PH1 but not SH. It is\n29"},{"paragraph_id":"p33","order":33,"text":"conjectured that in each of the lower bound problems under consideration,\nrobust obstructions exist (Section 7.6.6). In the lower bound problems un-\nder consideration, ultimately one is only interested in proving existence of\nobstructions. So one may as well search for only robust obstructions.\nTheorem 1.8.1 (cf. Chapter 7) Consider the P vs. NP or the NC vs.\nP #P problem over C [GCT1].\nAssume that the homogeneous coordinate\nrings of the relevant class varieties [GCT1, GCT2] in this context have ra-\ntional singularities.\nAlso assume that the structural constants associated\nwith these class varieties satisfy analogous PH1 as specified in Chapter 7.\nThen:\n(a) The problem of verifying a robust obstruction in each of these problems\nbelongs to P, so also the relaxed form of the problem of verifying any ob-\nstruction (not necessarily robust).\n(b) There exists an explicit family of robust obstructions in each of these\nproblems assuming an additional hypothesis OH specified in Chapter 7; the\nmeaning of the term explicit is also given there.\n(b) The problem of deciding existence of a geometric obstruction also belongs\nto P, assuming a stronger form of PH1 specified in Chapter 7. Here geomet-\nric obstruction is a simpler type of robust obstruction, defined in Chapter 7,\nwhich is conjectured to exist in the lower bound problems under considera-\ntion.\nFor a precise statement of this theorem, see Chapter 7.\nThis theorem needs only PH1, but not SH, which is only needed to\nargue why robust obstructions should exist (Section 7.6.6), and furthermore,\nit is only needed for Problems 1.1.1-1.1.3 and not for the GIT Problem\n1.1.4. Thus PH1 is the main positivity hypothesis of GCT in the context\nproving existence of (robust) obstructions for the lower bound problems\nunder consideration.\nA basic plan for implementing the flip suggested by the considerations\nabove is summarized in Figure 1.3. It is an elaboration of Figure 1.1. Ques-\ntion marks in the figure indicate open problems.\n1.9\nOrganization of the paper\nThe rest of this paper is organized as follows.\n30"},{"paragraph_id":"p34","order":34,"text":"Negative hypotheses in complexity theory (Lower bound problems)\n|\n|\nThe flip\n|\n↓\nPositive hypotheses in complexity theory (Upper bound problems)\n|\n|\nSaturated and positive integer programming, and\nthe quasi-polynomiality results in this paper\n|\n↓\nMathematical saturation and positivity hypotheses: PH1,SH (PH2,3)\n|\n|\nConstruction of the canonical models in this paper, and\nconstruction of the quantum groups in GCT4,7\n|\n??\n|\n↓\n(PH0): Construction of quantizations of the coordinate\nrings of the canonical models and their canonical bases\n|\n|\n|\n??\n|\n|\n↓\n(?): Problems related to the Riemann Hypothesis over finite\nfields, and their generalizations\nFigure 1.3: A basic plan for implementing the flip\n31"},{"paragraph_id":"p35","order":35,"text":"In Chapter 2 we describe the basic complexity theoretic notions that we\nneed in this paper and describe their significance in the context of represen-\ntation theory.\nIn Chapter 3, we give a polynomial time algorithm for saturated integer\nprogramming (Theorem 1.4.1), and give precise statements of the results\nand positivity hypotheses for Problems 1.1.3 and 1.1.4 (with X = G/P or\na class variety) mentioned in Section 1.5. These generalize the ones given\nin Section 1.6 for the plethysm constant. The framework of saturated in-\nteger programming in this paper may be applicable to many other struc-\ntural constants in representation theory and algebraic geometry, such as the\nKazhdan-Lusztig polynomials (cf. Sections 3.7).\nIn Chapter 4, we prove the basic quasi-polynomiality results–Theorem 1.6.1\nand its generalizations for Problems 1.1.3 and 1.1.4. We also define canonical\nmodels for the structural constants under consideration, and briefly describe\nthe relevance of the nonstandard quantum groups and the related results in\n[GCT4, GCT7, GCT8] in the context of quantizing the coordinate rings of\nthese canonical models and extending PH0 to them (Figure 1.2).\nIn Chapter 5, we prove the basic PSPACE results–Theorem 1.6.3 and\nits extensions for the various cases of Problem 1.1.3.\nIn Chapter 6, we give experimental evidence for the positivity hypotheses\nPH2 and PH3 in some special cases of the Problems 1.1.1-1.1.4.\nIn Chapter 7, we describe an application (Theorem 1.8.1) of the re-\nsults and positivity hypotheses in this paper to the problem of verifying or\ndiscovering a robust obstruction, i.e., a “proof of hardness” [GCT2] in the\ncontext of the P vs. NP and the permanent vs. determinant problems in\ncharacteristic zero.\n1.10\nNotation\nWe let ⟨X⟩denote the total bitlength of the specification of X. Here X can\nbe an integer, a partition, a classifying label of an irreducible representation\nof a reductive group, a polytope, and so on. The exact meaning of ⟨X⟩will\nbe clear from the context. The notation poly(n) means O(na), for some\nconstant a. The notation poly(n1, n2, . . .) similarly means bounded by a\npolynomial of a constant degree in n1, n2, . . .. Given a reductive group H,\nVλ(H) denotes the irreducible representation of H with the classifying label\nλ. The meaning depends on H. Thus if H = GLn(C), λ is a partition and\n32"},{"paragraph_id":"p36","order":36,"text":"Vλ(H) the Weyl module indexed by λ, if H = Sm, then λ is a partition of\nsize |λ| = m, and Vλ(H) the Specht module indexed by λ, and so on.\n33"},{"paragraph_id":"p37","order":37,"text":"Chapter 2\nPreliminaries in complexity\ntheory\nIn this chapter, we recall basic definitions in complexity theory, introduce\nadditional ones, and illustrate their significance in the context of represen-\ntation theory.\n2.1\nStandard complexity classes\nAs usual, P, NP and PSPACE are the classes of problems that can be\nsolved in polnomial time, nondeterministic polynomial time, and polyno-\nmial space, respectively. The class of functions that can be computed in\npolynomial time (space) is sometimes denoted by FP (resp. FPSPACE).\nBut, to keep the notation simple, we shall denote these classes by P and\nPSPACE again.\nLet SPACE(s(N)) denote the class of problems that can be solved in\nO(s(N)) space on inputs of bit length N; by convention s(N) counts only\nthe size of the work space. In other words, the size of the input, which is on\nthe read-only input tape, and the output, which is on the write-only output\ntape is not counted. Hence s(N) can be less than the size of the input or the\noutput, even logarithmic compared to these sizes. The class space(log(N))\nis denoted by LOGSPACE.\nAn algorithm is called strongly polynomial [GLS], if given an input x =\n(x1, . . . , xk),\n34"},{"paragraph_id":"p38","order":38,"text":"1. the total number of arithmetic steps (+, ∗, −and comparisones) in the\nalgorithm is polynomial in k, the total number of input parameters,\nbut does not depend ⟨x⟩, where ⟨x⟩= P\ni⟨xi⟩denotes the bitlength of\nx.\n2. the bit length of every intermediate operand in the computation is\npolynomial in ⟨x⟩.\nClearly, a strongly polynomial algorithm is also polynomial. let strong P ⊆\nP denote the subclass of problems with strongly polynomial time algorithms.\nThe counting class associated with NP is denoted by #P. Specifically,\na function f : Nk →N, where N is the set of nonnegative integers, is in #P\nif it has a formula of the form:\nf(x) = f(x1, · · · , xk) =\nX\ny∈Nl\nχ(x, y),\n(2.1)\nwhere χ is a polynomial-time computable function that takes values 0 or 1,\nand y runs over all tuples such that ⟨y⟩= poly(⟨x⟩). The formula (2.1) is\ncalled a #P-formula. An important feature of a #P-formula in the context\nof representation theory is that it is positive; i.e., it does not contain any\nalternating signs.\nThe formula (2.1) is called a strong #P-formula, if, in addition, l is\npolynomial in k and χ is a strongly polynomial-time computable function.\nLet strong #P be the class of functions with strong #P-fomulae.\nIt is known and easy to see that\n#P ⊆PSPACE.\n(2.2)\n2.1.1\nExample: Littlewood-Richardson coefficients\nBy the Littlewood-Richardson rule [FH], the coefficient cλ\nα,β (cf.\nProb-\nlem 1.2.1) in type A is given by:\ncλ\nα,β =\nX\nT\nχ(T),\n(2.3)\nwhere T runs over all numbering of the skew shape λ/α, and χ(T) is 1 if\nT is a Littlewood-Richardson skew tableau of content β, and zero, other-\nwise. The total number of entries in T is quadratic in the total number of\n35"},{"paragraph_id":"p39","order":39,"text":"nonzero parts in α, β, λ, and the number of arithmetic steps needed to com-\npute χ(T) is linear in this total number. Hence (2.3) is a strong #P-formula,\nand Littlewood-Richardson function c(α, β, λ) = cλ\nα,β belongs to strong #P.\nIt may be remarked that the character-based formulae for the Littlewood-\nRichardson coefficients are not #P-formulae, since they involve alternat-\ning signs. But the algorithms based on the these formulae for computing\nLittlewood-Richardson coefficients run in polynomial space. Thus, from the\nperspective of complexity theory, the main significance of the Littlewood-\nRichardson rule is that it puts the problem, which at the surface is only in\nPSPACE, in its smaller subclass (strong) #P.\nThough the Littlewood-Richardson rule is often called efficient in the\nrepresentation theory literature, it is not really so from the perspective of\ncomplexity theory. Because computation of cλ\nα,β using this formula takes\ntime that is exponential in both the total number of parts of α, β and λ, and\ntheir bit lengths. This is inevitable, since this problem is #P-complete [N].\nSpecifically, this means there is no polynomial time algorithm to compute\ncλ\nα,β, assuming P ̸= NP.\nAs remarked in earlier, nonzeroness (nonvanishing) of cλ\nα,β can be decided\nin poly(⟨α⟩, ⟨β⟩, ⟨λ⟩) time; [DM2, GCT3, KT1]. Furthermore, the algorithm\nin [GCT3] is strongly polynomial; i.e., the number of arithemtic steps in\nthis algorithm is a polynomial in the total number of parts of α, β, λ, and\ndoes not depend on the bit lengths of α, β, λ. Hence the problem of deciding\nnonvanishing of cλ\nα,β (type A) belongs to strong P.\nThe discussion above shows that the Littlewood-Richardson problem is\nakin to the problem of computing the permanent of an integer matrix with\nnonnegative coefficients. The latter is known to be #P-complete [V], but\nits nonvanishing can be decided in polynomial time, using the polynomial-\ntime algorithm for finding a perfect matching in bipartite graphs [Sc]. If\nthe positivity hypotheses in this paper hold, the situation would be similar\nfor many fundamental structural constants in representation theory and\nalgebraic geometry in a relaxed sense.\n2.2\nConvex #P\nNext we want to introduce a subclass of #P called convex #P.\nGiven a polytope P ⊆Rl, let χP denote the characteristic (membership)\nfunction of P: i.e., χP (y) = 1, if y ∈P, and zero otherwise. We say that\n36"},{"paragraph_id":"p40","order":40,"text":"f = f(x) = f(x1, . . . , xk) has a convex #P-formula if, for every x ∈Zk,\nthere exists a convex polytope (or, more generally, a convex body) Px ⊆Rl,\nsuch that\n1. The membership function χPx(y) can be computed in poly(⟨x⟩, ⟨y⟩)\ntime, each integer point in Px has O(poly(⟨x⟩)) bitlength, and\n2.\nf(x) = φ(Px),\n(2.4)\nwhere φ(Px) denotes the number of integer points in Px. Equivalently,\nf(x) =\nX\ny∈Zl\nχPx(y),\n(2.5)\nwhere y runs over tuples in Zl of poly(⟨x⟩) bitlength, and χPx denotes\nthe membership function of the polytope Px.\nEquation (2.5) is similar to eq.(2.1). The main difference is that χ is now\nthe membership function of a convex polytope. Clearly, eq.(2.5), and hence,\neq.(2.4) is a #P-formula, when χPx can be computed in polynomial time.\nLet convex #P be the subclass of #P consisting of functions with convex\n#P-formulae.\nWe say that eq.(2.4) is a strongly convex #P-formula, if the character-\nistic function of Px is computable in strongly polynomial time. Let strongly\nconvex #P be the subclass of #P consisting of functions with strongly con-\nvex #P-formulae.\nWe do not assume in eq.(2.4) that the polytope Px is explicitly specified\nby its defining constraints. Rather, we only assume, following [GLS], that\nwe are given a computer program, called a membership oracle, which, given\ninput parameters x and y, tells whether y ∈Px in poly(⟨x⟩, ⟨y⟩) time.\nIf the number of constraints defining Px is polynomial in ⟨x⟩, then it\nis possible to specify Px by simply writing down these constraints. In this\ncase the membership question can be trivially decided in polynomial time–in\nfact, even in LOGSPACE–by verifying each constraint one at a time. This\nwould not work if Px has exponentially many constraints. In good cases,\nit is possible to answer the membership question in polynomial time even\nif Px has exponentially many facets. Many such examples in combinatorial\noptimization are given in [GLS]. One such illustrative example in repre-\nsentation theory is given in Section 2.2.2. The polytopes that would arise\n37"},{"paragraph_id":"p41","order":41,"text":"in the plethysm and other problems of main interest in this paper are also\nexpected to be of this kind.\nWe now illustrate the notion of convex #P with a few examples in rep-\nresentation theory.\n2.2.1\nLittlewood-Richardson coefficients\nA geeneralized Littlewood-Richardson coefficient cλ\nα,β for arbitrary semisim-\nple Lie algebra (Problem 1.2.1) has a strong, convex #P-formula, because\ncλ\nα,β = φ(P λ\nα,β),\nwhere P λ\nα,β is the BZ-polytope [BZ] associated with the triple (α, β, λ).\nIt is easy to see from the description in [BZ] that the number of defin-\ning constraints of P λ\nα,β is polynomial in the total number of parts (coor-\ndinates) of α, β, λ.\nGiven α, β, λ, these constraints can be computed in\nstrongly polynomial time. Hence, the membership problem for P λ\nα,β belongs\nto LOGSPACE ⊆P. It follows that the Littlewood-Richardson function\nc(α, β, λ) = cλ\nα,β belongs to strongly convex #P.\n2.2.2\nLittlewood-Richardson cone\nWe now give a natural example of a polytope in representation theory, the\nnumber of whose defining constraints is exponential, but whose membership\nfunction can still be computed in polynomial time.\nGiven a complex, semisimple, simply connected group G, let the Littlewood-\nRichardson semigroup LR(G) be the set of all triples (α, β, λ) of dominant\nweights of G such that the irreducible module Vλ(G) appears in the tensor\nproduct Vα(G) ⊗Vβ(G) with nonzero multiplicity [Z]. Brion and Knop [El]\nhave shown that LR(G) is a finitely generated semigroup with respect to\naddition. This also follows from the polyhedral expression for Littlewood-\nRichardson coefficients in terms of BZ-polytopes [Z]. Let LRR(G) be the\npolyhedral cone generated by LR(G).\nWhen G = GLn(C), the facets of LRR(G) have an explicit description by\nthe affirmative solution to Horn’s conjecture in [Kl, KT1]. But their number\ncan be quite large (possibly exponential). Nevertheless, membership of any\nrational (α, β, λ) (not necessarily integral) in LRR(G) can be decided in\nstrongly polynomial time.\n38"},{"paragraph_id":"p42","order":42,"text":"This is because LRR(G) is the projection of a polytope P(G), the num-\nber of whose constraints is polynomial in the heights of α, β, λ [Z].\nIf\nφ : P(G) →LR(G) is this projection, we can choose P(G) so that for\nany integral (α, β, λ), φ−1(α, β, λ) is the BZ-polytope associated with the\ntriple (α, β, λ). To decide if (α, β, λ) ∈LR(G), we only have to decide if the\npolytope φ−1(α, β, λ) is nonempty. This can be done in strongly polynomial\ntime using Tardos’ linear programming algorithm [Ta].\n2.2.3\nEigenvalues of Hermitian matrices\nHere is another example of a polytope in representation theory with expo-\nnentially many facets, whose membership problem can still belong to P.\nFor a Hermitian matrix A, let λ(A) denote the sequence of eigenvalues\nof A arranged in a weakly decreasing order. Let HEr be the set of triple\n(α, β, λ) ∈Rr such that α = λ(A + B), β = λ(A), λ = λ(B) for some\nHermitian matrices A and B of dimension r. It is closely related to the\nLittlewood-Richardson semisgroup LRr = LR(GLr(C)): HEr ∩P 3\nr = LRr,\nwhere Pr is the semigroup of partitions of length ≤r. I. M. Gelfand asked\nfor an explicit description of HEr. Klyachko [Kl] showed that HEr is a\nconvex polyhedral cone. An explicit description of its facets is now known\nby the affirmative answer to Horn’s conjecture. But their number may be\nexponential. Hence, membership in HEr is still not easy to check using this\nexplicit description. This leads to the following complexity theoretic variant\nof Gelfand’s question:\nQuestion 2.2.1 Does the memembership problem for HEr belong to P?\nGiven that the answer is yes for the closely related LRr = LR(GLr(C))\n(Section 2.2.2), this may be so. If HEr were a projection of some polytope\nwith polynomially many facets, this would follow as in Section 2.2.2. But\nthis is not necessary. For example, Edmond’s perfect matching polytope for\nnon-bipartite graphs is not known to be a projection of any polytope with\npolynomially many constraints. Still the associated membership problem\nbelongs to P [Sc].\n2.3\nSeparation oracle\nSuppose P ⊆Rl is a convex polytope whose membership function χP is\npolynomial time computable. If χP(y) = 0 for some y ∈Rr, it is natural to\n39"},{"paragraph_id":"p43","order":43,"text":"ask, in the spirit of [GLS], for a “proof” of nonmembership in the form of a\nhyperplane that separates y from P.\nIn this paper, we assume that all polytopes are specified by the separation\noracle. This is a computer program, which given y, tells if y ∈P, and if\ny ̸∈P, returns such a separting hyperplane as a proof of nonmembership. We\nassume that the hyperplane is given in the form l = 0, where a linear function\nl such that P is contained in the half space l ≥0, but l(y) < 0. Furthermore.\nwe assume that P is a well-described polyhedron in the sense of [GLS]. This\nmeans P is specified in the form of a triple (χP , n, φ), where P ⊆Rn, χP\nis a program for computing the membership function given y ∈Rn, and\nthere exists a system of inequalities with rational coefficients having P as\nits solution set such that the encoding bit length of each inequality is at\nmost φ. We define the encoding length ⟨P⟩of P as n + φ. We also assume\nthat the separation oracle works in O(poly(⟨P⟩, ⟨y⟩) time.\nFor example, the polynomial time algorithm for the membership function\nof the Littlewood-Richardson cone (cf. Section 2.2.2) can be easily modified\nto return a separating hyperplane as a proof of nonmembership.\nIn what follows, we shall assume, as a part of the definition of a convex\n#P-formula, that Px in (2.4) is a well-described polyhedron specified by\na separation oracle that works in polynomial time with ⟨Px⟩= poly(⟨x⟩).\nThese additional requirements are needed for the saturated integer program-\nming algorithm in Chapter 3.\n40"},{"paragraph_id":"p44","order":44,"text":"Chapter 3\nSaturation and positivity\nIn this chapter we describe (Section 3.1) a polynomial time algorithm for\nsaturated and positive integer programming (Theorem 1.4.1). In Section 3.3\nwe state the main results and positivity hypotheses for the relaxed forms of\nProblem 1.1.3 and Problem 1.1.4, with X = G/P or a class variety therein.\nTogether they say that these relaxed decision problems can be efficiently\ntransformed into saturated (more strongly, positive) integer programming\nproblems, and hence can be solved in polynomial time.\n3.1\nSaturated and positive integer programming\nWe begin by proving Theorem 1.4.1.\nLet P ⊆Rn be a polytope given by a separation oracle (Section 2.3).\nLet ⟨P⟩be the encoding length of P as defined in Section 2.3. An oracle-\npolynomial time algorithm [GLS] is an algorithm whose running time is\nO(poly(⟨P⟩)), where each call to the separation oracle is computed as one\nstep.\nThus if the separation oracle works in polynomial time, then such\nan algorithm works in polynomial time in the usual sense.\nLet φ(P) be\nthe number of integer points in P.\nLet fP(n) = φ(nP) be the Ehrhart\nquasi-polynomial [St1] of P. Let l(P) be the least period of fP(n), if P\nis nonempty.\nLet fi,P(n), 1 ≤i ≤l(P), be the polynomials such that\nfP (n) = fi,P(n) if n = i modulo l(P). Let FP (t) = P\nn≥0 fP(n)tn denote\nthe Ehrhart series of P. It is a rational function.\nTheorem 3.1.1 (a) The index of fP (n), index(fP), can be computed in\noracle-polynomial time, and hence, in polynomial time, assuming that the\n41"},{"paragraph_id":"p45","order":45,"text":"oracle works in polynomial time. Furthermore, if index(fP) ̸= 0 (i.e. if P\nis nonempty), then fi,P(n) is not an identically zero polynomial for every i\ndivisible by index(fP).\n(b) The saturated, and hence, positive integer programming problem, as de-\nfined in Section 1.4, can be solved in oracle-polynomial time.\nHere it is\nassumed that the specification of P also contains the saturation index esti-\nmate sie(P), or the positivity index estimate pie(P), and that the bitlength\nof this estimate is O(poly(⟨P⟩)). Given a relaxation parameter c > sie(P)\n(or pie(P)), the problem is to determine if cP contains an integer point in\nO(poly(⟨P⟩, ⟨c⟩)) time.\n(c) Suppose {Px} is a family of polytopes, indexed by some parameter x,\nwith the following property: wherenver Px is nonempty, the Ehrhart quasi-\npolynomial fPx(n) is “almost always” strictly saturated.\nAlmost always\nmeans, the density of x’s of bitlength ≤N, with nonempty Px for which\nfPx(n) is not strictly saturated is less than 1/N c′′, for any positive c′′, as\nN →0. We also assume that Px is given by a separation oracle that works in\nO(poly(⟨x⟩)) time, where ⟨x⟩is the bitlength of x, and ⟨Px⟩= O(poly(⟨x⟩)).\nThen there exists a O(poly(⟨x⟩)) time algorithm for deciding if Px con-\ntains an integer point that works correctly “almost always”; i.e., on almost\nall x.\nProof:\n(a):\nNonemptyness of P can be decided in oracle-polynomial time using the\nalgorithm of Gr ̈otschel, Lov ́asz and Schrijver [GLS] (cf.\nTheorem 6.4.1\ntherein). An extension of this algorithm, furthermore, yields a specifica-\ntion of the affine space span(P) containing P if P is nonempty (cf. Theo-\nrems 6.4.9, and 6.5.5 in [GLS]). Specifically, it outputs an integral matrix\nC and an integral vector d such that span(P) is defined by Cx = d. This\nfinal specification is exact, even though the first part of the algorithm in\n[GLS] uses the ellipsoid method. Indeed, the use of simultneous diophan-\ntine approximation based on basis reduction in lattices is precisely to ensure\nthis exactness in the final answer. This is crucial for the next step of our\nalgorithm.\nIf P is empty, index(fP) = 0. So assume that it is nonempty. Let ̄C be\nthe Smith normal form of C; i.e., ̄C = ACB for some unimodular matrices\nA and B, where the leftmost principal submatrix of ̄C is a diagonal, integral\nmatrix, and all other columns are zero.\n42"},{"paragraph_id":"p46","order":46,"text":"The matrices ̄C, A and B can be computed in polynomial time using\nthe algorithm in [KB]. After a unimodular change of coordinates, by letting\nz = B−1x, span(P) is specified by the linear system ̄Cz = ̄d = Ad. The\nequations in this system are of the form:\n ̄cizi = ̄di,\n(3.1)\ni ≤codim(P), for some integers ̄ci and ̄di. By removing common factors if\nnecessary, we can assume that ̄ci and ̄di are relatively prime for each i. Let\n ̃c be the l.c.m. of ̄ci’s.\nThe statement (a) follows from:\nClaim 3.1.2 index(fP ) = ̃c and fi,P(n) is not an identically zero polyno-\nmial for every i divisible by ̃c.\nProof of the claim: Indeed, nP = {nz | z ∈P} contains no integer point\nunless ̃c divides n.\nHence, it is easy to see that FP (t) = F ̄P (t ̃c), where\nF ̄P (x) is the Ehrhart series of the dilated polytope ̄P = ̃cP. By eq.(3.1),\nthe equations defining ̄P are:\nzi = ̄di( ̃c/ ̄ci),\n(3.2)\nClearly, ̃c divides the least period l(P) of fP, and l( ̄P) = l(P)/ ̃c is the period\nof the Ehrhart quasipolynomial f ̄P(n). It suffices to show that the index of\nf ̄P (n) is one and that fj, ̄P(n) is not an identically zero polynomial for every\n1 ≤j ≤l( ̄P). This is equivalent to showing that ̄P contains a point z with\nwith zi = ai/b, for some integers ai’s and b such that b = j modulo l( ̄P).\nLet us call such a point j-admissible. Because of the form of the equations\n(3.2) defining span( ̄P), we can assume, without loss of generality, that ̄P is\nfull dimensional. This means the system (3.2) is empty. Then this follows\nfrom denseness of the set of j-admissible points. This proves the claim, and\nhence (a).\n(b): Let s = sie(P) be the given saturation index estimate. This means\nfP (n + s) is strictly saturated. This in conjunction with (a) implies that,\ngiven a relaxation parameter c > s, cP contains an integer point, iffc\nis divisible by index(fP ) (by letting n = c −s). This can be checked in\nO(poly(⟨P⟩, ⟨c⟩)) time since index(fP ) can be computed in polynomial time\nby (a).\n(c) The algorithm computes index(fPx) and says “Probably Yes”if the index\nis one, and “No” otherwise. Since the saturation index of fPx(n) is zero\n43"},{"paragraph_id":"p47","order":47,"text":"almost always, by the argument in (b) with s = 0 and c = 1, “Probably\nYes” really means “Yes” almost always. Q.E.D.\nThe algorithm in (c) has one drawback.\nIf the answer is “Probably\nYes”, we have no easy way of checking if Px really contains an integer point.\nIdeally, we would like an algorithm that says “Yes”, with an integer point\nin Px as a proof certificate, or “No”, or “Unsure”, and the density of x’s on\nwhich it says “Unsure” should be very small. This problem can be overcome\nif the family {Px} has the following stronger property, akin to the family of\nhive polytopes [KT1]: there is a linear function lx such that, for almost all x,\nif {Px} is nonempty, then the lx-optimum of Px is integral (this is stronger\nthan saying that fPx(n) is strictly saturated). In this case, the algorithm in\n(c) can be extended to yield the integral lx-optimum as a proof certificate. If\nthe lx-optimum is not integeral, the algorithm says “Unsure”. PH1 and SH\n(Section 1.6) for the plethysm (and more generally, the subgroup restriction)\nproblem may be strengthened by stipulating that the polytopes therein have\nthis property. But this is not needed in this paper.\nWe note down one corollary of the proof of Theorem 3.1.1 (this should\nbe well known):\nProposition 3.1.3 The rational function FP (t) = F ̄P (t ̃c), where F ̄P (x) is\nthe Ehrhart series of the dilated polytope ̄P = ̃cP, and ̃c is the index of\nfP (n).\nIf P is explicitly specified in the form a linear system\nAx ≤b,\n(3.3)\nwhere A is an m × n matrix, b an m vector and m = poly(n), then the\nfollowing stronger version of Theorem 3.1.1 holds. Let ⟨A⟩and ⟨A, b⟩denote\nthe bitlength of the specification of A and of the linear system (3.3).\nTheorem 3.1.4 Suppose P is specified in terms of an explicit linear system\n(3.3). Then the index of the Erhart quasi-polynomial fP(n) can be computed\nin poly(⟨A, b⟩) time, using poly(⟨A⟩) arithmetic operations.\nThus, saturated, and hence, positive integer programming problem spec-\nified in the form (3.3) can be solved in in poly(⟨A, b, c⟩) time, where c is the\nrelaxation parameter, using poly(⟨A⟩) arithmetic operations.\nProof: This is proved exactly as Theorem 3.1.1, but with Tardos’ strongly\npolynomial time algorithm for combinatorial linear programming [Ta] used\nin place of the algorithm in [GLS]. Q.E.D.\n44"},{"paragraph_id":"p48","order":48,"text":"3.1.1\nA general estimate for the saturation index\nNow we give a general estimate for the saturation index of any polytope P\nwith a specification of the form\nAx ≤b,\n(3.4)\nwhere A is an m × n matrix, m possibly exponential. Let ∥P∥= n + ψ,\nwhere ψ is the maximum bitlength of any entry of A. Trivially, ∥P∥≤⟨P⟩.\nWe do not assume that we know the specification (3.4) of P explicitly. We\nonly assume that it exists, and that we are told ∥P∥. Then:\nTheorem 3.1.5 The saturation index of P is O(2poly(∥P ∥)).\nThus the\nbitlength of the saturation index is O(poly(∥P∥)).\nConjecturally, this also holds for the positivity index. This estimate is\nvery conservative, but useful when no better estimate is available.\nProof: There exists a triangulation of P into simplices such that every vertex\nof any simplex is also a vertex of P. Then\nfP(n) =\nX\n∆\nf∆(n),\nwhere ∆ranges over all open simplices in this triangulation; a zero-dimensional\nopen simplex is a vertex. The saturation index of fP(n) is clearly bounded\nby the maximum of the saturation indices of f∆(n).\nHence, we can assume, without loss of generality, that P is an open sim-\nplex. Let v0, . . . , vn be its vertices. Then, by Ehrhart’s result (cf. Theorem\n1.3 in [st5]),\nFP (t) =\nP\ni hiti\nQn\nj=0(1 −taj),\n(3.5)\nwhere h0 = 1, hi’s are nonnegative, and aj is the least positive integer\nsuch that ajvj is integral. By Cramer’s rule, the bit length of each aj is\npoly(∥P∥).\nWithout loss of generality, we can also assume that aj’s are\nrelatively prime. Otherwise, the estimate on the saturation index below has\nto be multiplied by the g.c.d. of aj’s. Then the result follows by applying\nthe following lemma to FP (t), since ⟨aj⟩= O(poly(∥P∥)). Q.E.D.\n45"},{"paragraph_id":"p49","order":49,"text":"Lemma 3.1.6 Let f(n) be a quasipolynomial whose generating function\nF(t) has a positive form\nF(t) =\nP\ni hiti\nQn\nj=0(1 −taj),\n(3.6)\nwhere h0 = 1, hi’s are nonnegative, and aj’s are positive and relatively\nprime.\nLet a = max{aj}.\nThen the saturation index s(f) of f(n) is\nO(poly(a, n)).\nProof: Let g(n) be the quasi-polynomial whose generating function G(t) =\nP g(n)tn is 1/Qn\nj=0(1 −taj). It is known that this is the Ehrhart quasipoly-\nnomial of the polytope N(a0, . . . , an) defined by the linear system\nX\najxj = 1, xj > 0.\nThe saturation index s(g) of g(n) is bounded by the Frobenius number\nassociated with the set of integers {aj}–this is the largest positive integer m\nsuch that the diophantine equation\nX\nj\najxj = m\nhas no positive integeral solution (x0, . . . , xn). It is known (e.g. [BDR]) that\nthe Frobenius number is bounded by\nX\nj\naj +\np\na0a1a2(a0 + a1 + a2) = O(poly(a)),\nassumming that a0 ≤a1 . . .. Hence, s(g) = O(poly(a)).\nSince f(n) is a quasi-polynomial, the degree of the numerator of F(t) is\nless than the degree of the denominator. Thus the maximum value of i that\noccurs in (3.6) is an.\nLet gi(n), i ≤an, be the quasi-polynomial whose generating function is\nti/Qn\nj=0(1 −taj). Then\ns(gi) ≤i + s(g) = O(poly(a, n)).\nSince, hi’s in (3.6) are nonnegative, s(f) = max s(gi). The result follows.\nQ.E.D.\n46"},{"paragraph_id":"p50","order":50,"text":"3.1.2\nExtensions\nWe now mention a few straightforard extensions of Theorem 3.1.1.\nFirst, it is not necessary that P be a closed polytope. We can allow\nit to be half-closed.\nSpecifically, it can be a solution set of a system of\ninequalitites of the form:\nA1x ≤b1\nand\nA2x < b2,\n(3.7)\nwhere we have allowed strict inequalities. The function FP (n) = φ(nP), the\nnumber of integer points in nP, is again a quasi-polynomial. Hence, the\nnotions of saturation and positivity can be generalized to this setting in a\nnatural way.\nSecond, the algorithm in Theorem 3.1.1 (b) only needs a nonnegative\nnumber s(P) such that, for any positive integer c > s(P):\nSaturation guarantee: If the affine span of cP, contains an integer point,\nthen cP is guaranteed to contain an integer point.\nIf s(P) = sie(P), then this guarantee holds, as can be seen from the\nproof of Theorem 3.1.1.\n3.1.3\nIs there a simpler algorithm?\nThough the algorithm for saturated integer programming in Theorem 3.1.1\nis conceptually very simple, in reality it is quite intricate, because the work\nof Gr ̈otschel, Lov ́asz and Schrijver [GLS] needs a delicate extension of the el-\nlipsoid algorithm [Kh] and the polynomial-time algorithm for basis reduction\nin lattices due to Lenstra, Lenstra and Lov ́asz [LLL]. As has been empha-\nsized in [GLS], such a polynomial-time algorithm should only be taken as a\nproof of existence of an efficient algorithm for the problem under consider-\nation. It may be conjectured that for the problems under consideration in\nthis paper such simple, combinatorial algorithms exist. But for the design\nof such algorithms, saturation alone does not suffice. The stronger property\n(PH3), and more, is necessary. We shall address this issue in Section 3.6.\n3.2\nLittlewood-Richardson coefficients again\nTheorem 3.1.4 applied to the BZ-polytope [BZ], with saturation index esti-\nmate equal to zero, specializes to the following in the setting of the Littlewood-\n47"},{"paragraph_id":"p51","order":51,"text":"Richardson problem (Problem 1.2.1):\nTheorem 3.2.1 [GCT5] Assuming SH (Hypothesis 1.2.5), nonvanishing of\ncλ\nα,β, given α, β, λ, can be decided in strongly polynomial time (Section 2.1)\nfor any semisimple classical Lie algebra G.\nIt is assumed here that α, β, λ are specified by their coordinates in the\nbasis of fundamental weights.\nFor type A, this reduces to the result in\n[GCT3], which holds unconditionly.\nThe saturation conjecture for type A arose [Z] in the context of Horn’s\nconjecture and the related result of Klyachko [Kl]. We now turn to implica-\ntions of Theorem 3.2.1 in this context.\nGiven a complex, semisimple, simply connected, classical group G, let\nLR(G) be the Littlewood-Richardson semigroup as in Section 2.2.2. The\nfollowing is a natural generalization of the problem raised by Zelevinsky [Z]\nto this general setting:\nProblem 3.2.2 Give an efficient description of LR(G).\nZelevinsky asks for a mathematically explicit description. This is a com-\nputer scientist’s variant of his problem.\nLet LRR(G) be the polyhedral convex cone generated by LR(G). For\nG = GLn(C), by the saturation theorem, a triple (α, β, λ) of dominant\nweights belongs to LR(G) iffit belongs to LRR(G).\nAssuming SH (Hy-\npothesis 1.2.5), Theorem 3.2.1 provides the following efficient description\nfor LR(G) in general. Recall that the period of the Littlewood-Richardson\nstretching polynomial ̃cλ\nα,β(n) divides a fixed constant d(G), which only de-\npends on the types of simple factors of G [DM2, GCT5]. Let αi’s denote\nthe coordinates of α in the basis of fundamental weights.\nCorollary 3.2.3 (a) Assuming SH, whether a given (α, β, λ) belongs to\nLR(G) can be determined in strongly polynomial time.\n(b) There exists a decomposition of LRR(G) into a set of polyhedral cones,\nwhich form a cell complex C(G), and, for each chamber C in this complex,\na set M(C) of O(rank(G)2) modular equations, each of the form\nX\ni\naiαi +\nX\ni\nbiβi +\nX\ni\nciλi = 0\n(mod d),\nfor some d dividing d(G), such that\n48"},{"paragraph_id":"p52","order":52,"text":"1. SH (Hypothesis 1.2.5) is equivalent to saying that: (α, β, λ) ∈LR(G)\niff(α, β, λ) ∈LRR(G) and (α, β, λ) satisfies the modular equations in\nthe set M(Cα,β,λ) associated with the cone Cα,β,λ containing α, β, λ.\n2. Given (α, β, λ), whether (α, β, λ) ∈LRR(G) can be determined in\nstrongly polynomial time (cf. Section 1.2.5).\n3. If so, the cone Cα,β,λ and the associated set M(Cλ\nα,β) of modular equa-\ntions can also be determined in strongly polynomial time. After this,\nwhether (α, β, λ) satisfies the equations in M(Cλ\nα,β) can be trivially\ndetermined in strongly polynomial time.\nProof: (a) is a consequence of Theorem 3.2.1. (b) follows from a careful\nanalysis of the algorithm therein; see the proof of a more general result\n(Theorem 4.4.2) later. Q.E.D.\nWe call the labelled cell complex C(G), in which each cell C ∈C(G)\nis labelled with the set of modular equations M(C), the modular complex,\nassociated with LRR(G).\nWhen G = SLn(C), the modular complex is\ntrivial: it just consists of the whole cone LRR(G) with only one obvious\nmodular equation attached to it. But, for general G, the modular complex\nand the map C →M(C) are nontrivial.\nWe do not know their explicit\ndescription.\nCorollary 3.2.3 says that, given x = (α, β, λ), whether x ∈\nLRR(G), and whether the relevant modular equations are satisfied can be\nquickely verified on a computer, though the modular equations cannot be\neasily determined and verified by hand, as in type A.\nThis is the main\ndifference between type A and general types.\nThis naturally leads to:\nQuestion 3.2.4 Is there a mathematically explicit description of the mod-\nular complex C(G) for a general G?\n3.3\nThe saturation and positivity hypotheses\nNow let f(x), x ∈Nk, be a counting function associated with a structural\nconstant in representation theory or algebraic geometry.\nHere x denotes\nthe sequence of parameters associated with the constant. Let ⟨x⟩denote\nthe bitlength of x. Let ∥x∥and rank(x) denote its combinatorial size and\ncombinatorial rank–these measure complexity of the nonstretchable part in\n49"},{"paragraph_id":"p53","order":53,"text":"the specification of x and will be specified later for the f’s of interest in this\npaper.\nFor example, in the Littlewood-Richardson problem, x is the triple (α, β, λ),\nf(x) = f(α, β, λ) = cλ\nα,β, ⟨x⟩is the total bitlength of the coordinates of\nα, β, λ, ∥x∥is the total number of coordinates of α, β and λ, and rank(x) =\n∥x∥. The number of coordinates does not change during stretching, and\nhence, constitute the nonstretchable part of the input specification here.\nAssume that f(x) is nonnegative for all x ∈Nk, Then we can successively\nask the following questions:\n1. Does f ∈PSPACE? That is, can f(x) be computed in poly(⟨x⟩)\nspace?\n2. Does f ∈#P? (cf. Section 2.1)\n3. Does f ∈convex#P? (cf. Section 2.2)\n4. Can a stretching function ̃f(x, n) be associated with f(x) intrinsically\nso that ̃f(x, n) is quasi-polynomial?\n5. (PH1?): Is there a polytope Px, for every x, with ⟨Px⟩= O(poly(⟨x⟩))\nand ∥Px∥= O(poly(∥x∥)), such that ̃f(x, n) = fPx(n)?\n6. Are there good analogues of SH and/or PH2, PH3 for ̃f(x, n)?\nIf\nso, nonvanishing of f(x), modulo small relaxation, can be decided in\nO(poly(⟨Px⟩)) time by Theorem 3.1.1.\nIn the rest of this paper, we study these questions when f = f(x) is a\nnonnegative function associated with a structural constant in any of the deci-\nsion problems in Section 1.1. Exact specifications of x, ⟨x⟩, ∥x∥, rank(x), f(x),\nand ̃f(x, n) for these decision problems are given in Sections 3.4-3.5. It is\nshown in Chapter 5 that f(x) ∈PSPACE for Problem 1.1.2 and the special\ncases of Problem 1.1.3 that arise in the flip. This may be conjectured to be\nso for the f’s in Problem 1.1.4, with X therein a class variety; cf [GCT10]\nfor its justification. Quasipolynomiality of ̃f(x, n) is addressed in Chapter 4.\nThe hypotheses PH1, SH, PH2, and PH3 in these cases have the following\nunified form.\nHypothesis 3.3.1 (PH1) Let f = f(x) be the function associated with a\nstructural constant in\n50"},{"paragraph_id":"p54","order":54,"text":"1. Problem 1.1.1, or\n2. 1.1.2, or\n3. Problem 1.1.3, or\n4. Problem 1.1.4, with X being a class variety therein.\nThen the function f(x) has a convex #P-formula (cf. (2.4))\nf(x) = φ(Px),\nsuch that:\n1. for every fixed x, the Ehrhart quasi-polynomial fPx(n) of Px coincides\nwith ̃f(x, n).\n2. ⟨P⟩= O(poly(P)) and ∥P∥= O(poly(∥x∥)).\nHypothesis 3.3.2 (SH)\n(a) Suppose f(x) is a structural constant as in PH1 above. Then for every x,\nthe saturation index s( ̃f) of ̃f(x, n) is O(poly(rank(x))). This means there\nexist absolute nonnegative constants c, c′ such that s( ̃f) ≤c(rank(x))c′.\n(b) For f(x) in Problems 1.1.1-1.1.3, the saturation index of ̃f(x, n) is zero–\ni.e., ̃f(x, n) is strictly saturated–for almost all x. This means the density of\nx, with ⟨x⟩≤N and f(x) nonzero, for which the saturation index s( ̃f) is\nnonzero is ≤1/N c′′, for any positive costant c′′, as N →∞.\nMore strongly than (a),\nHypothesis 3.3.3 (PH2) For f(x) as in PH1, the positivity index of ̃f(x, n)\nis O(poly(rank(x))).\nHypothesis 3.3.4 (PH3) For f(x) as in PH1, the generating function\nF(x, t) = P\nn ̃f(x, n)tn has a positive rational form of modular index O(poly(rank(x))).\nMore specifically, the modular index of ̃f(x, n), as defined in Section 4.1.1\nfor f’s that arise in this paper, is O(poly(rank(x))).\nPH3 implies SH (a); this follows from Lemma 3.1.6.\nThe following conservative bound follows from Theorem 3.1.5.\n51"},{"paragraph_id":"p55","order":55,"text":"Theorem 3.3.5 (Weak SH)\nAssuming PH1 (Hypothesis 3.3.1), the saturation index of ̃f(x, n) is\nbounded by 2O(poly(∥x∥)); hence its bitlength is bounded by O(poly(∥x∥)).\nThe following result addresses the relaxed forms of the decision problems\nfor the structural constants under consideration (cf. Section 1.1).\nTheorem 3.3.6 Suppose f(x) is a structural constant as in PH1 above.\nThen PH1 (Hypothesis 3.3.1) and SH (Hypothesis 3.3.2) imply Hypothe-\nsis 1.1.6 (PHflip) in this case. Specifically:\n(a) For f(x) in Problems 1.1.1-1.1.4, nonvanishing of ̃f(x, a), for a given x\nand a relaxation parameter a > c(rank(x))c′, with c, c′ as in Hypothesis 3.3.2,\ncan be decided in poly(⟨x⟩, ⟨a⟩) time.\n(b) For f(x) as in Problems 1.1.1-1.1.3, there is a poly(⟨x⟩) time algorithm\nfor deciding nonvanishing of f(x) that works correctly on almost all x.\nThis follows from Theorem 3.1.1.\nThe following sections give precise descriptions of x, ⟨x⟩, ∥x∥, rank(x)\nand ̃f(x, n) for the structural constants under consideration.\n3.4\nThe subgroup restriction problem\nIn this section we consider the subgroup restriction problem (Problem 1.1.3).\nThe Kronecker and the plethysm problems (Problems 1.1.1, 1.1.2) are its\nspecial cases.\nLet G, H, ρ, λ, π, mπ\nλ be as in Problem 1.1.3. We shall define below an ex-\nplicit polynomial homomorphism ρ : H →G, as needed in the statement of\nProblem 1.1.3, and also the precise specifications [H], [ρ], [λ], [π] of H, ρ, λ, π,\nrespectively. We shall also define the bitlengths ⟨H⟩, ⟨ρ⟩, ⟨λ⟩, ⟨π⟩and the\ncombinatorial bit lengths ∥λ∥, ∥π∥. We let ∥H∥= ⟨H⟩and ∥ρ∥= ⟨ρ⟩, since\nH and ρ belong to the nonstretchable part of the input. On the other hand,\nλ and π will be stretched in the definition of ̃f(x, n), and hence their com-\nbinatorial bit lengths will differ from the usual bit lengths. The input x in\nthe subgroup restriction problem is the tuple ([H], [ρ], [λ], [π]). Its bitlength\n⟨x⟩is defined to be the sum of the bitlengths ⟨H⟩, ⟨ρ⟩, ⟨λ⟩, ⟨π⟩, and ∥x∥is\ndefined to be the sum of ∥H∥, ∥ρ∥, ∥λ∥and ∥π∥. Finally rank(x) is defined\nto the sum of the ranks of H and G and ∥λ∥and ∥π∥. Here that rank of\n52"},{"paragraph_id":"p56","order":56,"text":"a (reductive) group is defined in a standard way. For example, the rank of\nthe symmentric group Sn is n, that of GLn(C) is n. The rank of a general\nfinite or connected simple group can be defined similarly, and the rank of a\nmore complex reductive group is defined to be the sum of the ranks of its\nsimple components. With this terminology, we let f(x) = mπ\nλ, with x as\ndefined here in Hypotheses 3.3.1-3.3.4 and Theorem 3.3.6 for the subgroup\nrestriction problem. Here H and ρ are implicit in the definition of mπ\nλ.\nFor example, in the plethym problem (Problem 1.1.2), these specifi-\ncations are as follows.\nThe specification [H] is just the root system for\nH = GLn(C). Its bitlength ⟨H⟩is n. The specification [ρ] of the repre-\nsentation map ρ : H →G = GL(Vμ(H)) consists of just the partition μ\nspecified in terms of its nonzero parts. Its bitlength ⟨ρ⟩= ⟨μ⟩. The ranks\nof H and G are as usual. The partitions λ and μ are specified in terms of\ntheir nonzero parts. Their bitlength is the total bitlength of the parts, and\nthe combinatorial bit length is the total number of parts (the height). It\nis crucial here that only nonzero parts of λ are specified, because the rank\nof G can be exponential in the rank of H and the bitlength of μ. Hence,\nthe bitlength of this compact representation of λ can be polynomial in the\nrank of H and the bitlength of μ, even if the dimension of G is exponential.\nThe main difference between ⟨x⟩and ∥x∥is that the stretchable data λ and\nπ contribute their bitlengths to the former, and their heights to the latter.\nThe plethysm problem is the main prototype of the subgroup restriction\nproblem. If the reader wishes, (s)he can skip the rest of this subsection and\njump to Section 3.4.3 in the first reading.\nIn general, we assume that H in Problem 1.1.3 is a finite simple group, or\na complex simple, simply connected Lie group, or an algebraic torus (C∗)k,\nor a direct product of such groups. The results and hypotheses in this paper\nare also applicable if we allow simple types of semidirect products, such as\nwreath products, which is all that we need for the sake of the flip. But these\nextensions are routine, and hence, for the sake of simplicity, we shall confine\nourselves to direct products.\n3.4.1\nExplicit polynomial homomorphism\nNow let us define an explicit polynomial homomorphism. This will be done\nby defining basic explicit homomorphisms, and composing them functorially.\nBasic explicit homomorphisms:\nLet V be an irreducible polynomial representation of H (character-\n53"},{"paragraph_id":"p57","order":57,"text":"istic zero), or more generally, an explicit polynomial representation that\nis constructed functorially from the irreducible polynomial representations\nusing the operations ⊕and ⊗.\nThen the corresponding homomorphism\nρ : H →G = GL(V ) is an explicit polynomial homomorphism. The iden-\ntity map H →H is also an explicit polynomial homomorphism.\nThe polynomiality restriction here only applies to the torus component\nof H. If H is a finite simple group, or a complex semisimple group, then\nany irreducible representation of H is, by definition, polynomial. In general,\na representation is polynomial if its restriction to the torus component is\npolynomial; i.e., a sum of polynomial (one dimensional) characters.\nTo see why the polynomiality restriction is essential, let H be a torus,\nV its rational representation, and G = GL(V ).\nLet Vλ(G) = Symd(V ),\nthe symmetric representation of G, and let π be the label of the trivial\ncharacter of H. Then the multiplicity mπ\nλ is the number of H-invariants in\nSymd(V ). This is easily seen to be the number of nonnegative solutions of a\nsystem of linear diophontine equations. But the problem of deciding whether\na given system of linear diophontine equations has a nonnegative solution\nis, in general, NP-complete. Though the system that arises above is of a\nspecial form, it is not expected to be in P if V is allowed to be any rational\nrepresentation; the associated decision problem may be NP-complete even\nin this special case. If V is a polynomial representation of a torus H, then\nall coefficients of the system are nonnegative, and the decision problem is\ntrivially in P.\nComposition:\nWe can now compose the basic explicit (polynomial) homomorphisms\nabove functorially:\n1. If ρi : H →Gi are explicit, the product map ρ : H →Q\ni Gi is also\nexplicit.\n2. If ρi : Hi →Gi are explicit, the product map ρ : Q Hi →Q Gi is also\nexplicit.\nInstead of products, we can also allow simple semi-direct products such\nas wreath products here. We may also allow other functorial constructions\nsuch as induced representations and restrictions. For example, if ρ : H →G\nis an explicit polynomial homomorphism, and G′ ⊆G is an explicit subgroup\nof G such that ρ(H) ⊆G′, then the restricted homomorphism ρ′ : H →G′\ncan also be considered to be an explicit polynomial homomorphism. But\n54"},{"paragraph_id":"p58","order":58,"text":"for the sake of simplicity, we shall confine ourselves to the simple functorial\nconstructions above.\n3.4.2\nInput specification and bitlengths\nNow we describe the specifications [H], [ρ], [λ], [μ], their bitlengths. These\nare very similar to the ones in the plethysm problem.\nThe specification [H]:\nWe assume that H is specified as follows.\n(1) If H is a complex, simple, simply connected Lie group, then the specifica-\ntion [H] consists of the root system of H or the Dynkin diagram. Let ⟨H⟩be\nthe bitlength of this specification. Thus, if H = SLn(C), then ⟨H⟩= O(n).\n(2) If H is a simple group of Lie type (Chevalley group) then it has a similar\nspecification [Ca]. The only finite groups of Lie type that arise in GCT are\nSLn(Fpk) and GLn(Fpk). In this case the specification [H] is easy: we only\nhave to specify n, p, k. We define ⟨H⟩in this case to be n + k + log2 p; not\nlog2 n + log2 k + log2 n. As a rule, ⟨H⟩is defined to be the sum of the rank\nparameters (such as n and k here) and bit lengths of the weight parameters\n(such as p here) in the specification. This is equivalent to assuming that the\nrank parameters are specified in unary.\n(3) If H is the alternating group An, we only specify n. Let ⟨H⟩= n.\n(4) The torus is specified by its dimension. We define ⟨H⟩to be the dimen-\nsion.\n(5) If H is a product of such groups, its specification is composed from the\nspecifications of its factors, and the bitlength ⟨H⟩is defined to be the sum\nof the bitlengths of the constituent specifcations.\nThe specification [ρ]:\nLet us first assume that ρ is a basic explicit polynomial homomorphism.\nIn this case the specification of ρ : H →G = GL(V ) is a pair [ρ] = ([H], [V ])\nconsisting of the specification [H] of H as above, and the combinatorial\nspecification [V ] of the representation V as defined below:\n(1) If H is a semisimple, simply connected Lie group, and V = Vμ(H) its\nirrreducible representation for a dominant weight μ of H, then V is specified\nby simply giving the coordinates of μ in terms of the fundamental weights\nof H.\nThus [V ] = μ, and its bitlength ⟨V ⟩is the total bitlength of all\ncoordinates of μ, and the combinatorial bit length ∥V ∥is the total number\nof coordinates of μ.\n55"},{"paragraph_id":"p59","order":59,"text":"(2) If H = Sn, and V = Sγ its irreducible representation (Specht module),\nthen [V ] is the partition γ labelling this Specht module. We define ⟨V ⟩to\nbe the bitlength of this partition, and ∥V ∥= ⟨V ⟩.\n(3) If H is a finite general linear group GLn(Fpk), and V its irreducible rep-\nresentation, as classified by Green [Mc], then [V ] is the combinatorial clas-\nsifying label of V as given in [Mc]. It is a certain partition-valued function,\nwhich can be specified by listing the places where the function is nonzero\nand the nonzero partition values at these places. Let ⟨V ⟩be the bitlength\nof this specification; it is O(poly(n, k, ⟨p⟩)). We let ∥V ∥= ⟨V ⟩. More gener-\nally, if H is a finite group of Lie type, and V its irreducible representation,\nthen [V ] is the combinatorial classifying label of V as given by Lusztig [Lu1].\n(4) If H is a torus and V is a polynomial character, then [V ] is the speci-\nfication of the character. Its bitlength is the bitlength of the specification,\nand combinatorial bit length is the dimension of H.\n(5) If V is composed from irreducible representations, then [V ] is composed\nfrom the specifications of the irreducible representations in an obvious way.\nBitlengths and combinatorial bitlengths are defined additively.\nThe bitlength ⟨ρ⟩is defined to be ⟨H⟩+ ⟨V ⟩, where ⟨V ⟩is the bitlength\nof [V ].\nIf ρ is a composite homomorphism, its specification [ρ] is composed from\nthe specifications of its basic constituents in an obvious way. The bitlength\n⟨ρ⟩is defined to be the sum of the bitlengths of these basic specifications.\nThe specifications [λ] and [π]:\nVπ(H) is the tensor product of the irreducible representations of the\nfactors of H. We let [π] be the tuple of the combinatorial classifying labels\nof each of these irreducible representations, as specified above. Let ⟨π⟩be\ntheir total bit length, and ∥π∥the total combinatorial bit length. Similarly,\nVλ(G) is the tensor product of the irreducible representations of the factors\nof G. When G = GLm(C), λ is a partition, which we specify by only giving\nits nonzero parts, whose number is equal to the height of λ. This is crucial\nsince the height of λ can be much less than than the rank m of G, as in\nthe plethysm problem (Problem 1.1.2). We shall leave a similar compact\nspecification [λ] for a general connected, reductive G to the reader. Let ⟨λ⟩\nbe its bitlength and ∥λ∥its combinatorial bit length.\n56"},{"paragraph_id":"p60","order":60,"text":"3.4.3\nStretching function and quasipolynomiality\nLet f(x) = mπ\nλ as above, with x = ([H], [ρ], [λ], [π]). Here λ is the dominant\nweight of G. First, assume that H is connected, reductive. Then π is the\ndominant weight of H. For a given x, let us define the stretching function\nas\n ̃f(x, n) = ̃mπ\nλ(n) = mnπ\nnλ,\n(3.8)\nwhich is the multiplicity of Vnπ(H) in Vnλ(G), considered as an H-module\nvia ρ : H →G. Let Mπ\nλ (t) = P\nn≥0 ̃mπ\nλ(n)tn be the generating function of\nthis stretching quasi-polynomial.\nThe following is the generalization of Theorem 1.6.1 in this setting.\nTheorem 3.4.1 (a) (Rationality) The generating function Mπ\nλ (t) is ratio-\nnal.\n(b) (Quasi-polynomiality) The stretching function ̃mπ\nλ(n) is a quasi-polynomial\nfunction of n.\n(c) There exist graded, normal C-algebras S = S(mπ\nλ) = ⊕nSn and T =\nT(mπ\nλ) = ⊕nTn such that:\n1. The schemes spec(S) and spec(T) are normal and have rational singu-\nlarities.\n2. T = SH, the subring of H-invariants in S.\n3. The quasi-polynomial ̃mπ\nλ(n) is the Hilbert function of T.\n(d) (Positivity) The rational function Mπ\nλ (t) can be expressed in a positive\nform:\nMπ\nλ (t) = h0 + h1t + · · · + hdtd\nQ\nj(1 −ta(j))d(j)\n,\n(3.9)\nwhere a(j)’s and d(j)’s are positive integers, P\nj d(j) = d+1, where d is the\ndegree of the quasi-polynomial, h0 = 1, and hi’s are nonnegative integers.\nThe specific rings S(mπ\nλ) and T(mπ\nλ) constructed in the proof of this\nresult are called the canonical rings associated with the structrural con-\nstant mπ\nλ. The projective schemes Y (mπ\nλ) = Proj(S(mπ\nλ)), and Z(mπ\nλ) =\nProj(T(mπ\nλ)) are called the canonical models associated with mπ\nλ.\nTheorem 3.4.1 and its generalization, when H can be disconnected, is\nproved in Chapter 4; cf. Theorem 4.1.1.\n57"},{"paragraph_id":"p61","order":61,"text":"Finitely generated semigroup\nThe following is an analogue of Theorem 1.6.2.\nTheorem 3.4.2 Assume that H is connected. For a fixed ρ : H →G, let\nT(H, G) be the set of pairs (μ, λ) of dominant weights of H and G such that\nthe irreducible representation Vπ(H) of H occurs in the irreducible repre-\nsentation Vλ(G) of G with nonzero multiplicity. Then T(H, G) is a finitely\ngenerated semigroup with respect to addition.\nThis is proved in Section 4.4.\nPSPACE\nThe following is a generalization of Theorem 1.6.3.\nTheorem 3.4.3 Assume that H in Problem 1.1.3 is a direct product, whose\neach factor is a complex simple, simply connected Lie group, or an alternat-\ning (or symmetric) group, or SLn(Fpk) (or GLn(Fpk)), or a torus. Then\nf(x) = mπ\nλ can be computed in poly(⟨x⟩) space, with x as specified above.\nThis is proved in Chapter 5. It may be conjectured that Theorem 3.4.3\nholds even when the composition factors of H are allowed to be general\nfinite simple groups of Lie type. This will be so if Lusztig’s algorithm [Lu5]\nfor computing the characters of finite simple groups of Lie type can be\nparallelized; cf. Section 5.4.\nPositivity hypotheses\nTheorem 3.4.1-3.4.3, along with the experimental results in special cases\n(cf. Chapter 6), constitute the main evidence in support of the positivity\nHypotheses 3.3.1-3.3.4 for the subgroup restrition problem.\n3.5\nThe decision problem in geometric invariant\ntheory\nFinally, let us turn to the most general Problem 1.1.4.\n58"},{"paragraph_id":"p62","order":62,"text":"3.5.1\nReduction from Problem 1.1.3 to Problem 1.1.4\nFirst, let us note that the subgroup restriction problem (Problem 1.1.3)\nis a special case of Problem 1.1.4.\nTo see this, let H, ρ and G be as in\nProblem 1.1.3, and let X be the closed G-orbit of the point vλ corresponding\nto the highest weight vector of Vλ(G) in the projective space P(Vλ(G)). Then\nX = Gvλ ∼= G/Pλ,\n(3.10)\nwhere the P = Pλ = Gvλ is the parabolic stabilizer of vλ. We have a natural\naction of H on X via ρ. Let R be the homogeneous coordinate ring of X. By\n[Ha, MR, Rm, Sm], the singularities of spec(R) are rational. By Borel-Weil\n[FH], the degree one component R1 of the homogeneous coordinate ring R\nof X is Vλ(G). Hence, sπ\n1 in this special case of Problem 1.1.4 is precisely mπ\nλ\nin Problem 1.1.3. The results in Section 3.4 for sπ\n1 generalize in a natural\nway for sπ\nd.\n3.5.2\nInput specification\nThe variety X in the above example is completely specified by H, ρ and λ.\nHence its specification [X] can be given in the form a tuple ([H], [ρ], [λ]),\nwhere [H], [ρ] and [λ] are the specifications of H, ρ and λ as in Section 3.4,\nThe input specification x for Problem 1.1.4 in the special case above is the\ntuple ([X], d, [π]) = ([H], [ρ], [λ], d, [π]), where [π] is the specification of π as\nin Section 3.4.\nWe now describe a class of varieties X which have similar compact spec-\nifications.\nLet G be a connected, reductive group, H a reductive, possibly discon-\nnected, reductive group, and ρ : H →G an explicit polynomial homomor-\nphism as in Section 3.4. Let V = Vλ(G) be an irreducible representation of G\nfor a dominant weight λ. Let P(V ) be the projective space associated with\nV . It has a natural action of H via ρ. Let v ∈P(V ) be a point that is char-\nacterized by its stabilizer Gv ⊆G. This means it is the only point in P(V )\nthat is stabilized by Gv. For example, the point vλ above is characterized by\nits parabolic stabilier. We assume that we know the Levi decompositioon of\nGv explicity, and its compact specification [Gv], like that of H, and also an\nexplicit compact specification of the embedding ρ′ : Gv →G, aking to that\nof the explicit homomorphism ρ : H →G. Let X ⊆P(V ) be the projective\nclosure of the G-orbit of v in P(V ). Then X as well as the action of H on\nX are completely specified by λ, H, ρ, Gv and ρ′. Hence, we can let [X] be\n59"},{"paragraph_id":"p63","order":63,"text":"the tuple (λ, [H], [ρ], [Gv], [ρ′]). The input specification x for Problem 1.1.4\nwith the X of this form is the tuple ([X], d, [π]). The bitlengths ⟨x⟩and ∥x∥\nare defined additively. The rank(x) is defined to be the sum of the ranks of\nH and G, dim(V ) and ∥π∥. Since the point vλ above is characterized by its\nstabilizer, G/P is a variety of this form.\nThe class varieties [GCT1, GCT2] are either of this form, or a slight ex-\ntension of this form, and admit such compact specifications. The algebraic\ngeometry of an X of the above form is completely determined by the repre-\nsentation theories of the two homomorphisms ρ : H →G and ρ′ : Gv →G.\nFurthermore, the results in [GCT2] say that Problem 1.1.4 for a class variety\nis intimately linked with the subgroup restriction problem and its variants\nfor the homomomorphisms ρ and ρ′. Hence it is qualitatively similar to the\nsubgroup restriction problem in this case; cf. [GCT10] for further elabora-\ntion of the connection between these two problems.\n3.5.3\nStretching function and quasi-polynomiality\nNow let H, X, R and sπ\nd be as in Problem 1.1.4, with H therein assumed to\nbe connected. We associate with f(x) = sπ\nd the following stretching fucntion:\n ̃f(x, n) = ̃sπ\nd(n) = snπ\nnd,\n(3.11)\nwhere snπ\nnd is the multiplicity of the irrreducible representation Vnπ(H) of H\nin Rnd, the componenent of the homogeneous coordinate ring R of X with\ndegree nd. Let S(t) = P\nn≥0 ̃sπ\nd(n)tn.\nTheorem 3.5.1 Assume that the singularities of spec(R) are rational.\n(a) (Rationality) The generating function Sπ\nd (t) is rational.\n(b) (Quasi-polynomiality) The stretching function ̃sπ\nd(n) is a quasi-polynomial\nfunction of n.\n(c) There exist graded, normal C-algebras S = S(sπ\nd) = ⊕nSn and T =\nT(sπ\nd) = ⊕nTn such that:\n1. The schemes spec(S) and spec(T) are normal and have rational singu-\nlarities.\n2. T = SH, the subring of H-invariants in S.\n3. The quasi-polynomial ̃sπ\nd(n) is the Hilbert function of T.\n60"},{"paragraph_id":"p64","order":64,"text":"(d) (Positivity) The rational function Sπ\nd (t) can be expressed in a positive\nform:\nSπ\nd (t) = h0 + h1t + · · · + hktk\nQ\nj(1 −ta(j))k(j)\n,\n(3.12)\nwhere a(j)’s and k(j)’s are positive integers, P\nj k(j) = k + 1, where k is\nthe degree of the quasi-polynomial ̃sπ\nd(n), h0 = 1, and hi’s are nonnegative\nintegers.\nThis is proved in Chapter 4. Theorem 3.4.1 is a special case of this theorem,\nin view of the reduction in Section 3.5.1. Theorem 3.5.1 is applicable when\nX is a class variety, assuming that its singularities are rational.\n3.5.4\nPositivity hypotheses\nEven though Theorem 3.5.1 holds for any X, with spec(R) having ratio-\nnal singularities, the positivity hypotheses PH1, SH, PH2, and PH3 can\nbe expected to hold for only very special X’s. In general, characterizing\nthe X’s with compact specification for which these hypotheses hold is a\ndelicate problem.\nHypotheses 3.3.1-3.3.4 say that these hold when X in\nProblem 1.1.4 is G/P (as in Section 3.5.1) or a class variety, with the input\nspecification x as described above. For future reference, we shall reformulate\nthese hypotheses purely in geometric terms.\nFor this we need a definition.\nLet T = P\nn Tn be a graded complex C-algebra so that the singularities\nof spec(T) rational.\nLet Z = Proj(T).\nAssume that Z has a compact\nspecification [Z]; we shall specify it below for the Z’s of interest to us.\nWe let [T], the specification of T, to be [Z].\nThis will play the role of\nthe input in the definition below. Let ⟨T⟩denote its bitlength, and ∥T∥\ncombinatorial bit length.\nLet hT (n) = dim(Tn) be its Hilbert function,\nwhich is a quasipolynomial, since the singularities of spec(T) are rational;\ncf. Lemma 4.1.3.\nDefinition 3.5.2 We say that PH1 holds for T (or Z) if the Hilbert quasi-\npolynomial hT (n) is convex. This means there exists a polytope P = PT\ndepending on the input [T], whose Ehrhart quasipolynomial fP(n) coincides\nwith the Hilbert function hT (n), and whose membership function χP(y) can\nbe computed in poly(⟨T⟩, y) time. We assume that a separating hyperplane\ncan also be computed in polynomial time if y ̸∈P (Section 2.3).\n61"},{"paragraph_id":"p65","order":65,"text":"If PH1 holds we can also ask if analogues of SH, PH2, and PH3–whose\nformulation is similar and hence omitted–hold.\n3.5.5\nG/P and Schubert varieties\nLet us illustrate this definition with an example. Let X ∼= G/Pλ be as in\nSection 3.5.1 and R its homogeneous coordinate ring. We have already seen\nthat it has a compact specification: namely [X] = λ. Since singularities\nof spec(R) are rational, PH1 makes sense.\nFor G/P it follows from the\nBorel-Weil theorem. The Hilbert series of R is of the form\nh0 + · · · + hdtd\n(1 −t)d+1\n,\nwith h0 = 1 and hi’s nonnegative. This is so because R is Cohen-Macauley\n[Rm] and is generated by its degree one component. Hence, the modular\nindex of the Hilbert function is one (PH3). PH2 turns out to be nontriv-\nial. Experimental evidence in its support for the classical G/P is given in\nSection 6.3. Considerations for the Schubert subvarieties are similar. Ex-\nperimental evidence for PH2 for the classical Schubert varieties is also given\nin Section 6.3.\nNow let s = sπ\nd be the multiplicity as Problem 1.1.4, with X having a\ncompact specification [X] as above. Let T = T(s) be the ring associated\nwith s as in Theorem 3.5.1 (c).\nLet Z = Z(s) = Proj(T).\nWe let the\nspecification [Z] = ([X], d, π). Let ⟨Z⟩be its bitlength.\nSo Theorem 3.1.1 in this context implies:\nTheorem 3.5.3 If PH1 and SH holds for Z(s) then nonvanishing of s,\nmodulo small relaxation, can be decided in poly(⟨Z⟩) time.\nWe also have the following reformulation:\nProposition 3.5.4 Hypotheses 3.3.1-3.3.4 are equivalent to PH1,SH,PH2,PH3\nfor Z(s), where s is a stucture constant that corresponds the structure con-\nstant f(x) in Hypotheses 3.3.1. Thus, in the case of the subgroup restriction\nproblem, s = sπ\n1 = mπ\nλ as in Section 3.5.1.\nThis is just a consequence of definitions.\n62"},{"paragraph_id":"p66","order":66,"text":"3.6\nPH3 and existence of a simpler algorithm\nAs we remarked in Section 3.1.3, the use of the ellipsoid method and basis\nreduction in lattices makes the the algorithm for saturated integer program-\nming (cf. Theorem 3.1.1) fairly intricate. For the flip (cf. [GCTflip] and\nChapter 7), it is desirable to have simpler algorithms for the relaxed forms\nof the decision problems under consideration, akin to the the polynomial\ntime combinatorial algorithms in combinatorial optimization [Sc] that do\nnot rely on the elliposoid method or basis reduction. We briefly examine in\nthis section the role of PH3 in this context.\nThe simple combinatorial algorithms in combinatorial optimization work\nonly when the problem under consideration is unimodular–in which case the\nvertices of the underlying polytope P are integral–or almost unimodular–\ne.g. when the vertices of P are half integral. Edmond’s algorithm for finding\nminimum weight perfect matching in nonbipartite graphs [Sc] is a classic\nexample of the second case.\nIn the unimodular case, Stanley’s positivity result [St1] implies that the\nrational function FP (t) has a positive form\nFP (t) = h(d)td + · · · + h(0)\n(1 −t)d+1\n.\nIf PH3 (Hypothesis 3.3.4) holds for a structural function f(x) under con-\nsideration then the Ehrhart series FPx(t) of the polytope Px associated with\nx in PH1 (Hypothesis 3.3.1) has a minimal positive form in which each root\nof the denominator has O(poly(∥x∥)) order. Roughly, this says that the\nsituation is “close” to the unimodular case. Hence, in such a case we can\nexpect a purely combinatorial polynomial-time algorithm for deciding non-\nvanishing of f(x), modulo small relaxation, that does not need the ellipsoid\nmethod or basis reduction.\n3.7\nOther structural constants\nThe paradigm of saturated and positive integer programming in this paper,\nalong with appropriate analogues of PH1,SH,PH2,PH3, may be applicable\nseveral other fundamental structural constants in representation theory and\nalgebraic geometry, in addition to the ones in Problems 1.1.1-1.1.4 treated\nabove, such as\n63"},{"paragraph_id":"p67","order":67,"text":"1. the value of a Kazhdan-Lusztig polynomial at q = 1, [KL1];\n2. the values at q = 1 of the well behaved special cases of the parabolic\nKostka polynomials and their q-analogues [Ki];\n3. the structural coefficients of the multiplication of Schubert polynomi-\nals, and so on.\n64"},{"paragraph_id":"p68","order":68,"text":"Chapter 4\nQuasi-polynomiality and\ncanonical models\nIn this chapter we prove quasipolynomiality of the stretching functions as-\nsociated with the various structural constants under consideration (Sec-\ntion 4.1), describe the associated canonical models (Section 4.2), describe\nthe role of nonstandard quantum groups in [GCT4, GCT7, GCT8] in the\ndeeper study of these models (Section 4.3), prove finite generation of the\nsemigroup of weights (Theorem 3.4.2) (Section 4.4), and give an elementary\nproof of rationality in Theorem 3.4.1 (a) (Section 4.5).\n4.1\nQuasi-polynomiality\nHere we prove Theorem 3.5.1; Theorems 1.6.1 and 3.4.1 are its special cases\nin view of the reduction in Section 3.5.1. This, in turn, follows from the\nfollowing more general result.\nLet R = ⊕kRd be a normal graded C-algebra with an action of a reduc-\ntive group H. Assume that spec(R) has rational singularities. Let H0 be\nthe connected component of H containing the identity. Let HD = H/H0 be\nits discrete component. Given a dominant weight π of H0, we consider the\nmodule Vπ = Vπ(H0), an H-module with trivial action of HD. Let sπ\nd denote\nthe multiplicity of the H-module Vπ in Rd. Let ̃sπ\nd(n) be the multiplicity of\nthe H-module Vnπ in Rnd. This is a stretching function associated with the\nmulitplicity sπ\nd. Let Sπ\nd (t) = P\nn≥0 ̃sπ\nd(n)tn.\n65"},{"paragraph_id":"p69","order":69,"text":"Theorem 4.1.1 (a) (Rationality) The generating function Sπ\nd (t) is ratio-\nnal.\n(b) (Quasi-polynomiality) The stretching function ̃sπ\nd(n) is a quasi-polynomial\nfunction of n.\n(c) There exist graded, normal C-algebras S = S(sπ\nd) = ⊕nSn and T =\nT(sπ\nd) = ⊕nTn such that:\n1. The schemes spec(S) and spec(T) are normal and have rational singu-\nlarities.\n2. T = SH, the subring of H-invariants in S.\n3. The quasi-polynomial ̃sπ\nd(n) is the Hilbert function of T.\n(d) (Positivity) The rational function Sπ\nd (t) can be expressed in a positive\nform:\nSπ\nd (t) = h0 + h1t + · · · + hktk\nQ\nj(1 −ta(j))k(j)\n,\n(4.1)\nwhere a(j)’s and k(j)’s are positive integers, P\nj k(j) = k + 1, where k is\nthe degree of the quasi-polynomial ̃sπ\nd(n), h0 = 1, and hi’s are nonnegative\nintegers.\nTheorem 3.5.1 follows from this by letting R be the homogeneous coordinate\nring of X.\nMore generally, if W is an irreducible representation of HD, we can\nconsider the H-module Vπ ⊗W. Let sπ,W\nd\nbe its multiplicity in Rd. Let\n ̃sπ,W\nd\n(n) be the multiplicity of the trivial H-representation in the H-module\nRnd ⊗V ∗\nnπ ⊗Symn(W ∗). Then\nTheorem 4.1.2 Analogue of Theorem 4.1.1 holds for ̃sπ,W\nd\n(n).\nFor the purposes of the flip, Theorem 4.1.1 suffices.\nProof: We shall only prove Theorem 4.1.1, the proof of Theorem 4.1.2 be-\ning similar.\nThe proof is an extension of M. Brion’s proof (cf.\n[Dh]) of\nquasi-polynomiality of the stretching function associated with a Littlewood-\nRichardson coefficient of any semisimple Lie algebra.\nClearly (a) follows from (b); cf. [St1].\n66"},{"paragraph_id":"p70","order":70,"text":"(b) and (c):\nLet Cd be the cyclic group generated by the primitive root ζ of unity of\norder d. It has a natural action on R: x ∈Cd maps z ∈Rk to xkz. Let\nB = RCd = P\nn≥0 Rnd ⊆R be the subring of Cd-invariants. By Boutot\n[Bou], B is a normal C-algebra and spec(B) has rational singularities.\nAssume that H0 is semisimple; extension to the reductive case being easy.\nLet π∗be the dominant weight of H0 such that V ∗\nπ = Vπ∗. By Borel-Weil\n[FH],\nCπ∗= ⊕n≥0V ∗\nnπ = ⊕n≥0Vnπ∗,\nis the homogeneous coordinate ring of the H0-orbit of the point vπ∗∈P(Vπ∗)\ncorresponding to the highest weight vector. This H0-orbit is isomorphic to\nH0/Pπ∗, where Pπ∗⊆H0 is the parabolic stabilizer of vπ∗. Hence Cπ∗is\nnormal and spec(Cπ∗) has rational signularities; cf.\n[Ha, MR, Rm, Sm].\nIt follows that B ⊗Cπ∗is also normal, and spec(B ⊗Cπ∗) has rational\nsingularities. Consider the action of C∗on B ⊗Cπ∗given by:\nx(b ⊗c) = (x · b) ⊗(x−1 · c),\nwhere x ∈C∗maps b ∈Bn to xnb, the action on Cπ∗being similar. Consider\nthe invariant ring\nS = (B ⊗Cπ∗)C∗= ⊕nSn = ⊗n≥0Rnd ⊗V ∗\nnπ.\n(4.2)\nBy Boutot [Bou], it is a normal, and spec(D) has rational singularities.\nSince Vnπ is an H-module, the algebra S has an action of H. Let\nT = T(sπ\nd) = SH = ⊕n≥0Tn\n(4.3)\nbe its subring of H-invariants. By Boutot [Bou], it is normal, and spec(T)\nhas rational singularities–this is the crux of the proof. By Schur’s lemma, the\nmultiplicity of the trivial H-representation in Sn = Rnd⊗V ∗\nnπ is precisely the\nmultiplicity ̃sπ\nd(n) of the H-module Vnπ in Rnd. Hence, the Hilbert function\nof T, i.e., dim(Tn), is precisely ̃sπ\nd(n), and the Hilbert series P\nn≥0 dim(Tn)tn\nis Sπ\nd (t).\nQuasipolynomiality of ̃sπ\nd(n) follows by applying the following\nlemma:\nLemma 4.1.3 (cf. [Dh]) If T = ⊕∞\nn=0Tn is a graded C-algebra, such that\nspec(T) is normal and has rational simgularites, then dim(Tn), the Hilbert\nfunction of T, is a quasi-polynomial function of n.\n67"},{"paragraph_id":"p71","order":71,"text":"(d) Since spec(T) has rational singularities, T is Cohen-Macaualey.\nLet\nt1, . . . , tu be its homogeneous sequence of parameters (h.s.o.p.), where u =\nk + 1 is the Krull dimension of T. By the theory of Cohen-Macauley rings\n[St2], it follows that its Hilbert series Sπ\nd (t) is of the form\nh0 + h1t + · · · + hktk\nQk+1\ni=1 (1 −tdi)\n,\n(4.4)\nwhere (1) h0 = 1, (2) di is the degree of ti, and (3) hi’s are nonnegative\nintegers. This proves (d). Q.E.D.\nRemark 4.1.4 A careful examination of the proof above shows that ratio-\nnality of Sπ\nd (t), and more strongly, asymptotic quasi-polynomiality of ̃sπ\nd(n)\nas n →∞, can be proved using just Hilbert’s result on finite generation of\nthe algebra of invariants of a reductive-group action. Boutot’s result is nec-\nessary to prove quasi-polynomiality for all n. This is crucial for saturated\nand positive integer programming (Chapter 3).\n4.1.1\nThe minimal positive form and modular index\nThe form (4.4) of Sπ\nd (t) is not unique because it depends on the degrees di’s\nof the paramters ti’s. For future use, let us record the following consequences\nof the proof. Let T be the ring constructed in the proof above.\nCorollary 4.1.5 Suppose T has an h.s.o.p.\nt = (t1, . . . , tu) with di =\ndeg(ti). Then Sπ\nd (T) has a positive rational form (4.4) with di = deg(ti)\ntherein.\nThe proof above is lets us define a minimal positive form of the rational\nfunction Sπ\nd (t) associated with a structural constant s. For this, let us or-\nder h.s.o.p.’s of T lexicographically as per their degree sequences. Here the\ndegree seqeunce of an h.s.o.p. t = (t1, . . . , tu) is defined to be (d1, . . . , du),\nwhere di = deg(ti). The form (4.4) is the same for any h.s.o.p. of lexi-\ncographically minimum degree sequence. We call it the minimal positive\nform of Sπ\nd (t). The modular index of sπ\nd is defined to be max{di}, where\n(d1, . . . , du) is the degree sequence of a lexicographically minimal h.s.o.p.\nSince Problems 1.1.1, 1.1.2,1.1.3, 1.2.1 are special cases of Problem 1.1.4,\nthis defines minimal positive forms of the rational generating functions of the\nstretching quasi-polynomials (cf. Theorem 3.4.1) associated with the struc-\ntural constants in these problems, and also the modular indices of these\nstructural constants.\n68"},{"paragraph_id":"p72","order":72,"text":"4.1.2\nThe rings associated with a structural constant\nThe preceding proof also associates with the structural constant s a few\nrings which will be important later. Specifically, let S = S(s) and T = T(s)\nbe the rings as in Theorem 4.1.1 (c) associated with the structural constant\ns = sπ\nd.\nLet R = R(s) be the homogeneous coordinate ring of X as in\nTheorem 4.1.1. We call R(s), S(s) and T(s) the rings associated with the\nstructure constant s.\nWhen s = mπ\nλ, as in the subgroup restriction problem (Problem 1.1.3),\nX ∼= G/P as given in eq.(3.10. Then these rings are explicitly as follows:\nR(mπ\nλ)\n=\n⊕n≥0Vnλ(G),\nS(mπ\nλ)\n=\n⊕n≥0Vnλ(G) ⊗Vnπ(H)∗,\nT(mπ\nλ)\n=\n⊕n≥0(Vnλ(G) ⊗Vnπ(H)∗)H.\n(4.5)\nBy specializing the subgroup restriction problem further to the Littlewood-\nRichardson problem (Problem 1.2.1), we get the following rings associated\nby Brion (cf. [Dh]) with the Littlewood-Richardson coefficient cλ\nα,β:\nR(cλ\nα,β)\n=\n⊕n≥0Vnα(H) ⊗Vnβ(H),\nS(cλ\nα,β)\n=\n⊕n≥0Vnα(H) ⊗Vnβ(H) ⊗Vnλ(H)∗,\nT(cλ\nα,β)\n=\n⊕n≥0(Vnα(H) ⊗Vnβ(H) ⊗Vnλ(H)∗)H.\n(4.6)\n4.2\nCanonical models\nThere are several rings other than T(cλ\nα,β) whose Hilbert function coincides\nwith the Littlewood-Richardson stretching quasi-polynomial ̃cλ\nα,β(n).\nFor\nexample, let P = P λ\nα,β be the BZ-polytope [BZ] whose Ehrhart quasi-\npolynomial coincides with ̃cλ\nα,β(n).\nWe can associate with P a ring TP\nas in Stanley [St3] whose Hilbert function coincides with ̃cλ\nα,β(n).\nThere\nare many other choices for P. For example, in type A, we can consider a\nhive polytope or a honeycomb polytope [KT1] instead of the BZ-polytope.\nThe rings TP ’s associated with different P’s will, in general, be different,\nand there is nothing canonical about them. In contrast, the ring T(cλ\nα,β) is\nspecial because:\nProposition 4.2.1 (PH0) The rings R(cλ\nα,β), S(cλ\nα,β), T(cλ\nα,β) have quan-\ntizations Rq(cλ\nα,β), Sq(cλ\nα,β), Tq(cλ\nα,β) endowed with canonical bases in the ter-\nminology of Lusztig [Lu4]. Furthermore, the canonical bases of Rq(cλ\nα,β), Sq(cλ\nα,β)\n69"},{"paragraph_id":"p73","order":73,"text":"are compatible with the action of the Drinfeld-Jimbo quantum group associ-\nated with H = GLn(C), and the canonical basis of Sq(cλ\nα,β) is an extension\nof the canonical basis of Tq(cλ\nα,β) in a natural way.\nThis follows from the work of Lusztig (cf. [Lu3], Chapter 27 in [Lu4]) and\nKashiwara (cf.\nTheorem2 in [Kas3]).\nSpecializations of these canonical\nbases at q = 1 will be called canonical bases of R(cλ\nα,β), S(cλ\nα,β), T(cλ\nα,β).\nLusztig [Lu4] has conjectured that the structural constants associated with\nthe canonical bases in Proposition 4.2.1 are polynomials in q with nonnega-\ntive integral coefficients as in the case of the canonical basis of the (negative\npart of the) Drinfeld-Jimbo enveloping algebra. We refer to Proposition 4.2.1\nas PH0 in view of this (conjectural) positivity property.\nIn view of this proposition, we call the rings R(cλ\nα,β), S(cλ\nα,β) and T(cλ\nα,β)\nthe canonical rings associated with the Littlewood-Richardson coefficient\ncλ\nα,β, and X = Proj(R(cλ\nα,β)), Y = Proj(S(cλ\nα,β)) and Z = Proj(T(cλ\nα,β)) the\ncanonical models associated with cλ\nα,β.\n4.2.1\nFrom PH0 to PH1,3\nNow we study the relevance of PH0 above in the context of PH1,SH,PH2,\nand PH3 for Littlewood-Richardson coefficients (Section 1.2).\nPH1\nAs already remarked in Section 1.7, PH1 for Littlewood-Richardson coeffi-\ncients is a formal consequence of the properties of Kashiwara’s crystal oper-\nators on the canonical bases in PH0 (Proposition 4.2.1); [Dh, Kas2, Li, Lu4].\nSpecifically, the canonical basis of the ring Rq(cλ\nα,β) also yields a canon-\nical basis for the tensor product Vq,α ⊗Vq,β of the irreducible Hq modules\nwith highest weights α and β. The Littlwood-Richardson rule for arbitrary\ntypes follows from the study of Kashiwara’s crystal operators on this canon-\nical basis for the tensor product; [Lu4]. This rule is equivalent to the one\nin [Li] based on combinatorial interpretation of the crystal operators in the\npath model therein. The article [Dh] derives a convex polyhedral formula\nfor Littlewood-Richardson coefficients (of arbitrary type) using this com-\nbinatorial interpretation. Though the complexity-theoretic issues are not\naddressed in [Dh], it can be verified that the polyhedral formula therein is a\nconvex #P-formula. This yields PH1 for Littlewood-Richardson coefficients\nof arbitrary types using PH0.\n70"},{"paragraph_id":"p74","order":74,"text":"SH\nNow let us see the relevance of PH0 in the context of SH for Littlewood-\nRichardson coefficients of arbitrary type.\nThe polytope in [Dh], mentioned above, for type A is equivalent to the\nhive polytope in [KT1] in the sense that the number integer points in both\nthe polytopes is the same. Knutson and Tao prove SH for type A by show-\ning that the hive polytope always has in integral vertex. To extend this\nproof to an arbitrary type, one has to convert the polytope in [Dh] into a\npolytope that is guaranteed to contain an integral vertex if the index of the\nstretching quasipolynomial ̃cλ\nα,β(n) is one. The main difficulty here is that\nwe do not have a nice mathematical interpretation for the index. Algorithm\nin Theorem 3.1.1 applied to the polytope in [Dh] computes this index in\npolynomial time. But it does not give a nice interpretation that can be used\nin a proof as above.\nThis index is simply the largest integer dividing the degrees of all ele-\nments in any basis of the canonical ring T(cλ\nα,β)–in particular, the canon-\nical basis. This follows by applying Proposition 3.1.3 to the polytope in\n[Dh]. This leads us to ask: is there an interpretation for the index based on\nLusztig’s topological construction of the canonical basis in Proposition 4.2.1?\nIf so, this may be used to extend the known polyhedral proof for SH in type\nA to arbitrary types. Alternatively, it may be possible to prove SH using\ntopological properties of the canonical basis in the spirit of the topological\n(intersection-theoretic) proof [Bl] of SH in type A.\nPH3\nNow let us see the relevance of PH0 in the context of PH3 for Littlewood-\nRichardson coefficients.\nFirst, let us consider the minimal positive form (Section 4.1.1) associated\nwith a Littlewood-Richardson coefficient cλ\nα,β of type A. Let T = T(cλ\nα,β)\ndenote the ring that arises in this case; cf. eq.(4.6). Now we can ask:\nQuestion 4.2.2 Are all di’s occuring in the minimal positive form (cf.\n(4.4)) one in this special case?\nThis is equivalent to asking if the ring\nT = T(cλ\nα,β) in this case is integral over T1, the degree one component of T.\nIf so, this would provide an explanation for the conjecture of King at al\n[KTT] (cf. eq.(1.3)) in the theory of Cohen-Macauley rings:\n71"},{"paragraph_id":"p75","order":75,"text":"Proposition 4.2.3 Assuming yes, the conjecture of King et al [KTT] (Hy-\npothesis 1.2.6) holds.\nRemark 4.2.4 In contrast, the ring TP associated with the hive polytope\n(cf. beginning of Section 4.2) need not be integral over its degree one compo-\nnent, in view of the fact that the hive polytope can have nonintegral vertices\n[DM1].\nRemark 4.2.5 T = T(cλ\nα,β) need not be generated by its degree one compo-\nnent T1. If this were always so, the h-vector (hd, · · · , h0) in eq.(1.3) would\nbe an M-vector (Macauley-vector) [St2]. But one can construct α, β and λ\nfor which this does not hold.\nProof: (of the proposition) Since T is integral over T1, it has an h.s.o.p., all of\nwhose elements have degree 1. By Theorem 3.4.1, the singularities of spec(T)\nare rational.\nHence T is Cohen-Macaulay.\nNow the result immediately\nfollows from the theory of Cohen-Macauley rings [St2]. Q.E.D.\nIn view of this Proposition, the conjecture of King et al will follow if all\ncanonical basis elements of T(cλ\nα,β) can be shown to be integral over the basis\nelements of degree one. This requires a further study of the multiplicative\nstructure of this canonical basis. Considerations for PH3 (Hypothesis 1.2.8)\nfor Littlewood-Richardson coefficients of arbitrary type are similar.\nPH2\nSimilarly, the positivity property (PH2) of the stretching quasipolynomial\nassociated with Littlewood-Richardson coefficients may possibly follow from\na deep study of the multiplicative structure of the canonical basis as per\nPH0 (Proposition 4.2.1), just as positivity of the multiplicative structural\ncoefficients of the canonical basis for the (negative part of the) Drinfeld-\nJimbo enveloping algebra follows from a deep study of the multiplicative\nstructure of this basis [Lu4].\n4.2.2\nOn PH0 in general\nThe discussion above indicates that for Littlewood-Richardson coefficients\nPH1,SH,PH3, and plausibly PH2 as well are intimately related to PH0\n(Proposition 4.2.1).\nThis leads us to ask if the rings associated in Sec-\ntion 4.1.2 with other structural constants under consideration in this paper\n72"},{"paragraph_id":"p76","order":76,"text":"have quantizations which satisfy appropriate forms of PH0. If so, this PH0\nmay be used to derive PH1, SH, PH3, and PH2 (Hypotheses 3.3.1-3.3.4)\nfor these structural constants. Note that SH (a) follows from PH3 (see the\nremark after Hypothesis 3.3.4); PH2 may also follow from PH3. Thus PH1\nand PH3 are the ones to focus on.\nTo formalize this, let s be a structural constant which is either the Kro-\nnecker coefficient as in Problem 1.1.1, or the plethysm constant as in Prob-\nlem 1.1.2, or the multiplicity mπ\nλ in Problem 1.1.3, or the multiplicity sπ\nd, as\nin Problem 1.1.4, when X therein is a class variety. Let R(s), S(s), T(s) be\nthe rings associated with s (Section 4.1.2). Let X(s) = Proj(R(s)), Y (s) =\nProj(S(s)) and Z(s) = Proj(R(s)). We call R = R(s), S = S(s), T = T(s)\nthe canonical rings associated with s, and X(s), Y (s), Z(s) the canonical\nmodels associated with s, because we expect these rings and models to be\nspecial as in the case of the Littlewood-Richardson coefficients.\nLet H be as in Problem 1.1.3 or Problem 1.1.4.\nAssume that H is\nconnected. Let Hq denote the Drifeld-Jimbo quantization of H. Now we\nask:\nQuestion 4.2.6 (PH0??) Are there quatizations Rq, Sq of R, S, with Hq-\naction, and a quantization Tq of T with “canonical” bases (in some appro-\npriate sense) B(Rq), B(Sq), B(Tq), where B(Rq) and B(Sq) are compatible\nwith the Hq-action and B(Sq) is an extension of B(Tq)? Furthermore, do\nthese canonical bases have appropriate positivity properties?\nIn other words, are there quantizations of R, S and T for which PH0\n(Proposition 4.2.1) can be extended in a natural way?\nIf so, this extended PH0 may be used to prove PH1 and SH for s just as\nin the case of Littlewood-Richardson coefficients (of type A).\n4.3\nNonstandard quantum group for the Kronecker\nand the plethysm problems\nWe now consider this question when s is the kronecker or the plethysm\nconstant (cf. Problems 1.1.1 and 1.1.2). PH0 for Littlewood-Richardson\ncoefficients (Proposition 4.2.1) depends critically on the theory of Drinfeld-\nJimbo quantum groups. This is intimately related (in type A) [GrL] to the\nrepresentation theory of Hecke algebras. To extend PH0 in the context of the\nkronecker and the plethysm constants, one needs extensions of these theories\n73"},{"paragraph_id":"p77","order":77,"text":"in the context of Problems 1.1.1-1.1.2. In this section, we briefly review the\nresults in [GCT4, GCT8, GCT7] in this direction and the theoretical and\nexperimental evidence it provides in support of PH0–that is, affirmative\nanswer to Question 4.2.6–in this context.\nSo let us consider the generalized plethysm problem (Problem 1.1.2).\nAs expected, the representation theory of Drinfeld-Jimbo quantum groups\nand Hecke algebras does not work in the context of this general problem.\nBriefly, the problem is that if H is a connected, reductive group and V its\nrepresentation, then the homomorphism H →G = GL(V ) does not quan-\ntize in the setting of Drinfeld-Jimbo quantum groups. That is, there is no\nquantum group homomorphism from Hq, the Drinfeld-Jimbo quantization\nof H, to Gq, the Drinfeld-Jimbo quantization of G. In [GCT4, GCT7], a new\nnonstandard quantization GH\nq of G– called a nonstandard quantum group–is\nconstructed so that there is a quantum group homomorphism Hq →GH\nq .\nWhen H = G, GH\nq coincides with the Drinfeld-Jimbo quantum group. The\narticle [GCT8] gives a conjectural scheme for constructing a nonstandard\ncanonical basis for the matrix coordinate ring of GH\nq\nthat is akin to the\ncanonical basis for the matrix coordinate ring of the Drinfeld-Jimbo quan-\ntum group [Lu4, Kas3].\nIt is known that the Drinfeld-Jimbo quantum group Gq = GLq(V ) and\nthe Hecke algebra Hn(q) are dually paired: i.e., they have commuting ac-\ntions on V ⊗n\nq\nfrom the left and the right that determine each other, where Vq\ndenotes the standard quantization of V . Furthermore, the Kazhdan-Lusztig\nbasis for Hn(q) is intimately related to the canonical basis for Gq [GrL]. Sim-\nilarly, [GCT7] constructs a nonstandard generalization BH\nn (q) of the Hecke\nalgebra which is (conjecturally) dually paired to GH\nq . The article [GCT8]\ngives a conjectural scheme for constructing a nonstandard canonical basis of\nBH\nn (q) akin to the Kazhdan-Lusztig basis of the Hecke algebra Hn(q).\nThe nonstandard quantum group GH\nq and the nonstandard algebra BH\nn (q)\nturn out to be fundamentally different from the standard Drinfeld-Jimbo\nquantum group Gq and the Hecke algebra Hn(q). For example, the non-\nstandard quantum group GH\nq is a nonflat deformation of G in general. This\nmeans the Poincare series of the matrix coordinate ring of GH\nq is different\nfrom the Poincare series of the matrix coordinate ring of G. Specifically,\nthe terms of the first series can be smaller than the respective terms of the\nsecond series. Similarly, BH\nn (q) is a nonflat deformation of the group algebra\nC[Sn] of the symmetric group Sn; i.e., its dimension can be bigger than that\nof C[Sn].\n74"},{"paragraph_id":"p78","order":78,"text":"Nonflatness of GH\nq intuitively means that it is “smaller” than G in gen-\neral. Hence, it may seem that there is a loss of information when one goes\nfrom G to GH\nq . Fortunately, there is none, as per the reciprocity conjecture\nin [GCT7]. This roughly says that the information which is lost in the tran-\nsition from G to GH\nq simply gets transfered to BH\nn (q), which is bigger than\nHn(q). In other words, there is no information loss overall. Hence analogues\nof the properties in the standard setting should also hold in the nonstandard\nsetting, though in a far more complex way.\nThat is what seems to happen to positivity. Specifically, experimental ev-\nidence suggests that the conjectural nonstandard canonical bases in [GCT8]\nhave nonstandard positivity properties which are complex versions of the\npositivity properties in the standard setting. See [GCT7, GCT8, GCT10]\nfor a detailed story.\n4.4\nThe cone associated with the subgroup restric-\ntion problem\nIn this section, we prove Theorem 3.4.2, by extending the proof of Brion\nand Knop (cf. [El]) for the Littlewood-Richardson problem. The proof is in\nthe spirit of the proof of quasipolynomiality in Section 4.1.\nLet G be a connected, reductive group, H a connected, reductive sub-\ngroup, and ρ : H →G a homomorphism. Theorem 3.4.2 has the following\nequivalent formulation.\nLet S(H, G) be the set of pairs (μ, λ) such that\nVμ(H) ⊗Vλ(G) has a nonzero H-invariant. Then,\nTheorem 4.4.1 The set S(H, G) is a finitely generated semigroup with re-\nspect to addition.\nWhen G = H × H and the embedding H ⊆G is diagonal, this special-\nizes to the Brion-Knop result mentioned above. The proof follows by an\nextension the technique therein.\nProof: Let B be a Borel subgroup of G, U the unipotent radical of B and\nT the maximal torus in B. Similarly, let B′ be a Borel subgroup of H, U ′\nthe unipotent radical of B′ and T ′ the maximal torus in B′. Without loss\nof generality, we can assume that B′ ⊆B, U ′ ⊆U, T ′ ⊆T. Let A = C[G]U\nbe the algebra of regular functions on G that are invariant with respect to\nthe right multiplication by U. It is known to be finitely generated [El]. The\ngroups G and T act on A via left and right multiplication, respectively. As\n75"},{"paragraph_id":"p79","order":79,"text":"a G × T-module,\nA = ⊕λVλ(G),\n(4.7)\nwhere the torus T acts on Vλ(G) via multiplication by the highest weight\nλ∗of the dual module. Similarly,\nA′ = C[H]U′ = ⊕λVμ(H),\n(4.8)\nwhere the torus T ′ acts on Vμ(H) via multiplication by the highest weight\nμ∗of the dual module.\nNow A ⊗A′ is finitely generated since A and A′ are. Let X = (A ⊗A′)H\nbe the ring of invariants of H acting diagonally on A⊗A′. The torus T ×T ′\nacts on X from the right. Since H is reductive, X is finitely generated [PV].\nHence, the semigroup of the weights of the right action of T × T ′ on X is\nfinitely generated. We have\nX = (A ⊗A′)H = ((⊕Vλ(G)) ⊗(⊕Vμ(H)))H = ⊕(Vλ(G) ⊗Vμ(H))H,\nand the weights of the algebra X are of the form (λ∗, μ∗) such that Vλ(G) ⊗\nVμ(H) contains a nontrivial H-invariant.\nTherefore these pairs form a\nfinitely generated semigroup. Q.E.D.\nFor the sake of simplicity, assume that G and H are semisimple in what\nfollows.\nLet TR(H, G) denote the polyhedral convex cone in the weight\nspace of H × G generated by T(H, G), as defined in Theorem 3.4.2. This is\na generalization of the Littlewood-Richardson cone (Section 2.2.2).\nThe following generalization of Corollary 3.2.3 is a consequence of The-\norem 3.1.1 and its proof.\nTheorem 4.4.2 Assume that the positivity hypothesis PH1 (Section 3.3)\nholds for the subgroup restriction problem for the pair (H, G), where both H\nand G are classical. Given dominant weights μ, λ of H and G, the polytope\nPμ,λ as in PH1 has a specification of the form\nAx ≤b\n(4.9)\nwhere A depends only on H and G, but not on μ or λ, and b depends\nhomogeneously and linearly on μ, λ. Let n be the total number of columns\nin A.\nThen, there exists a decomposition of TR(H, G) into a set of polyhedral\ncones, which form a cell complex C(H, G), and, for each chamber C in this\n76"},{"paragraph_id":"p80","order":80,"text":"complex, a set M(C) of O(n) modular equations, each of the form\nX\ni\naiμi +\nX\ni\nbiλi = 0\n(mod d),\nsuch that\n1. Saturation hypothesis SH is equivalent to saying that: (μ, λ) ∈T(H, G)\niff(μ, λ) ∈TR(H, G) and (μ, λ) satisfies the modular equations in the\nset M(Cμ,λ) associated with the smallest cone Cμ,λ ∈C(H, G) contain-\ning (μ, λ).\n2. Given (μ, λ), whether (μ, λ) ∈TR(H, G) can be determined in polyno-\nmial time.\n3. If so, whether (μ, λ) satisfies the modular equations associated with\nthe smallest cone in C(H, G) containing it can also be determined in\npolynomial time.\nProof: Given a point p = (μ′, λ′) in the weight space of H ×G, where μ′ and\nλ′ are arbitrary rational points, let S(p) denote the constraints (half-spaces)\nin the sytem (4.9) whose bounding hyperplanes contain the polytope Pμ′,λ′.\nWe can decompose TR(H, G) into a conical, polyhedral cell complex, so that\ngiven a cone C in this complex, and a point p in its interior, the set S(p)\ndoes not depend on p. We shall denote this set by S(C). Thus the affine\nspan of Pμ,λ, for any (μ, λ) ∈C, is determined by the linear system\nA′x = b′,\nwhere [A′, b′] consists of the rows of [A, b] in (4.9) corresponding to the set\nS(C). By finding the Smith normal form of A′, we can associate with C a set\nof modular equations that the entries of b′ must satisfy for this affine span to\ncontain an integer point; see the proof of Theorem 3.1.1. Since the entries of\nA′ depend only on H and G, these equations depend only on C. If (μ, λ) ∈\nT(H, G), then (μ, λ) is integral, and hence these equations are satisfied.\nConversely, if (μ, λ) ∈TR(H, G) and these equations are satisfied, then the\nsaturation property implies that (μ, λ) ∈T(H, G), as seen by examining\nthe proof of Theorem 3.1.1. Furthermore, given (μ, λ), the algorithm in the\nproof of Theorem 3.1.1 implicitly determines if (μ, λ) ∈TR(H, G) and if\nthese modular equations are satisfied in polynomial time. Q.E.D.\n77"},{"paragraph_id":"p81","order":81,"text":"4.5\nElementary proof of rationality\nIn this section we give an elementary proof of rationality in Theorem 3.4.1\n(a), when H therein is connected–actually of a slightly stronger statement:\nnamely, the stretching function ̃mπ\nλ(n) is asymptotically a quasipolynomial,\nas n →∞; cf. Remark 4.1.4. But this proof cannot be extended to prove\nquasipolynomiality for all n. The proof here is motivated by the work of\nRassart [Rs], De Loera and McAllister on the stretching function associated\nwith a Littlewood-Richardson coefficient.\nFirst, we recall some standard results that we will need.\nVector partition functions\nGiven an integral s×n matrix B and integral n-vector c, consider the vector\nparitition function φB(c), which is the number of integer solutions to the\ninteger programming problem\nBy = c,\ny ≥0.\n(4.10)\nFor a fixed c, b, let\nφB,c(n) = φB(nc)\nφB,c,b(n) = φB(nc + b).\n(4.11)\nBy Sturmfels [Stm] and Szenes-Vergne residue formula [SV], φB(c) is a\npiecewise quasipolynomial function of c. That is, Rn can be decomposed into\npolyhedral cones, called chambers, so that the restriction of φB(c) to each\nchamber R is a multivariate quasipolynomial function of the coordinates of c.\nThis implies that φB,c(n) is a quasipolynomial function of n. It also implies\nthat the function φB,c,b(n) is asymptotically a quasipolynomial function of\nn, as n →∞, because the points nc + b, as n →∞, lie in just one chamber.\nThe Szenes-Verne residue formula [SV] for vector partition functions also\nimplies that there is a constant d(B), depending only on B, such that the\nperiod of φB,c(n), for any c, divides d(B).\nKlimyk’s formula\nLet H ⊆G and mπ\nλ be as in Theorem 3.4.1 (a), with H connected. Let us\nassume that H is semisimple, the general case being similar. Let H and G\nbe the Lie algebras of H and G respectively. We recall Klimyk’s formula for\nmπ\nλ. Without loss of generality, we can assume that the Cartan subalgebra\n78"},{"paragraph_id":"p82","order":82,"text":"C ⊆H is a subalgebra of the Cartan subalgebra D ⊆G.\nSo we have a\nrestriction from D∗to C∗, and we assume that the half-spaces determining\npositive roots are compatible. We denote weights of H by symbols such as μ\nand of G by symbols such as ̄μ. To be consistent, we shall use the notation\nmπ ̄λ instead of mπ\nλ in this proof. We write ̄μ ↓μ if the weight ̄μ of G restricts\nto the weight μ of H. We denote a typical element of the Weyl group of\nH by W, and a typical element of the Weyl group of G by ̄W. Given a\ndominant weight π of G and a weight ̄μ of G, let n ̄μ( ̄λ) denote the dimension\nof the weight space for ̄μ in B ̄λ = V ̄λ(G).\nWe assume that:\n(A): For any weight μ of H, the number of ̄μ’s such that ̄μ ↓μ is finite.\nFor example, this is so in the plethysm problem (Problem 1.1.2). We\nshall see later how this assumption can be removed.\nBy Klimyk’s formula (cf. page 428, [FH]),\nmπ ̄λ =\nX\nW\n(−1)W\nX\n ̄μ↓π−ρ−W (ρ)\nn ̄μ(V ̄λ),\n(4.12)\nwhere ρ is half the sum of positive roots of H. We allow ̄μ in the inner sum\nto range over all weights ̄μ of G such that ̄μ ↓π −ρ −W(ρ) by defining\nn ̄μ(V ̄λ) to be zero if ̄μ does not occur in V ̄λ.\nProof of Theorem 3.4.1 (a)\nThe goal is to express ̃mπ ̄λ(n) as a linear combination of vector partition\nfunctions φB,c,b(n)’s, for suitable B, c, b’s, using Klimyk’s formula for mπ ̄λ.\nAfter this, we can deduce asymptotic quasipolynomiality of ̃mπ ̄λ(n) from\nasymptotic quasipolynomiality of φB,c,b(n)’s.\nBy Kostant’s multiplicity formula (cf. page 421 [FH]),\nn ̄μ(V ̄λ) =\nX\n ̄\nW\n(−1)\n ̄\nW P( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ)),\n(4.13)\nwhere P( ̄λ), for a weight ̄λ of G, denotes the Kostant partition function;\ni.e., the number of ways to write ̄λ as a sum of positive roots of G. It is\nimportant for the proof that Kostant’s formula (4.13) holds even if ̄μ is not\na weight that occurs in the representation V ̄λ–in this case, n ̄μ(V ̄λ) = 0, and\nthe right hand side of (4.13) vanishes.\nBy eq.(4.12) and (4.13),\n79"},{"paragraph_id":"p83","order":83,"text":"mπ ̄λ =\nX\nW\nX\n ̄\nW\n(−1)W (−1)\n ̄\nW\nX\n ̄μ↓π−ρ−W (ρ)\nP( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ)).\n(4.14)\nLet D denote the dominant Weyl chamber in the weight space of G. Let\nC denote the Weyl chamber complex associated with the weight space of G.\nThe cells in this complex are closed polyhedral cones. Each cone is either\nthe chamber ̄W(D), for some Weyl group element ̄W, or a closed face of\n ̄W(D) of any dimension.\nUsing M ̈obius inversion, the inner sum\nX\n ̄μ↓π−ρ−W (ρ)\nP( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ))\nin eq.(4.14) can be written as a linear combination\nX\nC\na(C)\nX\n ̄μ∈C: ̄μ↓π−ρ−W (ρ)\nP( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ)),\nwhere C ranges over chambers in the Weyl chamber complex C, a(C) is an\nappropriate constant for each C.\nHence,\nmπ ̄λ =\nX\nW\nX\n ̄\nW\n(−1)W (−1)\n ̄\nW X\nC\na(C)\nX\n ̄μ∈C: ̄μ↓π−ρ−W (ρ)\nP( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ)).\n(4.15)\nNow think of π and ̄λ as variables. But H and G are fixed, and hence\nalso the quantities such as ρ and ̄ρ.\nClaim 4.5.1 For fixed Weyl group elements W, ̄\nW and a fixed C, the sum\nX\n ̄μ∈C: ̄μ↓π−ρ−W (ρ)\nP( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ))\n(4.16)\ncan be expressed as a vector partition function associated with an appropriate\nlinear system\nBy = c,\ny ≥0,\n(4.17)\nwhere the matrix\nB = BH,G,C,\n80"},{"paragraph_id":"p84","order":84,"text":"depends only on C and the root systems of H and G, but not on π and ̄λ,\nand the coordinates of the vector\nc = mW, ̄\nW,C( ̄λ, π, ρ, ̄ρ),\ndepend on W, ̄W, C, ρ, ̄ρ, π, π, and furthermore, their dependence on π, ̄λ, ρ, ̄ρ\nis linear.\nHere assumption (A) is crucial. Without it, the sum (4.16) can diverge. Of\ncourse, without assumption (A), we can still make the sum finite, by requir-\ning that ̄μ lie within the convex hull H ̄λ generated by the points { ̄W( ̄λ)},\nwhere ̄W ranges over all Weyl group elements. This means we have to add\nconstraints to the system (4.17) corresponding to the facets of H ̄λ.\nBut\nthe entries of the resulting B would depend on ̄λ, and the theory of vector\npartition functions will no longer apply.\nProof of the claim: Let ̄μi’s denote the integer coordinates of ̄μ in the basis\nof fundamental weights.\nWe denote the integer vector ( ̄μ1, ̄μ2, · · · ) by ̄μ\nagain. The Kostant partition function P(ν) is a vector partition function\nassociated with an integer programming problem:\nBPv = ν,\nv ≥0,\nwhere the columns of BP correspond to positive roots of G. The sum in\n(4.16) is equal to the number of integral pairs ( ̄μ, v) such that\n1. ̄μ ∈C,\n2. ̄μ ↓π −ρ −W(ρ),\n3. BPv = ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ), v ≥0.\nThe first two condititions here can be expressed in terms of linear con-\nstraints (equalities and inequalities) on the coordinates ̄μi’s. Thus the three\nconditions together can be expressed in terms of linear constraints on ( ̄μ, v).\nBy the finiteness assumption (A), the polytope determined by these con-\nstraints is a bounded polytope.\nThe number of integer points in such a\npolytope can be expressed as a vector partition function (cf. [BBCV]). This\nproves the claim.\nLet us denote the vector partition associated with the integer program-\nming problem (4.17) in the claim by φW, ̄\nW,C(c( ̄λ, π, ρ, ̄ρ)). Then\nmπ ̄λ =\nX\nW\nX\n ̄\nW\n(−1)W (−1)\n ̄\nW X\nC\na(C)φW, ̄\nW,C(c( ̄λ, π, ρ, ̄ρ)).\n(4.18)\n81"},{"paragraph_id":"p85","order":85,"text":"Hence,\n ̃mπ ̄λ(n) = mn ̄λ\nnπ =\nX\nW\nX\n ̄\nW\n(−1)W (−1)\n ̄\nW X\nC\na(C)φW, ̄\nW,C(c(n ̄λ, nπ, ρ, ̄ρ)).\n(4.19)\nIt follows from Claim 4.5.1 and the standard results on vector partition\nfunctions mentioned in the begining of this section that\ngW, ̄\nW,C(n) = φW, ̄\nW,C(c(n ̄λ, nπ, ρ, ̄ρ)),\nis asymptitically a quasipolynomial function of n.\nHence, ̃mπ ̄λ(n) is also\nasymptotically a quasipolynomial function of n.\nThis implies (cf.\n[St1])\nthat\nMπ ̄λ (t) =\nX\nn≥0\n ̃mπ ̄λ(n)tn\n(4.20)\nis rational function of t.\nThis proves Theorem 3.4.1 (a) under the finiteness assumption (A).\nIt remains to remove the assumption (A). Let G′ ⊇H be the smallest\nLevi subalgebra of G containing H. Then\nmπ ̄λ =\nX\nπ′\nmπ′\n ̄λ mπ\nπ′,\n(4.21)\nwhere π′ ranges over dominant weights of G′, mπ′\n ̄λ denotes the multiplicity of\nVπ′(G′) in V ̄λ(G), and mπ\nπ′ the multiplicity of Vπ(H) in Vπ′(G′). Furthermore,\n1. the finiteness asssumption (A) is now satisfied for the pair (G′, H): i.e.,\nfor any weight μ of H, the number of weights μ′’s of G′ such that μ′ ↓μ\nis finite.\n2. There is a polyhedral expression for mπ′\n ̄λ ; this follows from [Li, Dh].\nBy the first condition and the argument above, we get an expression for\nmπ\nπ′ akin to (4.18). Substituting this expression and the polyhedral expres-\nsion for mπ′\n ̄λ in (4.21), leads to a formula for ̃mπ ̄λ(n) as a linear combination\nof φB,c,b(n)’s for appropriate B, c, b’s. After this, we proceed as before.\nThis proves Theorem 3.4.1 (a). Q.E.D.\nWe also note down the following consequence of the proof.\nProposition 4.5.2 There is a constant D depending only G and H, such\nthat for any ̄λ, π, orders of the poles of Mπ ̄λ (t) (cf. (4.20), as roots of unity,\ndivide D.\n82"},{"paragraph_id":"p86","order":86,"text":"A bound on D provided by the proof below is very weak: D = O(2O(rank(G))).\nProof: It suffices to to bound the period of the quasipolynomial ̃mπ ̄λ(n). For\nthis, it suffices to let n →∞. For a fixed W, ̄\nW , C, the chamber containing\nc(n ̄λ, nπ, ρ, ̄ρ)) is completely determined by ̄λ and π as n →∞. Under these\nconditions, the degree of φW, ̄\nW,C(c(n ̄λ, nπ, ρ, ̄ρ)) is equal to the dimension of\nthe polytope associated with this vector partition function. This dimension\nis clearly O(rank(G)2).\nBy Szenes-Vergne residue formula [SV], there is a constant D depending\non only G, H, W, ̄\nW , C, such that the period of the quasipolynomial h(n) =\nφW, ̄\nW,C(c(n ̄λ, nπ, 0, 0)) divides D for every ̄λ, π; here we are putting ρ and\n ̄ρ equal to zero, since we are interested in what happens as n →∞. Q.E.D.\n83"},{"paragraph_id":"p87","order":87,"text":"Chapter 5\nParallel and PSPACE\nalgorithms\nIn this chapter we give PSPACE algorithms (cf. Theorem 3.4.3) for com-\nputing the various structural constants under consideration . We shall only\nprove Theorem 3.4.3, when H is therein is either a complex, semisimple\ngroup, or a symmetric group, or a general linear group over a finite field,\nthe extension to the general case being routine.\nWe recall two standard results in parallel complexity theory [KR], which\nwill be used repeatedly.\nLet NC(t(N), p(N) denote the class of problems that can be solved\nin O(t(N)) parallel time using O(p(N)) processors, where N denotes the\nbitlength of the input. Let\nNC = ∪iNC(logi(N), poly(N)).\nThis is the class of problems having efficient parallel algorithms.\nProposition 5.0.3 [Cs, KR] Let A be an n × n-matrix with entries in a\nring R of characteristic zero. Then the determinant of A, and A−1, if A\nis nonsingular, can be computed in O(log2 n) parallel steps using poly(n)\nprocessors; here each operation in the ring is considered one step. Hence, if\nR = Q, the problems of computing the determinant, the inverse and solving\nlinear systems belong to NC.\nProposition 5.0.4 The class NC(t(N), 2t(N)) ⊆SPACE(O(t(N))).\nIn\nparticular, NC(poly(N), 2O(poly(N))) ⊆PSPACE.\n84"},{"paragraph_id":"p88","order":88,"text":"5.1\nComplex semisimple Lie group\nIn this section we prove a special case of Theorem 3.4.3 for the general-\nized plethym problem (Problem 1.1.2). Accordingly, let H be a complex,\nsemisimple, simply connected Lie group, G = GL(V ), where V = Vμ(H) is\nan irreducible representation of H with dominant weight μ, ρ : H →G the\nhomomorphism corresponding to the representation, and mπ\nλ the multiplic-\nity of Vπ(H) in Vλ(G), considered as an H-module via ρ; cf. Problem 1.1.3.\nThen:\nTheorem 5.1.1 The multiplicity mπ\nλ can be computed in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, dim(H))\nspace.\nHere it is assumed that the partition λ = λ1 ≥λ2 ≥· · · λr > 0 is rep-\nresented in a compact form by specifying only its nonzero parts λ1, . . . , λr.\nThis is important since dim(G) can be exponential in dim(H) and ⟨μ⟩. A\ncompact representation allows ⟨λ⟩to be small, say poly(dim(H), ⟨μ⟩), in this\ncase.\nWe begin with a simpler special case.\nProposition 5.1.2 If dim(V ) = poly(dim(H)), then mπ\nλ can be computed\nin PSPACE; i.e., in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, dim(H)) space.\nThis implies that the Kronecker coefficient (Problem 1.1.1) can be computed\nin PSPACE.\nProof: Let us use the notation ̄λ instead of λ to be consistent with the\nnotation used in Klimyk’s formula (4.12). By the latter, mπ ̄λ can be computed\nin PSPACE if n ̄μ(V ̄λ) in that formula can be computed in PSPACE for every\n ̄μ and ̄λ. In type A, this is just the number of Gelfand-Tsetlin tableau with\nthe shape ̄λ and weight ̄μ.\nIf dim(V ) = poly(dim(H)), the size of such\na tableau is O(dim(V )2) = poly(dim(H)).\nSo we can count the number\nof such tableu in PSPACE as follows: Begin with a zero count, and cycle\nthrough all tableaux of shape ̄λ in polynomial space one by one, increasing\nthe count by one everytime the tableau satisfies all constraints for Gelfand-\nTsetlin tableau and has weight ̄μ. In general, the role of Gelfand-Tsetlin\ntableaux is played by Lakshmibai-Seshadri (LS) paths [Li, Dh]. Q.E.D.\nThe argument above does not work if dim(V ) is not poly(dim(H)), as\nin the plethym problem (Problem 1.1.2), where dim(V ) = dim(Vμ) can\nbe exponential in n = dim(H) and the bitlength of μ. In this case, the\n85"},{"paragraph_id":"p89","order":89,"text":"algorithm cannot even afford to write down a tableau since its size need not\nbe polynomial.\nNext we turn to Theorem 5.1.1.\nFor the sake of simplicity, we shall\nprove it only for H = SLn(C), or rather GLn(C)–i.e., the usual plethysm\nproblem. This illustrates all the basic ideas. The general case is similar. We\nshall prove a slightly stronger result in this case:\nTheorem 5.1.3 The plethysm constant aπ\nλ,μ can be can be computed in\npoly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) space.\nHere the dependence on n = dim(H) is not there. This makes a difference\nif the heights of μ and π are less than n = dim(H)–remember that we are\nusing a compact representation of a partition in which only nonzero parts\nare specified. This is really not a big issue. Because aπ\nλ,μ depends only on\nthe partitions λ, μ, π and not n. Hence, without loss of generality, we can\nassume that n is the maximum of the heights of μ and π. It is possible to\nstrengthen Theorem 5.1.1 similarly.\nTo prove Theorem 5.1.3, we shall give an efficient parallel algorithm to\ncompute ̃aπ\nλ,μ that works in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) parallel time using O(2poly(⟨λ⟩,⟨μ⟩,⟨π⟩))\nprocessors. This will show that the problem of computing ̃aπ\nλ,μ is in the com-\nplexity class NC(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩), 2poly(⟨λ⟩,⟨μ⟩,⟨π⟩)), which is contained in\nPSPACE by Proposition 5.0.4. The basic idea is to parallelize the classical\ncharacter-based algorithm for computing aπ\nλ,μ by using efficient parallel algo-\nrithm for inverting a matrix and solving a linear system (Proposition 5.0.3).\nWe begin by recalling the standard facts concerning the characters of\nthe general linear group.\nGiven a representation W of GLm(C), let ρ :\nGLm(C) →GL(W) be the representation map.\nLet χρ(x1, . . . , xm) de-\nnote the formal character of this representation W. This is the trace of\nρ(diag(x1, . . . , xm)), where diag(x1, . . . , xn) denotes the generic diagonal ma-\ntrix with variable entries x1, . . . , xm on its diagonal. If W is an irreducible\nrepresentation Vλ(GLm(C)), then χρ(x1, . . . , xm) is the Schur polynomial\nSλ(x1, . . . , xm). By the Weyl character formula,\nSλ =\n|xλi+m−i\nj\n|\n|xm−i\nj\n| ,\n(5.1)\nwhere |ai\nj| denotes the determinant of an m×m-matrix a. The Schur polyno-\nmials form a basis of the ring of symmetric polynomials in x1, . . . , xm. The\n86"},{"paragraph_id":"p90","order":90,"text":"simplest basis of this ring consists of the complete symmetric polynomials\nMβ(x1, . . . , xm) defined by\nMβ(x1, . . . , xm) =\nX\nγ\ntγ,\nwhere γ ranges over all permutations of β and tγ = Q\ni xγi\ni . Schur polyno-\nmials are related to Mβ by:\nSλ =\nX\nβ\nkβ\nλMβ,\n(5.2)\nwhere kβ\nλ is the Kostka number. This is the number of semistandard tableau\nof shape λ and weight β.\nIf the representation W is reducible, its decomposition into irreducibles\nis given by:\nW =\nX\nπ\nm(π)Vπ(GLn(C)),\n(5.3)\nwhere m(π)’s are the coefficients of the formal character χρ(x1, . . . , xm) in\nthe Schur basis:\nχρ =\nX\nπ\nm(π)Sπ.\nProof of Theorem 5.1.3\nLet λ, μ, π be as in Theorem 5.1.3. Let H = GLn(C), V = Vμ(H), G =\nGL(V ). Let sλ(x1, . . . , xm) be the formal character of the representation\nVλ(G) of G. Here m = dim(Vμ) can be exponential in n and ⟨μ⟩. The basis\nof Vμ(H) is indexed by semistandard tableau of shape μ with entries in [1, n].\nLet us order these tableau, say lexicographically, and let Ti, 1 ≤i ≤m,\ndenote the i-th tableau in this order. With each tableau T, we associate a\nmonomial\nt(T) =\nn\nY\ni=1\ntwi(T)\ni\n,\nwhere wi(T) denotes the number of i’s in T. Given a polynomial f(x1, . . . , xm),\nlet us define fμ = fμ(t1, . . . , tn) to be the polynomial obtained by substi-\ntuting xi = t(Ti) in f(x1, . . . , xm). Then the formal character of Vλ(G),\nconsidered as an H-representation of via the homomorphism H →G =\n87"},{"paragraph_id":"p91","order":91,"text":"GL(Vμ(H)), is the symmetric polynomial Sλ,μ(t1, . . . , tn) = (Sλ)μ.\nThe\nplethysm constant aπ\nλ,μ is defined by:\nSλ,μ(t1, . . . , tn) =\nX\nπ\naπ\nλ,μSπ(t1, . . . , tn).\n(5.4)\nAn efficient parallel algorithm to compute aπ\nλ,μ is as follows. Here by an\nefficient parallel algorithm, we mean an algorithm that works in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)\ntime using 2poly(⟨λ⟩,⟨μ⟩,⟨π⟩) processors.\nWe will repeatedly use Proposi-\ntion 5.0.3.\nAlgorithm\n(1) Compute Sλ,μ(t1, . . . , tn). By the Weyl character formula (5.1),\nSλ,μ(t1, . . . , tn) = Aλ,μ(t1, . . . , tn)\nBλ,μ(t1, . . . , tn),\nwhere Aλ(x1, . . . , xm) and Bλ(x1, . . . , xm) denote the numerator and denom-\ninator in (5.1), and Aλ,μ = (Aλ)μ, and Bλ,μ = (Bλ)μ. Let R = C[t1, . . . , tn].\nThen\nAλ,μ(t1, . . . , tn) = |t(Tj)λi+m−i|.\nThis is the determinant of an m × m matrix with entries in R, where m =\ndim(V ) can be exponential in n and ⟨μ⟩. It can be evaluated in O(log2 m)\nparallel ring operations using poly(m) processors. Each ring element that\narises in the course of this algorithm is a polynomial in t1, . . . , tn of total\ndegree O(|λ|m), where |λ| denotes the size of λ. The total number of its\ncoefficients is r = O((|λ|m)n). Hence each ring operation can be carried\nout efficiently in O(log2(r)) parallel time using poly(r) processors. Since\nlog m = poly(n, ⟨μ⟩) and log r = poly(n, ⟨λ⟩, ⟨μ⟩), it follows that Aλ,μ can\nbe evaluated in poly(n, ⟨μ⟩, ⟨λ⟩) parallel time using 2poly(n,⟨μ,λ⟩) processors.\nThe determinant Bλ,μ can also be computed efficiently in parallel in a similar\nfashion. To compute Sλ,μ, we have to divide Aλ,μ by Bλ,μ. This can be done\nby solving an r × r linear system, which, again, can be done efficiently in\nparallel. This computation yields representation of Sλ,μ in the monomial\nbasis {Mβ} of the ring of symmetric polynomials in t1, . . . , tn.\n(2) To get the coefficients aπ\nλ,μ, we have to get the representation of Sλ,μ(t)\nin the Schur basis. This change of basis requires inversion of the matrix\nin the linear system (5.2). The entries of the matrix K occuring in this\n88"},{"paragraph_id":"p92","order":92,"text":"linear system are Kostka numbers. Each Kostka number can be computed\nefficiently in parallel.\nHence, all entries of this matrix can be computed\nefficiently in parallel. After this, the matrix can be inverted efficiently in\nparallel, and the coefficients aπ\nλ,μ’s of Sλ,μ in the Schur basis can be computed\nefficiently in parallel. Finally, we use Proposition 5.0.4 to conclude that aπ\nλ,μ\ncan be computed in PSPACE. Q.E.D.\n5.2\nSymmetric group\nNext we prove Theorem 3.4.3 when H = Sm.\nLet X = Vμ(Sm) be an\nirreducible representation (the Specht module) of Sm corresponding to a\npartition μ of size m.\nLet ρ : H →G = GL(X) be the corresponding\nhomomorphism.\nTheorem 5.2.1 Given partitions λ, μ, π, where μ and π have size m, the\nmultiplicity mπ\nλ,μ of the Specht module Vπ(Sm) in Vλ(G) can be computed in\npoly(m, ⟨λ⟩) space.\nRemark 5.2.2 The bitlengths ⟨μ⟩and ⟨π⟩are not mentioned in the com-\nplexity bound because they are bounded by m.\nFor this, we need three lemmas.\nLemma 5.2.3 The character of a symmetric group can be computed in\nPSPACE.\nSpecifically, given a partition π of size m, and a sequence\ni = (i1, i2, . . .) of nonnegative integers such that P jij = m, the value of\nthe character χπ of Sm on the conjugacy class Ci of permutations indexed\nby i can be computed in poly(m) parallel time using 2poly(m) processors.\nHence it can be computed in poly(m) space (cf. Proposition 5.0.4).\nHere the conjugacy class Ci consists of those permutations that have i1\n1-cycles, i2 2-cycles, and so on.\nProof: Let k be the height of the partition π. Let x = (x1, . . . , xk) be the\ntuple of variables xi’s. Given a formal series f(x) and a tuple (l1, . . . , lk) of\nnonnegative integers, let [f(x)](l1,...,lk) denote the coefficient of xl1\n1 · · · xlk\nk in\nf.\nBy the Frobenius character formula [FH],\nχλ(Ci) = [f(x)](l1,...,lk),\n(5.5)\n89"},{"paragraph_id":"p93","order":93,"text":"where\nl1 = π1 + k −1, l2 = π2 + k −2, . . . , lk = πk,\nand\nf(x) = ∆(x)\nm\nY\nj=1\nPj(x)ij,\nwith\n∆(x)\n=\nQ\ni<j(xi −xj),\nPj(x)\n=\nxj\n1 + · · · + xj\nk.\n(5.6)\nSince deg(f) = poly(m) and k ≤m, the total number of coefficients of\nf(x) is 2poly(m). Hence, we can evaluate f(x) in PSPACE by setting up\nappropriate recurrence relations.\nAlternatively, we can easily evaluate f(x) in poly(m) parallel time using\n2poly(m) processors, and then extract its required coefficient. After this, the\nresult follows from Proposition 5.0.4. Q.E.D.\nLemma 5.2.4 Suppose φ is a character of Sm whose value on any conju-\ngacy class Ci can be computed in O(s) space, for some parameter s. Then,\nthe multiplicity of the representation Vπ(Sm) in the representation Vφ(Sm)\ncorresponding φ can be computed in O(poly(m) + s) space.\nProof: The multiplicity is given by the inner product\n⟨φ, χπ⟩= 1\nm!\nX\nσ∈Sm\n ̄\nφ(σ)χπ(σ).\n(5.7)\nBy assumption, φ(σ) can be computed in O(s) space, and by Lemma 5.2.3,\nχπ(σ) can be computed in poly(m) space. Hence, the result follows from\nthe preceding formula. Q.E.D.\nGiven an irreducible representation X = Vμ(Sm) and an irreducible rep-\nresentation W = Vλ(G) of G = GL(X)), let ρμ denote the representation\nmap Sm →G, ρλ the representation map G →GL(W), and\nρ : Sm →G →GL(W)\ntheir composition. This is a representation of Sm. Let χρ be the character\nof ρ.\nLemma 5.2.5 For any σ ∈Sm, χρ(σ) can be computed in poly(m, ⟨λ⟩) in\npoly(m, ⟨λ⟩) space.\n90"},{"paragraph_id":"p94","order":94,"text":"The bitlength ⟨μ⟩is not mentioned in the complexity bound because it\nis bounded by m.\nProof: Let r = dim(X). The formal character of the representation Vλ(G)\nof G = GL(X) is the Schur polynomial Sλ(x1, . . . , xr), r = dim(X). Hence,\nχρ(σ) = Sλ(α)\nwhere α = (α1, . . . , αr) is the tuple of eigenvalues of ρμ(σ). We shall compute\nthe right hand side fast in parallel–i.e., in poly(m, ⟨λ⟩) parallel time using\n2poly(m,⟨λ⟩) processors–and then use Proposition 5.0.4 to conclude the proof.\nThis is done as follows.\n(1) Let χμ denote the character of the representation ρμ. Let pi(α) = αi\n1 +\n· · · + αi\nr denote the i-th power sum of the eigenvalues. For any i,\npi(α) = χμ(σi).\nWe can compute σi, for i ≤|λ|, where |λ| denotes the size of λ, in poly(log i, m) =\npoly(m, ⟨λ⟩) time using repeated squaring. After this χμ(σi) can be com-\nputed fast in parallel in poly(m) time using Lemma 5.2.3. Thus each pi(α)\ncan be computed in poly(m, ⟨λ⟩) time in parallel using 2poly(m,⟨λ⟩) proces-\nsors. We calculate pi(α) in parallel for all i ≤|λ|, and all pγ(α) = Q\nj pγj(α)\nin parallel for all partitions γ of size at most m.\n(2) After this, we calculate the complete symmetric function hi(α), for\neach i ≤|λ|, fast in parallel, by using the relation [Mc]:\nhi =\nX\n|γ|=i\nz−1\nγ pγ,\nwhere zγ = Q\ni≥1 imimi!, and mi = mi(γ) denotes the number of parts of γ\nequal to i. Thus we can calculate hγ(α) = Q\nj hγj(α), for all partitions γ of\nsize m, fast in parallel.\n(3) To compute Sλ(α), we recall that the transition matrix between\nthe Schur basis {Sλ} and the complete symmetric basis {hγ} of the ring\nof symmetric functions is K∗, the transpose inverse of the Kostka matrix\nK = [Kλ,γ], where Kλ,γ denote the Kostka number; cf. [Mc]. As we noted in\nthe proof of Theorem 5.1.3, each Kostka number can be computed in fast in\nparallel. Hence, K can be computed fast in parallel. After this, its inverse\nK−1 can be computed fast in parallel by Proposition 5.0.3–this is the crux\nof the proof–and finally K∗as well. Thus Sλ(α) can be computed fast in\nparallel, since each hγ(α) can be computed fast in parallel. Q.E.D.\nTheorem 5.2.1 follows from Lemma 5.2.3,5.2.4 and 5.2.5. Q.E.D.\n91"},{"paragraph_id":"p95","order":95,"text":"5.3\nGeneral linear group over a finite field\nIn this section we prove Theorem 3.4.3, when H therein is the general lin-\near group GLn(Fpk) over a finite field Fpk. Irreducible representations of\nH = GLn(Fpk) have been classified by Green [Mc]. They are labelled by\ncertain partition-valued functions. See [Mc] for a precise description of these\nlabelling functions. It is clear from the description therein that each labelling\nfunction has a compact representation of bitlength O(n + k + ⟨p⟩), where\n⟨p⟩= log2 p; we specify a function by giving its partition values at the places\nwhere it is nonzero. Let μ denote any such label. Let X = Vμ(H) be the\ncorresponding irreducible representation of H, and ρ : H →G = GL(X)\nthe corresponding homomorphism.\nTheorem 5.3.1 Given a partition λ and labelling functions μ and π as\nabove, the multiplicity mπ\nλ,μ of the irreducible representation Vπ(H) in Vλ(G)\ncan be computed in poly(n, k, ⟨p⟩, ⟨λ⟩) space.\nThe proof is similar to that of Theorem 5.2.1 for the symmetric group\nwith the following result playing the role of Lemma 5.2.3:\nLemma 5.3.2 Given a label γ of an irreducible character χγ of H = GLn(Fpk)\nand a label δ of a conjugacy class in H, the value χγ(δ) can be computed\nin poly(n, k, ⟨p⟩) parallel time using 2poly(n,k,⟨p⟩) processors, and hence by\nProposition 5.0.4, in poly(n, k, ⟨p⟩) space.\nThe label δ of a conjugacy class in H is also a partition valued function\n[Mc], which admits a compact representation of bitlength poly(n, k, ⟨p⟩).\nProof: We shall parallelize Green’s algorithm [Mc] for computing the charac-\nter values, and then conclude by Proposition 5.0.3. Green shows that χγ(δ)’s\nare entries of a transition matrix between a two polynomial bases: the first\nconstructed using Hall-Littlewood polynomials, and the second using Schur\npolynomials. We have construct this transition matrix fast in parallel. We\nshall only indicate here how the transition matrix between the basis of Hall-\nLittlewood polynomials and the Schur basis for the ring symmetric functions\nover Z[t] can be constructed fast in parallel. This idea can then be easily\nextended to complete the proof.\nFirst, we recall the definition of the Hall-Littlewood polynomial Pπ(x; t) =\nPπ(x1, . . . , xk; t) [Mc].\nThis is a symmetric polynomial in xi’s with co-\nefficients in Z[t].\nIt interpolates between the Schur function sπ(x) and\n92"},{"paragraph_id":"p96","order":96,"text":"the monomial symmetric function mπ(x) because Pπ(x; 0) = sπ(x) and\nPπ(x; 1) = mπ(x). The formal definition is as follows:\nFor a given partition π, let vπ(t) = Q\ni≥0 vmi(t), where mi is the number\nof parts of π equal to i, and\nvm(t) =\nm\nY\ni=1\n1 −ti\n1 −t .\nThen\nPπ(x; t) = Aπ(x, t)\nBπ(x, t),\n(5.8)\nwhere\nAπ(x, t) = P\nσ∈Sk sgn(σ)σ(xπ1\n1 · · · xπk\nk\nQ\ni<j xi −txj,\nBπ(x, t) = vπ(t) Q\ni<j(xi −xj).\n(5.9)\nHere sgn(σ) denotes the sign of σ.\nLet wπ,α(t)’s be the coeffcients of Pπ(x, t) in the Schur basis:\nPπ(x; t) =\nX\nα\nwπ,α(t)sα(x).\n(5.10)\nWe want to calculate the matrix W = [wπ,α] fast in parallel.\nUsing\nformula (5.9), we calculate Aπ(x; t) fast in parallel; i.e., we calculate the\ncoefficients of Aπ(x; t) in the basis of monomials in x and t. We calculate\nBπ(x; t) similarly.\nAfter this the division in (5.8) can be carried out by\nsolving a an appropriate linear system. This can be done fast in parallel\nby Proposition 5.0.3.\nSince, Pπ(x; t) is symmetric in xi’s, this yields its\ncoefficients in the monomial symmetric basis {mα(x)} with the coefficients\nbeing in Z[t]. The transition matrix [Mc] from the monomial symmetric\nbasis to the Schur basis is given by the inverse of the Kostka matrix. This\ninverse can be computed fast in parallel by Proposition 5.0.3. After this,\nthe coefficients wπ,α’s of Pπ(x; t) in the Schur basis can be computed fast in\nparallel.\nFurthermore, the inverse of W = [Wπ,α] can also be computed fast in\nparallel by Proposition 5.0.3. Q.E.D.\n5.3.1\nTensor product problem\nAnalogue of the Kronecker problem (Problem 1.1.1) for H = GLn(Fpk) is:\n93"},{"paragraph_id":"p97","order":97,"text":"Problem 5.3.3 Given partition valued functions λ, μ, π, decide if the mul-\ntiplicity bπ\nλ,μ of Vπ(H) in the tensor product Vλ(H) ⊗Vμ(H) is nonzero.\nIn this context:\nTheorem 5.3.4 The multiplicity mπ\nλ,μ can be computed in PSPACE; i.e.,\nin poly(n, k, ⟨p⟩) space.\nProof: This follows from Lemma 5.3.2 and analogues of Lemmas 5.2.4 and\n5.2.5 in this setting. Q.E.D.\nA possible canditate for a stretching function assoociated with bπ\nλ,μ is:\n ̃bπ\nλ,μ(n) = bnπ\nnλ,nμ,\nwhere nλ denotes the stretched partition-valed function obtained by stretch-\ning each partition value of λ by a factor of n. In other words ̃bπ\nλ,μ(n) is\nthe multiplicity of Vnπ(H(n)) in Vnλ(H(n)) ⊗Vnμ(H(n)), where H(n) =\nGLnm(Fpk) is the stretched group. Is it a quasi-polynomial? If so, we can\nalso ask for a good bound on its saturation and positivity indices.\n5.4\nFinite simple groups of Lie type\nThe work of Deligne-Lusztig [DL] and Lusztig[Lu5] yield an algorithm for\ncomputing the character values for finite simple groups of Lie type.\nQuestion 5.4.1 Can this algorithm be parallelized?\nIf so, Lemma 5.3.2, and hence Theorem 5.3.1, can be extended to finite\nsimple groups of Lie type.\n94"},{"paragraph_id":"p98","order":98,"text":"Chapter 6\nExperimental evidence for\npositivity\nIn this chapter we give experimental evidence for positivity (PH2,3).\n6.1\nLittlewood-Richardson problem\nExperimental evidence for PH2 in the context of the Littlewood-Richardson\nproblem (Problem 1.2.1) has been given in [DM2], and for PH3 in type A\nin [KTT]. We give experimental evidence for PH3 in types B, C, D here.\nLet Cλ\nα,β(t) be as in eq.(1.2). Its reduced positive form for various values\nof α, β, λ is shown in Figure 6.1 for type B, in Figure 6.2 for type C, and\nFigure 6.3 for type D. The rank of the Lie algebra is three in all cases.\nIn these types, the period of the stretching quasipolynomial ̃cλ\nα,β(n) is at\nmost two. Accordingly, the period of every pole of Cλ\nα,β(t) is at most two.\nThe tables were computed from the tables in [DM2] for the stretching quasi-\npolynomial ̃cλ\nα,β(n) in these cases.\n6.2\nKronecker problem, n = 2\nLet kπ\nλ,μ be the Kronecker coefficient in Problem 1.1.1. Let ̃kπ\nλ,μ(n) = ̃knπ\nnλ,nμ\nbe the associated stretching quasi-polynomial, and\nKπ\nλ,μ(t) =\nX\nn≥0\n ̃kπ\nλ,μ(n)tn,\n95"},{"paragraph_id":"p99","order":99,"text":"the associated rational function. An explicit formula (with alternating signs)\nfor the Kronecker coefficient, when n = 2, has given by Remmel and White-\nhead [RW] and Rosas [Ro], and a positive formula in [GCT9]. This case\nturns out to be nontrivial. For example, the number of chambers (domains)\nof quasi-polynomiality in this case turns out to be more than a million. Their\nexplicit description can be found out using the formula for the Kronecker\ncoefficient in [RW].\nWe implemented Rosas’ formula to check PH2 for the quasipolynomial\n ̃kπ\nλ,μ(n) for a few thousand values of μ, ν and λ with the help of a computer.\nA large number of samples was chosen to ensure that a significant fraction\nof the chambers were sampled. The quasi-polynomial ̃kπ\nλ,μ(n) and a positive\nform of the rational function Cπ\nλ,μ(t) are shown Figures 6.4 and 6.5 for few\nsample values of λ = (λ1, λ2), μ = (μ1, μ2), and π = (π1, π2, π3, π4).\nIt\nmay be noted that ̃kπ\nλ,μ(n) need not be a polynomial; this answers Kirillov’s\nquestion [Ki] in the negative. But its period is at most two for n = 2. This\nfollows from the formula in [RW].\nFor the λ, μ and π that we sampled,\npositivity index of ̃kπ\nλ,μ(n) is always zero.\nBut it turns out [BOR] that\nthere are some λ, μ and π for which the saturation and positivity indices of\n ̃kπ\nλ,μ(n) are nonzero (one), but very small and thus consistent with SH and\nPH2 (Hypothesis1.6.6) in this paper; in the earlier version of this paper, SH\nand PH2 stipulated that the saturation and positivity indices are always\nzero. These (λ, μ, π) escaped our random sampling, because their density is\nextremely small [BOR].\n6.3\nG/P and Schubert varieties\nLet V = Vλ(G) be an irreducible representation of G = SLk(C) correspond-\ning to a partition λ.\nLet vλ be the point in P(V ) corresponding to the\nhighest weight vector, and X = Gvλ ∼= G/Pλ its closed orbit. Let hk,λ(n)\nbe the Hilbert function of the homogeneous coordinate ring R of X. It is\na quasipolynomial since spec(R) has rational singularities. In fact, it is a\npolynomial, since t = 1 is the only pole of the Hilbert series\nHk,λ(t) =\nX\nn≥0\nhk,λ(n)tn.\nFigure 6.6 gives experimental evidence for strict positivity (PH2) of hk,λ(n)\n(as discussed in Section 3.5.5) for a few sample values of k and λ. Figure 6.7\ngives experimental evidence for strict positivity of the Hilbert polynomial\n96"},{"paragraph_id":"p100","order":100,"text":"of the Schubert subvarieties of the Grassmanian; there Gn,k denotes the\nGrassmannian of k-planes in V = Cn, and Ωa, a = (a(1), . . . , a(d)) its\nSchubert subvariety consisting of the k-subspaces W such that dim(W ∩\nVn−k+i−a(i)) ≥i for all i, where V = Vn ⊃· · · V1 ⊃0 is a complete flag of\nsubspaces in V . The Hilbert polynomials were computed using the explicit\npolyhedral interpretation for them deduced from the theory of algebras with\nstraightening laws (Hodge algebras) [DEP2].\n6.4\nThe ring of symmetric functions\nLet V = Ck, G = GL(V ), H = Sk, with the natural embedding H →\nG.\nLet us consider the spacial case of the subgroup restriction problem\n(Problem 1.1.3), with Vλ(G) = V , and Vπ(H) the trivial representation of\nH.\nThen s = mπ\nλ, the multiplicity of the trivial representation in V , is\none. Though the decision problem (Problem 1.1.3) is trivial in this case, the\ncanonical model associated with s is nontrivial.\nThe canonical rings R = R(s) and S = S(s) associated with s in this\ncase coincide with C[V ] = C[x1, . . . , xk].\nThe ring T = T(s) = SH =\nC[x1, . . . , xk]Sk is the subring of symmetric functions. Its Hilbert function\nh(n) is a quasipolynomial. PH1 and PH3 for Z = Proj(T), as per Defini-\ntion 3.5.2, follow easily, the latter from the well known rational generating\nfunction for the partition function [St1]. But PH2 turns out to be nontriv-\nial. Figures 6.8-6.13 give experimental evidence for strict positivity of h(n)\n(PH2). In these figures, the i-th row of the table for a given k shows hi(n),\nwhere hi(n), 1 ≤i ≤l, are such that h(n) = hi(n), when n = i modulo the\nperiod l of h(n).\n97"},{"paragraph_id":"p101","order":101,"text":"α\nβ\nλ\nCλ\nα,β(t)\n(0, 15, 5)\n(12, 15, 3)\n(6, 15, 6)\n350 t8+19121 t7+123576 t6+297561 t5+342064 t4+192779 t3+46992 t2+2641 t+1\n(1−t)3(1−t2)3\n(4, 8, 11)\n(3, 15, 10)\n(10, 1, 3)\n1+5 t+6 t2+t3\n(1−t2)3\n(8, 1, 3)\n(11, 13, 3)\n(8, 6, 14)\n2 t8+45 t7+259 t6+591 t5+773 t4+522 t3+198 t2+29 t+1\n(1−t)3(1−t2)4\n(8, 9, 14)\n(8, 4, 5)\n(1, 5, 15)\n136 t9+3422 t8+20204 t7+53608 t6+76076 t5+60986 t4+26674 t3+5568 t2+345 t+1\n(1−t)3(1−t2)4\n(10, 5, 6)\n(5, 4, 10)\n(0, 7, 12)\n219 t8+12135 t7+79231 t6+193003 t5+223919 t4+127907 t3+31704 t2+1870 t+1\n(1−t)6(1+t)3"},{"paragraph_id":"p102","order":102,"text":"Figure 6.1: Cλ\nα,β(t) for B3\n98"},{"paragraph_id":"p103","order":103,"text":"α\nβ\nλ\nCλ\nα,β(t)\n(1, 13, 6)\n(14, 15, 5)\n(5, 11, 7)\n18145 t8+267151 t7+1070716 t6+1917716 t5+1735692 t4+778184 t3+144596 t2+5538 t+1\n(1−t)4(1−t2)3\n(4, 15, 14)\n(12, 12, 10)\n(4, 9, 8)\n2280 t9+267658 t8+2746131 t7+9276935 t6+14682332 t5+11903923 t4+4746803 t3+751126 t2+21249 t+1\n(1−t)4(1−t2)3\n(9, 0, 8)\n(8, 12, 9)\n(7, 7, 3)\n3 t2+4 t+1\n(1−t)6\n(10, 2, 7)\n(8, 10, 1)\n(7, 5, 5)\n8984 t8+132826 t7+534183 t6+960491 t5+873227 t4+394045 t3+74067 t2+2941 t+1\n(1−t)4(1−t2)3\n(10, 10, 15)\n(11, 3, 15)\n(10, 7, 15)\n7162 t9+736327 t8+7178960 t7+23540366 t6+36359642 t5+28788904 t4+11166361 t3+1693696 t2+43515 t+1\n(1−t)7(1+t)3"},{"paragraph_id":"p104","order":104,"text":"Figure 6.2: Cλ\nα,β(t) for C3\n99"},{"paragraph_id":"p105","order":105,"text":"α\nβ\nλ\nCλ\nα,β(t)\n(0, 15, 5)\n(12, 15, 3)\n(6, 15, 6)\n633 t7+24259 t6+142236 t5+252113 t4+168220 t3+36131 t2+1414 t+1\n(1−t)7(1−t2)\n(4, 8, 11)\n(3, 15, 10)\n(10, 1, 3)\n7962 t8+503679 t7+4525372 t6+11944350 t5+12218255 t4+4879052 t3+586370 t2+10862 t+1\n(1−t)8(1−t2)\n(8, 1, 3)\n(11, 13, 3)\n(8, 6, 14)\n81 t7+19407 t6+211964 t5+513585 t4+426652 t3+110317 t2+4609 t+1\n(1−t)6(1−t2)\n(8, 9, 14)\n(8, 4, 5)\n(1, 5, 15)\n9 t2+8 t+1+2 t3\n(1−t)3\n(10, 5, 6)\n(5, 4, 10)\n(0, 7, 12)\n3647 t7+111208 t6+570739 t5+920201 t4+560336 t3+106748 t2+3435 t+1\n(1−t)8(1+t)"},{"paragraph_id":"p106","order":106,"text":"Figure 6.3: Cλ\nα,β(t) for D3\n100"},{"paragraph_id":"p107","order":107,"text":"λ1\nλ2\nμ1\nμ2\nπ1\nπ2\nπ3\nπ4\n ̃kπ\nλ,μ(n); n odd\n ̃kπ\nλ,μ(n); n even\nKπ\nλ,μ(t)\n87\n62\n97\n52\n64\n39\n24\n22\n1/2 + 4 n + 11/2 n2\n1 + 4 n + 11/2 n2\n1+8 t+11 t2+2 t3\n(1−t)2(1−t2)\n104\n95\n149\n50\n95\n78\n15\n11\n1/2 + 13/2 n + 18 n2\n1 + 13/2 n + 18 n2\n1+23 t+36 t2+12 t3\n(1−t)2(1−t2)\n101\n85\n102\n84\n78\n72\n24\n12\n17/2 n + 71\n2 n2\n1 + 17/2 n + 71\n2 n2\n1+42 t+72 t2+27 t3\n(1−t)2(1−t2)\n79\n63\n93\n49\n88\n37\n14\n3\n3/4 + 27\n2 n + 303\n4 n2\n1 + 27\n2 n + 303\n4 n2\n1+88 t+151 t2+63 t3\n(1−t)2(1−t2)\n97\n93\n114\n76\n77\n66\n47\n0\n1/2 + 15/2 n + 21 n2\n1 + 15/2 n + 21 n2\n1+27 t+42 t2+14 t3\n(1−t)2(1−t2)\n88\n56\n113\n31\n99\n35\n7\n3\n1/2 + 11/2 n + 10 n2\n1 + 11/2 n + 10 n2\n1+14 t+20 t2+5 t3\n(1−t)2(1−t2)\n134\n82\n140\n76\n91\n72\n49\n4\n3/4 + 21 n + 669\n4 n2\n1 + 21 n + 669\n4 n2\n1+187 t+334 t2+147 t3\n(1−t)2(1−t2)\n133\n69\n149\n53\n98\n55\n43\n6\n1 + 6 n + 8 n2\n1 + 6 n + 8 n2\n15 t2+13 t+1+3 t3\n(1−t)3\n80\n63\n111\n32\n88\n38\n10\n7\n1\n1\n1+t\n1−t\n118\n69\n151\n36\n95\n63\n20\n9\n1 + 4 n + 4 n2\n1 + 4 n + 4 n2\n7 t2+7 t+1+t3\n(1−t)3\n96\n51\n103\n44\n90\n53\n3\n1\n1/2 + 39\n2 n + 36 n2\n1 + 39\n2 n + 36 n2\n1+54 t+72 t2+17 t3\n(1−t)2(1−t2)\n117\n72\n133\n56\n82\n57\n41\n9\n1 + 9 n + 18 n2\n1 + 9 n + 18 n2\n35 t2+26 t+1+10 t3\n(1−t)3\n72\n63\n77\n58\n49\n38\n28\n20\n1/2 + 7 n + 55\n2 n2\n1 + 7 n + 55\n2 n2\n1+33 t+55 t2+21 t3\n(1−t)2(1−t2)\n48\n37\n49\n36\n34\n24\n16\n11\n1/2 + 6 n + 37\n2 n2\n1 + 6 n + 37\n2 n2\n1+23 t+37 t2+13 t3\n(1−t)2(1−t2)\n108\n56\n113\n51\n73\n50\n29\n12\n1 + 4 n + 4 n2\n1 + 4 n + 4 n2\n7 t2+7 t+1+t3\n(1−t)3"},{"paragraph_id":"p108","order":108,"text":"Figure 6.4: The quasipolynomial ̃kπ\nλ,μ and the rational function Kπ\nλ,μ(t) for the Kronecker problem, n = 2.\n101"},{"paragraph_id":"p109","order":109,"text":"λ1\nλ2\nμ1\nμ2\nπ1\nπ2\nπ3\nπ4\n ̃kπ\nλ,μ(n); n odd\n ̃kπ\nλ,μ(n); n even\nKπ\nλ,μ(t)\n77\n40\n78\n39\n58\n29\n24\n6\n1 + 19/2 n + 57\n2 n2\n1 + 19/2 n + 57\n2 n2\n56 t2+37 t+1+20 t3\n(1−t)3\n153\n81\n157\n77\n96\n63\n61\n14\n1 + 3 n + 2 n2\n1 + 3 n + 2 n2\n3 t2+4 t+1\n(1−t)3\n90\n89\n102\n77\n90\n42\n30\n17\n1/2 + 13/2 n + 6 n2\n1 + 13/2 n + 6 n2\n1+11 t+12 t2\n(1−t)2(1−t2)\n145\n102\n160\n87\n96\n84\n39\n28\n1 + 10 n + 25 n2\n1 + 10 n + 25 n2\n49 t2+34 t+1+16 t3\n(1−t)3\n109\n95\n136\n68\n78\n60\n46\n20\n1 + 3 n + 2 n2\n1 + 3 n + 2 n2\n3 t2+4 t+1\n(1−t)3\n100\n42\n104\n38\n85\n27\n27\n3\n1 + 8 n\n1 + 8 n\n8 t+1+7 t2\n(1−t)2\n74\n51\n86\n39\n52\n34\n26\n13\n1\n1\n1+t\n1−t\n98\n90\n124\n64\n92\n67\n22\n7\n1/2 + 23/2 n + 60 n2\n1 + 23/2 n + 60 n2\n1+70 t+120 t2+49 t3\n(1−t)2(1−t2)\n57\n38\n75\n20\n52\n25\n17\n1\n1 + 3 n + 2 n2\n1 + 3 n + 2 n2\n3 t2+4 t+1\n(1−t)3\n159\n140\n170\n129\n89\n82\n73\n55\n1 + 3/2 n + 1/2 n2\n1 + 3/2 n + 1/2 n2\n1+t\n(1−t)3\n144\n122\n157\n109\n88\n86\n74\n18\n3/4 + n + 1/4 n2\n1 + n + 1/4 n2\n1\n(1−t)2(1−t2)\n90\n68\n92\n66\n88\n37\n23\n10\n1/4 + 12 n + 351\n4 n2\n1 + 12 n + 351\n4 n2\n1+98 t+176 t2+76 t3\n(1−t)2(1−t2)\n89\n42\n100\n31\n76\n28\n19\n8\n1 + 6 n + 8 n2\n1 + 6 n + 8 n2\n15 t2+13 t+1+3 t3\n(1−t)3\n88\n56\n107\n37\n71\n39\n20\n14\n1 + 9/2 n + 9/2 n2\n1 + 9/2 n + 9/2 n2\n8 t2+8 t+1+t3\n(1−t)3\n124\n111\n133\n102\n98\n89\n27\n21\n1/2 + 7 n + 53\n2 n2\n1 + 7 n + 53\n2 n2\n1+32 t+53 t2+20 t3\n(1−t)2(1−t2)"},{"paragraph_id":"p110","order":110,"text":"Figure 6.5: Continuation of Figure 6.4\n102"},{"paragraph_id":"p111","order":111,"text":"k\nλ\nhk,λ(n)\n3\n(21, 19)\n399 n3 + 35527969472513\n137438953472 n2 + 4329327034365\n137438953472 n + 1\n5\n(21, 19)\n3700378042361\n4194304\nn7 + 575575719967\n524288\nn6 + 2157156441\n4096\nn5 + 266554253\n2048\nn4 + 4643843\n256\nn3 + 1468423\n1024\nn2 + 7619\n128 n + 1\n3\n(21, 9, 6)\n270 n3 + 40819369181185\n274877906944 n2 + 3092376453119\n137438953472 n + 1\n3\n(12, 9, 5)\n42 n3 + 40132174413825\n1099511627776 n2 + 11544872091645\n1099511627776 n + 1\n3\n(21, 9, 6)\n27396522639355\n536870912\nn6 + 463063744509\n8388608\nn5 + 6265700353\n262144\nn4 + 5577375771\n1048576\nn3 + 84246529\n131072 n2 + 20971505\n524288 n + 1048573\n1048576\n3\n(21, 19, 16)\n15 n3 + 81363860455425\n4398046511104 n2 + 8246337208319\n1099511627776 n + 1\n4\n(9, 7, 5)\n7215545057279\n17179869184 n6 + 4183298146289\n4294967296\nn5 + 247765925897\n268435456\nn4 + 1914699777\n4194304\nn3 + 4160749567\n33554432 n2 + 587202553\n33554432 n + 67108863\n67108864\n4\n(21, 12, 9)\n16437913583613\n268435456\nn6 + 132498063359\n2097152\nn5 + 109509083155\n4194304\nn4 + 1462763527\n262144\nn3 + 171442179\n262144\nn2 + 10485755\n262144 n + 524287\n524288\n4\n(21, 9, 5)\n32469952757755\n536870912\nn6 + 129805320191\n2097152\nn5 + 108129157137\n4194304\nn4 + 2926313487\n524288\nn3 + 86638593\n131072 n2 + 10616825\n262144 n + 262143\n262144\n4\n(21, 9, 6)\n27396522639355\n536870912\nn6 + 463063744509\n8388608\nn5 + 6265700353\n262144\nn4 + 5577375771\n1048576\nn3 + 84246529\n131072 n2 + 20971505\n524288 n + 1048573\n1048576\n4\n(31, 19, 5)\n35969680015355\n33554432\nn6 + 1424674346311\n2097152\nn5 + 22705493343\n131072\nn4 + 46973953\n2048\nn3 + 3423915\n2048\nn2 + 65365\n1024 n + 16383\n16384\nFigure 6.6: Hilbert polynomial for G/Pλ, G = SLk(C). There is a slight rounding error caused by interpolation–\ne.g., the constant term of each polynomial should be one.\n103"},{"paragraph_id":"p112","order":112,"text":"n\nk\na\n7\n3\n(1, 3, 5)\n1/3 n3 + 59373627899905\n39582418599936 n2 + 28587302322173\n13194139533312 n + 1\n7\n3\n(1, 2, 4)\nn + 1\n7\n3\n(1, 4, 6)\n22265110462465\n534362651099136 n5 + 4638564679679\n11132555231232 n4 + 13\n8 n3 + 105942526633\n34359738368 n2 + 146028888073\n51539607552 n + 34359738361\n34359738368\n6\n2\n(1, 4, 5)\n15637498706143\n2251799813685248 n6 + 3665038759245\n35184372088832 n5 + 1389660529559\n2199023255552 n4 + 272014595421\n137438953472 n3\n+ 230973796809\n68719476736 n2 + 100215903571\n34359738368 n + 1\n6\n2\n(1, 4, 6)\n69578470195\n25048249270272 n7 + 1217623228439\n25048249270272 n6 + 372534725887\n1043677052928 n5 + 30953963537\n21743271936 n4 + 12044363351\n3623878656 n3\n+ 683671553\n150994944 n2 + 1335466297\n402653184 n + 268435457\n268435456\n7\n3\n(1, 4, 6)\n23456248059223\n562949953421312 n5 + 7330077518505\n17592186044416 n4 + 1786706395137\n1099511627776 n3 + 423770106525\n137438953472 n2 + 24338148015\n8589934592 n + 17179869169\n17179869184\n6\n3\n(1, 3, 6)\n1/8 n4 + 16126170540715\n17592186044416 n3 + 19\n8 n2 + 710101259605\n274877906944 n + 1\n8\n3\n(1, 3, 6)\n171798691840001\n1374389534720000 n4 + 31496426837333\n34359738368000 n3 + 4080218931199\n1717986918400 n2 + 443813287253\n171798691840 n + 1\nFigure 6.7: Hilbert polynomial of the Schubert subvariety Ωa, a = (a(1), . . . , a(k)), of the Grassmannian Gn,k.\n104"},{"paragraph_id":"p113","order":113,"text":"k = 2\n\"\n1/2 n + 1/2\n1/2 n + 1\n#\nk = 3"},{"paragraph_id":"p114","order":114,"text":"1/12 n2 + 1/2 n + 5\n12\n1/12 n2 + 1/2 n + 2/3\n1/12 n2 + 1/2 n + 3/4\n1/12 n2 + 1/2 n + 46912496118443\n70368744177664\n1/12 n2 + 1/2 n + 58640620148053\n140737488355328\n1/12 n2 + 1/2 n + 1"},{"paragraph_id":"p115","order":115,"text":"k = 4"},{"paragraph_id":"p116","order":116,"text":"1\n144 n3 + 5\n48 n2 + 61572651155457\n140737488355328 n + 15881834623431\n35184372088832\n1\n144 n3 + 117281240296107\n1125899906842624 n2 + 140737488355325\n281474976710656 n + 19\n36\n1\n144 n3 + 234562480592215\n2251799813685248 n2 + 123145302310909\n281474976710656 n + 19791209299969\n35184372088832\n1\n144 n3 + 234562480592215\n2251799813685248 n2 + 70368744177667\n140737488355328 n + 62549994824587\n70368744177664\n1\n144 n3 + 5\n48 n2 + 61572651155453\n140737488355328 n + 748278746681\n2199023255552\n1\n144 n3 + 117281240296107\n1125899906842624 n2 + 70368744177665\n140737488355328 n + 26388279066621\n35184372088832\n1\n144 n3 + 117281240296107\n1125899906842624 n2 + 7\n16 n + 7940917311717\n17592186044416\n1\n144 n3 + 117281240296107\n1125899906842624 n2 + 35184372088831\n70368744177664 n + 6841405683939\n8796093022208\n1\n144 n3 + 29320310074027\n281474976710656 n2 + 30786325577729\n70368744177664 n + 9\n16\n1\n144 n3 + 58640620148053\n562949953421312 n2 + 35184372088831\n70368744177664 n + 5619726097523\n8796093022208\n1\n144 n3 + 117281240296105\n1125899906842624 n2 + 7\n16 n + 2993114986727\n8796093022208\n1\n144 n3 + 58640620148055\n562949953421312 n2 + 1/2 n + 1"},{"paragraph_id":"p117","order":117,"text":"Figure 6.8: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk; k = 2, 3, 4.\n105"},{"paragraph_id":"p118","order":118,"text":"1\n2880 n4 +\n46912496118441\n4503599627370496 n3 + 3787206717893\n35184372088832 n2 + 469583091025\n1099511627776 n + 499743305817\n1099511627776\n1\n2880 n4 +\n46912496118445\n4503599627370496 n3 + 3787206717891\n35184372088832 n2 + 503942829399\n1099511627776 n + 310001195055\n549755813888\n1\n2880 n4 +\n46912496118441\n4503599627370496 n3 + 3787206717897\n35184372088832 n2 + 469583091031\n1099511627776 n + 30279519437\n68719476736\n400319966877379\n1152921504606846976 n4 +\n23456248059221\n2251799813685248 n3 + 3787206717893\n35184372088832 n2 + 503942829403\n1099511627776 n + 47340083975\n68719476736\n400319966877379\n1152921504606846976 n4 +\n5864062014805\n562949953421312 n3 + 3787206717895\n35184372088832 n2 + 117395772755\n274877906944 n + 89955703917\n137438953472\n400319966877379\n1152921504606846976 n4 +\n5864062014805\n562949953421312 n3 + 3787206717893\n35184372088832 n2 + 31496426837\n68719476736 n + 92771293595\n137438953472\n400319966877379\n1152921504606846976 n4 +\n46912496118447\n4503599627370496 n3 + 1893603358947\n17592186044416 n2 + 234791545515\n549755813888 n + 5661005505\n17179869184\n1\n2880 n4 +\n23456248059223\n2251799813685248 n3 + 1893603358949\n17592186044416 n2 + 62992853675\n137438953472 n + 94680167945\n137438953472\n400319966877379\n1152921504606846976 n4 +\n46912496118441\n4503599627370496 n3 + 3787206717891\n35184372088832 n2 + 117395772757\n274877906944 n + 38869454029\n68719476736\n1\n2880 n4 +\n46912496118441\n4503599627370496 n3 + 3787206717897\n35184372088832 n2 + 62992853675\n137438953472 n + 52494044729\n68719476736\n1\n2880 n4 +\n5864062014805\n562949953421312 n3 + 946801679473\n8796093022208 n2 + 234791545515\n549755813888 n + 2830502753\n8589934592\n400319966877379\n1152921504606846976 n4 +\n23456248059219\n2251799813685248 n3 + 1893603358949\n17592186044416 n2 + 251971414695\n549755813888 n + 27487790695\n34359738368\n400319966877379\n1152921504606846976 n4 +\n23456248059223\n2251799813685248 n3 + 1893603358945\n17592186044416 n2 + 234791545509\n549755813888 n + 31233956605\n68719476736\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 251971414699\n549755813888 n + 19375074691\n34359738368\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 29348943189\n68719476736 n + 11005853695\n17179869184\n400319966877379\n1152921504606846976 n4 +\n23456248059219\n2251799813685248 n3 + 473400839737\n4398046511104 n2 + 251971414699\n549755813888 n + 5917510497\n8589934592\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 234791545511\n549755813888 n + 15616978311\n34359738368\n1\n2880 n4 +\n23456248059223\n2251799813685248 n3 + 1893603358947\n17592186044416 n2 + 62992853673\n137438953472 n + 23192823403\n34359738368\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 946801679475\n8796093022208 n2 + 117395772757\n274877906944 n + 2830502755\n8589934592\n400319966877379\n1152921504606846976 n4 +\n11728124029609\n1125899906842624 n3 + 1893603358949\n17592186044416 n2 + 31496426837\n68719476736 n + 30541989663\n34359738368\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 117395772759\n274877906944 n + 9717363505\n17179869184\n1\n2880 n4 +\n2932031007403\n281474976710656 n3 + 1893603358945\n17592186044416 n2 + 125985707353\n274877906944 n + 4843768673\n8589934592\n400319966877379\n1152921504606846976 n4 +\n23456248059223\n2251799813685248 n3 + 59175104967\n549755813888 n2 + 117395772755\n274877906944 n + 2830502755\n8589934592\n400319966877379\n1152921504606846976 n4 +\n23456248059221\n2251799813685248 n3 + 946801679475\n8796093022208 n2 + 62992853675\n137438953472 n + 3435973837\n4294967296\n400319966877379\n1152921504606846976 n4 +\n23456248059223\n2251799813685248 n3 + 1893603358947\n17592186044416 n2 + 58697886379\n137438953472 n + 11244462985\n17179869184\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 946801679475\n8796093022208 n2 + 15748213419\n34359738368 n + 9687537337\n17179869184\n1\n2880 n4 +\n11728124029609\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 117395772753\n274877906944 n + 1892469965\n4294967296\n1\n2880 n4 +\n23456248059221\n2251799813685248 n3 + 946801679473\n8796093022208 n2 + 7874106709\n17179869184 n + 11835020991\n17179869184\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 117395772753\n274877906944 n + 3904244579\n8589934592\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 125985707349\n274877906944 n + 1879048193\n2147483648"},{"paragraph_id":"p119","order":119,"text":"Figure 6.9: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 5; the\nfirst 30 rows.\n106"},{"paragraph_id":"p120","order":120,"text":"1\n2880 n4 +\n11728124029609\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 58697886375\n137438953472 n + 88453211\n268435456\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 15748213419\n34359738368 n + 2958755247\n4294967296\n400319966877379\n1152921504606846976 n4 +\n23456248059223\n2251799813685248 n3 + 946801679475\n8796093022208 n2 + 58697886375\n137438953472 n + 4858681755\n8589934592\n400319966877379\n1152921504606846976 n4 +\n11728124029609\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 62992853673\n137438953472 n + 151367771\n268435456\n1\n2880 n4 +\n23456248059221\n2251799813685248 n3 + 946801679473\n8796093022208 n2 + 58697886375\n137438953472 n + 2274244835\n4294967296\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 236700419869\n2199023255552 n2 + 62992853671\n137438953472 n + 858993459\n1073741824\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 14674471595\n34359738368 n + 3904244571\n8589934592\n400319966877379\n1152921504606846976 n4 +\n23456248059219\n2251799813685248 n3 + 59175104967\n549755813888 n2 + 7874106709\n17179869184 n + 2421884337\n4294967296\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 59175104967\n549755813888 n2 + 29348943189\n68719476736 n + 3784939927\n8589934592\n400319966877379\n1152921504606846976 n4 +\n23456248059223\n2251799813685248 n3 + 473400839737\n4398046511104 n2 + 62992853673\n137438953472 n + 1908874355\n2147483648\n400319966877379\n1152921504606846976 n4 +\n5864062014805\n562949953421312 n3 + 946801679473\n8796093022208 n2 + 29348943189\n68719476736 n + 1952122291\n4294967296\n400319966877379\n1152921504606846976 n4 +\n11728124029609\n1125899906842624 n3 + 946801679471\n8796093022208 n2 + 62992853675\n137438953472 n + 2899102929\n4294967296\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 473400839735\n4398046511104 n2 + 58697886381\n137438953472 n + 707625689\n2147483648\n400319966877379\n1152921504606846976 n4 +\n11728124029613\n1125899906842624 n3 + 59175104967\n549755813888 n2 + 15748213419\n34359738368 n + 2958755253\n4294967296\n1\n2880 n4 +\n11728124029609\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 58697886377\n137438953472 n + 3288334339\n4294967296\n100079991719345\n288230376151711744 n4 +\n5864062014805\n562949953421312 n3 + 59175104967\n549755813888 n2 + 62992853675\n137438953472 n + 1210942169\n2147483648\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 946801679477\n8796093022208 n2 + 29348943193\n68719476736 n + 1415251375\n4294967296\n1\n2880 n4 +\n5864062014805\n562949953421312 n3 + 59175104967\n549755813888 n2 + 31496426839\n68719476736 n + 3435973835\n4294967296\n100079991719345\n288230376151711744 n4 +\n1466015503701\n140737488355328 n3 + 473400839737\n4398046511104 n2 + 29348943195\n68719476736 n + 976061147\n2147483648\n100079991719345\n288230376151711744 n4 +\n11728124029611\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 31496426839\n68719476736 n + 3280877793\n4294967296\n1\n2880 n4 +\n5864062014805\n562949953421312 n3 + 946801679473\n8796093022208 n2 + 29348943191\n68719476736 n + 946234983\n2147483648\n100079991719345\n288230376151711744 n4 +\n2932031007403\n281474976710656 n3 + 946801679475\n8796093022208 n2 + 15748213419\n34359738368 n + 1479377623\n2147483648\n1\n2880 n4 +\n5864062014805\n562949953421312 n3 + 59175104967\n549755813888 n2 + 29348943195\n68719476736 n + 122007643\n268435456\n100079991719345\n288230376151711744 n4 +\n2932031007403\n281474976710656 n3 + 473400839737\n4398046511104 n2 + 3937053355\n8589934592 n + 1449551461\n2147483648\n100079991719345\n288230376151711744 n4 +\n2932031007403\n281474976710656 n3 + 236700419869\n2199023255552 n2 + 29348943193\n68719476736 n + 71070151\n134217728\n1\n2880 n4 +\n1466015503701\n140737488355328 n3 + 59175104967\n549755813888 n2 + 15748213419\n34359738368 n + 739688813\n1073741824\n100079991719345\n288230376151711744 n4 +\n11728124029611\n1125899906842624 n3 + 473400839739\n4398046511104 n2 + 14674471597\n34359738368 n + 607335219\n1073741824\n1\n2880 n4 +\n2932031007403\n281474976710656 n3 + 236700419869\n2199023255552 n2 + 3937053355\n8589934592 n + 605471085\n1073741824\n100079991719345\n288230376151711744 n4 +\n11728124029611\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 29348943193\n68719476736 n + 88453211\n268435456\n100079991719345\n288230376151711744 n4 +\n1466015503701\n140737488355328 n3 + 473400839737\n4398046511104 n2 + 3937053355\n8589934592 n + 2147483647\n2147483648"},{"paragraph_id":"p121","order":121,"text":"Figure 6.10: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 5;\nthe last 30 rows.\n107"},{"paragraph_id":"p122","order":122,"text":"53375995583651\n4611686018427387904 n5 +\n21892498188609\n36028797018963968 n4 +\n418085902907\n35184372088832 n3 + 115486898397\n1099511627776 n2 + 26847522421\n68719476736 n + 8448724291\n17179869184\n53375995583651\n4611686018427387904 n5 +\n10946249094305\n18014398509481984 n4 +\n836171805815\n70368744177664 n3 + 7396888121\n68719476736 n2 + 15302809397\n34359738368 n + 9853503363\n17179869184\n106751991167299\n9223372036854775808 n5 +\n21892498188599\n36028797018963968 n4 +\n1672343611625\n140737488355328 n3 + 57743449207\n549755813888 n2 + 14060052663\n34359738368 n + 975427339\n2147483648\n13343998895913\n1152921504606846976 n5 +\n10946249094301\n18014398509481984 n4 +\n1672343611641\n140737488355328 n3 + 59175104961\n549755813888 n2 + 15302809423\n34359738368 n + 610309549\n1073741824\n106751991167305\n9223372036854775808 n5 +\n21892498188611\n36028797018963968 n4 +\n1672343611623\n140737488355328 n3 + 115486898411\n1099511627776 n2 + 13423761203\n34359738368 n + 4461910959\n8589934592\n13343998895913\n1152921504606846976 n5 +\n21892498188605\n36028797018963968 n4 +\n1672343611631\n140737488355328 n3 + 29587552483\n274877906944 n2 + 15939100865\n34359738368 n + 1927366573\n2147483648\n213503982334599\n18446744073709551616 n5 +\n10946249094305\n18014398509481984 n4 +\n418085902909\n35184372088832 n3 + 57743449199\n549755813888 n2 + 13423761217\n34359738368 n + 835973461\n2147483648\n106751991167299\n9223372036854775808 n5 +\n10946249094305\n18014398509481984 n4 +\n836171805821\n70368744177664 n3 + 59175104963\n549755813888 n2 + 7651404713\n17179869184 n + 320001575\n536870912\n106751991167299\n9223372036854775808 n5 +\n10946249094299\n18014398509481984 n4 +\n1672343611637\n140737488355328 n3 + 115486898395\n1099511627776 n2 + 14060052667\n34359738368 n + 255936429\n536870912\n213503982334597\n18446744073709551616 n5 +\n2736562273577\n4503599627370496 n4 +\n1672343611629\n140737488355328 n3 + 7396888121\n68719476736 n2 + 7651404703\n17179869184 n + 2859997491\n4294967296\n106751991167299\n9223372036854775808 n5 +\n342070284197\n562949953421312 n4 + 104521475727\n8796093022208 n3 + 57743449199\n549755813888 n2 + 1677970153\n4294967296 n + 1790721319\n4294967296\n1\n86400 n5 +\n10946249094299\n18014398509481984 n4 +\n836171805815\n70368744177664 n3 + 59175104959\n549755813888 n2 + 3984775219\n8589934592 n + 987842475\n1073741824\n1\n86400 n5 +\n5473124547153\n9007199254740992 n4 +\n209042951453\n17592186044416 n3 + 57743449193\n549755813888 n2 + 3355940307\n8589934592 n + 884291849\n2147483648\n106751991167303\n9223372036854775808 n5 +\n10946249094301\n18014398509481984 n4 +\n836171805817\n70368744177664 n3 + 118350209919\n1099511627776 n2 + 1912851173\n4294967296 n + 264972307\n536870912\n213503982334605\n18446744073709551616 n5 +\n2736562273577\n4503599627370496 n4 +\n836171805827\n70368744177664 n3 + 57743449201\n549755813888 n2 + 7030026327\n17179869184 n + 1233125369\n2147483648\n53375995583651\n4611686018427387904 n5 +\n10946249094299\n18014398509481984 n4 +\n836171805819\n70368744177664 n3 + 118350209933\n1099511627776 n2 + 1912851177\n4294967296 n + 1478317113\n2147483648\n106751991167297\n9223372036854775808 n5 +\n1368281136787\n2251799813685248 n4 +\n836171805817\n70368744177664 n3 + 57743449195\n549755813888 n2 + 1677970151\n4294967296 n + 117959881\n268435456\n1667999861989\n144115188075855872 n5 +\n10946249094303\n18014398509481984 n4 + 104521475727\n8796093022208 n3 + 59175104961\n549755813888 n2 + 3984775215\n8589934592 n + 109722991\n134217728\n106751991167297\n9223372036854775808 n5 +\n342070284197\n562949953421312 n4 +\n836171805815\n70368744177664 n3 + 28871724597\n274877906944 n2 + 1677970153\n4294967296 n + 332087389\n1073741824\n213503982334607\n18446744073709551616 n5 +\n10946249094307\n18014398509481984 n4 +\n836171805821\n70368744177664 n3 + 59175104963\n549755813888 n2 + 7651404709\n17179869184 n + 384426083\n536870912"},{"paragraph_id":"p123","order":123,"text":"Figure 6.11: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 6; the first 20 rows.\n108"},{"paragraph_id":"p124","order":124,"text":"213503982334615\n18446744073709551616 n5 +\n10946249094307\n18014398509481984 n4 +\n836171805813\n70368744177664 n3 + 28871724597\n274877906944 n2 + 7030026337\n17179869184 n + 640721865\n1073741824\n106751991167309\n9223372036854775808 n5 +\n10946249094297\n18014398509481984 n4 +\n209042951455\n17592186044416 n3 + 29587552479\n274877906944 n2 + 1912851179\n4294967296 n + 157275009\n268435456\n26687997791827\n2305843009213693952 n5 +\n342070284197\n562949953421312 n4 +\n836171805825\n70368744177664 n3 + 57743449199\n549755813888 n2 + 3355940301\n8589934592 n + 90445245\n268435456\n13343998895913\n1152921504606846976 n5 +\n5473124547153\n9007199254740992 n4 + 104521475729\n8796093022208 n3 + 29587552481\n274877906944 n2 + 1992387605\n4294967296 n + 450971557\n536870912\n106751991167303\n9223372036854775808 n5 +\n10946249094307\n18014398509481984 n4 +\n836171805827\n70368744177664 n3 + 28871724597\n274877906944 n2 + 209746269\n536870912 n + 17843591\n33554432\n106751991167303\n9223372036854775808 n5 +\n10946249094297\n18014398509481984 n4 +\n836171805819\n70368744177664 n3 + 924611015\n8589934592 n2 + 239106397\n536870912 n + 5146825\n8388608\n1667999861989\n144115188075855872 n5 +\n5473124547153\n9007199254740992 n4 +\n418085902911\n35184372088832 n3 + 7217931151\n68719476736 n2 + 54922081\n134217728 n + 265331665\n536870912\n1667999861989\n144115188075855872 n5 +\n10946249094291\n18014398509481984 n4 +\n836171805825\n70368744177664 n3 + 59175104963\n549755813888 n2 + 478212795\n1073741824 n + 326629601\n536870912\n1667999861989\n144115188075855872 n5 +\n10946249094297\n18014398509481984 n4 +\n418085902909\n35184372088832 n3 + 14435862303\n137438953472 n2 + 3355940305\n8589934592 n + 24121261\n67108864\n53375995583655\n4611686018427387904 n5 +\n10946249094291\n18014398509481984 n4 +\n209042951453\n17592186044416 n3 + 59175104973\n549755813888 n2 + 3984775217\n8589934592 n + 503316475\n536870912\n26687997791827\n2305843009213693952 n5 +\n10946249094285\n18014398509481984 n4 +\n836171805825\n70368744177664 n3 + 57743449207\n549755813888 n2 + 3355940305\n8589934592 n + 115234099\n268435456\n106751991167303\n9223372036854775808 n5 +\n2736562273573\n4503599627370496 n4 +\n209042951459\n17592186044416 n3 + 7396888121\n68719476736 n2 + 3825702363\n8589934592 n + 42684549\n67108864\n106751991167301\n9223372036854775808 n5 +\n10946249094305\n18014398509481984 n4 +\n209042951453\n17592186044416 n3 + 28871724605\n274877906944 n2 + 1757506587\n4294967296 n + 277411267\n536870912\n106751991167303\n9223372036854775808 n5 +\n10946249094299\n18014398509481984 n4 +\n418085902905\n35184372088832 n3 + 924611015\n8589934592 n2 + 3825702353\n8589934592 n + 33950041\n67108864\n106751991167307\n9223372036854775808 n5 +\n1368281136787\n2251799813685248 n4 +\n52260737865\n4398046511104 n3 + 28871724601\n274877906944 n2 + 209746269\n536870912 n + 61328751\n134217728\n53375995583651\n4611686018427387904 n5 +\n2736562273575\n4503599627370496 n4 + 104521475727\n8796093022208 n3 + 29587552487\n274877906944 n2 + 249048451\n536870912 n + 128849017\n134217728\n106751991167301\n9223372036854775808 n5 +\n5473124547145\n9007199254740992 n4 +\n209042951459\n17592186044416 n3 + 7217931151\n68719476736 n2 + 1677970157\n4294967296 n + 30318473\n67108864\n106751991167303\n9223372036854775808 n5 +\n342070284197\n562949953421312 n4 +\n418085902919\n35184372088832 n3 + 14793776243\n137438953472 n2 + 1912851181\n4294967296 n + 143223581\n268435456\n53375995583651\n4611686018427387904 n5 +\n5473124547153\n9007199254740992 n4 +\n209042951459\n17592186044416 n3 + 1804482787\n17179869184 n2 + 878753295\n2147483648 n + 13898875\n33554432\n106751991167303\n9223372036854775808 n5 +\n5473124547147\n9007199254740992 n4 +\n418085902907\n35184372088832 n3 + 29587552479\n274877906944 n2 + 956425585\n2147483648 n + 195527051\n268435456"},{"paragraph_id":"p125","order":125,"text":"Figure 6.12: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 6; the middle 20 rows.\n109"},{"paragraph_id":"p126","order":126,"text":"13343998895913\n1152921504606846976 n5 +\n5473124547145\n9007199254740992 n4 +\n418085902919\n35184372088832 n3 + 7217931149\n68719476736 n2 + 209746269\n536870912 n + 64348647\n134217728\n53375995583651\n4611686018427387904 n5 +\n5473124547143\n9007199254740992 n4 + 104521475729\n8796093022208 n3 + 14793776237\n137438953472 n2 + 498096901\n1073741824 n + 28772927\n33554432\n106751991167301\n9223372036854775808 n5 +\n5473124547141\n9007199254740992 n4 +\n209042951457\n17592186044416 n3 + 3608965575\n34359738368 n2 + 1677970153\n4294967296 n + 11719905\n33554432\n13343998895913\n1152921504606846976 n5 +\n5473124547151\n9007199254740992 n4 +\n418085902905\n35184372088832 n3 + 3698444059\n34359738368 n2 + 478212797\n1073741824 n + 74631683\n134217728\n106751991167311\n9223372036854775808 n5 +\n5473124547147\n9007199254740992 n4 +\n418085902907\n35184372088832 n3 + 28871724591\n274877906944 n2 + 878753299\n2147483648 n + 10682367\n16777216\n106751991167313\n9223372036854775808 n5 +\n5473124547151\n9007199254740992 n4 + 104521475729\n8796093022208 n3 + 14793776237\n137438953472 n2 + 478212797\n1073741824 n + 84006215\n134217728\n106751991167307\n9223372036854775808 n5 +\n2736562273573\n4503599627370496 n4 +\n209042951459\n17592186044416 n3 + 28871724589\n274877906944 n2 + 104873135\n268435456 n + 3161957\n8388608\n53375995583655\n4611686018427387904 n5 +\n5473124547149\n9007199254740992 n4 +\n209042951451\n17592186044416 n3 + 29587552485\n274877906944 n2 + 996193803\n2147483648 n + 118111589\n134217728\n106751991167309\n9223372036854775808 n5 +\n1368281136787\n2251799813685248 n4 +\n418085902907\n35184372088832 n3 + 28871724601\n274877906944 n2 + 419492539\n1073741824 n + 49899533\n134217728\n106751991167305\n9223372036854775808 n5 +\n2736562273573\n4503599627370496 n4 +\n52260737863\n4398046511104 n3 + 29587552479\n274877906944 n2 + 1912851173\n4294967296 n + 87717907\n134217728\n106751991167309\n9223372036854775808 n5 +\n1368281136787\n2251799813685248 n4 + 104521475729\n8796093022208 n3 + 28871724595\n274877906944 n2 + 878753303\n2147483648 n + 8962701\n16777216\n106751991167305\n9223372036854775808 n5 +\n2736562273571\n4503599627370496 n4 +\n209042951453\n17592186044416 n3 + 29587552499\n274877906944 n2 + 956425585\n2147483648 n + 10878263\n16777216\n53375995583651\n4611686018427387904 n5 +\n5473124547157\n9007199254740992 n4 + 104521475727\n8796093022208 n3 + 3608965575\n34359738368 n2 + 838985083\n2147483648 n + 53611225\n134217728\n13343998895913\n1152921504606846976 n5 +\n5473124547145\n9007199254740992 n4 +\n418085902907\n35184372088832 n3 + 14793776249\n137438953472 n2 + 498096903\n1073741824 n + 104354293\n134217728\n106751991167299\n9223372036854775808 n5 +\n2736562273575\n4503599627370496 n4 +\n209042951447\n17592186044416 n3 + 3608965575\n34359738368 n2 + 838985087\n2147483648 n + 7873221\n16777216\n106751991167303\n9223372036854775808 n5 +\n5473124547151\n9007199254740992 n4 +\n418085902899\n35184372088832 n3 + 14793776243\n137438953472 n2 + 956425599\n2147483648 n + 90737805\n134217728\n53375995583655\n4611686018427387904 n5 +\n5473124547155\n9007199254740992 n4 + 104521475727\n8796093022208 n3 + 14435862303\n137438953472 n2 + 878753295\n2147483648 n + 18680385\n33554432\n106751991167305\n9223372036854775808 n5 +\n2736562273573\n4503599627370496 n4 + 104521475725\n8796093022208 n3 + 3698444063\n34359738368 n2 + 478212799\n1073741824 n + 36634403\n67108864\n106751991167311\n9223372036854775808 n5 +\n684140568395\n1125899906842624 n4 +\n209042951463\n17592186044416 n3 + 7217931153\n68719476736 n2 + 52436567\n134217728 n + 19926951\n67108864\n26687997791827\n2305843009213693952 n5 +\n1368281136789\n2251799813685248 n4 +\n6532592233\n549755813888 n3 + 14793776243\n137438953472 n2 + 498096903\n1073741824 n + 67108871\n67108864"},{"paragraph_id":"p127","order":127,"text":"Figure 6.13: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 6; the last 20 rows.\n110"},{"paragraph_id":"p128","order":128,"text":"Chapter 7\nOn verification and discovery\nof obstructions\nIn this chapter we give applications of the results and positivity hypotheses\nin this paper to the problem of verifying or discovering an obstruction, i.e.,\na “proof of hardness” [GCT2] in the context of the P vs.\nNP and the\npermanent vs. determinant problems in characteristic zero.\n7.1\nObstruction\nAn obstruction in an abstract setting of Problem 1.1.4 is defined as follows.\nLet X and Y be H-varieties with compact specifications (Section 3.5),\nH a connected reductive group. Let ⟨X⟩and ⟨Y ⟩denote the bit lengths of\ntheir specifications (Section 3.5). Suppose we wish to show that X cannot\nbe embedded as an H-subvariety of Y . Pictorially:\nX ̸֒→Y.\n(7.1)\nFor example, in the context of the P vs. NP problem in characteristic\nzero [GCT1, GCT2], X is a class variety XNP (n, l) associated with the\ncomplexity class NP for the given input size parameter n and the circuit\nsize parameter l. The variety Y is the class variety XP (l) associated with\nthe class P for given l. And H is SLl(C). If NP ⊆P (over C) to the\ncontrary, then it would turn out that\nXNP (n, l) ֒→XP (l)\n111"},{"paragraph_id":"p129","order":129,"text":"as an H-subvariety, for every l = poly(n). The goal is to show that this\nembedding cannot exist when l = poly(n) and n →∞.\nLet R(X) and R(Y ) be the homogeneous coordinate rings of X and\nY , respectively. Let R(X)d and R(Y )d denote their degree d-compoenents.\nSuppose to the contrary that an H-embedding as in (7.1) exists. Then there\nexists a degre preserving H-equivariant surjection from R(Y )d to R(X)d for\nevery d, and hence, a degree-preserving H-equivariant injection from R(X)∗\nd\nto R(Y )∗\nd. Hence, every irreducible H-module Vλ(G) that occurs in R(X)∗\nd\nalso occurs in R(Y )∗\nd. This leads to:\nDefinition 7.1.1 An irreducible representation Vλ(H) is called an obstruc-\ntion for the pair (X,Y) if it occurs (as an H-submodule) in R(X)∗\nd but not\nin R(Y )∗\nd, for some d. We say that Vλ(H) is an obstruction of degree d.\nRemark 7.1.2 The obstruction as defined here is dual to the obstrution as\ndefined in [GCT2].\nExistence of such an obstruction implies that X cannot be embedded in\nY as an H-subvariety.\nLet us assume that X and Y are H-subvarieties of P(V ), where V is an\nH-module, and that we are given a point y ∈Y ⊆P(V ) that is distiguished\nin the following sense. Let Hy ⊆H be the stabilizer of y. Then Cy, the line\nin V corresponding to y, is invariant under Hy. Let [y] be the set of points\nin P(V ) stabilized by Hy. We say that y is characterized by its stabilizer Hy\nif y = [y]; i.e., y is the only point in P(V ) stabilized by Hy. Let\nH[y] = ∪z∈[y]Hz\nbe the union of the H-orbits of all points in [y]. We say that y is a distin-\nguished point of Y if Y equals the projective closure of H[y] in P(V ). If y\nis characterized by its stabilizer, this means Y is the projective closure of\nthe orbit Hy of y.\nIf Vλ(H) occurs in R(Y )∗\nd, then it can be shown (cf. Proposition 4.2 in\n[GCT2]) that Vλ(H) contains an Hy-submodule isomorphic to (Cy)d, the\nd-th tensor power of Cy.\nThis leads to the following stronger notion of\nobstruction:\nDefinition 7.1.3 [GCT2] We say that Vλ(H) is a strong obstruction for\nthe pair (X, Y ) if, for some d, it occurs in R(X)∗\nd, but it does not contain\nan Hy-module isomorphic to (Cy)d.\n112"},{"paragraph_id":"p130","order":130,"text":"Existence of a strong obstruction also implies that X cannot be embedded\nin Y as an H-subvariety. The results in [GCT2] suggest that strong obstruc-\ntions exist in the context of the lower bound problems under cosideration.\nThe goal then is to show their existence.\n7.2\nDecision problems\nThe decisions problems that arise in this context are the following. Let sλ\nd\nbe the multiplicity of Vλ(H) in R(X)∗\nd, and md\nλ the multiplicity of the Hy-\nmodule (Cy)d in Vλ(H), considered an Hy-module via the the embedding\nρ : Hy ֒→H. Thus λ is a strong obstruction of degree d iffsλ\nd is nonzero\nand md\nλ is zero.\nProblem 7.2.1 (Decision Problems)\n(a) Given d, λ and the specification of X, decide if sλ\nd is nonzero.\n(b) Given d, λ and the specifications of H, Hy and ρ, decide if md\nλ is nonzero.\n(c) Given d, λ and the specifications of X, H, Hy and ρ, decide if λ is a\nstrong obstruction of degree d.\nThe first is an instance of the decision Problem 1.1.4 in geometric in-\nvariant theory, and the second of the subgroup restriction Problem 1.1.3.\nBy the results in Chapter 3-4, relaxed forms of the decision problems in (a)\nand (b) belong to P assuming appropriate PH1 and SH; this implies that\na relaxed form of the decision problem in (c) also belongs to P assumming\nPH1 and SH. We will only need a weak relaxed form of (c), for which the\nweak form of SH that is implied by PH1 (cf. Theorem 3.3.5) will suffice.\n7.3\nVerification of obstructions\nThe relevant PH1 are as follows.\nAssume that that singularities of spec(R(X)) are rational.\nBy Theo-\nrem 3.5.1, the stretching function ̃sλ\nd(k) = skλ\nkd is a quasipolynomial. Hence\nPH1 for sλ\nd (Hypothesis 3.3.1, or rather its slight variant obtained by replac-\ning R(X)d with R(X)∗\nd) is well defined. It is:\nHypothesis 7.3.1 (PH1):\nThere exists a polytope P λ\nd such that:\n113"},{"paragraph_id":"p131","order":131,"text":"1. The number of integer points in P λ\nd is equal to sλ\nd.\n2. The Ehrhart quasi-polynomial of P λ\nd coincides with the stretching quasi-\npolynomial ̃sλ\nd(n) (cf. Theorem 3.5.1).\n3. The polytope P λ\nd is given by a separating oracle, as in Section 2.3. Its\nencoding bitlength ⟨P λ\nd ⟩is poly(⟨d⟩, ⟨λ⟩, ⟨X⟩), and the combinatorial\nsize ∥P λ\nd ∥is poly(ht(λ), ∥X∥), where ∥X∥is the combinatorial size of\nX (Section 3.5), and ht(λ) is the height of λ.\nSimilarly by Theorem 3.4.1 (or rather its slight variant which can be\nproved similarly), the stretching function mkd\nkλ is a quasipolynomial. Hence\nPH1 for md\nλ (cf. Hypothesis 3.3.1 and Section 3.4) is also well defined. It is:\nHypothesis 7.3.2 (PH1:)\nThere exists a polytope Qd\nλ such that:\n1. The number of integer points in Qd\nλ is equal to md\nλ.\n2. The Ehrhart quasi-polynomial of Qd\nλ coincides with the stretching quasi-\npolynomial ̃md\nλ(n) (Theorem 3.5.1).\n3. The polytope Qd\nλ is given by a separating oracle. Its encoding bitlength\n⟨Qd\nλ⟩is poly(⟨d⟩, ⟨λ⟩, ⟨ρ⟩, ⟨Hy⟩, ⟨H⟩), and the combinatorial size ∥Qd\nλ∥\nis O(poly(ht(λ), ⟨H⟩, ⟨Hy⟩, ⟨ρ⟩)). Here ⟨H⟩, ⟨Hy⟩and ⟨ρ⟩denote the\nbitlengths of H, Hy and ρ (Section 3.4).\nTheorem 7.3.3 (Weak SH:)\n(a) Assuming PH1 (Hypothesis 7.3.1), the saturation index of ̃sλ\nd(n) is at\nmost apoly(∥P λ\nd ∥), for some explicit constant a > 0.\n(b) Assuming PH1 (Hypothesis 7.3.2), the saturation index of ̃md\nλ(n) is at\nmost bpoly(∥Qd\nλ∥), for some explicit constant b > 0.\nThis follows from Theorem 3.3.5.\nTheorem 7.3.4 Assume PH1 (Hypotheses 7.3.1-7.3.2). Then, given d, λ,\nthe specifications of X, H, Hy and ρ, and a relaxation parameter c greater\nthan the explicit bounds on the saturation indices in Theorem 7.3.3, whether\ncλ is an obstruction of degree d can be decided in\npoly(⟨d⟩, ⟨λ⟩, ⟨X⟩, ⟨H⟩, ⟨Hy⟩, ⟨ρ⟩, ⟨c⟩)\ntime.\n114"},{"paragraph_id":"p132","order":132,"text":"This follows by applying Theorem 3.1.1 to the polytopes P λ\nd and Qd\nλ with\nthe saturation index estimates in Theorem 7.3.3.\n7.4\nRobust obstruction\nWe now define a notion of obstruction that is well behaved with respect to\nrelaxation.\nDefinition 7.4.1 Assume PH1 for both sλ\nd and md\nλ (Hypotheses 7.3.1-7.3.2).\nWe say that Vλ(H) is a robust obstruction for the pair (X, Y ) if one of the\nfollowing hold:\n1. Qd\nλ is empty, and P λ\nd is nonempty.\n2. Both Qdλ and P λ\nd are nonempty, the affine span of Qd\nλ does not contain\nan integer point and the affine span of P λ\nd contains an integer point.\nIf Vλ(H) is a robust obstruction, so is Vlλ(H), for all or most positive\nintegral l, hence the name robust.\nProposition 7.4.2 Assume PH1 for both sλ\nd and md\nλ as above. If Vλ(H)\nis a robust obstruction for the pair (X, Y ), then for some positive integer\nk–called a relaxation parameter–Vkλ(H) is a strong obstruction for (X, Y ).\nIn fact, this is so for most large enough k.\nProof:\n(1) Suppose Qd\nλ is empty, and P λ\nd is nonempty. Let k be a large enough\npositive integer k such that kP λ\nd = P kλ\ndk contains an integer point. Then skλ\nkd\nis nonzero. But mkd\nkλ is zero since Qkd\nkλ = kQd\nλ is empty. Thus kλ is a strong\nobstruction.\n(2) Suppose both Qdλ and P λ\nd are nonempty, the affine span of Qd\nλ does not\ncontain an integer point and the affine span of P λ\nd contains an integer point.\nWe can choose a positive integer k such that the affine span of kQd\nλ = Qdk\ndλ\ndoes not contain an integer point, but kP λ\nd = P kλ\ndk contains an integer point;\nmost large enough k have this property. This means skλ\nkd is nonzero, but mkd\nkλ\nis zero. Thus kλ is a strong obstruction. Q.E.D.\n115"},{"paragraph_id":"p133","order":133,"text":"7.5\nVerification of robust obstructions\nTheorem 7.5.1 Assume that the singularities of spec(R(X)) are rational.\nAssume PH1 for both sλ\nd and md\nλ as above. Then, given λ, d and the speci-\nfications of ρ : Hy ֒→H and X, whether Vλ(H) is a robust obstruction can\nbe verified in poly(⟨ρ⟩, ⟨Hy⟩, ⟨H⟩, ⟨X⟩, ⟨d⟩, ⟨λ⟩) time. Furthermore, a positive\nintegral relaxation parameter k such that Vkλ(G) is a strong obstruction can\nalso be found in the same time.\nThe crucial result used implicitly here is the quasipolynomiality theorem\n(Theorem 4.1.1) because of which PH1 for both sλ\nd and md\nλ are well defined.\nProof: By linear programming [GLS], whether Qd\nλ is nonempty or not can\nbe determined in poly(⟨Qd\nλ⟩) = poly(⟨ρ⟩, ⟨Hy⟩, ⟨H⟩, ⟨d⟩, ⟨λ⟩) time.\nIf it\nis nonempty, the linear programming algorithm also gives its affine span.\nWhether this contains an integer point can be determined in polynomial\ntime, using the polynomial time algorithm for computing the Smith normal\nform, as in the proof of Theorem 3.1.1.\nSimilarly, whether P λ\nd is nonempty or not can be determined in poly(⟨P λ\nd ⟩) =\npoly(⟨X⟩, ⟨d⟩, ⟨λ⟩) time. If it is nonempty, whether its affine span contains\nan integer point can be determined in polynomial time similarly. Further-\nmore, the algorithm can also be made to return a vertex v of the polytope\nP λ\nd if it is nonempty.\nUsing these observations, whether Vλ(G) is a robust obstruction can be\ndetermined in polynomial time.\nAs far as the computation of the relaxation parameter k is concerned,\nlet us consider the second case in Definition 7.4.1–when both Qd\nλ and P λ\nd are\nnonempty, the affine span of Qd\nλ does not contain an integer point and the\naffine span of P λ\nd contains an integer point–the first case being simpler. In\nthis case, by examining the Smith normal forms of the defining equations\nof the affine spans of P λ\nd and Qd\nλ and the rational coordinates of a vertex\nv ∈P λ\nd , we can find a large enough k so that the affine span of Qkd\nkλ does not\ncontain an integer point, the affine span of P kλ\nkd contains an integer point,\nand P kλ\nkd contains an integer point that is some multiple of v. Q.E.D.\nThe value of the relaxation parameter k computed above is rather con-\nservative.\nOne may wish to compute as small value of k as possible for\nwhich Vkλ(G) is a strong obstruction (though in our application this is not\nnecessary).\nIf SH for holds for the structural constant sλ\nd (cf.\nHypothe-\nsis 3.3.2 and Section 3.5), then we can let k be the smallest integer larger\n116"},{"paragraph_id":"p134","order":134,"text":"than the saturation index (estimate) for P λ\nd such that affine span of Qkd\nkλ (if\nnonempty) does not contain an integer point (as can be ensured by looking\nat the Smith normal of the defining equations of the affine span).\n7.6\nArithemetic version of the P #P vs. NC prob-\nlem in characteristric zero\nWe now specialize the discussion in the preceding sections in the context\nof the arithmetic form of the P #P vs. NC problem in characteristric zero\n[V]. In concrete terms, the problem is to show that the permanent of an\nn × n complex matrix X cannot be expressed as a determinant of an m × m\ncomplex matrix, whose entries are (possibly nonhomogeneous) linear com-\nbinations of the entries of X.\n7.6.1\nClass varieties\nThe class varieties in this context are as follows [GCT1]. Let Y be an m×m\nvariable matrix, which can also be thought of as a variable l-vector, l = m2.\nLet X be its, say, principal bottom-right n × n submatrix, n < m, which\ncan be thought of as a variable k-vector, k = n2. Let V = Symm(Y ) be the\nspace of homogeneous forms of degree m in the variable entries of Y . The\nspace V , and hence P(V ), has a natural action of G = GL(Y ) = GLl(C)\ngiven by\n(σf)(Y ) = f(σ−1Y ),\nfor any f ∈V , σ ∈G, and thinking of Y as an l-vector. Let W = Symn(X)\nbe the space of homogeneous forms of degree n in the variable entries of\nX. The space W, and also P(W), has a similar action of K = GL(X) =\nGLk(C). We use any entry y of Y not in X as the homogenizing variable\nfor embedding W in V via the map φ : W →V defined by:\nφ(h)(Y ) = ym−nh(X),\n(7.2)\nfor any h(X) ∈W. We also think of φ as a map from P(W) to P(V ).\nLet g = det(Y ) ∈P(V ) be the determinant form, and f = φ(h), where\nh = perm(X) ∈P(W). Let ∆V [g], ∆V [f] ⊆P(V ) be the projective closures\nof the orbits Gg and Gf, respectively, in P(V ). Let ∆W [h] ⊆P(W) be the\nprojective closure of the K-orbit Kh of h in P(W). Then ∆V [g] is called the\nclass variety associated with NC and ∆V [f] the class variety associated with\n117"},{"paragraph_id":"p135","order":135,"text":"P #P ; ∆W[h] is called the base class variety associated with P #P . (The base\nclass variety is not used in what follows. Rather its variant, called a reduced\nclass variety defined below, will be used.) These class varieties depend on\nthe lower bound parameters n and m. If we wish to make these explicit, we\nwould write ∆V [f, n, m] and ∆V [g, m] instead of ∆V [f] and ∆V [g].\nThe class varieties ∆V [g] = ∆V [g, m] and ∆V [f] = ∆V [f, n, m] are\nG-subvarieties of P(V ), and their homogeneous coordinate rings RV [g] =\nRV [g, m] and RV [f] = RV [f, n, m] have natural degree-preserving G-action.\nIt is conjectured in [GCT1] that, if m = poly(n) and n →∞, then\nf ̸∈∆V [g]; this is equivalent to saying that the class variety ∆V [f, n, m]\ncannot be embedded in the class variety ∆V [g, m] (as a subvariety). This\nimplies the arithmetic form of the P #P ̸= NC conjecture in characteristic\nzero.\n7.6.2\nObstructions\nThe obstruction in this context is defined as follows. A G-module Vλ(G) is\ncalled an obstruction for the pair (f, g) if it occurs in RV [f, n, m]∗\nd but not\nRV [g, m]∗\nd for some d. It is called a strong obstruction if, for some d, it occurs\nin RV [f, n, m]∗\nd but it does not contain (Cg)d as a Gg-submodule, where\n(Cg) ⊆V denotes the one dimensional line corresponding to g, and Gg ⊆G\nis the stabilizer of g = det(Y ) ∈P(V ). If Vλ(G) is a (strong) obstruction\nof degree d, then the size |λ| = dm; hence d is completely determined by λ\nand m.\nExistence of an obstruction or a strong obstruction implies that the\nclass variety ∆V [f, n, m] cannot be embedded in the class variety ∆V [g, m],\nas sought. The main algebro-geometric results of [GCT1, GCT2] suggest\nthat strong obstructions should indeed exist for all n →∞, assuming m =\npoly(n); cf. Section 4, Conjecture 2.10 and Theorem 2.11 in [GCT2]. The\ngoal then is to prove existence of strong obstructions for all n.\nThe definition of a strong obstruction can be simplified further as follows.\nLet X′ denote the set of variables, which consists of the variable entries in\nX and the homogenizing variable y above. Let W ′ = Symm(X′) ⊆V =\nSymm(Y ) be the space of homogeneous forms of degree m in the variables\nof X′.\nWe have a natural action of H = GL(X′) = GLn2+1(C) on W ′\nand hence on P(W ′).\nWe have a natural map φ′ : W →W ′ given by\nφ′(h)(X′) = ym−nh(X). The map φ in (7.2) is φ′ followed by the inclusion\nfrom W ′ to V . We also think of φ′ as a map from P(W) to P(W ′).\n118"},{"paragraph_id":"p136","order":136,"text":"Let f ′ = φ′(h), for h = perm(X) ∈P(W). Let ∆W ′[f ′] ⊆P(W ′) be\nthe orbit closure of Hf ′.\nIt is an H-subvariety of P(W ′), and hence its\nhomogeneous coordinate ring RW ′[f ′] has the natural degree preserving H-\naction. We call ∆W ′[f ′] the reduced class variety for P #P . It is known (cf.\nTheorem 8.2 in [GCT2]) that Vλ(G) occurs in RV [f]∗\nd iffVλ(H) occurs in\nRW ′[f ′]∗\nd. Here the dominant weight λ of G is considered a dominant weight\nof H by restriction from G to H.\nHence Vλ(G) is a strong obstruction for the pair (f, g), ifffor some d,\nVλ(H) occurs in RW ′[f ′]∗\nd as an H-submodule and Vλ(G) does not contain\n(Cg)d as a Gg-submodule.\nIn particular, we can assume without loss of\ngenerality that the height of the Young diagram for λ is at most n2 + 1;\notherwise Vλ(H) would be zero.\n7.6.3\nRobust obstructions\nIt is known that the stabilizer Gg of g = det(Y ) ∈P(V ) consists of lin-\near transformations in G of the form Y →AY ∗B−1, thinking of Y as an\nm × m matrix, where Y ∗is either Y or Y T , A, B ∈GLm(C). Thus the con-\nnected component of Gg is essentially GLm(C) × GLm(C) ⊆G = GLl(C) =\nGLm2(C). This means the subgroup restriction problem for the embedding\nρ : Gg ֒→G is essentially the Kronecker problem (Problem 1.1.1).\nAssume PH1 (Hypothesis 7.3.2) for the subgroup restriction ρ : Gg ֒→G;\nwhich is essentially PH1 for the Kronecker problem. It now assumes the\nfollowing concrete form. Let md\nλ denote the multiplicity of the Gg-module\n(Cg)d in Vλ(G). Assume that the height of λ is at most n2+1 for the reasons\ngive above.\nHypothesis 7.6.1 (PH1:)\nThere exists a polytope Qd\nλ such that:\n1. The number of integer points in Qd\nλ is equal to md\nλ.\n2. The Ehrhart quasi-polynomial of Qd\nλ coincides with the stretching quasi-\npolynomial ̃md\nλ(n) (cf. Theorem 3.5.1).\n3. The polytope Qd\nλ is given by a separating oracle, and its encoding\nbitlength ⟨Qd\nλ⟩is poly(n, ⟨m⟩, ⟨d⟩, ⟨λ⟩) time.\nWe have to explain why ⟨Qd\nλ⟩is stipulated to depend polynomially on\nn and ⟨m⟩, rather than m. After all, the bitlengths ⟨G⟩, ⟨Gg⟩and ⟨ρ⟩are\n119"},{"paragraph_id":"p137","order":137,"text":"O(poly(m2)) as per the definitions in Section 3.4. So, as per PH1 for sub-\ngroup restriction in Section 3.4.3, ⟨Qd\nλ⟩should depend polynomially on m.\nWe are stipulating a stronger condition for the following reason. First, as we\nalready mentioned, the above hypothesis is essentially PH1 for the Kronecker\nproblem, which is obtained by specializing PH1 for the plethysm problem\n(Hypothesis 1.6.4). In Hypothesis 1.6.4, the encoding bitlength of the poly-\ntope depends polynomially on the bitlengths of the various partition param-\neters λ, π, μ of the plethysm constant aπ\nλ,μ, but is independent of the rank of\nthe group G therein. (As explained in the remarks after Hypothesis 1.6.4,\nthis is justified because the bound in Theorem 1.6.3 is also independent of\nthe rank of G). For the same reason, the encoding bitlength of the polytope\nhere should be independent of the rank of G (which is m2), but should de-\npend polynomiallly on the total bit length of the partitions parametrizing\nthe representations Vλ(G) and (Cy)d. This is O(n + ⟨m⟩+ ⟨d⟩+ ⟨λ⟩). (Note\nthat the one dimensional representation (Cy)d of Gg is essentially the d-th\npower of the determinant representation of Gg, since the connected compo-\nnent of Gg is isomorphic to GLm(C)×GLm(C). The Young diagram for the\npartition corresponding to the d-th power of the determinant representation\nof GLm(C) is a rectangle of height m and width d. It can be specified by\nsimply giving m and d–this specification has bit length ⟨m⟩+ ⟨d⟩.)\nNext let us specialize PH1 as per Hypothesis 7.3.1. The class variety\n∆V [f] = ∆[f, n, m] will now play the role of X in Hypothesis 7.3.1. But,\nfor the reasons explained in the proof of Proposition 7.6.4 below, we shall\ninstead specialize Hypothesis 7.3.1 to the (simpler) reduced class variety\nZ = ∆W ′[f ′]. It now assumes that following concrete form. Let sλ\nd denote\nthe multiplicity of Vλ(H) in RW ′[f ′]∗\nd. Putting Z in place of X in Hypothe-\nsis 7.3.1, we get:\nHypothesis 7.6.2 (PH1):\nThere exists a polytope P λ\nd such that:\n1. The number of integer points in P λ\nd is equal to sλ\nd.\n2. The Ehrhart quasi-polynomial of P λ\nd coincides with the stretching quasi-\npolynomial ̃sλ\nd(n) (cf. Theorem 3.5.1).\n3. The polytope P λ\nd is given by a separating oracle, and its encoding\nbitlength ⟨P λ\nd ⟩is\npoly(⟨d⟩, ⟨λ⟩, ⟨Z⟩) = poly(⟨d⟩, ⟨λ⟩, n, ⟨m⟩).\n(7.3)\n120"},{"paragraph_id":"p138","order":138,"text":"Here (7.3) follows because ⟨Z⟩= n+⟨m⟩. To see why, let us observe that\nZ = ∆W ′[f ′] is completely specified once the point f ′ = ym−nh ∈P(W ′) is\nspecified. To specify f ′, it sufficies to specify m, n and the point h ∈P(W).\nIt is known [GCT2] that the point h = perm(X) ∈P(W) is completely\ncharacterized by its stabilizer Kh ⊆K = GL(X) = GLk(C). Furthermore,\nKh is explicitly known [Mc]. It is generated by the linear transformation in\nK of the form X →λXμ−1, thinking of X as an n × n matrix, where λ and\nμ are either diagonal or permutation matrices. So to specifiy h, it suffices\nto specify Kh, K and the embedding ρ′ : Kh ֒→K. The bit length of this\nspecification is O(n) (cf. Section 3.4). To specify f ′, and hence Z, it suffices\nto specify m, n, K, Kh and ρ′. The total bit length of this specification is\nO(n + ⟨m⟩).\nAssume PH1 for both md\nλ and sλ\nd, i.e., Hypotheses 7.6.1 and 7.6.2.\nDefinition 7.6.3 We say that Vλ(G) is a robust obstruction for the pair\n(f, , g) if one of the following hold:\n1. Qd\nλ is empty, and P λ\nd is nonempty.\n2. Both Qd\nλ and P λ\nd are nonempty, the affine span of Qd\nλ does not contain\nan integer point and the affine span of P λ\nd contains an integer point.\nIf the first condition holds, we say that Vλ(G) is a geometric obstruction.\nIf the second condition holds, it is called a modular obstruction.\nProposition 7.6.4 Assume PH1 for both md\nλ and sλ\nd (Hypotheses 7.6.1 and\n7.6.2). If Vλ(G) is a robust obstruction for the pair (f, g), then for some\npositive integral relaxation parameter k, Vkλ(G) is a strong obstruction for\n(f, g). In fact, this is so for most large enough k.\nProof: This essentially follows from Proposition 7.4.2. It only remains to\nclarify why we can use PH1 for the reduced class variety ∆W ′[f ′]–as we are\ndoing here– in place of PH1 for the class variety ∆V [f]. This is because,\nas already mentioned, Vλ(G) occurs in RV [f]∗\nd iffVλ(H) occurs in RW ′[f ′]∗\nd.\nQ.E.D.\n7.6.4\nVerification of robust obstructions\nTheorem 7.6.5 Assume that the singularities of spec(RW ′[f ′]) are ratio-\nnal. Assume PH1 for both md\nλ and sλ\nd as above (Hypotheses 7.6.1 and 7.6.2).\n121"},{"paragraph_id":"p139","order":139,"text":"Then, given n, m, λ and d, whether Vλ(H) is a robust obstruction can be\nverified in poly(n, ⟨m⟩, ⟨d⟩, ⟨λ⟩) time. Furthermore, a positive integral relax-\nation parameter k such that kλ is a strong obstruction can also be computed\nin this much time.\nOnce n and m are specified, the various class varieties and K, Kh, ρ′, G, Gg, ρ\nabove are automatically specified implicitly.\nProof: This follows from Theorem 7.5.1; cf. also the remark following its\nproof. Q.E.D.\nTheorem 7.3.4 can be similarly specialized in this context; we leave that\nto the reader.\n7.6.5\nOn explicit construction of obstructions\nTheorem 7.6.6 Assume that m = poly(n) or even 2polylog(n), and:\n1. (RH) [Rationality Hypothesis]: The singularities of spec(RW ′[f ′]) are\nrational.\n2. PH1 for both md\nλ and sλ\nd (Hypotheses 7.6.1 and 7.6.2).\n3. OH [Obstruction Hypothesis]: For every (large enough) n, there exists\nλ of poly(n) bit length such that |λ| is divisible by m and one of the\nfollowing holds (with d = |λ|/m):\n(a) Qd\nλ is empty, and P λ\nd is nonempty.\n(b) Both Qdλ and P λ\nd are nonempty, the affine span of Qd\nλ does not\ncontain an integer point and the affine span of P λ\nd,x contains an\ninteger point.\nThen there exists an explicit family {λn} of robust obstructions.\nHere we say that {λn} is an explicit family of robust obstructions if each\nλn is short and easy to verify. Short means ⟨λn⟩is O(poly(n)). Easy to verify\nmeans whether λn is a robust obstruction can be verified in O(poly(n)) time.\nThe poly(n) bound here and in OH is meant to be independent of m,\nas long as m << 2n; i.e., it should hold even when m = 2polylog(n). In\nother words, the family {λn} should continue to remain an explicit robust\nobstruction family, as we vary m over all values ≤2polylog(n), and perhaps\n122"},{"paragraph_id":"p140","order":140,"text":"even values ≤2o(n), but will cease to be an obstruction family for some large\nenough m = 2Ω(n). This is an important uniformity condition.\nProof: OH basically says that there exists a short robust obstruction λn for\nevery n. By Theorem 7.6.5, it is easy to verify. Q.E.D.\n7.6.6\nWhy should robust obstructions exist?\nThe main question now is: why should OH hold?\nThat is, why should\n(short) robust obstructions exist?\nAs we already mentioned, the results in [GCT1, GCT2] indicate that\nstrong obstructions should exist for every n, assuming m = poly(n). We\nshall give a heuristic argument for existence of robust obstructions assuming\nthat strong obstructions exist. This will crucially depend on the following SH\nfor md\nλ, which is essentially SH for the Kronecker problem (i.e. specialization\nof Hypothesis 1.6.5 to the Kronecker problem), good experimental evidence\nfor which is provided in [BOR].\nHypothesis 7.6.7 (SH:) (a): The saturation index of ̃md\nλ(k) is bounded by\na polynomial in m. (Observe that the rank of G is poly(m) and the height of\nλ is at most n2 + 1). (b): The quasi-polynomial ̃md\nλ(n) is strictly saturated,\ni.e. the saturation index is zero, for almost all λ (and d).\nIf Vλ(G) is a strong obstruction, sλ\nd is nonzero but md\nλ is zero. Thus,\nassumming PH1, there are three possibilities:\n1. Qd\nλ is empty, and P λ\nd is nonempty and contains an integer point.\n2. Both Qdλ and P λ\nd are nonempty, the affine span of Qd\nλ does not contain\nan integer point and P λ\nd contains an integer point.\n3. Both Qd\nλ and P λ\nd are nonempty. The affine span of Qd\nλ contains an\ninteger point, but Qd\nλ does not. And P λ\nd contains an integer point.\nIn the first two cases, λ is a robust obstruction. As per SH (Hypothe-\nsis 7.6.7), for almost all λ, the Ehrhart quasipolynomial of Qd\nλ is saturated:\nthis means (cf. the proof of Theorem 3.1.1), if the affine span of Qd\nλ contains\nan integer point then Qd\nλ also contains an integer point. And hence, with\na high probability, the third case should not occur. In other words, strong\nobstructions can be expected to be robust with a high probability.\n123"},{"paragraph_id":"p141","order":141,"text":"Let us call a strong obstruction λ fragile if it is not robust; this means\nthe affine span of Qd\nλ contains an integer point, but Qd\nλ does not. By SH\n(Hypothesis 7.6.7), if λ is fragile, then for some k = poly(m), Qkd\nkλ contains\nan intger point, and hence, kλ is not obstruction. Thus fragile obstructions\nare close to not being obstructions, and furthermore, are expected to be\nrare, as argued above. This is why we are focussing on robust obstructions.\nIt may be remarked that the only SH needed in the argument above\nis the one (Hypothesis 7.6.7) for the structural constant md\nλ.\nThis is a\nspecial case of the SH for the subgroup restriction problem (cf. Section 3.4)\nspecialized to the embedding Gg ֒→G. In particular, we do not need SH\nfor the structural constant sλ\nd; i.e., for the more difficult decision problem in\ngeometric invariant theory (cf. Problem 1.1.4 and Section 3.5).\n7.6.7\nOn discovery of robust obstructions\nIt may be conjectured that not just the verification (cf.\nTheorem 7.6.5)\nbut also the discovery of robust obstructions is easy for the problem under\nconsideration. In this section we shall give an argument in support of this\nconjecture for geometric (robust) obstructions (which may be conjectured to\nexist in the problem under consideration). For this we need to reformulate\nthe notions of strong and robust obstructions (Definition 7.6.3) as follows.\nLet TZ be the set of pairs (d, λ) such that sλ\nd is nonzero and SZ the set\nof pairs (d, λ) such that md\nλ is nonzero.\nProposition 7.6.8 Assuming PH1 above (Hypotheses 7.6.1 and 7.6.2), TZ\nand SZ are finitely generated semigroups with respect to addition.\nThese semi-groups are analogues of the Littlewood-Richardson semigroup\n(Section 2.2.2) in this setting.\nProof: The proof is similar to that for the Littlewood-Richardson semigroup\n[Z].\nFor given d and λ, the polytope P λ\nd in PH1 for sλ\nd (Hypothesis 7.6.2) has\na specification of the form\nAx ≤b\n(7.4)\nwhere A depends only the variety Z = ∆W ′[f ′], but not on d or λ, and\nb depends homogeneously and linearly on d and λ. Let P be the polytope\ndefined by the inequalities (7.4) where both d and λ are treated as variables.\nThen P is a polyhedral cone (through the origin) in the ambient space\n124"},{"paragraph_id":"p142","order":142,"text":"containing P with the coordinates x, d and λ. Let PZ be the set of integer\npoints in P. It is a finitely generated semigroup since P is a polyhedral cone.\nLet TR be the orthogonal projection of P on the hyperplane corresponding\nto the coordinates d and λ. Now TZ is simply the projection of PZ. Hence\nit is a finitely generated semigroup as well.\nThe proof for SZ is similar, with SR defined similarly. Q.E.D.\nThe polyhedral cones TR and SR here are analogues of the Littlewood-\nRichardson cone (Section 2.2.2) in this setting. Note that (d, λ) ∈TR iffP λ\nd\nis nonempty; similarly for SR.\nA Weyl module Vλ(G) is a strong obstruction for the pair (f, g) of degree\nd iff(d, λ) occurs in TZ but not in SZ. It is a robust obstruction iffit occurs\nin TR but not in SZ. It is a geometric obstruction iffit occurs in TR but not\nin SR. It is a modular obstruction iffit occurs in TR and also in SR but not\nin SZ.\nAssuming PH1 (Hypothesis 7.6.2), whether (d, λ) belongs to TR can be\ndetermined in polynomial time by linear programming, since (d, λ) ∈TR\niffP λ\nd is nonempty. Similarly, assuming PH1 (Hypothesis 7.6.1), whether\n(d, λ) ∈SR can be determined in polynomial time.\nThe following is a stronger complement to PH1.\nHypothesis 7.6.9 (PH1*)\nWhether TR\\SR is nonempty can be determined in polynomial time; i.e.,\npoly(n, ⟨m⟩) time. If so, the algorithm can also output (d, λ) ∈TR \\ SR of\npolynomial bit length.\nProposition 7.6.10 Assuming PH1*, given n and m, the problem of decid-\ning if a geometric obstruction exists for the pair (f, g), and finding one if one\nexists, belongs to the complexity class P; i.e., it can be done in poly(n, ⟨m⟩)\ntime.\nThis immediately follows from Hypothesis 7.6.9 since (d, λ) is a geometric\nobstruction iff(d, λ) ∈TR \\ SR.\nHypothesis 7.6.9 is supported by the following:\nProposition 7.6.11 Assuming PH1 (Hypotheses 7.6.1 and 7.6.2), Hypoth-\nesis 7.6.9 holds if TR and SR have polynomially many explicitly given con-\nstraints with the specification of polynomial bit length; here polynomial means\npoly(n, ⟨m⟩).\n125"},{"paragraph_id":"p143","order":143,"text":"The proposition holds even if the polytope SR has exponentially many\nconstraints, as long as it is given by a separation oracle that works in poly-\nnomial time.\nProof: It suffices to check if SR satisfies each constraint of TR. This can be\ndone in polynomial time using the linear programming algorithm in [GLS].\nSpecifically, let l(y) ≥0 be a constraint of TR. Then we just need to minimize\nl(y) on SR and check if the minimum exceeds zero. Q.E.D.\nBut this method does not work when the number of constraints of TR is\nexponential, as expected in the context of the lower bound problems under\nconsideration. In fact, no generic black-box-type algorithm, like the one in\n[GLS] based on just a membership or separation oracle for TR, can be used\nto prove (4) when the number of constraints of TR is exponential.\nFortunately, this is not a serious problem. A basic principle in combi-\nnatorial optimization, as illustrated in [GLS], is that a complexity theoretic\nproperty that holds for polytopes with polynomially many constraints will\nalso hold for polytopes with exponentially many constraints, provided these\nconstraints are sufficiently well-behaved.\nFor example, Edmond’s perfect\nmatching polytope for nonbipartite graphs has complexity-theoretic proper-\nties similar to the perfect matching polytope for bipartite graphs, though it\ncan have exponentially many constraints. We have already remarked that\nTR and SR are analogues of the Littlewood-Richardson cones. The facets of\nthe Littlewood-Richardson cone have a very nice explicit description [Kl, Z].\nThe cones TR, SR here are expected to have similar nice explicit descrip-\ntion.\nThis is why Hypothesis 7.6.9 can be expected to hold even if the\nnumber of constraints of TR is exponential, just as it holds even when SR\nhas exponentially many constraints. But a polynomial-time algorithm as in\nHypothesis 7.6.9 would have to depend crucially on the specific nature of\nthe facets (constraints) of TR in the spirit of the linear-programming-based\nalgorithm for the construction of a maximum-weight perfect matching in\nnonbipartite graphs [Ed], where too the number of constraints is exponen-\ntial but the algorithm still works because of the structure theorems based\non the specific nature of the constraints.\n7.7\nArithmetic form of the P vs NP problem in\ncharacteristic zero\nWe turn now to the arithemetic form of the P vs. NP problem in character-\nistic zero. The arguments are essentially verbatim translations of those for\n126"},{"paragraph_id":"p144","order":144,"text":"the arithmetic form of the P #P vs. NC problem in the preceding section.\nHence we shall be brief.\nIn the preceding section h(X) was perm(X) and g(Y ) was det(Y ). Now\nh(X) and g(Y ) would be explicit (co)-NP-complete and P-complete func-\ntions E(X) and H(Y ) constructed in [GCT1]. They can be thought of as\npoints in suitable W = Symk(X) and V = Syml[Y ], k = O(n2), l = O(m2),\nwith the natural action of GL(X) and G = GL(Y ), where n denotes the\nnumber of input parameters and m denotes the circuit size parameter in\nthe lower bound problem. These functions are extremely special like the\ndeterminant and the permanent in the sense that they are “almost” char-\nacterized by their stabilizers as explained in [GCT1]–and this is enough for\nour purposes.\nWe again have a natural embedding φ : P(W) →P(V ), which lets us\ndefine f = φ(h). The class variety for NP is defined to be ∆V [f] ⊆P(V ),\nthe projective closure of the orbit Gf. The class variety for P is ̃∆V [g] ⊆\nP(V ), which is defined to be the projective closure of G[g], where [g] denotes\nthe set of points in P(V ) that are stabilized by Gg ⊆G, the stabilizer of g.\nAn explicit description of Gg is given in [GCT1]; cf. Section 7 therein. To\nshow P ̸= NP in characteristic zero, it suffices to show that ∆V [f] is not a\nsubvariety of ̃∆V [g] for all large enough n, if m = poly(n) (cf. Conjecture\n7.4. in [GCT1]). For this, in turn, it suffices to show existence of strong\nobstructions, defined very much as in Section 7.6, for all n, assumming\nm = poly(n).\nWe can then formulate PH1 for the new h(X) and g(Y ) just as in Hy-\npotheses 7.6.1 and 7.6.2, and the notion of a robust obstruction as in Defi-\nnition 7.6.3. We then have:\nTheorem 7.7.1 (Verification of obstructions)\nAnalogues of Theorems 7.6.5 and 7.6.6 holds for h(X) = E(X) and\ng(Y ) = H(Y ).\nFurthermore, even discovery of robust obstructions can be conjectured\nto be easy (poly-time)–this would follow from the obvious analogue of Hy-\npothesis 7.6.9 here.\nHeuristic argument for existence of robust obstructions is very similar\nto the one in Section 7.6.6. It needs SH for the special case of the subgroup\nrestriction problem for the embedding Gg ֒→G. The group Gg, as described\nin [GCT1], is a product of some copies of the algebraic torus and the sym-\n127"},{"paragraph_id":"p145","order":145,"text":"metric group. The subgroup restriction problem in this case is akin to but\nharder than the plethysm problem.\n128"},{"paragraph_id":"p146","order":146,"text":"Bibliography\n[BGS]\nT. Baker, J. Gill, R. Soloway, Relativization of the P =?NP ques-\ntion, SIAM J. Comput. 4, 431-442, 1975.\n[BBCV] M. Baldoni, M. Beck, C. Cochet, M. 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Sohoni, Geometric complexity theory, P vs. NP\nand explicit obstructions, in “Advances in Algebra and Geometry”,\nEdited by C. Musili, the proceedings of the International Confer-\nence on Algebra and Geometry, Hyderabad, 2001.\n[GCTflip] K. Mulmuley, On P vs. NP, geometric complexity theory, and the\nflip I: a high level view, Technical report TR-2007-16, computer\nscience department, The university of Chicago, September 2007.\nAvailable at http://ramakrishnadas.cs.uchicago.edu.\n[GCT1] K. Mulmuley, M. Sohoni, Geometric complexity theory I: an ap-\nproach to the P vs. NP and related problems, SIAM J. Comput.,\nvol 31, no 2, pp 496-526, 2001.\n[GCT2] K. Mulmuley, M. Sohoni, Geometric complexity theory II: towards\nexplicit obstructions for embeddings among class varieties, SIAM\nJ. Comput., Vol. 38, Issue 3, June 2008.\n[GCT3] K. Mulmuley, M. Sohoni, Geometric complexity theory III, on de-\nciding positivity of Littlewood-Richardson coefficients, cs. ArXiv\npreprint cs. CC/0501076 v1 26 Jan 2005.\n[GCT4] K. Mulmuley, M. Sohoni, Geometric complexity theory IV: quan-\ntum group for the Kronecker problem, cs. ArXiv preprint cs.\nCC/0703110, March, 2007.\n131"},{"paragraph_id":"p149","order":149,"text":"[GCT5] K. Mulmuley, H. Narayanan, Geometric complexity theory V:\non deciding nonvanishing of a generalized Littlewood-Richardson\ncoefficient, Technical Report TR-2007-05, computer science de-\npartment, The University of Chicago, May, 2007. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu\n[GCT6erratum] K. Mulmuley, Erratum to the saturation hypothesis in\n“Geometric Complexity Theory VI”, technical report TR-2008-10,\ncomputer science department, the university of Chicago, October\n2008. Available at: http://ramakrishnadas.cs.uchicago.edu.\n[GCT7] K.\nMulmuley,\nGeometric\ncomplexity\ntheory\nVII:\nNonstan-\ndard\nquantum\ngroup\nfor\nthe\nplethysm\nproblem,\nTech-\nnical\nReport\nTR-2007-14,\ncomputer\nscience\ndepartment,\nThe University of Chicago,\nSeptember,\n2007.\nAvailable at:\nhttp://ramakrishnadas.cs.uchicago.edu.\n[GCT8] K. Mulmuley, Geometric complexity theory VIII: On canon-\nical\nbases\nfor\nthe\nnonstandard\nquantum\ngroups,\nTech-\nnical\nReport\nTR\n2007-15,\ncomputer\nscience\ndepartment,\nThe\nuniversity\nof\nChicago,\nSeptember\n2007.\nAvailable\nat:\nhttp://ramakrishnadas.cs.uchicago.edu.\n[GCT9] B. Adsul, M. Sohoni, K. Subrahmanyam, Geometric complexity\ntheory IX: algbraic and combinatorial aspects of the Kronecker\nproblem, under preparation.\n[GCT10] K. Mulmuley, Geometric complexity theory X: On class varieties\nand nonstandard quantum groups, under preparation.\n[GCT11] K. 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Valiant, The complexity of computing the permanent, Theoret-\nical Computer Science 8, pp 189-201, 1979.\n[Z]\nA.\nZelevinsky,\nLittlewood-Richardson\nsemigroups,\narXiv:math.CO/9704228 v1 30 Apr 1997.\n136"}],"pages":[{"page":1,"text":"arXiv:0704.0229v4 [cs.CC] 22 Jan 2009\nGeometric Complexity Theory VI: the flip via\nsaturated and positive integer programming in\nrepresentation theory and algebraic geometry\nDedicated to Sri Ramakrishna\nKetan D. Mulmuley\nThe University of Chicago\nhttp://ramakrishnadas.cs.uchicago.edu\n(Technical report TR 2007-04, Comp. Sci. Dept.,\nThe University of Chicago, May, 2007)\nRevised version\nOctober 23, 2018"},{"page":2,"text":"Abstract\nThis article belongs to a series on geometric complexity theory (GCT), an\napproach to the P vs. NP and related problems through algebraic geometry\nand representation theory. The basic principle behind this approach is called\nthe flip. In essence, it reduces the negative hypothesis in complexity theory\n(the lower bound problems), such as the P vs. NP problem in characteristic\nzero, to the positive hypothesis in complexity theory (the upper bound prob-\nlems): specifically, to showing that the problems of deciding nonvanishing of\nthe fundamental structural constants in representation theory and algebraic\ngeometry, such as the well known plethysm constants [Mc, FH], belong to the\ncomplexity class P. In this article, we suggest a plan for implementing the\nflip, i.e., for showing that these decision problems belong to P. This is based\non the reduction of the preceding complexity-theoretic positive hypotheses to\nmathematical positivity hypotheses: specifically, to showing that there exist\npositive formulae–i.e. formulae with nonnegative coefficients–for the struc-\ntural constants under consideration and certain functions associated with\nthem. These turn out be intimately related to the similar positivity proper-\nties of the Kazhdan-Lusztig polynomials [KL1, KL2] and the multiplicative\nstructural constants of the canonical (global crystal) bases [Kas2, Lu2] in\nthe theory of Drinfeld-Jimbo quantum groups. The known proofs of these\npositivity properties depend on the Riemann hypothesis over finite fields\n(Weil conjectures proved in [Dl]) and the related results [BBD]. Thus the\nreduction here, in conjunction with the flip, in essence, says that the validity\nof the P ̸= NP conjecture in characteristic zero is intimately linked to the\nRiemann hypothesis over finite fields and related problems.\nThe main ingradients of this reduction are as follows.\nFirst, we formulate a general paradigm of saturated, and more strongly,\npositive integer programming, and show that it has a polynomial time al-\ngorithm, extending and building on the techniques in [DM2, GCT3, GCT5,\nGLS, KB, KTT, Ki, KT1].\nSecond, building on the work of Boutot [Bou] and Brion (cf. [Dh]), we\nshow that the stretching functions associated with the structural constants\nunder consideration are quasipolynomials, generalizing the known result that\nthe stretching function associated with the Littlewood-Richardson coefficient\nis a polynomial for type A [Der, Ki] and a quasi-polynomial for general types"},{"page":3,"text":"[BZ, Dh]. In particular, this proves Kirillov’s conjecture [Ki] for the plethysm\nconstants.\nThird, using these stretching quasi-polynomials, we formulate the math-\nematical saturation and positivity hypotheses for the plethysm and other\nstructural constants under consideration, which generalize the known sat-\nuration and conjectural positivity properties of the Littlewood-Richardson\ncoefficients [KT1, DM2, KTT]. Assuming these hypotheses, it follows that\nthe problem of deciding nonvanishing of any of these structural constants,\nmodulo a small relaxation, can be transformed in polynomial time into a\nsaturated, and more strongly, positive integer programming problem, and\nhence, can be solved in polynomial time.\nFourth, we give theoretical and experimental results in support of these\nhypotheses.\nFinally, we suggest an approach to prove these positivity hypotheses\nmotivated by the works on Kazhdan-Lusztig bases for Hecke algebras [KL1,\nKL2] and the canonical (global crystal) bases of Kashiwara and Lusztig [Lu2,\nLu4, Kas2] for representations of Drinfeld-Jimbo quantum groups [Dri, Ji].\nSteps in this direction are taken [GCT4, GCT7, GCT8].\nSpecifically, in [GCT4, GCT7] are constructed nonstandard quantum\ngroups, with compact real forms, which are generalizations of the Drinfeld-\nJimbo quantum group, and also associated nonstandard algebras, whose re-\nlationship with the nonstandard quantum groups is conjecturally similar\nto the relationship of the Hecke algebra with the Drinfeld-Jimbo quantum\ngroup. The article [GCT8] gives conjecturally correct algorithms to con-\nstruct canonical bases of the matrix coordinate rings of the nonstandard\nquantum groups and of nonstandard algebras that have conjectural posi-\ntivity properties analogous to those of the canonical (global crystal) bases,\nas per Kashiwara and Lusztig, of the coordinate ring of the Drinfeld-Jimbo\nquantum group, and the Kazhdan-Lusztig basis of the Hecke algebra. These\npositivity conjectures (hypotheses) lie at the heart of this approach. In view\nof [KL2, Lu2], their validity is intimately linked to the Riemann hypothesis\nover finite fields and the related works mentioned above.\n2"},{"page":4,"text":"Contents\n1\nIntroduction\n4\n1.1\nThe decision problems . . . . . . . . . . . . . . . . . . . . . .\n7\n1.2\nDeciding nonvanishing of Littlewood-Richardson coefficients .\n12\n1.3\nBack to the general decision problems\n. . . . . . . . . . . . .\n16\n1.4\nSaturated and positive integer programming . . . . . . . . . .\n16\n1.5\nQuasi-polynomiality, positivity hypotheses, and the canonical\nmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n19\n1.6\nThe plethysm problem . . . . . . . . . . . . . . . . . . . . . .\n20\n1.7\nTowards PH1, SH, PH2,PH3 via canonial bases and canonical\nmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n27\n1.8\nBasic plan for implementing the flip\n. . . . . . . . . . . . . .\n29\n1.9\nOrganization of the paper . . . . . . . . . . . . . . . . . . . .\n30\n1.10 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n32\n2\nPreliminaries in complexity theory\n34\n2.1\nStandard complexity classes . . . . . . . . . . . . . . . . . . .\n34\n2.1.1\nExample: Littlewood-Richardson coefficients\n. . . . .\n35\n2.2\nConvex #P . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n36\n2.2.1\nLittlewood-Richardson coefficients\n. . . . . . . . . . .\n38\n2.2.2\nLittlewood-Richardson cone . . . . . . . . . . . . . . .\n38\n2.2.3\nEigenvalues of Hermitian matrices\n. . . . . . . . . . .\n39\n2.3\nSeparation oracle . . . . . . . . . . . . . . . . . . . . . . . . .\n39\n3\nSaturation and positivity\n41\n1"},{"page":5,"text":"3.1\nSaturated and positive integer programming . . . . . . . . . .\n41\n3.1.1\nA general estimate for the saturation index . . . . . .\n45\n3.1.2\nExtensions\n. . . . . . . . . . . . . . . . . . . . . . . .\n47\n3.1.3\nIs there a simpler algorithm? . . . . . . . . . . . . . .\n47\n3.2\nLittlewood-Richardson coefficients again . . . . . . . . . . . .\n47\n3.3\nThe saturation and positivity hypotheses\n. . . . . . . . . . .\n49\n3.4\nThe subgroup restriction problem . . . . . . . . . . . . . . . .\n52\n3.4.1\nExplicit polynomial homomorphism\n. . . . . . . . . .\n53\n3.4.2\nInput specification and bitlengths . . . . . . . . . . . .\n55\n3.4.3\nStretching function and quasipolynomiality . . . . . .\n57\n3.5\nThe decision problem in geometric invariant theory . . . . . .\n58\n3.5.1\nReduction from Problem 1.1.3 to Problem 1.1.4 . . . .\n59\n3.5.2\nInput specification . . . . . . . . . . . . . . . . . . . .\n59\n3.5.3\nStretching function and quasi-polynomiality . . . . . .\n60\n3.5.4\nPositivity hypotheses . . . . . . . . . . . . . . . . . . .\n61\n3.5.5\nG/P and Schubert varieties . . . . . . . . . . . . . . .\n62\n3.6\nPH3 and existence of a simpler algorithm\n. . . . . . . . . . .\n63\n3.7\nOther structural constants . . . . . . . . . . . . . . . . . . . .\n63\n4\nQuasi-polynomiality and canonical models\n65\n4.1\nQuasi-polynomiality\n. . . . . . . . . . . . . . . . . . . . . . .\n65\n4.1.1\nThe minimal positive form and modular index\n. . . .\n68\n4.1.2\nThe rings associated with a structural constant . . . .\n69\n4.2\nCanonical models . . . . . . . . . . . . . . . . . . . . . . . . .\n69\n4.2.1\nFrom PH0 to PH1,3 . . . . . . . . . . . . . . . . . . .\n70\n4.2.2\nOn PH0 in general . . . . . . . . . . . . . . . . . . . .\n72\n4.3\nNonstandard quantum group for the Kronecker and the plethysm\nproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n73\n4.4\nThe cone associated with the subgroup restriction problem\n.\n75\n4.5\nElementary proof of rationality . . . . . . . . . . . . . . . . .\n78\n5\nParallel and PSPACE algorithms\n84\n2"},{"page":6,"text":"5.1\nComplex semisimple Lie group\n. . . . . . . . . . . . . . . . .\n85\n5.2\nSymmetric group . . . . . . . . . . . . . . . . . . . . . . . . .\n89\n5.3\nGeneral linear group over a finite field . . . . . . . . . . . . .\n92\n5.3.1\nTensor product problem . . . . . . . . . . . . . . . . .\n93\n5.4\nFinite simple groups of Lie type . . . . . . . . . . . . . . . . .\n94\n6\nExperimental evidence for positivity\n95\n6.1\nLittlewood-Richardson problem . . . . . . . . . . . . . . . . .\n95\n6.2\nKronecker problem, n = 2 . . . . . . . . . . . . . . . . . . . .\n95\n6.3\nG/P and Schubert varieties . . . . . . . . . . . . . . . . . . .\n96\n6.4\nThe ring of symmetric functions\n. . . . . . . . . . . . . . . .\n97\n7\nOn verification and discovery of obstructions\n111\n7.1\nObstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111\n7.2\nDecision problems\n. . . . . . . . . . . . . . . . . . . . . . . . 113\n7.3\nVerification of obstructions\n. . . . . . . . . . . . . . . . . . . 113\n7.4\nRobust obstruction . . . . . . . . . . . . . . . . . . . . . . . . 115\n7.5\nVerification of robust obstructions\n. . . . . . . . . . . . . . . 116\n7.6\nArithemetic version of the P #P vs. NC problem in charac-\nteristric zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117\n7.6.1\nClass varieties . . . . . . . . . . . . . . . . . . . . . . . 117\n7.6.2\nObstructions\n. . . . . . . . . . . . . . . . . . . . . . . 118\n7.6.3\nRobust obstructions . . . . . . . . . . . . . . . . . . . 119\n7.6.4\nVerification of robust obstructions\n. . . . . . . . . . . 121\n7.6.5\nOn explicit construction of obstructions . . . . . . . . 122\n7.6.6\nWhy should robust obstructions exist? . . . . . . . . . 123\n7.6.7\nOn discovery of robust obstructions\n. . . . . . . . . . 124\n7.7\nArithmetic form of the P vs NP problem in characteristic zero126\n3"},{"page":7,"text":"Chapter 1\nIntroduction\nThis article belongs to a series of papers, [GCT1] to [GCT11], on geomet-\nric complexity theory (GCT), which is an approach to the P vs. NP and\nrelated problems in complexity theory through algebraic geometry and rep-\nresentation theory. We assume here that the underlying field of computation\nis of characteristic zero. The usual P vs. NP problem is over a finite field.\nThe characteristic zero version is its weaker, formal implication, and philo-\nsophically, the crux.\nThe basic principle underlying GCT is called the flip [GCTflip]. The\nflip, in essence, reduces the negative hypotheses (lower bound problems) in\ncomplexity theory, such as the P ̸=?NP problem in characteristic zero, to\npositive hypotheses in complexity theory (upper bound problems): specifi-\ncally, to the problem of showing that a series of decision problems in rep-\nresentation theory and algebraic geometry belong to the complexity class\nP.\nEach of these decision problem is of the form: Given a nonnegative\nstructural constant in representation theory or geometric invariant theory,\nsuch as the well known plethysm constant, decide if it is nonzero (nonvan-\nishing), or rather, if is nonzero after a small relaxation. This flip from the\nnegative to the positive may be considered to be a nonrelativizable form of\nthe flip–from the undecidable to the decidable–that underlies the proof of\nG ̈odel’s incompleteness theorem. But the classical diagonalization technique\nin G ̈odel’s result is relativizable [BGS], and hence, not applicable to the P\nvs. NP problem. The flip, in contrast, is nonrelativizable. It is furthermore\nnonnaturalizable [GCT10]); i.e., it crosses the natural proof barrier [RR]\nthat any approach to the P vs. NP problem must cross.\nWe suggest here a plan for implementating the flip; i.e., for showing that\n4"},{"page":8,"text":"the decision problems above belong to P. This is based on the reduction\nin this paper of the complexity-theoretic positivity hypotheses mentioned\nabove to mathematical positivity hypotheses: specifically, to showing that\nthere exist positive formulae for the structural constants under consideration\nand certain functions associated with them. We also give theoretical and\nexperimental evidence in support of the latter hypotheses.\nHere we say that a formula is positive if its coefficients are nonegative.\nThe problem finding the positive formulae as above turns out be intimately\nrelated to the analogous problem for the Kazhdan-Lusztig polynomials [KL1]\nand the multiplicative structural constants of the canonical (global crystal)\nbases [Kas2, Lu2] in the theory of Drinfeld-Jimbo quantum groups. The\nknown solution to the latter problem [KL2, Lu2] depends on the Riemann\nhypothesis over finite fields, proved in [Dl], and the related results in [BBD].\nThus the flip and the reduction here together roughly say that the valid-\nity of the P ̸= NP conjecture in characteristic zero is intimately linked\nto the Riemann hypothesis over finite fields and related problems. This is\nillustrated in Figure 1.1; the question marks there indicate unsolved prob-\nlems. It seems that substantial extension of the techniques related to the\nRiemann hypothesis over finite fields may be needed to prove the required\nmathematical positivity hypotheses here.\nWe do not have the necessary\nmathematical expertize for this task. But it is our hope that the experts in\nalgebraic geometry and representation theory will have something to say on\nthis matter.\nIt may be conjectured that the flip paradigm would also work in the\ncontext of the usual P vs. NP problem over F2 (the boolean field) or the\nfinite field Fp. But implementation of the flip over a finite field is expected\nto be much harder than in characteristic zero. That is why we focus on\ncharacteristic zero here, deferring discussion of the problems that arise over\nfinite field to [GCT11].\nNow we turn to a more detailed exposition of the main results in this\npaper and of Figure 1.1.\nAcknoledgements\nWe are grateful to the authors of [BOR] for pointing out an error in the\nsaturation hypothesis (SH) in the earlier version of this paper. It has been\ncorrected in this version with appropriate relaxation without affecting the\noverall approach of GCT (cf. Section 1.6 and also [GCT6erratum]). We\nare also grateful to Peter Littelmann for bringing the reference [Dh] to our\n5"},{"page":9,"text":"Complexity theoretic negative hypotheses (lower bound problems)\n|\n|\nThe flip\n|\n↓\nComplexity theoretic positive hypotheses (upper bound problems)\n|\n|\nThe reduction in this paper|\n|\n↓\nMathematical positivity hypotheses |\n|\n?\n|\n↓\n(?) The Riemann hypothesis over finite fields, related problems and their extensions\nFigure 1.1: Pictorial depiction of the basic plan for implementing the flip\n6"},{"page":10,"text":"attention, to H. Narayanan for suggesting the use of [KB] in the proof of\nTheorem 3.1.1 and bringing the positivity conjecture in [DM2] to our atten-\ntion, and to Madhav Nori for a helpful discussion. The experimental results\nin Chapter 6 were obtained using Latte [DHHH].\n1.1\nThe decision problems\nWe now describe the relevant decision problems in representation theory\nand algebraic geometry. The actual decision problems that arise in the flip\n(cf. the second box in Figure 1.1) are relaxed versions of these problems\ndescribed later (cf. Hypothesis 1.1.6).\nProblem 1.1.1 (Decision version of the Kronecker problem)\nGiven partitions λ, μ, π, decide nonvanishing of the Kronecker coefficient\nkπ\nλ,μ. This is the multiplicity of the irreducible representation (Specht mod-\nule) Sπ of the symmetric group Sn in the tensor product Sλ ⊗Sμ.\nEquivalently [FH], let H = GLn(C) × GLn(C) and ρ : H →G =\nGL(Cn ⊗Cn) = GLn2(C) the natural embedding. Then kπ\nλ,μ is the multi-\nplicity of the H-module Vλ(GLn(C)) ⊗Vμ(GLn(C)) in the G-module Vπ(G),\nconsidered as an H-module via the embedding ρ.\nHere Vλ(GLn(C)) denotes the irreducible representation (Weyl module)\nof GLn(C) corresponding to the partition λ; Vπ(G) is the Weyl module of\nG = GLn2(C).\nProblem 1.1.1 is a special case of the following generalized plethysm\nproblem.\nProblem 1.1.2 (Decision version of the plethysm problem)\nGiven partitions λ, μ, π, decide nonvanishing of the plethysm constant\naπ\nλ,μ. This is the multiplicity of the irreducible representation Vπ(H) of H =\nGLn(C) in the irreducible representation Vλ(G) of G = GL(Vμ), where Vμ =\nVμ(H) is an irreducible representation H. Here Vλ(G) is considered an H-\nmodule via the representation map ρ : H →G = GL(Vμ).\n(Decision version of the generalized plethysm problem)\nThe same as above, allowing H to be any connected reductive group.\nThis is a special case of the following fundamental problem of represen-\ntation theory (characteristic zero):\n7"},{"page":11,"text":"Problem 1.1.3 (Decision version of the subgroup restriction problem)\nLet G be connected reductive group, H a reductive group, possibly discon-\nnected, and ρ : H →G an explicit, polynomial homomorphism (as defined\nin Section 3.4). Here H will generally be a subgroup of G, and ρ its em-\nbedding. Let Vπ(H) be an irreducible representation of H, and Vλ(G) an\nirreducible representation of G. Here π and λ denote the classifying labels\nof the irreducible representations Vπ(H) and Vλ(G), respectively. Let mπ\nλ be\nthe multiplicity of Vπ(H) in Vλ(G), considered as an H-module via ρ.\nGiven specifications of the embedding ρ and the labels λ, π, as described\nin Section 3.4, decide nonvanishing of the multiplicity mπ\nλ.\nAll reductive groups in this paper are over C.\nThe reductive groups\nthat arise in GCT in characteristic zero are: the general and special linear\ngroups GLn(C) and SLn(C), algebraic tori, the symmetric group Sn, and\nthe groups formed from these by (semidirect) products. The reader may\nwish to focus on just these concrete cases, since all main ideas in this paper\nare illustrated therein.\nProblem 1.1.3 is, in turn, a special case of the following most general\nproblem.\nProblem 1.1.4 (Decision problem in geometric invariant theory)\nLet H be a reductive group, possibly disconnected, X a projective H-\nvariety (H-scheme), i.e., a variety with H-action.\nLet ρ denote this H-\naction. Let R = ⊕dRd be the homogeneous coordinate ring of X. Assume\nthat the singularities of spec(R) are rational.\nWe assume that X and ρ have special properties (as described in Sec-\ntion 3.5), so that, in particular, they have short specifications. Let Vπ(H)\nbe an irreducible representation of H. Let sπ\nd be the multiplicity of Vπ(H) in\nRd, considered as an H-module via the action ρ.\nGiven d, π, the specifications of X and ρ, decide nonvanishing of the\nmultiplicity sπ\nd.\nThis last problem is hopeless for general X. Indeed the usual specifi-\ncation of X, say in terms of the generators of the ideal of its appropriate\nembedding, is so large as to make this problem meaningless for a general\nX. But the instances of this decision problem that arise in GCT are for the\nfollowing very special kinds of projective H-varieties X, which, in particular,\nhave small specifications (Section 3.5):\n8"},{"page":12,"text":"1. G/P, where G is a connected, reductive group, P ⊆G its parabolic\nsubgroup, and H ⊆G a reductive subgroup with an explicit polyno-\nmial embedding. Problem 1.1.3 reduces to this special case of Prob-\nlem 1.1.4; cf. Section 3.5.\n2. Class varieties [GCT1, GCT2], which are associated with the funda-\nmental complexity classes such as P and NP. They are very special\nlike G/P, with conjecturally rational singularities [GCT10]. Each class\nvariety is specified by the complexity class and the parameters of the\nlower bound problem under consideration. Briefly, the P vs. NP prob-\nlem in characteristic zero is reduced in [GCT1, GCT2] to showing that\nthe class variety corresponding to the complexity class NP and the pa-\nrameters of the lower bound problem (such as the input size) cannot\nbe embedded in the class variety corresponding to the complexity class\nP and the same parameters. Efficient criteria for the decision prob-\nlems stated above are needed to construct explicit obstructions [GCT2]\nto such embeddings, thereby proving their nonexistence. Specifically,\nProblems 1.1.3 and 1.1.4 are the decision problems associated with\nProblems 2.5 and 2.6 in [GCT2], respectively. See Sections 7.6-7.7 for\na brief review of this story.\nFor these varieties Problem 1.1.4 turns out to be qualitatively similar to\nProblem 1.1.3 (cf. Section 3.5 and [GCT2, GCT10]). For this reason, the\nKronecker and the plethysm problems, which lie at the heart of the subgroup\nrestriction problem, can be taken as the main prototypes of the decision\nproblems that arise here.\nOne can now ask:\nQuestion 1.1.5 Do the decision problems above (Problems 1.1.1-1.1.3 and\nProblem1.1.4, when X therein is G/P or a class variety) belong to P? That\nis, can the nonvanishing of any of structural constants in these problems\nbe decided in poly(⟨x⟩) time, where x denotes the input-specification of the\nstructural constant and ⟨x⟩its bitlength?\nFor Problem 1.1.2, the input specification for the plethysm constant aπ\nλ,μ\nis given in the form of a triple x = (λ, μ, π). Here the partition λ is specified\nas a sequence of positive integers λ1 ≥λ2 ≥· · · λk > 0 (the zero parts of\nthe partition are suppressed); k is called the height or length of λ, and is\ndenoted by ht(λ). The bitlength ⟨λ⟩is defined to be the total bitlength of\nthe integers λr’s. The bitlength ⟨x⟩is defined to be ⟨λ⟩+ ⟨μ⟩+ ⟨π⟩. A\n9"},{"page":13,"text":"detailed specification of the input specification x and its bitlength ⟨x⟩for\nthe other problems is given in Section 3.3.\nFor the reasons described in Section 1.6, Question 1.1.5 may not have\nan affirmative answer in general; i.e., these problems may not be in P in\ntheir strict form stated above. The following main conjectural complexity-\ntheoretic positivity hypothesis governing the flip says that the relaxed forms\nof these decision problems described in Section 3.3 belong to P. As we shall\nsee in Chapter 7, these relaxed forms suffice for the purposes of the flip.\nHypothesis 1.1.6 (PHflip) The relaxed forms (cf. Section 3.3) of Prob-\nlems 1.1.1, 1.1.2, 1.1.3, and the special cases of Problem 1.1.4, when X\ntherein is G/P or a class variety–which together include all decision prob-\nlems that arise in the flip–belong to the complexity class P.\nThis means nonvanishing of any of these structural constants, modulo a\nsmall relaxation (as described in Section 3.3), can be decided in poly(⟨x⟩)\ntime, where x denotes the input-specification of the structural constant and\n⟨x⟩its bitlength.\nThe phrase “modulo a small relaxation” in the relaxed form of the\nplethysm problem means the following:\n(a) Let h = dim G + htλ + htπ, where dim(G) is the dimension of the\ngroup G in Problem 1.1.2. Then there exist absolute nonnegative constants\nc and c′, independent of λ, μ and π, such that nonvanishing of the relaxed\n(stretched) plethysm constant abπ\nbλ,bμ, for any positive integral relaxation\nparameter b > chc′, can be decided in O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, ⟨b⟩)) time, where\n⟨b⟩denotes the bitlength b. The notation poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, ⟨b⟩) here means\nbounded by a polynomial of constant degree in ⟨λ⟩, ⟨μ⟩, ⟨π⟩and ⟨b⟩.\nIn\nparticular, the time is O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) if the relaxation parameter b is\nsmall; i.e. if its bitlength ⟨b⟩is O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)). (Observe that the bit\nlength of h is O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)).)\n(b) There exists a polynomial time algorithm for deciding nonvanishing of\naπ\nλ,μ, which works correctly on almost all λ, μ and π. Here polynomial time\nmeans O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) time. The meaning of “correctly on almost all”\nis specified in Hypothesis 1.6.5 below.\nA detailed specification of the relaxation, i.e., the meaning of the phrase\n“modulo a small relaxation” for the other problems is given in Section 3.3.\nThe structural constants in Problems 1.1.1-1.1.3 are of fundamental im-\nportance in representation theory. The kronecker and the plethysm con-\n10"},{"page":14,"text":"stants in Problems 1.1.1 and 1.1.2, in particular, have been studied inten-\nsively; see [FH, Mc, St4] for their significance.\nThere are many known\nformulae for these structural constants based on on the character formulae\nin representation theory. Several formulae for the characters of connected,\nreductive groups are known by now [FH], starting with the Weyl character\nformula. For the symmetric group, there is the Frobenius character formula\n[FH], for the general linear group over a finite field, Green’s formula [Mc],\nand for finite simple groups of Lie type, the character formula of Deligne-\nLusztig [DL], and Lusztig [Lu1]. (Finite simple groups of Lie type, other\nthan GLn(Fq), are not needed in GCT.)\nOne obvious method for deciding nonvanishing of the structural con-\nstants in Problems 1.1.1-1.1.4 is to compute them exactly. But all known al-\ngorithms for exact computation of the structural constants in Problems 1.1.1-\n1.1.3 take exponential time. This is expected, since this problem is #P-\ncomplete.\nIn fact, even the problem of exact computation of a Kostka\nnumber, which is a very special case of these structural constants, is #P-\ncomplete [N]. This means there is no polynomial time algorithm for com-\nputing any of them, assuming P ̸= NP.\nOf course, there are #P-complete quantities–e.g. the permanent of a\nnonnegative matrix [V]–whose nonvanishing can still be decided in polyno-\nmial time [Sc]. But the decision problems above are of a totally different\nkind and, at the surface, appear to have inherently exponential complexity.\nThis is because the dimensions of the irreducible representations that occur\nin their statements can be exponential in the ranks of the groups involved\nand the bit lengths of the classifying labels of these representations. For\nexample, the dimension of the Weyl module Vλ(GLn(C)) can be exponen-\ntial in n and the bit length of the partition λ. Furthermore, the number\nof terms in any of the preceding character formulae is also exponential. All\nthese decisions problems ask if one exponential dimensional representation\ncan occur within another exponential dimensional representation. To solve\nthem, it may seem necessary to take a detailed look into these representa-\ntions and/or the character formulae of exponential complexity. Hence, it\nseemed hard to believe that nonvanishing of these structural constants can,\nnevertheless, be decided in polynomial time (modulo a small relaxation).\nThis constituted the main philosophical obstacle in the course of GCT.\n11"},{"page":15,"text":"1.2\nDeciding nonvanishing of Littlewood-Richardson\ncoefficients\nThe first result, which indicated that this obstacle may be removable, came\nin the wake of the saturation theorem of Knutson and Tao [KT1].\nThis\nconcerns the following special case of Problem 1.1.3, with G = H × H, the\nembedding ρ : H →G being diagonal.\nProblem 1.2.1 (Littlewood-Richardson problem)\nGiven a complex semisimple, simply connected Lie group H, and its\ndominant weights α, β, λ, decide nonvanishing of a generalized Littelwood-\nRichardson coefficient cλ\nα,β. This is the multiplicity of the irreducible repre-\nsentation Vλ(H) of H in the tensor product Vα(H) ⊗Vβ(H).\nIt was shown in [GCT3, KT2, DM2] independently that nonvanishing of\nthe Littlewood-Richardson coefficient of type A can be decided in polyno-\nmial time; i.e., polynomial in the bit lengths of α, β, λ. Furthermore, the\nalgorithm in [GCT3] works in strongly polynomial time in the terminology\nof [GLS]; cf. Section 2.1. The three main ingradients in this result are:\n1. PH1: The Littlewood-Richarson rule, which goes back to 1940’s, and\nwhose most important feature is that it is positive–i.e., it involves no\nalternating signs as in character-based formulae–and its strengthening\nin [BZ], which gives a positive, polyhedral formula for the Littlewood-\nRichardson coefficient as the number of integer points in a polytope;\nthis can be the BZ-polytope [BZ] or the hive polytope [KT1].\nWe\nshall refer to this positivity property as the first positivity hypothesis\n(PH1).\n2. The polynomial and strongly polynomial time algorithms for linear\nprogramming [Kh, Ta], and\n3. SH: The saturation theorem of Knutson and Tao [KT1]. This says\nthat cλ\nα,β is nonzero if cnλ\nnα,nβ is nonzero for any n ≥1. We shall refer\nto this saturation property as the saturation hypothesis (SH).\nBrion [Z] observed that the verbatim translation of the saturation prop-\nerty in [KT1] fails to hold for the the generalized Littlewood-Richardson\ncoefficients of types B, C, D (it also fails for the Kronecker coefficients, as\nwell as the plethysm constants [Ki]). Hence, the algorithms in [GCT3, KT2,\n12"},{"page":16,"text":"DM2] do not work in types B, C and D. Fortunately, this situation can\nbe remedied. It is shown in [GCT5] that nonvanishing of the generalized\nLittewood-Richardson coefficient cλ\nα,β of arbitrary type can be decided in\n(strongly) polynomial time, assuming the positivity conjecture of De Loera\nand McAllister [DM2]. This conjectural hypothesis, based on considerable\nexperimental evidence, is as follows. Let\n ̃cλ\nα,β(n) = cnλ\nnα,nβ\n(1.1)\nbe the stretching function associated with the Littlewood-Richardson co-\nefficient cλ\nα,β. It is known to be a polynomial in type A [Der, Ki], and a\nquasi-polynomial, in general [BZ, Dh, DM2]. Recall that a fuction f(n) is\ncalled a quasi-polynomial if there exist l polynomials fj(n), 1 ≤j ≤l, such\nthat f(n) = fj(n) if n = j mod l. Here l is supposed to be the smallest such\ninteger, and is called the period of f(n). The period of ̃cλ\nα,β(n) for types\nB, C, D is either 1 or 2 [DM2]. In general, it is bounded by a fixed constant\ndepending on the types of the simple factors the Lie algebra.\nDefinition 1.2.2 We say that the quasi-polynomial f(n) is strictly posi-\ntive, if all coefficients of fj(n), for all j, are nonnegative; i.e., the nonzero\ncoefficients are positive. In general, we define the positivity index p(f) of\nf to be the smallest nonnegative integer such that f(n + p(f)) is strictly\npositive. We also say that f(n) is positive with index p(f).\nThus f(n) is strictly positive, iffits positivity index is zero.\nWith this terminology, the hypothesis mentioned above is the following.\nWe say a connected reductive group H is classical, if each simple factor of\nits Lie algebra H is of type A, B, C or D. We also say that the type of H\nor H is classical.\nHypothesis 1.2.3 (PH2): [KTT, DM2] Assume that H in Problem 1.2.1\nis classical.\nThen the Littlewood-Richardson stretching quasi-polynomial\n ̃cλ\nα,β(n) is strictly positive.\nWe shall refer to this as the second positivity hypothesis (PH2). This\nwas conjectured by King, Tollu and Toumazet [KTT] for type A, and De\nLoera and McAllister for types B, C, D. Since the stretching function above\nis a polynomial in type A, the positivity conjecture of King et al clearly\nimplies the saturation theorem of Knutson and Tao. That is, PH2 implies\nSH for type A.\n13"},{"page":17,"text":"We can formulate an analogue of SH for a Lie algerbra of arbitrary clas-\nsical type so that PH2 implies SH for an arbitrary type. For this, we need\nto formulate the notion of a saturated quasi-polynomial, which is not con-\ntradicted by the counterexamples, mentioned above, to verbatim translation\nof the saturation property in [KT1, Ki] to the setting of quasi-polynomials.\nSpecifically, the notion of saturation in [KT1, Ki] works well if the stretching\nfunction is a polynomial, but not so if it is a quasipolynomial. Let f(n) be\na quasi-polynomial with period l. Let fj(n), 1 ≤j ≤l, be the polynomials\nsuch that f(n) = fj(n) if n = j mod l. The index of f, index(f), is defined\nto be the smallest j such that the polynomial fj(n) is not identically zero.\nIf f(n) is identically zero, we let index(f) = 0. If f(1) ̸= 0, then clearly\nindex(f) = 1.\nDefinition 1.2.4 We say that f(n) is strictly saturated if for any i: fi(n) >\n0 for every n ≥1 whenever fi(n) is not identically zero. The saturation in-\ndex s(f) of f is defined to be the smallest nonnegative integer such that\nf(n + s(f)) is strictly saturated. We also say that f(n) is saturated with\nindex s(f).\nThus f(n) is strictly saturated iffits saturation index is zero. Clearly\nthe saturation idex is bounded above by the positivity index. Thus if f(n)\nis strictly positive, it is strictly saturated. Hence, PH2 (Hypothesis 1.2.3)\nimplies:\nHypothesis 1.2.5 (SH): The Littlewood-Richardson stretching quasi-polynomial\ncλ\nα,β(n) of arbitary classical type is strictly saturated.\nThe polynomial time algorithm in [GCT5] works assuming SH as well.\nFor the Littlewood-Richardson coefficient of type A, the notion of strict\nsaturation here coincides with the notion of saturation in [KT1] since cλ\nα,β(n)\nis a polynomial in that case.\nKnutson and Tao [KT1] also conjectured\na generalized saturation property for arbitrary types. But that property,\nunlike the one defined above, is only conjectured to be sufficient, but not\nclaimed to be, or expected to be necessary. For this reason, it cannot be\nused in the complexity-theoretic applications in this paper.\nThere is another positivity conjecture for Littlewood-Richardson coeffi-\ncients that also implies the saturation theorem of Knutson and Tao. For\nthis consider the generating function\nCλ\nα,β(t) =\nX\nn≥0\n ̃cλ\nα,β(n)tn.\n(1.2)\n14"},{"page":18,"text":"It is a rational function since ̃cλ\nα,β(n) is a quasi-polynomial [St1]. For type\nA, if ̃cλ\nα,β(n) is not identically zero, then Cλ\nα,β(t) is a rational function of\nform\nhdtd + · · · + h0\n(1 −t)d+1\n,\n(1.3)\nsince ̃cλ\nα,β(n) is a polynomial [St1]. It is conjectured in [KTT] that:\nHypothesis 1.2.6 (PH3:) The coefficients hi’s in eq.(1.3) are nonnegative\n(and h0 = 1).\nWe shall call this the third positivity hypothesis (PH3). It clearly implies SH\nfor Littlewood-Richardson coefficients of type A. To describe its analogue\nfor arbitrary classical type we need a definition.\nLet F(t) = P\nn f(n)tn be the generating function associated with the\nquasi-polynomial f(n). It is a rational function [St1].\nDefinition 1.2.7 We say that F(t) has a positive form, if, when f(n) is\nnot identically zero, it can be expressed in the form\nF(t) = hdtd + · · · + h0\nQk\ni=0(1 −tai)di ,\n(1.4)\nwhere (1) h0 = 1, and hi’s are nonnegative integers, (2) ai’s and di’s are\npositive integers, (3) P\ni di = d + 1, where d = max deg(fj(n)) is the degree\nof f(n).\nWe define the modular index of this positive form to be max{ai}.\nIf F(t) has a positive form with a0 = 1, then f(n) is strictly saturated\n(Definition 1.2.4); this easily follows from the power series expansion of the\nright hand side of eq.(1.4).\nThe analogue of Hypothesis 1.2.6 for arbitrary classical type is:\nHypothesis 1.2.8 (PH3:) The rational function Cλ\nα,β(t) has a positive\nform, with a0 = 1, of modular index bounded by a constant depending only\non the types of the simple factors of the Lie algebra of H.\nThis too implies SH for arbitrary classical type. For types B, C, D, the\nconstant above is 2. Experimental evidence for this hypothesis is given in\nSection 6.1.\n15"},{"page":19,"text":"The analogue of the PH3, even in the more general q-setting, is known to\nhold for the generating function of the Kostant partition function of type A,\nand more generally, for a parabolic Kostant partition function; cf. Kirillov\n[Ki]. This also gives a support for the PH3 above, given a close relationship\nbetween Littlewood-Richardson coefficients and Kostant partition functions\n[FH].\n1.3\nBack to the general decision problems\nIt may be remarked that the Littlewood-Richardson problem actually never\narises in the flip. It is only used as a simplest proptotype of the actual (much\nharder) problems that arise–namely relaxed forms of Problems 1.1.1-1.1.4.\nNow we turn to these problems. The goal is to generalize the preced-\ning results and hypotheses for the Littlewood-Richardson coefficients to the\nstructural constants that arise in these problems. The problem of finding a\npositive, combinatorial formula for the plethysm constant (Problem 1.1.2),\nakin to the positive Littlewood-Richardson rule, has already been recog-\nnized as an outstanding, classical problem in representation theory [St4]–\nthe known formulae based on character theory mentioned in Section 1.1\nare not positive, because they involve alternating signs. Indeed, existence\nof such a formula is a part of the first positivity hypothesis (PH1) below\nfor the plethysm constant, and this problem is the main focus of the work\nin [GCT4, GCT7, GCT8, GCT9].\nIn view of the intensive work on the\nplethym constant in the literature, it has now become clear that the com-\nplexity of the plethysm problem (Problem 1.1.2) is far higher than that of\nthe Littlewood-Richardson problem (Problem 1.2.1). This gap in the com-\nplexity is the main source of difficulties that has to be addressed. We now\nstate the main ingradients in the plan in this paper to show that the relaxed\nforms of Problems 1.1.1, 1.1.2, 1.1.3, and 1.1.4, with X = G/P or a class\nvariety, belong to P.\n1.4\nSaturated and positive integer programming\nFirst, we formulate a general algorithmic paradigm of saturated and positive\ninteger programming that can be applied in the context of these problems.\nLet A be an m×n integer matrix, and b an integral m-vector. An integer\nprogramming problem asks if the polytope P : Ax ≤b contains an integer\n16"},{"page":20,"text":"point. In general, it is NP-complete. We want to define its relaxed version,\nwhich will turn out to have a polynomial time algorithm.\nWe allow m, the number of constraints, to be exponential in n. Hence,\nwe cannot assume that A and b are explicitly specified. Rather, it is assumed\nthat the polytope P is specified in the form of a (polynomial-time) separation\noracle in the spirit of Gr ̈otschel, Lov ́asz and Schrijver [GLS]; cf. Section 2.3.\nGiven a point x ∈Rn, the separation oracle tells if x ∈P, and if not, gives\na hyperplane that separates x from P.\nLet fP(n) be the Ehrhart quasi-polynomial of P [St1]. By definition,\nfP (n) is the number of integer points in the dilated polytope nP.\nAn integer programming problem is called saturated, if\n1. The specification of P also contains a number sie(P), called the sat-\nuration index estimate, with the guarantee that the saturation in-\ndex s(fP) ≤sie(P); cf. Definition 1.2.4. In particular, this means\nfP(n + sie(P)) is strictly saturated.\n2. the goal of the problem is to give an efficient algorithm to decide if,\ngiven an integral relaxation parameter c > sie(P), if cP contains an\ninteger point.\nThe algorithm has to work only for relaxation parameters c > sie(P). In\nparticular, if sie(P) ≥1, the algorithm problem does not have to determine\nif P contains an integer point.\nAn integer programming problem is called positive, if\n1. the specification of P also contains a number pie(P), called the pos-\nitivity index estimate, with the guarantee that the positivity index\np(fP) ≤pie(P); cf. Definition 1.2.2. In particular, this means fP(n +\npie(P)) is strictly positive.\n2. the goal of the problem is to give an efficient algorithm to decide if,\ngiven an integral relaxation parameter c > pie(P), if cP contains an\ninteger point.\nAgain, the algorithm has to work only for relaxation parameters c > pie(P).\nSince s(fP) ≤p(fP), a positive integer programming problem is also satu-\nrated.\nThe following is the main complexity-theoretic result in this paper.\n17"},{"page":21,"text":"Theorem 1.4.1 (cf. Section 3.1)\n1. Index of the Ehrhart quasi-polynomial fP(n) of a polytope P presented\nby a separation oracle can be computed in oracle-polynomial time, and\nhence, in polynomial time, assuming that the oracle works in polyno-\nmial time.\n2. A saturated, and hence positive, integer programming problem has a\npolynomial time algorithm.\n3. Suppose the polytopes P’s that arise in a specific decision problem have\nthe following property: whenever P is nonempty, the Ehrhart quasi-\npolynomial fP(n) is “almost always” strictly saturated.\nThen there\nexists a polynomial time algorithm for deciding if P contains an integer\npoint that works correctly “almost always”.\nThe meaning of the phrase “almost always” in the context of the decision\nproblems in this paper will be specified later (cf. Theorem 3.1.1).\nIt may be remarked that the index as well as the period of the Ehrhart\nquasi-polynomial can be exponential in the bit length of the specification\nof P. In contrast to the polynomial time algorithm above to compute the\nindex, the known algorithms to compute the period (e.g. [W]) take time\nthat is exponential in the dimension of P. It may be conjectured that one\ncannot do much better: i.e., the period, unlike the index here, cannot be\ncomputed in polynomial time, in fact, even in 2o(dim(P )) time.\nThe algorithm in Theorem 1.4.1 is based on the separation-oracle-based\nlinear programming algorithm of Gr ̈otschel, Lov ́asz and Schrijver [GLS], and\na polynomial time algorithm for computing the Smith normal form [KB].\nThe paradigm of saturated integer programming is useful when one\nknows, a priori, a good estimate for the saturation index of the polytope\nunder consideration, or when the saturation index is almost always zero.\nFor example, if P is the hive polypolype for the Littlewood-Richardson co-\nefficient (type A), then sie(P) = 0, by the saturation theorem [KT1], and\npie(P) = 0, by PH2 (Hypothesis 1.2.3). For the polytopes P that would\narise in this paper, sie(P) and pie(P) would in general be nonzero, but con-\njecturally always small, and sie(P) would conjecturally be almost always\nzero.\n18"},{"page":22,"text":"1.5\nQuasi-polynomiality, positivity hypotheses, and\nthe canonical models\nThe basic goal now is to use Theorem 1.4.1 to get polynomial time algorithms\nto decide nonvanishing, modulo small relaxation, of the structural constants\nin Problems 1.1.1, 1.1.2, 1.1.3 and 1.1.4, with X = G/P or a class variety.\nThe main results in this paper which go towards this goal are as follows.\nQuasi-polynomiality\nWe associate stretching functions with the structural constants in Prob-\nlems 1.1.1-1.1.4, akin to the stretching function ̃cλ\nα,β(n) in eq.(1.1) associ-\nated with the Littlewwod-Richardson coefficient, and show that they are\nquasipolynomials; cf. Chapter 4. (But their periods need not be constants,\nas in the case of Littlewood-Richardson coefficients; in fact, they may be\nexponential in general.) In particular, this proves Kirillov’s conjecture [Ki]\nfor the plethysm constants. The proof is an extension of Brion’s remarkable\nproof (cf.\n[Dh]) of quasi-polynomiality of the stretching function associ-\nated with the Littlewood-Richardson coefficient.\nThe main ingradient in\nthe proof is Boutot’s result [Bou] that singularities of the quotient of an\naffine variety with rational singularities with respect to the action of a re-\nductive group are also rational. This is a generalization of an earlier result\nof Hochster and Roberts [Ho] in the theory of Cohen-Macauley rings.\nSaturation and positivity hypotheses\nUsing the stretching quasipolynomials above, we formulate (cf. Section 3.3)\nanalogues of the saturation and positivity hypotheses SH, PH1,PH2,PH3 in\nSection 1.2 for the structural constants in Problems 1.1.1-1.1.3 and Prob-\nlem 1.1.4, with X = G/P or a class variety. As for Littlewood-Richardson\ncoefficients, it turns out that PH2 implies SH. The hypotheses PH1 and SH\n(more strongly, PH2) together imply that the problem of deciding nonvan-\nishing of the structural constant in any of these problems, modulo a small\nrelaxation, can be transformed in polynomial time into a saturated (more\nstrongly, positive) integer programming problem, and hence, can be solved\nin polynomial time by Theorem 1.4.1.\nIn particular, this shows that all\nthe relaxed decision problems that arise in flip (cf. Hypothesis 1.1.6) have\npolynomial time algorithms, assuming these positivity hypotheses. Though\nthese algorithms are elementary, the positivity hypotheses on which their\n19"},{"page":23,"text":"correctness depends turn out to be nonelementary.\nThey are intimately\nlinked to the fundamental phenomena in algebraic geometry and the theory\nof quantum groups, as we shall see.\nWe also give theoretical and experimental results in support of these\nhypotheses; cf. Chapter 4-6.\nCanonical models\nThe proofs of quasi-polynomiality mentioned above also associate with each\nstructural constant under consideration a projective scheme, called the canon-\nical model, whose Hilbert function coicides with the stretching quasi-polynomial\nassociated with that structural constant, akin to the model associated by\nBrion [Dh] with the Littlewood-Richardson coefficient.\nThese canonical\nmodels play a crucial role in the approach to the posivity hypotheses sug-\ngested in Section 1.7.\n1.6\nThe plethysm problem\nWe now give precise statements of these results and hypotheses for the\nplethysm problem (Problem 1.1.2). It is the main prototype in this paper,\nwhich illustrates the basic ideas. Precise statements for the more general\nProblems 1.1.3 and 1.1.4 appear in Section 3.3.\nAs for the Littlewood-Richardson coefficients (cf.(1.1)), Kirillov [Ki] as-\nsociates with a plethysm constant aπ\nλ,μ a stretching function\n ̃aπ\nλ,μ(n) = anπ\nnλ,μ,\n(1.5)\nand a generating function\nAπ\nλ,μ(t) =\nX\nn≥0\nanπ\nnλ,μtn.\n(Note that μ is not stretched in these definitions.)\nHe conjectured that Aπ\nλ,μ(t) is a rational function. This is verified here\nin a stronger form:\nTheorem 1.6.1 (a) (Rationality) The generating function Aπ\nλ,μ(t) is ratio-\nnal.\n20"},{"page":24,"text":"(b) (Quasi-polynomiality) The stretching function ̃aπ\nλ,μ(n) is a quasi-polynomial\nfunction of n. This is equivalent to saying that all poles of Aπ\nλ,μ(t) are roots\nof unity, and the degree of the numerator of Aπ\nλ,μ(t) is strictly smaller than\nthat of the denominator.\n(c) There exist graded, normal C-algebras S = S(aπ\nλ,μ) = ⊕nSn, and T =\nT(aπ\nλ,μ) = ⊕nTn such that:\n1. The schemes spec(S) and spec(T) are normal and have rational singu-\nlarities.\n2. T = SH, the subring of H-invariants in S, where H = GLn(C) as in\nProblem 1.1.2,\n3. The quasi-polynomial ̃aπ\nλ,μ(n) is the Hilbert function of T. In other\nwords, it is the Hilbert function of the homogeneous coordinate ring of\nthe projective scheme Proj(T).\n(d) (Positivity) The rational function Aπ\nλ,μ(t) can be expressed in a positive\nform:\nAπ\nλ,μ(t) = h0 + h1t + · · · + hdtd\nQ\nj(1 −ta(j))d(j)\n,\n(1.6)\nwhere a(j)’s and d(j)’s are positive integers, P\nj d(j) = d + 1, where d is\nthe degree of the quasi-polynomial ̃aπ\nλ,μ(n), h0 = 1, and hi’s are nonnegative\nintegers.\nThe specific rings S(aπ\nλ,μ) and T(aπ\nλ,μ) constructed in the proof of The-\norem 1.6.1 are very special. We call them canonical rings associated with\nthe plethysm constant aπ\nλ,μ. We call Y (aπ\nλ,μ) = Proj(S(aπ\nλ,μ)), and Z(aπ\nλ,μ) =\nProj(T(aπ\nλ,μ)) the canonical models associated with aπ\nλ,μ. The canonical rings\nare their homogenous coordinate rings.\nIt may be remarked that the analogue of Theorem 1.6.1 (b) for Littlewood-\nRichardson coefficients has an elementary polyhedral proof. Specifically, the\nLittlewood-Richardson stretching function ̃cλ\nα,β(n) of any type is a quasi-\npolynomial since it coincides with the Ehrhart quasi-polynomial of the BZ-\npolytope [BZ]. Similarly, the analogue of Theorem 1.6.1 (d) for Littlewood-\nRichardson coefficients follows from Stanley’s positivity theorem for the\nEhrhart series of a rational polytope (which is implicit in [St3]).\nThese\npolyhderal proofs cannot be extended to the plethysm constant at this point,\n21"},{"page":25,"text":"since no polyhedral expression for them is known so far–in fact, this is a part\nof the conjectural positivity hypothesis PH1 below.\nIn contrast, Brion’s\nproof in [Dh] of quasi-polynomiality of ̃cλ\nα,β(n) can be extended to prove\nTheorem 1.6.1 since it does not need a polyhedral interpretation for aπ\nλ,μ.\nBut Boutot’s result [Bou] that it relies on is nonelementary (because it needs\nresolution of singularities in characteristic zero [Hi], among other things).\nWe also give an elementary (nonpolyhedral proof) for Theorem 1.6.1 (a) (ra-\ntionality). But this does not extend to a proof of quasipolynomiality for all\nn, which turns out to be a far delicate problem. It is crucial in the context\nof saturated integer programming.\nTheorem 1.6.2 (Finitely generated cone)\nFor a fixed partition μ, let Tμ be the set of pairs (π, λ) such that the\nirreducible representation Vπ(H) of H = GLn(C) occurs in the irreducible\nrepresentation Vλ(G) of G = GL(Vμ(H)) with nonzero multiplicity. Then\nTμ is a finitely generated semigroup with respect to addition.\nThis is proved by an extension of Brion and Knop’s proof of the analogous\nresult for Littlewood-Richardson coefficients based on invariant theory. In\nthe case of Littlewood-Richardson coefficients, this again has an elementary\npolyhedral proof [Z].\nTheorem 1.6.3 (PSPACE)\nGiven partitions λ, μ, π, the plethysm constant aπ\nλ,μ can be computed in\npoly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) space.\nThe main observation in the proof of Theorem 1.6.3 is that the oldest\nalgorithm for computing the plethysm constant, based on the Weyl character\nformula, can be efficiently parallelized so as to work in polynomial parallel\ntime using exponentially many processors. After this, the result follows from\nthe relationship between parallel and space complexity classes. It may be\nremarked that the known algorithms for computing aπ\nλ,μ in the literature–\ne.g., the one based on Klimyk’s formula [FH]–take exponential time as well\nas space.\nTheorems 1.6.1, 1.6.2 and 1.6.3 lead to the following conjectural sat-\nuration and positivity hypotheses for the plethysm constant.\nThese are\nanalogues of PH1,PH2,PH3, SH in Section 1.2 for Littlewood-Richardson\ncoefficients.\n22"},{"page":26,"text":"Hypothesis 1.6.4 (PH1)\nFor every (λ, μ, π) there exists a polytope P = P π\nλ,μ ⊆Rm such that:\n(1) The Ehrhart quasi-polynomial of P coincides with the stretching quasi-\npolynomial ̃aπ\nλ,μ(n) in Theorem 1.6.1. (This means P is given by a linear\nsystem of the form\nAx ≤b,\n(1.7)\nwhere A does not depend on λ and π and b depends only on λ and π in a\nhomogeneous, linear fashion.) In particular,\naπ\nλ,μ = φ(P),\n(1.8)\nwhere φ(P) is equal to the number of integer points in P.\n(2) The dimension m of the ambient space, and hence the dimension of P\nas well, and the bitlength of every entry in A are polynomial in the bitlength\nof μ and the heights of λ and π.\n(3) Whether a point x ∈Rm lies in P can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, ⟨x⟩)\ntime. That is, the membership problem belongs to the complexity class P.\nIf x does not lie in P, then this membership algorithm also outputs, in the\nspirit of [GLS], the specification of a hyperplane separating x from P.\nThe first statement here, in particular, would imply a positive, polyhedral\nformula for aμ\nλ,α, in the spirit of the known positive polyhedral formulae for\nthe Littlewood-Richardson coefficients in terms of the BZ- [BZ], hive [KT1]\nor other types of polytopes [Dh]. It would also imply polyhedral proofs for\nTheorem 1.6.1 (a), (b), (d), and Theorem 1.6.2. Conversely, Theorem 1.6.1\n(a), (b), (d), and Theorem 1.6.2 constitute a theoretical evidence for exis-\ntence of such a positive polyhedral formula.\nThe second statement in PH1 is justified by Theorem 1.6.3.\nSpecifi-\ncally, it should be possible to compute the number of integer points in P\nin PSPACE in view of Theorem 1.6.3. If dim(P) and m were exponential,\nthen the usual algorithms for this problem, e.g. Barvinok [Bar], cannot be\nmade to work in PSPACE. Indeed, it may be conjectured that the number\nof integer points in a general polytope P ⊆Rm can not be computed in\no(m) space.\nThe number of constraints in the hive [KT1] or the BZ-polytope [BZ]\nfor the Littlewood-Richardson coefficient cλ\nα,β is polynomial in the number\nof parts of α, β, λ. In contrast, the number of constraints defining P π\nλ,μ may\nbe exponential in the ⟨μ⟩and the number of parts of λ and π. But this is\n23"},{"page":27,"text":"not a serious problem. As long as the faces of the polytope P have a nice\ndescription, the third statement in PH1 is a reasonable assumption. This\nhas been demonstrated in [GLS] for the well-behaved polytopes in combina-\ntorial optimization with exponentially many constraints. The situation in\nrepresentation theory should be similar, or even better. For example, the\nfacets of the hive polytope [KT1] are far nicer than the facets of a typical\npolytope in combinatorial optimization.\nIt is known that membership in a polytope is a “very easy” problem.\nFormally, if a polytope has polynomially many constraints, this problem\nbelongs to the complexity class NC ⊆P [KR], the subclass of problems\nwith efficient parallel algorithms, which is very low in the usual complexity\nhierarchy. Even if the number of constraints of P π\nλ,μ in PH1 is exponen-\ntial, the membership problem may still be conjectured to be in NC (cf.\nRemarkrnc)–which would be “very easy” compared to the decision problem\nwe began with (Problem 1.1.2). For this reason, PH1 is primarily a mathe-\nmatical positivity hypothesis as against PHflip (Hypothesis 1.1.6), and the\npositive, polyhedral formula for aπ\nλ,μ in (1.8) is its main content.\nThe remaining positivity hypotheses are purely mathematical.\nThey\ngeneralize SH,PH2 and PH3 for the Littlewood-Richardson coefficients to\nthe plethysm constants. We turn their specification next. We can begin\nby asking if the stretching quasipolynomial ̃aπ\nλ,μ(n) is strictly saturated or\npositive.\nThis need not be so.\nThe recent article [Ro] shows that strict\nsaturation need not hold for the Kronecker coefficients, as was conjectured\nin the earlier version of this paper. A similar phenomenon was also reported\nin [GCT7, GCT8], where it was observed that the structural constants of\nthe nonstandard quantum groups associated with the plethysm problem (of\nwhich the Kronecker problem is a special case) need not satisfy an analogue\nof PH2. But it was observed there that the positivity (and hence saturation)\nindices of these structural constants are small, though not always zero; eg.\nsee Figures 30,33,35 in [GCT8]. The same can be expected here. This is\nalso supported by the experimental evidence in [BOR] where too it may be\nobserved that the positivity index is small. Furthermore, in the special case\n(n = 2) of the Kronecker problem analysed in [BOR], the saturation index\nis zero for almost all Kronecker coefficients.\nThese considerations suggest:\nHypothesis 1.6.5 (SH)\n(a): The saturation index (Definition 1.2.4) of ̃aπ\nλ,μ(n) is bounded by a poly-\nnomial in the dimension of G in Problem 1.1.2 and the heights of λ and π.\n24"},{"page":28,"text":"This means there exist absolute nonnegative constants c and c′, independent\nof n, λ, μ and π, such that the saturation index is bounded above by chc′,\nwhere h = dim G + htλ + htπ.\n(b): The quasi-polynomial ̃aπ\nλ,μ(n) is strictly saturated, i.e. the saturation\nindex is zero, for almost all λ, μ, π. Specifically, the density of the triples\n(λ, μ, π) of total bit length N with nonzero aπ\nλ,μ for which the saturation index\nis not zero is less than 1/N c′′, for any positive constant c′′, as N →∞.\nA stronger form of (a) is:\nHypothesis 1.6.6 (PH2) The positivity index (Definition 1.2.2) of the\nstretching quasi-polynomial ̃aπ\nλ,μ(n) is bounded by a polynomial in the di-\nmension of G and the heights of λ and π.\nThe following is another stronger form of SH (a). For this, we observe\nthat the positive rational form in Theorem 1.6.1 (d) is not unique. Indeed,\nthere is one such form for every h.s.o.p. (homogeneous sequence of param-\neters) of the homogenenous coordinate ring S; the a(j)’s in eq.(1.6) are the\ndegrees of these parameters.\nKirillov asked if the only possible pole of Aπ\nλ,μ is at t = 1–i.e. if aμ\nλ,α(n) is\na polynomial. This is not so (cf. Section 6.2). But it may be conjectured that\nthe structural constants a(j)’s are small. Specifcally, consider an h.s.o.p. of\nS with a (lexicographically) minimum degree sequence, and call the (unique)\npositive rational form in Theorem 1.6.1 (d) associated with such an h.s.o.p.\nminimal. The modular index χ(aπ\nλ,μ) of the plethysm constant is defined to\nbe the modular index (Definition 1.2.7) of this minimal positive form. Then:\nHypothesis 1.6.7 (PH3)\nThe function Aπ\nλ,μ(t) associated with aπ\nλ,μ has a positive rational form\nwith modular index bounded by a polynomial in the dimension of G and the\nheights of λ and π.\nMore specifically, this is so for the minimial positive rational form of\nAπ\nλ,μ(t) as above; i.e., the modular index χ(aπ\nλ,μ) is itself bounded by a poly-\nnomial in the dimension of G and the heights of λ and π.\nThis is a conjectural analogue of a stronger form of PH3 for Littlewood-\nRichardson coefficients (Hypothesis 1.2.6), which says that the modular in-\ndex of a Littlewood-Richardson coefficient, defined similarly, is one. PH3\n25"},{"page":29,"text":"here would imply that the period of Aπ\nλ,μ(t) is smooth–i.e. has small prime\nfactors–though it may be exponential in the heights of λ, μ, π. It can be\nshown that PH3 implies SH (a) (Section 3.3).\nThe following result addresses the second arrow in Figure 1.1 in the\ncontext of the relaxed decision problem for the plethysm constant:\nTheorem 1.6.8 The complexity theoretic positivity hypothesis PHflip (Hy-\npothesis 1.1.6) for the plethysm constant is implied by the mathematical\npositivity hypotheses PH1 and SH above. Specifically, assuming PH1 and\nSH:\n(a) Nonvanishing of abπ\nbλ,bμ for any b > chc′, with c, c′, h as in SH, can be\ndecided in O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, ⟨b⟩)) time.\n(b) There is an O(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)) time algorithm for deciding if aπ\nλ,μ is\nnonvanishing, which works correctly on almost all λ, μ and π; almost all\nmeans the same as in SH.\nHere (a) follows by applying Theorem 1.4.1 (2) to the polytope P π\nλ,μ in\nPH1, and letting the positivity index estimate for this polytope be chc′; (b)\nfollows from Theorem 1.4.1 (3).\nEvidence for the positivity hypotheses in special cases\nLittlewood-Richardson coefficients are special cases of (generalized) plethym\nconstants. We have already seen that PH1 holds in this case, and that there\nis considerable experimental evidence for PH2 and PH3 (Section 1.2). An-\nother crucial special case of the plethym problem is the Kronecker prob-\nlem (Problem 1.1.1)–in fact, this may be considered to be the crux of the\nplethysm problem. It follows from the results in [GCT9] that PH1 holds for\nthe Kronecker problem when n = 2; the earlier known formulae [RW, Ro]\nfor the Kronecker coefficient in this case are not positive. It can also be seen\nfrom the experimental evidence in [BOR] that the saturation and positivity\nindices of the Kronecker coefficient, for n = 2, are very small, and almost\nalways zero. We also give in Chapter 6 additional experimental evidence for\nPH2 for another basic special case of Problem 1.1.3, with H therein being\nthe symmetric group.\n26"},{"page":30,"text":"1.7\nTowards PH1, SH, PH2,PH3 via canonial bases\nand canonical models\nIn this section, we suggest an approach to prove PH1, SH, PH2 and PH3 for\nthe plethysm constant and the analogous hypotheses for the other structural\nconstants in Problems 1.1.3, and 1.1.4, with X = G/P or a class variety.\nIn the case of Littlewood-Richardson coefficients of type A, PH1 and SH\nhave purely combinatorial proofs. But it seems unrealistic to expect such\nproofs of the saturation and positivity hypotheses for the plethysm and\nother structural constants under consideration here given their substantially\nhigher complexity.\nThe approach that we suggest is motivated by the proof of PH1 for\nLittlewood-Richardson coefficients of arbitrary types based on the canonical\n(local/global crystal) bases of Kashiwara and Lusztig for representations of\nDrinfeld-Jimbo quantum groups [Dh, Kas2, Li, Lu2, Lu4]. By a Drinfeld-\nJimbo quantum group we shall mean in this paper quantization Gq of a\ncomplex, semisimple group G as in [RTF] that is dual to the Drinfeld-Jimbo\nquantized enveloping algebra [Dri]. Canonical bases for representions of a\nDrinfeld-Jimbo quantum group in type A are intimately linked [GrL] to the\nKazhdan-Lusztig basis for Hecke algebras [KL1, KL2]. A starting point for\nthe approach suggested here is:\nObservation 1.7.1 (PH0) The homogeneous coordinate rings of the canon-\nical models associated by Brion with the Littlewood-Richardson coefficients\nhave quantizations endowed with canonical bases as per Kashiwara and Lusztig.\nThis is a consequence of the work of Kashiwara [Kas3] and Lusztig [Lu3,\nLu4]; see Proposition 4.2.1 for its precise statment. This is why we call the\nmodels here canonical models.\nWe shall refer to the property above as the zeroeth positivity hypothesis\nPH0. Positivity here refers to the deep characteristic positivity property of\nthe canonical basis proved by Lusztig: namely its multiplicative and comul-\ntiplicative structure constants are nonnegative. For this reason, we say that\na canonical basis is positive. Similar positivity property is also known for\nthe Kazhdan-Lusztig basis [KL2]. The proofs of these positivity properties\nare based on the Riemann hypothesis over finite fields (Weil conjectures)\n[Dl] and the related work of Beilinson, Bernstein, Deligne [BBD].\nThe property above is called PH0 because it implies PH1 for Littlewood-\nRichardson coefficients of arbitrary types. Specifically, the latter is a formal\n27"},{"page":31,"text":"consequence of the abstract properties of these canonical bases and is inti-\nmately related to their positivity; cf. Section 4.2.1, and [Dh, Kas2, Li, Lu4].\nThe saturation hypothesis SH in type A [KT1] is a refined property of the\npolyhedral formulae in PH1. In Section 4.2 we suggest an approach to prove\nSH, PH2 and PH3 for arbitrary types based on the properties of these canon-\nical bases. All this indicates that for the Littlewood-Richardson problem\nPH1, SH, PH2 and PH3 are intimately linked to PH0.\nThis suggests the following approach for proving PH1, SH, PH2 and PH3\nfor the plethysm and other structural constants under consideration in this\npaper (cf. Section 4.2.2):\n1. Construct quantizations of the homogeneous coordinate rings of the\ncanonical models associated with these structural constants,\n2. Show that they have canonical bases in some appropriate sense thereby\nextending PH0 to this general setting.\n3. Prove PH1, SH, PH2, and PH3 by a detailed analysis and study of\nthese canonical bases as per this extended PH0, just as in the case of\nLittlewood-Richardson coefficients.\nPictorially, this is depicted in Figure 1.2.\nQuantizations of the homogeneous coordinate rings of the canonical\nmodels associated with Littlewood-Richardson coefficients and their posi-\ntive canonical bases are constructed using the theory Drinfeld-Jimbo quan-\ntum group. In type A, it is intimately related to the theory of Hecke al-\ngebras. But, as expected, the theories of Drinfeld-Jimbo quantum groups\nand Hecke algebras do not work for the plethysm problem. What is needed\nis a quantum group and a quantized algebra that can play the same role\nin the plethysm problem that the Drinfeld-Jimbo quantum group and the\nHecke algebra play in the Littlewood-Richardson problem. These have been\nconstructed in [GCT4] for the Kronecker problem (Problem 1.1.1) and in\n[GCT7] for the generalized plethysm problem (Problem 1.1.2). We shall call\nthem nonstandard quantum groups and nonstandard quantized algebras; cf.\nSection 4.3 for their brief overview. In the special case of the Littlewood-\nRichardson problem, these specialize to the Drinfeld-Jimbo quantum group\nand the Hecke algebra, respectively. The article [GCT8] gives conjecturally\ncorrect algorithms to construct canonical bases of the matrix coordinate\nrings of the nonstandard quantum groups and of nonstandard algebras that\nhave conjectural positivity properties analogous to those of the canonical\n28"},{"page":32,"text":"Construction of quantizations of the coordinate rings of canonical models\n|\n|\n|\n↓\nConstruction of canonical bases for these quantizations (PH0)\n|\n|\n|\n↓\nPositivity and saturation hypotheses PH1, SH |\n|\n|\n↓\nPolynomial-time algorithms for the relaxed decision problems\nFigure 1.2: Pictorial depiction of the approach\n(global crystal) bases, as per Kashiwara and Lusztig, of the coordinate ring\nof the Drinfeld-Jimbo quantum group, and the Kazhdan-Lusztig basis of the\nHecke algebra. These conjectures lie at the heart of the approach suggested\nhere, since they are crucial for the extension of PH0 (cf. Figure 1.2) to the\ngeneral setting here. Their verification seems to need substantial extension\nof the work surrounding the Riemann hypothesis over finite fields mentioned\nabove.\n1.8\nBasic plan for implementing the flip\nThe main application of the results and hypotheses in this paper in the\ncontext of the flip is the following result. As mentioned in Section 1.1, and\ndescribed in more detail in Sections 7.6-7.7, each lower bound problem, such\nas the P vs. NP problem over C, is reduced in [GCT1, GCT2] to the prob-\nlem of proving obstructions to embeddings among the class varieties that\narise in the problem. In Chapter 7 we define a robust obstruction, which is\nan obstruction that is well behaved with respect to relaxation, and whose\nvalidity (correctness) depends only on an appropriate PH1 but not SH. It is\n29"},{"page":33,"text":"conjectured that in each of the lower bound problems under consideration,\nrobust obstructions exist (Section 7.6.6). In the lower bound problems un-\nder consideration, ultimately one is only interested in proving existence of\nobstructions. So one may as well search for only robust obstructions.\nTheorem 1.8.1 (cf. Chapter 7) Consider the P vs. NP or the NC vs.\nP #P problem over C [GCT1].\nAssume that the homogeneous coordinate\nrings of the relevant class varieties [GCT1, GCT2] in this context have ra-\ntional singularities.\nAlso assume that the structural constants associated\nwith these class varieties satisfy analogous PH1 as specified in Chapter 7.\nThen:\n(a) The problem of verifying a robust obstruction in each of these problems\nbelongs to P, so also the relaxed form of the problem of verifying any ob-\nstruction (not necessarily robust).\n(b) There exists an explicit family of robust obstructions in each of these\nproblems assuming an additional hypothesis OH specified in Chapter 7; the\nmeaning of the term explicit is also given there.\n(b) The problem of deciding existence of a geometric obstruction also belongs\nto P, assuming a stronger form of PH1 specified in Chapter 7. Here geomet-\nric obstruction is a simpler type of robust obstruction, defined in Chapter 7,\nwhich is conjectured to exist in the lower bound problems under considera-\ntion.\nFor a precise statement of this theorem, see Chapter 7.\nThis theorem needs only PH1, but not SH, which is only needed to\nargue why robust obstructions should exist (Section 7.6.6), and furthermore,\nit is only needed for Problems 1.1.1-1.1.3 and not for the GIT Problem\n1.1.4. Thus PH1 is the main positivity hypothesis of GCT in the context\nproving existence of (robust) obstructions for the lower bound problems\nunder consideration.\nA basic plan for implementing the flip suggested by the considerations\nabove is summarized in Figure 1.3. It is an elaboration of Figure 1.1. Ques-\ntion marks in the figure indicate open problems.\n1.9\nOrganization of the paper\nThe rest of this paper is organized as follows.\n30"},{"page":34,"text":"Negative hypotheses in complexity theory (Lower bound problems)\n|\n|\nThe flip\n|\n↓\nPositive hypotheses in complexity theory (Upper bound problems)\n|\n|\nSaturated and positive integer programming, and\nthe quasi-polynomiality results in this paper\n|\n↓\nMathematical saturation and positivity hypotheses: PH1,SH (PH2,3)\n|\n|\nConstruction of the canonical models in this paper, and\nconstruction of the quantum groups in GCT4,7\n|\n??\n|\n↓\n(PH0): Construction of quantizations of the coordinate\nrings of the canonical models and their canonical bases\n|\n|\n|\n??\n|\n|\n↓\n(?): Problems related to the Riemann Hypothesis over finite\nfields, and their generalizations\nFigure 1.3: A basic plan for implementing the flip\n31"},{"page":35,"text":"In Chapter 2 we describe the basic complexity theoretic notions that we\nneed in this paper and describe their significance in the context of represen-\ntation theory.\nIn Chapter 3, we give a polynomial time algorithm for saturated integer\nprogramming (Theorem 1.4.1), and give precise statements of the results\nand positivity hypotheses for Problems 1.1.3 and 1.1.4 (with X = G/P or\na class variety) mentioned in Section 1.5. These generalize the ones given\nin Section 1.6 for the plethysm constant. The framework of saturated in-\nteger programming in this paper may be applicable to many other struc-\ntural constants in representation theory and algebraic geometry, such as the\nKazhdan-Lusztig polynomials (cf. Sections 3.7).\nIn Chapter 4, we prove the basic quasi-polynomiality results–Theorem 1.6.1\nand its generalizations for Problems 1.1.3 and 1.1.4. We also define canonical\nmodels for the structural constants under consideration, and briefly describe\nthe relevance of the nonstandard quantum groups and the related results in\n[GCT4, GCT7, GCT8] in the context of quantizing the coordinate rings of\nthese canonical models and extending PH0 to them (Figure 1.2).\nIn Chapter 5, we prove the basic PSPACE results–Theorem 1.6.3 and\nits extensions for the various cases of Problem 1.1.3.\nIn Chapter 6, we give experimental evidence for the positivity hypotheses\nPH2 and PH3 in some special cases of the Problems 1.1.1-1.1.4.\nIn Chapter 7, we describe an application (Theorem 1.8.1) of the re-\nsults and positivity hypotheses in this paper to the problem of verifying or\ndiscovering a robust obstruction, i.e., a “proof of hardness” [GCT2] in the\ncontext of the P vs. NP and the permanent vs. determinant problems in\ncharacteristic zero.\n1.10\nNotation\nWe let ⟨X⟩denote the total bitlength of the specification of X. Here X can\nbe an integer, a partition, a classifying label of an irreducible representation\nof a reductive group, a polytope, and so on. The exact meaning of ⟨X⟩will\nbe clear from the context. The notation poly(n) means O(na), for some\nconstant a. The notation poly(n1, n2, . . .) similarly means bounded by a\npolynomial of a constant degree in n1, n2, . . .. Given a reductive group H,\nVλ(H) denotes the irreducible representation of H with the classifying label\nλ. The meaning depends on H. Thus if H = GLn(C), λ is a partition and\n32"},{"page":36,"text":"Vλ(H) the Weyl module indexed by λ, if H = Sm, then λ is a partition of\nsize |λ| = m, and Vλ(H) the Specht module indexed by λ, and so on.\n33"},{"page":37,"text":"Chapter 2\nPreliminaries in complexity\ntheory\nIn this chapter, we recall basic definitions in complexity theory, introduce\nadditional ones, and illustrate their significance in the context of represen-\ntation theory.\n2.1\nStandard complexity classes\nAs usual, P, NP and PSPACE are the classes of problems that can be\nsolved in polnomial time, nondeterministic polynomial time, and polyno-\nmial space, respectively. The class of functions that can be computed in\npolynomial time (space) is sometimes denoted by FP (resp. FPSPACE).\nBut, to keep the notation simple, we shall denote these classes by P and\nPSPACE again.\nLet SPACE(s(N)) denote the class of problems that can be solved in\nO(s(N)) space on inputs of bit length N; by convention s(N) counts only\nthe size of the work space. In other words, the size of the input, which is on\nthe read-only input tape, and the output, which is on the write-only output\ntape is not counted. Hence s(N) can be less than the size of the input or the\noutput, even logarithmic compared to these sizes. The class space(log(N))\nis denoted by LOGSPACE.\nAn algorithm is called strongly polynomial [GLS], if given an input x =\n(x1, . . . , xk),\n34"},{"page":38,"text":"1. the total number of arithmetic steps (+, ∗, −and comparisones) in the\nalgorithm is polynomial in k, the total number of input parameters,\nbut does not depend ⟨x⟩, where ⟨x⟩= P\ni⟨xi⟩denotes the bitlength of\nx.\n2. the bit length of every intermediate operand in the computation is\npolynomial in ⟨x⟩.\nClearly, a strongly polynomial algorithm is also polynomial. let strong P ⊆\nP denote the subclass of problems with strongly polynomial time algorithms.\nThe counting class associated with NP is denoted by #P. Specifically,\na function f : Nk →N, where N is the set of nonnegative integers, is in #P\nif it has a formula of the form:\nf(x) = f(x1, · · · , xk) =\nX\ny∈Nl\nχ(x, y),\n(2.1)\nwhere χ is a polynomial-time computable function that takes values 0 or 1,\nand y runs over all tuples such that ⟨y⟩= poly(⟨x⟩). The formula (2.1) is\ncalled a #P-formula. An important feature of a #P-formula in the context\nof representation theory is that it is positive; i.e., it does not contain any\nalternating signs.\nThe formula (2.1) is called a strong #P-formula, if, in addition, l is\npolynomial in k and χ is a strongly polynomial-time computable function.\nLet strong #P be the class of functions with strong #P-fomulae.\nIt is known and easy to see that\n#P ⊆PSPACE.\n(2.2)\n2.1.1\nExample: Littlewood-Richardson coefficients\nBy the Littlewood-Richardson rule [FH], the coefficient cλ\nα,β (cf.\nProb-\nlem 1.2.1) in type A is given by:\ncλ\nα,β =\nX\nT\nχ(T),\n(2.3)\nwhere T runs over all numbering of the skew shape λ/α, and χ(T) is 1 if\nT is a Littlewood-Richardson skew tableau of content β, and zero, other-\nwise. The total number of entries in T is quadratic in the total number of\n35"},{"page":39,"text":"nonzero parts in α, β, λ, and the number of arithmetic steps needed to com-\npute χ(T) is linear in this total number. Hence (2.3) is a strong #P-formula,\nand Littlewood-Richardson function c(α, β, λ) = cλ\nα,β belongs to strong #P.\nIt may be remarked that the character-based formulae for the Littlewood-\nRichardson coefficients are not #P-formulae, since they involve alternat-\ning signs. But the algorithms based on the these formulae for computing\nLittlewood-Richardson coefficients run in polynomial space. Thus, from the\nperspective of complexity theory, the main significance of the Littlewood-\nRichardson rule is that it puts the problem, which at the surface is only in\nPSPACE, in its smaller subclass (strong) #P.\nThough the Littlewood-Richardson rule is often called efficient in the\nrepresentation theory literature, it is not really so from the perspective of\ncomplexity theory. Because computation of cλ\nα,β using this formula takes\ntime that is exponential in both the total number of parts of α, β and λ, and\ntheir bit lengths. This is inevitable, since this problem is #P-complete [N].\nSpecifically, this means there is no polynomial time algorithm to compute\ncλ\nα,β, assuming P ̸= NP.\nAs remarked in earlier, nonzeroness (nonvanishing) of cλ\nα,β can be decided\nin poly(⟨α⟩, ⟨β⟩, ⟨λ⟩) time; [DM2, GCT3, KT1]. Furthermore, the algorithm\nin [GCT3] is strongly polynomial; i.e., the number of arithemtic steps in\nthis algorithm is a polynomial in the total number of parts of α, β, λ, and\ndoes not depend on the bit lengths of α, β, λ. Hence the problem of deciding\nnonvanishing of cλ\nα,β (type A) belongs to strong P.\nThe discussion above shows that the Littlewood-Richardson problem is\nakin to the problem of computing the permanent of an integer matrix with\nnonnegative coefficients. The latter is known to be #P-complete [V], but\nits nonvanishing can be decided in polynomial time, using the polynomial-\ntime algorithm for finding a perfect matching in bipartite graphs [Sc]. If\nthe positivity hypotheses in this paper hold, the situation would be similar\nfor many fundamental structural constants in representation theory and\nalgebraic geometry in a relaxed sense.\n2.2\nConvex #P\nNext we want to introduce a subclass of #P called convex #P.\nGiven a polytope P ⊆Rl, let χP denote the characteristic (membership)\nfunction of P: i.e., χP (y) = 1, if y ∈P, and zero otherwise. We say that\n36"},{"page":40,"text":"f = f(x) = f(x1, . . . , xk) has a convex #P-formula if, for every x ∈Zk,\nthere exists a convex polytope (or, more generally, a convex body) Px ⊆Rl,\nsuch that\n1. The membership function χPx(y) can be computed in poly(⟨x⟩, ⟨y⟩)\ntime, each integer point in Px has O(poly(⟨x⟩)) bitlength, and\n2.\nf(x) = φ(Px),\n(2.4)\nwhere φ(Px) denotes the number of integer points in Px. Equivalently,\nf(x) =\nX\ny∈Zl\nχPx(y),\n(2.5)\nwhere y runs over tuples in Zl of poly(⟨x⟩) bitlength, and χPx denotes\nthe membership function of the polytope Px.\nEquation (2.5) is similar to eq.(2.1). The main difference is that χ is now\nthe membership function of a convex polytope. Clearly, eq.(2.5), and hence,\neq.(2.4) is a #P-formula, when χPx can be computed in polynomial time.\nLet convex #P be the subclass of #P consisting of functions with convex\n#P-formulae.\nWe say that eq.(2.4) is a strongly convex #P-formula, if the character-\nistic function of Px is computable in strongly polynomial time. Let strongly\nconvex #P be the subclass of #P consisting of functions with strongly con-\nvex #P-formulae.\nWe do not assume in eq.(2.4) that the polytope Px is explicitly specified\nby its defining constraints. Rather, we only assume, following [GLS], that\nwe are given a computer program, called a membership oracle, which, given\ninput parameters x and y, tells whether y ∈Px in poly(⟨x⟩, ⟨y⟩) time.\nIf the number of constraints defining Px is polynomial in ⟨x⟩, then it\nis possible to specify Px by simply writing down these constraints. In this\ncase the membership question can be trivially decided in polynomial time–in\nfact, even in LOGSPACE–by verifying each constraint one at a time. This\nwould not work if Px has exponentially many constraints. In good cases,\nit is possible to answer the membership question in polynomial time even\nif Px has exponentially many facets. Many such examples in combinatorial\noptimization are given in [GLS]. One such illustrative example in repre-\nsentation theory is given in Section 2.2.2. The polytopes that would arise\n37"},{"page":41,"text":"in the plethysm and other problems of main interest in this paper are also\nexpected to be of this kind.\nWe now illustrate the notion of convex #P with a few examples in rep-\nresentation theory.\n2.2.1\nLittlewood-Richardson coefficients\nA geeneralized Littlewood-Richardson coefficient cλ\nα,β for arbitrary semisim-\nple Lie algebra (Problem 1.2.1) has a strong, convex #P-formula, because\ncλ\nα,β = φ(P λ\nα,β),\nwhere P λ\nα,β is the BZ-polytope [BZ] associated with the triple (α, β, λ).\nIt is easy to see from the description in [BZ] that the number of defin-\ning constraints of P λ\nα,β is polynomial in the total number of parts (coor-\ndinates) of α, β, λ.\nGiven α, β, λ, these constraints can be computed in\nstrongly polynomial time. Hence, the membership problem for P λ\nα,β belongs\nto LOGSPACE ⊆P. It follows that the Littlewood-Richardson function\nc(α, β, λ) = cλ\nα,β belongs to strongly convex #P.\n2.2.2\nLittlewood-Richardson cone\nWe now give a natural example of a polytope in representation theory, the\nnumber of whose defining constraints is exponential, but whose membership\nfunction can still be computed in polynomial time.\nGiven a complex, semisimple, simply connected group G, let the Littlewood-\nRichardson semigroup LR(G) be the set of all triples (α, β, λ) of dominant\nweights of G such that the irreducible module Vλ(G) appears in the tensor\nproduct Vα(G) ⊗Vβ(G) with nonzero multiplicity [Z]. Brion and Knop [El]\nhave shown that LR(G) is a finitely generated semigroup with respect to\naddition. This also follows from the polyhedral expression for Littlewood-\nRichardson coefficients in terms of BZ-polytopes [Z]. Let LRR(G) be the\npolyhedral cone generated by LR(G).\nWhen G = GLn(C), the facets of LRR(G) have an explicit description by\nthe affirmative solution to Horn’s conjecture in [Kl, KT1]. But their number\ncan be quite large (possibly exponential). Nevertheless, membership of any\nrational (α, β, λ) (not necessarily integral) in LRR(G) can be decided in\nstrongly polynomial time.\n38"},{"page":42,"text":"This is because LRR(G) is the projection of a polytope P(G), the num-\nber of whose constraints is polynomial in the heights of α, β, λ [Z].\nIf\nφ : P(G) →LR(G) is this projection, we can choose P(G) so that for\nany integral (α, β, λ), φ−1(α, β, λ) is the BZ-polytope associated with the\ntriple (α, β, λ). To decide if (α, β, λ) ∈LR(G), we only have to decide if the\npolytope φ−1(α, β, λ) is nonempty. This can be done in strongly polynomial\ntime using Tardos’ linear programming algorithm [Ta].\n2.2.3\nEigenvalues of Hermitian matrices\nHere is another example of a polytope in representation theory with expo-\nnentially many facets, whose membership problem can still belong to P.\nFor a Hermitian matrix A, let λ(A) denote the sequence of eigenvalues\nof A arranged in a weakly decreasing order. Let HEr be the set of triple\n(α, β, λ) ∈Rr such that α = λ(A + B), β = λ(A), λ = λ(B) for some\nHermitian matrices A and B of dimension r. It is closely related to the\nLittlewood-Richardson semisgroup LRr = LR(GLr(C)): HEr ∩P 3\nr = LRr,\nwhere Pr is the semigroup of partitions of length ≤r. I. M. Gelfand asked\nfor an explicit description of HEr. Klyachko [Kl] showed that HEr is a\nconvex polyhedral cone. An explicit description of its facets is now known\nby the affirmative answer to Horn’s conjecture. But their number may be\nexponential. Hence, membership in HEr is still not easy to check using this\nexplicit description. This leads to the following complexity theoretic variant\nof Gelfand’s question:\nQuestion 2.2.1 Does the memembership problem for HEr belong to P?\nGiven that the answer is yes for the closely related LRr = LR(GLr(C))\n(Section 2.2.2), this may be so. If HEr were a projection of some polytope\nwith polynomially many facets, this would follow as in Section 2.2.2. But\nthis is not necessary. For example, Edmond’s perfect matching polytope for\nnon-bipartite graphs is not known to be a projection of any polytope with\npolynomially many constraints. Still the associated membership problem\nbelongs to P [Sc].\n2.3\nSeparation oracle\nSuppose P ⊆Rl is a convex polytope whose membership function χP is\npolynomial time computable. If χP(y) = 0 for some y ∈Rr, it is natural to\n39"},{"page":43,"text":"ask, in the spirit of [GLS], for a “proof” of nonmembership in the form of a\nhyperplane that separates y from P.\nIn this paper, we assume that all polytopes are specified by the separation\noracle. This is a computer program, which given y, tells if y ∈P, and if\ny ̸∈P, returns such a separting hyperplane as a proof of nonmembership. We\nassume that the hyperplane is given in the form l = 0, where a linear function\nl such that P is contained in the half space l ≥0, but l(y) < 0. Furthermore.\nwe assume that P is a well-described polyhedron in the sense of [GLS]. This\nmeans P is specified in the form of a triple (χP , n, φ), where P ⊆Rn, χP\nis a program for computing the membership function given y ∈Rn, and\nthere exists a system of inequalities with rational coefficients having P as\nits solution set such that the encoding bit length of each inequality is at\nmost φ. We define the encoding length ⟨P⟩of P as n + φ. We also assume\nthat the separation oracle works in O(poly(⟨P⟩, ⟨y⟩) time.\nFor example, the polynomial time algorithm for the membership function\nof the Littlewood-Richardson cone (cf. Section 2.2.2) can be easily modified\nto return a separating hyperplane as a proof of nonmembership.\nIn what follows, we shall assume, as a part of the definition of a convex\n#P-formula, that Px in (2.4) is a well-described polyhedron specified by\na separation oracle that works in polynomial time with ⟨Px⟩= poly(⟨x⟩).\nThese additional requirements are needed for the saturated integer program-\nming algorithm in Chapter 3.\n40"},{"page":44,"text":"Chapter 3\nSaturation and positivity\nIn this chapter we describe (Section 3.1) a polynomial time algorithm for\nsaturated and positive integer programming (Theorem 1.4.1). In Section 3.3\nwe state the main results and positivity hypotheses for the relaxed forms of\nProblem 1.1.3 and Problem 1.1.4, with X = G/P or a class variety therein.\nTogether they say that these relaxed decision problems can be efficiently\ntransformed into saturated (more strongly, positive) integer programming\nproblems, and hence can be solved in polynomial time.\n3.1\nSaturated and positive integer programming\nWe begin by proving Theorem 1.4.1.\nLet P ⊆Rn be a polytope given by a separation oracle (Section 2.3).\nLet ⟨P⟩be the encoding length of P as defined in Section 2.3. An oracle-\npolynomial time algorithm [GLS] is an algorithm whose running time is\nO(poly(⟨P⟩)), where each call to the separation oracle is computed as one\nstep.\nThus if the separation oracle works in polynomial time, then such\nan algorithm works in polynomial time in the usual sense.\nLet φ(P) be\nthe number of integer points in P.\nLet fP(n) = φ(nP) be the Ehrhart\nquasi-polynomial [St1] of P. Let l(P) be the least period of fP(n), if P\nis nonempty.\nLet fi,P(n), 1 ≤i ≤l(P), be the polynomials such that\nfP (n) = fi,P(n) if n = i modulo l(P). Let FP (t) = P\nn≥0 fP(n)tn denote\nthe Ehrhart series of P. It is a rational function.\nTheorem 3.1.1 (a) The index of fP (n), index(fP), can be computed in\noracle-polynomial time, and hence, in polynomial time, assuming that the\n41"},{"page":45,"text":"oracle works in polynomial time. Furthermore, if index(fP) ̸= 0 (i.e. if P\nis nonempty), then fi,P(n) is not an identically zero polynomial for every i\ndivisible by index(fP).\n(b) The saturated, and hence, positive integer programming problem, as de-\nfined in Section 1.4, can be solved in oracle-polynomial time.\nHere it is\nassumed that the specification of P also contains the saturation index esti-\nmate sie(P), or the positivity index estimate pie(P), and that the bitlength\nof this estimate is O(poly(⟨P⟩)). Given a relaxation parameter c > sie(P)\n(or pie(P)), the problem is to determine if cP contains an integer point in\nO(poly(⟨P⟩, ⟨c⟩)) time.\n(c) Suppose {Px} is a family of polytopes, indexed by some parameter x,\nwith the following property: wherenver Px is nonempty, the Ehrhart quasi-\npolynomial fPx(n) is “almost always” strictly saturated.\nAlmost always\nmeans, the density of x’s of bitlength ≤N, with nonempty Px for which\nfPx(n) is not strictly saturated is less than 1/N c′′, for any positive c′′, as\nN →0. We also assume that Px is given by a separation oracle that works in\nO(poly(⟨x⟩)) time, where ⟨x⟩is the bitlength of x, and ⟨Px⟩= O(poly(⟨x⟩)).\nThen there exists a O(poly(⟨x⟩)) time algorithm for deciding if Px con-\ntains an integer point that works correctly “almost always”; i.e., on almost\nall x.\nProof:\n(a):\nNonemptyness of P can be decided in oracle-polynomial time using the\nalgorithm of Gr ̈otschel, Lov ́asz and Schrijver [GLS] (cf.\nTheorem 6.4.1\ntherein). An extension of this algorithm, furthermore, yields a specifica-\ntion of the affine space span(P) containing P if P is nonempty (cf. Theo-\nrems 6.4.9, and 6.5.5 in [GLS]). Specifically, it outputs an integral matrix\nC and an integral vector d such that span(P) is defined by Cx = d. This\nfinal specification is exact, even though the first part of the algorithm in\n[GLS] uses the ellipsoid method. Indeed, the use of simultneous diophan-\ntine approximation based on basis reduction in lattices is precisely to ensure\nthis exactness in the final answer. This is crucial for the next step of our\nalgorithm.\nIf P is empty, index(fP) = 0. So assume that it is nonempty. Let ̄C be\nthe Smith normal form of C; i.e., ̄C = ACB for some unimodular matrices\nA and B, where the leftmost principal submatrix of ̄C is a diagonal, integral\nmatrix, and all other columns are zero.\n42"},{"page":46,"text":"The matrices ̄C, A and B can be computed in polynomial time using\nthe algorithm in [KB]. After a unimodular change of coordinates, by letting\nz = B−1x, span(P) is specified by the linear system ̄Cz = ̄d = Ad. The\nequations in this system are of the form:\n ̄cizi = ̄di,\n(3.1)\ni ≤codim(P), for some integers ̄ci and ̄di. By removing common factors if\nnecessary, we can assume that ̄ci and ̄di are relatively prime for each i. Let\n ̃c be the l.c.m. of ̄ci’s.\nThe statement (a) follows from:\nClaim 3.1.2 index(fP ) = ̃c and fi,P(n) is not an identically zero polyno-\nmial for every i divisible by ̃c.\nProof of the claim: Indeed, nP = {nz | z ∈P} contains no integer point\nunless ̃c divides n.\nHence, it is easy to see that FP (t) = F ̄P (t ̃c), where\nF ̄P (x) is the Ehrhart series of the dilated polytope ̄P = ̃cP. By eq.(3.1),\nthe equations defining ̄P are:\nzi = ̄di( ̃c/ ̄ci),\n(3.2)\nClearly, ̃c divides the least period l(P) of fP, and l( ̄P) = l(P)/ ̃c is the period\nof the Ehrhart quasipolynomial f ̄P(n). It suffices to show that the index of\nf ̄P (n) is one and that fj, ̄P(n) is not an identically zero polynomial for every\n1 ≤j ≤l( ̄P). This is equivalent to showing that ̄P contains a point z with\nwith zi = ai/b, for some integers ai’s and b such that b = j modulo l( ̄P).\nLet us call such a point j-admissible. Because of the form of the equations\n(3.2) defining span( ̄P), we can assume, without loss of generality, that ̄P is\nfull dimensional. This means the system (3.2) is empty. Then this follows\nfrom denseness of the set of j-admissible points. This proves the claim, and\nhence (a).\n(b): Let s = sie(P) be the given saturation index estimate. This means\nfP (n + s) is strictly saturated. This in conjunction with (a) implies that,\ngiven a relaxation parameter c > s, cP contains an integer point, iffc\nis divisible by index(fP ) (by letting n = c −s). This can be checked in\nO(poly(⟨P⟩, ⟨c⟩)) time since index(fP ) can be computed in polynomial time\nby (a).\n(c) The algorithm computes index(fPx) and says “Probably Yes”if the index\nis one, and “No” otherwise. Since the saturation index of fPx(n) is zero\n43"},{"page":47,"text":"almost always, by the argument in (b) with s = 0 and c = 1, “Probably\nYes” really means “Yes” almost always. Q.E.D.\nThe algorithm in (c) has one drawback.\nIf the answer is “Probably\nYes”, we have no easy way of checking if Px really contains an integer point.\nIdeally, we would like an algorithm that says “Yes”, with an integer point\nin Px as a proof certificate, or “No”, or “Unsure”, and the density of x’s on\nwhich it says “Unsure” should be very small. This problem can be overcome\nif the family {Px} has the following stronger property, akin to the family of\nhive polytopes [KT1]: there is a linear function lx such that, for almost all x,\nif {Px} is nonempty, then the lx-optimum of Px is integral (this is stronger\nthan saying that fPx(n) is strictly saturated). In this case, the algorithm in\n(c) can be extended to yield the integral lx-optimum as a proof certificate. If\nthe lx-optimum is not integeral, the algorithm says “Unsure”. PH1 and SH\n(Section 1.6) for the plethysm (and more generally, the subgroup restriction)\nproblem may be strengthened by stipulating that the polytopes therein have\nthis property. But this is not needed in this paper.\nWe note down one corollary of the proof of Theorem 3.1.1 (this should\nbe well known):\nProposition 3.1.3 The rational function FP (t) = F ̄P (t ̃c), where F ̄P (x) is\nthe Ehrhart series of the dilated polytope ̄P = ̃cP, and ̃c is the index of\nfP (n).\nIf P is explicitly specified in the form a linear system\nAx ≤b,\n(3.3)\nwhere A is an m × n matrix, b an m vector and m = poly(n), then the\nfollowing stronger version of Theorem 3.1.1 holds. Let ⟨A⟩and ⟨A, b⟩denote\nthe bitlength of the specification of A and of the linear system (3.3).\nTheorem 3.1.4 Suppose P is specified in terms of an explicit linear system\n(3.3). Then the index of the Erhart quasi-polynomial fP(n) can be computed\nin poly(⟨A, b⟩) time, using poly(⟨A⟩) arithmetic operations.\nThus, saturated, and hence, positive integer programming problem spec-\nified in the form (3.3) can be solved in in poly(⟨A, b, c⟩) time, where c is the\nrelaxation parameter, using poly(⟨A⟩) arithmetic operations.\nProof: This is proved exactly as Theorem 3.1.1, but with Tardos’ strongly\npolynomial time algorithm for combinatorial linear programming [Ta] used\nin place of the algorithm in [GLS]. Q.E.D.\n44"},{"page":48,"text":"3.1.1\nA general estimate for the saturation index\nNow we give a general estimate for the saturation index of any polytope P\nwith a specification of the form\nAx ≤b,\n(3.4)\nwhere A is an m × n matrix, m possibly exponential. Let ∥P∥= n + ψ,\nwhere ψ is the maximum bitlength of any entry of A. Trivially, ∥P∥≤⟨P⟩.\nWe do not assume that we know the specification (3.4) of P explicitly. We\nonly assume that it exists, and that we are told ∥P∥. Then:\nTheorem 3.1.5 The saturation index of P is O(2poly(∥P ∥)).\nThus the\nbitlength of the saturation index is O(poly(∥P∥)).\nConjecturally, this also holds for the positivity index. This estimate is\nvery conservative, but useful when no better estimate is available.\nProof: There exists a triangulation of P into simplices such that every vertex\nof any simplex is also a vertex of P. Then\nfP(n) =\nX\n∆\nf∆(n),\nwhere ∆ranges over all open simplices in this triangulation; a zero-dimensional\nopen simplex is a vertex. The saturation index of fP(n) is clearly bounded\nby the maximum of the saturation indices of f∆(n).\nHence, we can assume, without loss of generality, that P is an open sim-\nplex. Let v0, . . . , vn be its vertices. Then, by Ehrhart’s result (cf. Theorem\n1.3 in [st5]),\nFP (t) =\nP\ni hiti\nQn\nj=0(1 −taj),\n(3.5)\nwhere h0 = 1, hi’s are nonnegative, and aj is the least positive integer\nsuch that ajvj is integral. By Cramer’s rule, the bit length of each aj is\npoly(∥P∥).\nWithout loss of generality, we can also assume that aj’s are\nrelatively prime. Otherwise, the estimate on the saturation index below has\nto be multiplied by the g.c.d. of aj’s. Then the result follows by applying\nthe following lemma to FP (t), since ⟨aj⟩= O(poly(∥P∥)). Q.E.D.\n45"},{"page":49,"text":"Lemma 3.1.6 Let f(n) be a quasipolynomial whose generating function\nF(t) has a positive form\nF(t) =\nP\ni hiti\nQn\nj=0(1 −taj),\n(3.6)\nwhere h0 = 1, hi’s are nonnegative, and aj’s are positive and relatively\nprime.\nLet a = max{aj}.\nThen the saturation index s(f) of f(n) is\nO(poly(a, n)).\nProof: Let g(n) be the quasi-polynomial whose generating function G(t) =\nP g(n)tn is 1/Qn\nj=0(1 −taj). It is known that this is the Ehrhart quasipoly-\nnomial of the polytope N(a0, . . . , an) defined by the linear system\nX\najxj = 1, xj > 0.\nThe saturation index s(g) of g(n) is bounded by the Frobenius number\nassociated with the set of integers {aj}–this is the largest positive integer m\nsuch that the diophantine equation\nX\nj\najxj = m\nhas no positive integeral solution (x0, . . . , xn). It is known (e.g. [BDR]) that\nthe Frobenius number is bounded by\nX\nj\naj +\np\na0a1a2(a0 + a1 + a2) = O(poly(a)),\nassumming that a0 ≤a1 . . .. Hence, s(g) = O(poly(a)).\nSince f(n) is a quasi-polynomial, the degree of the numerator of F(t) is\nless than the degree of the denominator. Thus the maximum value of i that\noccurs in (3.6) is an.\nLet gi(n), i ≤an, be the quasi-polynomial whose generating function is\nti/Qn\nj=0(1 −taj). Then\ns(gi) ≤i + s(g) = O(poly(a, n)).\nSince, hi’s in (3.6) are nonnegative, s(f) = max s(gi). The result follows.\nQ.E.D.\n46"},{"page":50,"text":"3.1.2\nExtensions\nWe now mention a few straightforard extensions of Theorem 3.1.1.\nFirst, it is not necessary that P be a closed polytope. We can allow\nit to be half-closed.\nSpecifically, it can be a solution set of a system of\ninequalitites of the form:\nA1x ≤b1\nand\nA2x < b2,\n(3.7)\nwhere we have allowed strict inequalities. The function FP (n) = φ(nP), the\nnumber of integer points in nP, is again a quasi-polynomial. Hence, the\nnotions of saturation and positivity can be generalized to this setting in a\nnatural way.\nSecond, the algorithm in Theorem 3.1.1 (b) only needs a nonnegative\nnumber s(P) such that, for any positive integer c > s(P):\nSaturation guarantee: If the affine span of cP, contains an integer point,\nthen cP is guaranteed to contain an integer point.\nIf s(P) = sie(P), then this guarantee holds, as can be seen from the\nproof of Theorem 3.1.1.\n3.1.3\nIs there a simpler algorithm?\nThough the algorithm for saturated integer programming in Theorem 3.1.1\nis conceptually very simple, in reality it is quite intricate, because the work\nof Gr ̈otschel, Lov ́asz and Schrijver [GLS] needs a delicate extension of the el-\nlipsoid algorithm [Kh] and the polynomial-time algorithm for basis reduction\nin lattices due to Lenstra, Lenstra and Lov ́asz [LLL]. As has been empha-\nsized in [GLS], such a polynomial-time algorithm should only be taken as a\nproof of existence of an efficient algorithm for the problem under consider-\nation. It may be conjectured that for the problems under consideration in\nthis paper such simple, combinatorial algorithms exist. But for the design\nof such algorithms, saturation alone does not suffice. The stronger property\n(PH3), and more, is necessary. We shall address this issue in Section 3.6.\n3.2\nLittlewood-Richardson coefficients again\nTheorem 3.1.4 applied to the BZ-polytope [BZ], with saturation index esti-\nmate equal to zero, specializes to the following in the setting of the Littlewood-\n47"},{"page":51,"text":"Richardson problem (Problem 1.2.1):\nTheorem 3.2.1 [GCT5] Assuming SH (Hypothesis 1.2.5), nonvanishing of\ncλ\nα,β, given α, β, λ, can be decided in strongly polynomial time (Section 2.1)\nfor any semisimple classical Lie algebra G.\nIt is assumed here that α, β, λ are specified by their coordinates in the\nbasis of fundamental weights.\nFor type A, this reduces to the result in\n[GCT3], which holds unconditionly.\nThe saturation conjecture for type A arose [Z] in the context of Horn’s\nconjecture and the related result of Klyachko [Kl]. We now turn to implica-\ntions of Theorem 3.2.1 in this context.\nGiven a complex, semisimple, simply connected, classical group G, let\nLR(G) be the Littlewood-Richardson semigroup as in Section 2.2.2. The\nfollowing is a natural generalization of the problem raised by Zelevinsky [Z]\nto this general setting:\nProblem 3.2.2 Give an efficient description of LR(G).\nZelevinsky asks for a mathematically explicit description. This is a com-\nputer scientist’s variant of his problem.\nLet LRR(G) be the polyhedral convex cone generated by LR(G). For\nG = GLn(C), by the saturation theorem, a triple (α, β, λ) of dominant\nweights belongs to LR(G) iffit belongs to LRR(G).\nAssuming SH (Hy-\npothesis 1.2.5), Theorem 3.2.1 provides the following efficient description\nfor LR(G) in general. Recall that the period of the Littlewood-Richardson\nstretching polynomial ̃cλ\nα,β(n) divides a fixed constant d(G), which only de-\npends on the types of simple factors of G [DM2, GCT5]. Let αi’s denote\nthe coordinates of α in the basis of fundamental weights.\nCorollary 3.2.3 (a) Assuming SH, whether a given (α, β, λ) belongs to\nLR(G) can be determined in strongly polynomial time.\n(b) There exists a decomposition of LRR(G) into a set of polyhedral cones,\nwhich form a cell complex C(G), and, for each chamber C in this complex,\na set M(C) of O(rank(G)2) modular equations, each of the form\nX\ni\naiαi +\nX\ni\nbiβi +\nX\ni\nciλi = 0\n(mod d),\nfor some d dividing d(G), such that\n48"},{"page":52,"text":"1. SH (Hypothesis 1.2.5) is equivalent to saying that: (α, β, λ) ∈LR(G)\niff(α, β, λ) ∈LRR(G) and (α, β, λ) satisfies the modular equations in\nthe set M(Cα,β,λ) associated with the cone Cα,β,λ containing α, β, λ.\n2. Given (α, β, λ), whether (α, β, λ) ∈LRR(G) can be determined in\nstrongly polynomial time (cf. Section 1.2.5).\n3. If so, the cone Cα,β,λ and the associated set M(Cλ\nα,β) of modular equa-\ntions can also be determined in strongly polynomial time. After this,\nwhether (α, β, λ) satisfies the equations in M(Cλ\nα,β) can be trivially\ndetermined in strongly polynomial time.\nProof: (a) is a consequence of Theorem 3.2.1. (b) follows from a careful\nanalysis of the algorithm therein; see the proof of a more general result\n(Theorem 4.4.2) later. Q.E.D.\nWe call the labelled cell complex C(G), in which each cell C ∈C(G)\nis labelled with the set of modular equations M(C), the modular complex,\nassociated with LRR(G).\nWhen G = SLn(C), the modular complex is\ntrivial: it just consists of the whole cone LRR(G) with only one obvious\nmodular equation attached to it. But, for general G, the modular complex\nand the map C →M(C) are nontrivial.\nWe do not know their explicit\ndescription.\nCorollary 3.2.3 says that, given x = (α, β, λ), whether x ∈\nLRR(G), and whether the relevant modular equations are satisfied can be\nquickely verified on a computer, though the modular equations cannot be\neasily determined and verified by hand, as in type A.\nThis is the main\ndifference between type A and general types.\nThis naturally leads to:\nQuestion 3.2.4 Is there a mathematically explicit description of the mod-\nular complex C(G) for a general G?\n3.3\nThe saturation and positivity hypotheses\nNow let f(x), x ∈Nk, be a counting function associated with a structural\nconstant in representation theory or algebraic geometry.\nHere x denotes\nthe sequence of parameters associated with the constant. Let ⟨x⟩denote\nthe bitlength of x. Let ∥x∥and rank(x) denote its combinatorial size and\ncombinatorial rank–these measure complexity of the nonstretchable part in\n49"},{"page":53,"text":"the specification of x and will be specified later for the f’s of interest in this\npaper.\nFor example, in the Littlewood-Richardson problem, x is the triple (α, β, λ),\nf(x) = f(α, β, λ) = cλ\nα,β, ⟨x⟩is the total bitlength of the coordinates of\nα, β, λ, ∥x∥is the total number of coordinates of α, β and λ, and rank(x) =\n∥x∥. The number of coordinates does not change during stretching, and\nhence, constitute the nonstretchable part of the input specification here.\nAssume that f(x) is nonnegative for all x ∈Nk, Then we can successively\nask the following questions:\n1. Does f ∈PSPACE? That is, can f(x) be computed in poly(⟨x⟩)\nspace?\n2. Does f ∈#P? (cf. Section 2.1)\n3. Does f ∈convex#P? (cf. Section 2.2)\n4. Can a stretching function ̃f(x, n) be associated with f(x) intrinsically\nso that ̃f(x, n) is quasi-polynomial?\n5. (PH1?): Is there a polytope Px, for every x, with ⟨Px⟩= O(poly(⟨x⟩))\nand ∥Px∥= O(poly(∥x∥)), such that ̃f(x, n) = fPx(n)?\n6. Are there good analogues of SH and/or PH2, PH3 for ̃f(x, n)?\nIf\nso, nonvanishing of f(x), modulo small relaxation, can be decided in\nO(poly(⟨Px⟩)) time by Theorem 3.1.1.\nIn the rest of this paper, we study these questions when f = f(x) is a\nnonnegative function associated with a structural constant in any of the deci-\nsion problems in Section 1.1. Exact specifications of x, ⟨x⟩, ∥x∥, rank(x), f(x),\nand ̃f(x, n) for these decision problems are given in Sections 3.4-3.5. It is\nshown in Chapter 5 that f(x) ∈PSPACE for Problem 1.1.2 and the special\ncases of Problem 1.1.3 that arise in the flip. This may be conjectured to be\nso for the f’s in Problem 1.1.4, with X therein a class variety; cf [GCT10]\nfor its justification. Quasipolynomiality of ̃f(x, n) is addressed in Chapter 4.\nThe hypotheses PH1, SH, PH2, and PH3 in these cases have the following\nunified form.\nHypothesis 3.3.1 (PH1) Let f = f(x) be the function associated with a\nstructural constant in\n50"},{"page":54,"text":"1. Problem 1.1.1, or\n2. 1.1.2, or\n3. Problem 1.1.3, or\n4. Problem 1.1.4, with X being a class variety therein.\nThen the function f(x) has a convex #P-formula (cf. (2.4))\nf(x) = φ(Px),\nsuch that:\n1. for every fixed x, the Ehrhart quasi-polynomial fPx(n) of Px coincides\nwith ̃f(x, n).\n2. ⟨P⟩= O(poly(P)) and ∥P∥= O(poly(∥x∥)).\nHypothesis 3.3.2 (SH)\n(a) Suppose f(x) is a structural constant as in PH1 above. Then for every x,\nthe saturation index s( ̃f) of ̃f(x, n) is O(poly(rank(x))). This means there\nexist absolute nonnegative constants c, c′ such that s( ̃f) ≤c(rank(x))c′.\n(b) For f(x) in Problems 1.1.1-1.1.3, the saturation index of ̃f(x, n) is zero–\ni.e., ̃f(x, n) is strictly saturated–for almost all x. This means the density of\nx, with ⟨x⟩≤N and f(x) nonzero, for which the saturation index s( ̃f) is\nnonzero is ≤1/N c′′, for any positive costant c′′, as N →∞.\nMore strongly than (a),\nHypothesis 3.3.3 (PH2) For f(x) as in PH1, the positivity index of ̃f(x, n)\nis O(poly(rank(x))).\nHypothesis 3.3.4 (PH3) For f(x) as in PH1, the generating function\nF(x, t) = P\nn ̃f(x, n)tn has a positive rational form of modular index O(poly(rank(x))).\nMore specifically, the modular index of ̃f(x, n), as defined in Section 4.1.1\nfor f’s that arise in this paper, is O(poly(rank(x))).\nPH3 implies SH (a); this follows from Lemma 3.1.6.\nThe following conservative bound follows from Theorem 3.1.5.\n51"},{"page":55,"text":"Theorem 3.3.5 (Weak SH)\nAssuming PH1 (Hypothesis 3.3.1), the saturation index of ̃f(x, n) is\nbounded by 2O(poly(∥x∥)); hence its bitlength is bounded by O(poly(∥x∥)).\nThe following result addresses the relaxed forms of the decision problems\nfor the structural constants under consideration (cf. Section 1.1).\nTheorem 3.3.6 Suppose f(x) is a structural constant as in PH1 above.\nThen PH1 (Hypothesis 3.3.1) and SH (Hypothesis 3.3.2) imply Hypothe-\nsis 1.1.6 (PHflip) in this case. Specifically:\n(a) For f(x) in Problems 1.1.1-1.1.4, nonvanishing of ̃f(x, a), for a given x\nand a relaxation parameter a > c(rank(x))c′, with c, c′ as in Hypothesis 3.3.2,\ncan be decided in poly(⟨x⟩, ⟨a⟩) time.\n(b) For f(x) as in Problems 1.1.1-1.1.3, there is a poly(⟨x⟩) time algorithm\nfor deciding nonvanishing of f(x) that works correctly on almost all x.\nThis follows from Theorem 3.1.1.\nThe following sections give precise descriptions of x, ⟨x⟩, ∥x∥, rank(x)\nand ̃f(x, n) for the structural constants under consideration.\n3.4\nThe subgroup restriction problem\nIn this section we consider the subgroup restriction problem (Problem 1.1.3).\nThe Kronecker and the plethysm problems (Problems 1.1.1, 1.1.2) are its\nspecial cases.\nLet G, H, ρ, λ, π, mπ\nλ be as in Problem 1.1.3. We shall define below an ex-\nplicit polynomial homomorphism ρ : H →G, as needed in the statement of\nProblem 1.1.3, and also the precise specifications [H], [ρ], [λ], [π] of H, ρ, λ, π,\nrespectively. We shall also define the bitlengths ⟨H⟩, ⟨ρ⟩, ⟨λ⟩, ⟨π⟩and the\ncombinatorial bit lengths ∥λ∥, ∥π∥. We let ∥H∥= ⟨H⟩and ∥ρ∥= ⟨ρ⟩, since\nH and ρ belong to the nonstretchable part of the input. On the other hand,\nλ and π will be stretched in the definition of ̃f(x, n), and hence their com-\nbinatorial bit lengths will differ from the usual bit lengths. The input x in\nthe subgroup restriction problem is the tuple ([H], [ρ], [λ], [π]). Its bitlength\n⟨x⟩is defined to be the sum of the bitlengths ⟨H⟩, ⟨ρ⟩, ⟨λ⟩, ⟨π⟩, and ∥x∥is\ndefined to be the sum of ∥H∥, ∥ρ∥, ∥λ∥and ∥π∥. Finally rank(x) is defined\nto the sum of the ranks of H and G and ∥λ∥and ∥π∥. Here that rank of\n52"},{"page":56,"text":"a (reductive) group is defined in a standard way. For example, the rank of\nthe symmentric group Sn is n, that of GLn(C) is n. The rank of a general\nfinite or connected simple group can be defined similarly, and the rank of a\nmore complex reductive group is defined to be the sum of the ranks of its\nsimple components. With this terminology, we let f(x) = mπ\nλ, with x as\ndefined here in Hypotheses 3.3.1-3.3.4 and Theorem 3.3.6 for the subgroup\nrestriction problem. Here H and ρ are implicit in the definition of mπ\nλ.\nFor example, in the plethym problem (Problem 1.1.2), these specifi-\ncations are as follows.\nThe specification [H] is just the root system for\nH = GLn(C). Its bitlength ⟨H⟩is n. The specification [ρ] of the repre-\nsentation map ρ : H →G = GL(Vμ(H)) consists of just the partition μ\nspecified in terms of its nonzero parts. Its bitlength ⟨ρ⟩= ⟨μ⟩. The ranks\nof H and G are as usual. The partitions λ and μ are specified in terms of\ntheir nonzero parts. Their bitlength is the total bitlength of the parts, and\nthe combinatorial bit length is the total number of parts (the height). It\nis crucial here that only nonzero parts of λ are specified, because the rank\nof G can be exponential in the rank of H and the bitlength of μ. Hence,\nthe bitlength of this compact representation of λ can be polynomial in the\nrank of H and the bitlength of μ, even if the dimension of G is exponential.\nThe main difference between ⟨x⟩and ∥x∥is that the stretchable data λ and\nπ contribute their bitlengths to the former, and their heights to the latter.\nThe plethysm problem is the main prototype of the subgroup restriction\nproblem. If the reader wishes, (s)he can skip the rest of this subsection and\njump to Section 3.4.3 in the first reading.\nIn general, we assume that H in Problem 1.1.3 is a finite simple group, or\na complex simple, simply connected Lie group, or an algebraic torus (C∗)k,\nor a direct product of such groups. The results and hypotheses in this paper\nare also applicable if we allow simple types of semidirect products, such as\nwreath products, which is all that we need for the sake of the flip. But these\nextensions are routine, and hence, for the sake of simplicity, we shall confine\nourselves to direct products.\n3.4.1\nExplicit polynomial homomorphism\nNow let us define an explicit polynomial homomorphism. This will be done\nby defining basic explicit homomorphisms, and composing them functorially.\nBasic explicit homomorphisms:\nLet V be an irreducible polynomial representation of H (character-\n53"},{"page":57,"text":"istic zero), or more generally, an explicit polynomial representation that\nis constructed functorially from the irreducible polynomial representations\nusing the operations ⊕and ⊗.\nThen the corresponding homomorphism\nρ : H →G = GL(V ) is an explicit polynomial homomorphism. The iden-\ntity map H →H is also an explicit polynomial homomorphism.\nThe polynomiality restriction here only applies to the torus component\nof H. If H is a finite simple group, or a complex semisimple group, then\nany irreducible representation of H is, by definition, polynomial. In general,\na representation is polynomial if its restriction to the torus component is\npolynomial; i.e., a sum of polynomial (one dimensional) characters.\nTo see why the polynomiality restriction is essential, let H be a torus,\nV its rational representation, and G = GL(V ).\nLet Vλ(G) = Symd(V ),\nthe symmetric representation of G, and let π be the label of the trivial\ncharacter of H. Then the multiplicity mπ\nλ is the number of H-invariants in\nSymd(V ). This is easily seen to be the number of nonnegative solutions of a\nsystem of linear diophontine equations. But the problem of deciding whether\na given system of linear diophontine equations has a nonnegative solution\nis, in general, NP-complete. Though the system that arises above is of a\nspecial form, it is not expected to be in P if V is allowed to be any rational\nrepresentation; the associated decision problem may be NP-complete even\nin this special case. If V is a polynomial representation of a torus H, then\nall coefficients of the system are nonnegative, and the decision problem is\ntrivially in P.\nComposition:\nWe can now compose the basic explicit (polynomial) homomorphisms\nabove functorially:\n1. If ρi : H →Gi are explicit, the product map ρ : H →Q\ni Gi is also\nexplicit.\n2. If ρi : Hi →Gi are explicit, the product map ρ : Q Hi →Q Gi is also\nexplicit.\nInstead of products, we can also allow simple semi-direct products such\nas wreath products here. We may also allow other functorial constructions\nsuch as induced representations and restrictions. For example, if ρ : H →G\nis an explicit polynomial homomorphism, and G′ ⊆G is an explicit subgroup\nof G such that ρ(H) ⊆G′, then the restricted homomorphism ρ′ : H →G′\ncan also be considered to be an explicit polynomial homomorphism. But\n54"},{"page":58,"text":"for the sake of simplicity, we shall confine ourselves to the simple functorial\nconstructions above.\n3.4.2\nInput specification and bitlengths\nNow we describe the specifications [H], [ρ], [λ], [μ], their bitlengths. These\nare very similar to the ones in the plethysm problem.\nThe specification [H]:\nWe assume that H is specified as follows.\n(1) If H is a complex, simple, simply connected Lie group, then the specifica-\ntion [H] consists of the root system of H or the Dynkin diagram. Let ⟨H⟩be\nthe bitlength of this specification. Thus, if H = SLn(C), then ⟨H⟩= O(n).\n(2) If H is a simple group of Lie type (Chevalley group) then it has a similar\nspecification [Ca]. The only finite groups of Lie type that arise in GCT are\nSLn(Fpk) and GLn(Fpk). In this case the specification [H] is easy: we only\nhave to specify n, p, k. We define ⟨H⟩in this case to be n + k + log2 p; not\nlog2 n + log2 k + log2 n. As a rule, ⟨H⟩is defined to be the sum of the rank\nparameters (such as n and k here) and bit lengths of the weight parameters\n(such as p here) in the specification. This is equivalent to assuming that the\nrank parameters are specified in unary.\n(3) If H is the alternating group An, we only specify n. Let ⟨H⟩= n.\n(4) The torus is specified by its dimension. We define ⟨H⟩to be the dimen-\nsion.\n(5) If H is a product of such groups, its specification is composed from the\nspecifications of its factors, and the bitlength ⟨H⟩is defined to be the sum\nof the bitlengths of the constituent specifcations.\nThe specification [ρ]:\nLet us first assume that ρ is a basic explicit polynomial homomorphism.\nIn this case the specification of ρ : H →G = GL(V ) is a pair [ρ] = ([H], [V ])\nconsisting of the specification [H] of H as above, and the combinatorial\nspecification [V ] of the representation V as defined below:\n(1) If H is a semisimple, simply connected Lie group, and V = Vμ(H) its\nirrreducible representation for a dominant weight μ of H, then V is specified\nby simply giving the coordinates of μ in terms of the fundamental weights\nof H.\nThus [V ] = μ, and its bitlength ⟨V ⟩is the total bitlength of all\ncoordinates of μ, and the combinatorial bit length ∥V ∥is the total number\nof coordinates of μ.\n55"},{"page":59,"text":"(2) If H = Sn, and V = Sγ its irreducible representation (Specht module),\nthen [V ] is the partition γ labelling this Specht module. We define ⟨V ⟩to\nbe the bitlength of this partition, and ∥V ∥= ⟨V ⟩.\n(3) If H is a finite general linear group GLn(Fpk), and V its irreducible rep-\nresentation, as classified by Green [Mc], then [V ] is the combinatorial clas-\nsifying label of V as given in [Mc]. It is a certain partition-valued function,\nwhich can be specified by listing the places where the function is nonzero\nand the nonzero partition values at these places. Let ⟨V ⟩be the bitlength\nof this specification; it is O(poly(n, k, ⟨p⟩)). We let ∥V ∥= ⟨V ⟩. More gener-\nally, if H is a finite group of Lie type, and V its irreducible representation,\nthen [V ] is the combinatorial classifying label of V as given by Lusztig [Lu1].\n(4) If H is a torus and V is a polynomial character, then [V ] is the speci-\nfication of the character. Its bitlength is the bitlength of the specification,\nand combinatorial bit length is the dimension of H.\n(5) If V is composed from irreducible representations, then [V ] is composed\nfrom the specifications of the irreducible representations in an obvious way.\nBitlengths and combinatorial bitlengths are defined additively.\nThe bitlength ⟨ρ⟩is defined to be ⟨H⟩+ ⟨V ⟩, where ⟨V ⟩is the bitlength\nof [V ].\nIf ρ is a composite homomorphism, its specification [ρ] is composed from\nthe specifications of its basic constituents in an obvious way. The bitlength\n⟨ρ⟩is defined to be the sum of the bitlengths of these basic specifications.\nThe specifications [λ] and [π]:\nVπ(H) is the tensor product of the irreducible representations of the\nfactors of H. We let [π] be the tuple of the combinatorial classifying labels\nof each of these irreducible representations, as specified above. Let ⟨π⟩be\ntheir total bit length, and ∥π∥the total combinatorial bit length. Similarly,\nVλ(G) is the tensor product of the irreducible representations of the factors\nof G. When G = GLm(C), λ is a partition, which we specify by only giving\nits nonzero parts, whose number is equal to the height of λ. This is crucial\nsince the height of λ can be much less than than the rank m of G, as in\nthe plethysm problem (Problem 1.1.2). We shall leave a similar compact\nspecification [λ] for a general connected, reductive G to the reader. Let ⟨λ⟩\nbe its bitlength and ∥λ∥its combinatorial bit length.\n56"},{"page":60,"text":"3.4.3\nStretching function and quasipolynomiality\nLet f(x) = mπ\nλ as above, with x = ([H], [ρ], [λ], [π]). Here λ is the dominant\nweight of G. First, assume that H is connected, reductive. Then π is the\ndominant weight of H. For a given x, let us define the stretching function\nas\n ̃f(x, n) = ̃mπ\nλ(n) = mnπ\nnλ,\n(3.8)\nwhich is the multiplicity of Vnπ(H) in Vnλ(G), considered as an H-module\nvia ρ : H →G. Let Mπ\nλ (t) = P\nn≥0 ̃mπ\nλ(n)tn be the generating function of\nthis stretching quasi-polynomial.\nThe following is the generalization of Theorem 1.6.1 in this setting.\nTheorem 3.4.1 (a) (Rationality) The generating function Mπ\nλ (t) is ratio-\nnal.\n(b) (Quasi-polynomiality) The stretching function ̃mπ\nλ(n) is a quasi-polynomial\nfunction of n.\n(c) There exist graded, normal C-algebras S = S(mπ\nλ) = ⊕nSn and T =\nT(mπ\nλ) = ⊕nTn such that:\n1. The schemes spec(S) and spec(T) are normal and have rational singu-\nlarities.\n2. T = SH, the subring of H-invariants in S.\n3. The quasi-polynomial ̃mπ\nλ(n) is the Hilbert function of T.\n(d) (Positivity) The rational function Mπ\nλ (t) can be expressed in a positive\nform:\nMπ\nλ (t) = h0 + h1t + · · · + hdtd\nQ\nj(1 −ta(j))d(j)\n,\n(3.9)\nwhere a(j)’s and d(j)’s are positive integers, P\nj d(j) = d+1, where d is the\ndegree of the quasi-polynomial, h0 = 1, and hi’s are nonnegative integers.\nThe specific rings S(mπ\nλ) and T(mπ\nλ) constructed in the proof of this\nresult are called the canonical rings associated with the structrural con-\nstant mπ\nλ. The projective schemes Y (mπ\nλ) = Proj(S(mπ\nλ)), and Z(mπ\nλ) =\nProj(T(mπ\nλ)) are called the canonical models associated with mπ\nλ.\nTheorem 3.4.1 and its generalization, when H can be disconnected, is\nproved in Chapter 4; cf. Theorem 4.1.1.\n57"},{"page":61,"text":"Finitely generated semigroup\nThe following is an analogue of Theorem 1.6.2.\nTheorem 3.4.2 Assume that H is connected. For a fixed ρ : H →G, let\nT(H, G) be the set of pairs (μ, λ) of dominant weights of H and G such that\nthe irreducible representation Vπ(H) of H occurs in the irreducible repre-\nsentation Vλ(G) of G with nonzero multiplicity. Then T(H, G) is a finitely\ngenerated semigroup with respect to addition.\nThis is proved in Section 4.4.\nPSPACE\nThe following is a generalization of Theorem 1.6.3.\nTheorem 3.4.3 Assume that H in Problem 1.1.3 is a direct product, whose\neach factor is a complex simple, simply connected Lie group, or an alternat-\ning (or symmetric) group, or SLn(Fpk) (or GLn(Fpk)), or a torus. Then\nf(x) = mπ\nλ can be computed in poly(⟨x⟩) space, with x as specified above.\nThis is proved in Chapter 5. It may be conjectured that Theorem 3.4.3\nholds even when the composition factors of H are allowed to be general\nfinite simple groups of Lie type. This will be so if Lusztig’s algorithm [Lu5]\nfor computing the characters of finite simple groups of Lie type can be\nparallelized; cf. Section 5.4.\nPositivity hypotheses\nTheorem 3.4.1-3.4.3, along with the experimental results in special cases\n(cf. Chapter 6), constitute the main evidence in support of the positivity\nHypotheses 3.3.1-3.3.4 for the subgroup restrition problem.\n3.5\nThe decision problem in geometric invariant\ntheory\nFinally, let us turn to the most general Problem 1.1.4.\n58"},{"page":62,"text":"3.5.1\nReduction from Problem 1.1.3 to Problem 1.1.4\nFirst, let us note that the subgroup restriction problem (Problem 1.1.3)\nis a special case of Problem 1.1.4.\nTo see this, let H, ρ and G be as in\nProblem 1.1.3, and let X be the closed G-orbit of the point vλ corresponding\nto the highest weight vector of Vλ(G) in the projective space P(Vλ(G)). Then\nX = Gvλ ∼= G/Pλ,\n(3.10)\nwhere the P = Pλ = Gvλ is the parabolic stabilizer of vλ. We have a natural\naction of H on X via ρ. Let R be the homogeneous coordinate ring of X. By\n[Ha, MR, Rm, Sm], the singularities of spec(R) are rational. By Borel-Weil\n[FH], the degree one component R1 of the homogeneous coordinate ring R\nof X is Vλ(G). Hence, sπ\n1 in this special case of Problem 1.1.4 is precisely mπ\nλ\nin Problem 1.1.3. The results in Section 3.4 for sπ\n1 generalize in a natural\nway for sπ\nd.\n3.5.2\nInput specification\nThe variety X in the above example is completely specified by H, ρ and λ.\nHence its specification [X] can be given in the form a tuple ([H], [ρ], [λ]),\nwhere [H], [ρ] and [λ] are the specifications of H, ρ and λ as in Section 3.4,\nThe input specification x for Problem 1.1.4 in the special case above is the\ntuple ([X], d, [π]) = ([H], [ρ], [λ], d, [π]), where [π] is the specification of π as\nin Section 3.4.\nWe now describe a class of varieties X which have similar compact spec-\nifications.\nLet G be a connected, reductive group, H a reductive, possibly discon-\nnected, reductive group, and ρ : H →G an explicit polynomial homomor-\nphism as in Section 3.4. Let V = Vλ(G) be an irreducible representation of G\nfor a dominant weight λ. Let P(V ) be the projective space associated with\nV . It has a natural action of H via ρ. Let v ∈P(V ) be a point that is char-\nacterized by its stabilizer Gv ⊆G. This means it is the only point in P(V )\nthat is stabilized by Gv. For example, the point vλ above is characterized by\nits parabolic stabilier. We assume that we know the Levi decompositioon of\nGv explicity, and its compact specification [Gv], like that of H, and also an\nexplicit compact specification of the embedding ρ′ : Gv →G, aking to that\nof the explicit homomorphism ρ : H →G. Let X ⊆P(V ) be the projective\nclosure of the G-orbit of v in P(V ). Then X as well as the action of H on\nX are completely specified by λ, H, ρ, Gv and ρ′. Hence, we can let [X] be\n59"},{"page":63,"text":"the tuple (λ, [H], [ρ], [Gv], [ρ′]). The input specification x for Problem 1.1.4\nwith the X of this form is the tuple ([X], d, [π]). The bitlengths ⟨x⟩and ∥x∥\nare defined additively. The rank(x) is defined to be the sum of the ranks of\nH and G, dim(V ) and ∥π∥. Since the point vλ above is characterized by its\nstabilizer, G/P is a variety of this form.\nThe class varieties [GCT1, GCT2] are either of this form, or a slight ex-\ntension of this form, and admit such compact specifications. The algebraic\ngeometry of an X of the above form is completely determined by the repre-\nsentation theories of the two homomorphisms ρ : H →G and ρ′ : Gv →G.\nFurthermore, the results in [GCT2] say that Problem 1.1.4 for a class variety\nis intimately linked with the subgroup restriction problem and its variants\nfor the homomomorphisms ρ and ρ′. Hence it is qualitatively similar to the\nsubgroup restriction problem in this case; cf. [GCT10] for further elabora-\ntion of the connection between these two problems.\n3.5.3\nStretching function and quasi-polynomiality\nNow let H, X, R and sπ\nd be as in Problem 1.1.4, with H therein assumed to\nbe connected. We associate with f(x) = sπ\nd the following stretching fucntion:\n ̃f(x, n) = ̃sπ\nd(n) = snπ\nnd,\n(3.11)\nwhere snπ\nnd is the multiplicity of the irrreducible representation Vnπ(H) of H\nin Rnd, the componenent of the homogeneous coordinate ring R of X with\ndegree nd. Let S(t) = P\nn≥0 ̃sπ\nd(n)tn.\nTheorem 3.5.1 Assume that the singularities of spec(R) are rational.\n(a) (Rationality) The generating function Sπ\nd (t) is rational.\n(b) (Quasi-polynomiality) The stretching function ̃sπ\nd(n) is a quasi-polynomial\nfunction of n.\n(c) There exist graded, normal C-algebras S = S(sπ\nd) = ⊕nSn and T =\nT(sπ\nd) = ⊕nTn such that:\n1. The schemes spec(S) and spec(T) are normal and have rational singu-\nlarities.\n2. T = SH, the subring of H-invariants in S.\n3. The quasi-polynomial ̃sπ\nd(n) is the Hilbert function of T.\n60"},{"page":64,"text":"(d) (Positivity) The rational function Sπ\nd (t) can be expressed in a positive\nform:\nSπ\nd (t) = h0 + h1t + · · · + hktk\nQ\nj(1 −ta(j))k(j)\n,\n(3.12)\nwhere a(j)’s and k(j)’s are positive integers, P\nj k(j) = k + 1, where k is\nthe degree of the quasi-polynomial ̃sπ\nd(n), h0 = 1, and hi’s are nonnegative\nintegers.\nThis is proved in Chapter 4. Theorem 3.4.1 is a special case of this theorem,\nin view of the reduction in Section 3.5.1. Theorem 3.5.1 is applicable when\nX is a class variety, assuming that its singularities are rational.\n3.5.4\nPositivity hypotheses\nEven though Theorem 3.5.1 holds for any X, with spec(R) having ratio-\nnal singularities, the positivity hypotheses PH1, SH, PH2, and PH3 can\nbe expected to hold for only very special X’s. In general, characterizing\nthe X’s with compact specification for which these hypotheses hold is a\ndelicate problem.\nHypotheses 3.3.1-3.3.4 say that these hold when X in\nProblem 1.1.4 is G/P (as in Section 3.5.1) or a class variety, with the input\nspecification x as described above. For future reference, we shall reformulate\nthese hypotheses purely in geometric terms.\nFor this we need a definition.\nLet T = P\nn Tn be a graded complex C-algebra so that the singularities\nof spec(T) rational.\nLet Z = Proj(T).\nAssume that Z has a compact\nspecification [Z]; we shall specify it below for the Z’s of interest to us.\nWe let [T], the specification of T, to be [Z].\nThis will play the role of\nthe input in the definition below. Let ⟨T⟩denote its bitlength, and ∥T∥\ncombinatorial bit length.\nLet hT (n) = dim(Tn) be its Hilbert function,\nwhich is a quasipolynomial, since the singularities of spec(T) are rational;\ncf. Lemma 4.1.3.\nDefinition 3.5.2 We say that PH1 holds for T (or Z) if the Hilbert quasi-\npolynomial hT (n) is convex. This means there exists a polytope P = PT\ndepending on the input [T], whose Ehrhart quasipolynomial fP(n) coincides\nwith the Hilbert function hT (n), and whose membership function χP(y) can\nbe computed in poly(⟨T⟩, y) time. We assume that a separating hyperplane\ncan also be computed in polynomial time if y ̸∈P (Section 2.3).\n61"},{"page":65,"text":"If PH1 holds we can also ask if analogues of SH, PH2, and PH3–whose\nformulation is similar and hence omitted–hold.\n3.5.5\nG/P and Schubert varieties\nLet us illustrate this definition with an example. Let X ∼= G/Pλ be as in\nSection 3.5.1 and R its homogeneous coordinate ring. We have already seen\nthat it has a compact specification: namely [X] = λ. Since singularities\nof spec(R) are rational, PH1 makes sense.\nFor G/P it follows from the\nBorel-Weil theorem. The Hilbert series of R is of the form\nh0 + · · · + hdtd\n(1 −t)d+1\n,\nwith h0 = 1 and hi’s nonnegative. This is so because R is Cohen-Macauley\n[Rm] and is generated by its degree one component. Hence, the modular\nindex of the Hilbert function is one (PH3). PH2 turns out to be nontriv-\nial. Experimental evidence in its support for the classical G/P is given in\nSection 6.3. Considerations for the Schubert subvarieties are similar. Ex-\nperimental evidence for PH2 for the classical Schubert varieties is also given\nin Section 6.3.\nNow let s = sπ\nd be the multiplicity as Problem 1.1.4, with X having a\ncompact specification [X] as above. Let T = T(s) be the ring associated\nwith s as in Theorem 3.5.1 (c).\nLet Z = Z(s) = Proj(T).\nWe let the\nspecification [Z] = ([X], d, π). Let ⟨Z⟩be its bitlength.\nSo Theorem 3.1.1 in this context implies:\nTheorem 3.5.3 If PH1 and SH holds for Z(s) then nonvanishing of s,\nmodulo small relaxation, can be decided in poly(⟨Z⟩) time.\nWe also have the following reformulation:\nProposition 3.5.4 Hypotheses 3.3.1-3.3.4 are equivalent to PH1,SH,PH2,PH3\nfor Z(s), where s is a stucture constant that corresponds the structure con-\nstant f(x) in Hypotheses 3.3.1. Thus, in the case of the subgroup restriction\nproblem, s = sπ\n1 = mπ\nλ as in Section 3.5.1.\nThis is just a consequence of definitions.\n62"},{"page":66,"text":"3.6\nPH3 and existence of a simpler algorithm\nAs we remarked in Section 3.1.3, the use of the ellipsoid method and basis\nreduction in lattices makes the the algorithm for saturated integer program-\nming (cf. Theorem 3.1.1) fairly intricate. For the flip (cf. [GCTflip] and\nChapter 7), it is desirable to have simpler algorithms for the relaxed forms\nof the decision problems under consideration, akin to the the polynomial\ntime combinatorial algorithms in combinatorial optimization [Sc] that do\nnot rely on the elliposoid method or basis reduction. We briefly examine in\nthis section the role of PH3 in this context.\nThe simple combinatorial algorithms in combinatorial optimization work\nonly when the problem under consideration is unimodular–in which case the\nvertices of the underlying polytope P are integral–or almost unimodular–\ne.g. when the vertices of P are half integral. Edmond’s algorithm for finding\nminimum weight perfect matching in nonbipartite graphs [Sc] is a classic\nexample of the second case.\nIn the unimodular case, Stanley’s positivity result [St1] implies that the\nrational function FP (t) has a positive form\nFP (t) = h(d)td + · · · + h(0)\n(1 −t)d+1\n.\nIf PH3 (Hypothesis 3.3.4) holds for a structural function f(x) under con-\nsideration then the Ehrhart series FPx(t) of the polytope Px associated with\nx in PH1 (Hypothesis 3.3.1) has a minimal positive form in which each root\nof the denominator has O(poly(∥x∥)) order. Roughly, this says that the\nsituation is “close” to the unimodular case. Hence, in such a case we can\nexpect a purely combinatorial polynomial-time algorithm for deciding non-\nvanishing of f(x), modulo small relaxation, that does not need the ellipsoid\nmethod or basis reduction.\n3.7\nOther structural constants\nThe paradigm of saturated and positive integer programming in this paper,\nalong with appropriate analogues of PH1,SH,PH2,PH3, may be applicable\nseveral other fundamental structural constants in representation theory and\nalgebraic geometry, in addition to the ones in Problems 1.1.1-1.1.4 treated\nabove, such as\n63"},{"page":67,"text":"1. the value of a Kazhdan-Lusztig polynomial at q = 1, [KL1];\n2. the values at q = 1 of the well behaved special cases of the parabolic\nKostka polynomials and their q-analogues [Ki];\n3. the structural coefficients of the multiplication of Schubert polynomi-\nals, and so on.\n64"},{"page":68,"text":"Chapter 4\nQuasi-polynomiality and\ncanonical models\nIn this chapter we prove quasipolynomiality of the stretching functions as-\nsociated with the various structural constants under consideration (Sec-\ntion 4.1), describe the associated canonical models (Section 4.2), describe\nthe role of nonstandard quantum groups in [GCT4, GCT7, GCT8] in the\ndeeper study of these models (Section 4.3), prove finite generation of the\nsemigroup of weights (Theorem 3.4.2) (Section 4.4), and give an elementary\nproof of rationality in Theorem 3.4.1 (a) (Section 4.5).\n4.1\nQuasi-polynomiality\nHere we prove Theorem 3.5.1; Theorems 1.6.1 and 3.4.1 are its special cases\nin view of the reduction in Section 3.5.1. This, in turn, follows from the\nfollowing more general result.\nLet R = ⊕kRd be a normal graded C-algebra with an action of a reduc-\ntive group H. Assume that spec(R) has rational singularities. Let H0 be\nthe connected component of H containing the identity. Let HD = H/H0 be\nits discrete component. Given a dominant weight π of H0, we consider the\nmodule Vπ = Vπ(H0), an H-module with trivial action of HD. Let sπ\nd denote\nthe multiplicity of the H-module Vπ in Rd. Let ̃sπ\nd(n) be the multiplicity of\nthe H-module Vnπ in Rnd. This is a stretching function associated with the\nmulitplicity sπ\nd. Let Sπ\nd (t) = P\nn≥0 ̃sπ\nd(n)tn.\n65"},{"page":69,"text":"Theorem 4.1.1 (a) (Rationality) The generating function Sπ\nd (t) is ratio-\nnal.\n(b) (Quasi-polynomiality) The stretching function ̃sπ\nd(n) is a quasi-polynomial\nfunction of n.\n(c) There exist graded, normal C-algebras S = S(sπ\nd) = ⊕nSn and T =\nT(sπ\nd) = ⊕nTn such that:\n1. The schemes spec(S) and spec(T) are normal and have rational singu-\nlarities.\n2. T = SH, the subring of H-invariants in S.\n3. The quasi-polynomial ̃sπ\nd(n) is the Hilbert function of T.\n(d) (Positivity) The rational function Sπ\nd (t) can be expressed in a positive\nform:\nSπ\nd (t) = h0 + h1t + · · · + hktk\nQ\nj(1 −ta(j))k(j)\n,\n(4.1)\nwhere a(j)’s and k(j)’s are positive integers, P\nj k(j) = k + 1, where k is\nthe degree of the quasi-polynomial ̃sπ\nd(n), h0 = 1, and hi’s are nonnegative\nintegers.\nTheorem 3.5.1 follows from this by letting R be the homogeneous coordinate\nring of X.\nMore generally, if W is an irreducible representation of HD, we can\nconsider the H-module Vπ ⊗W. Let sπ,W\nd\nbe its multiplicity in Rd. Let\n ̃sπ,W\nd\n(n) be the multiplicity of the trivial H-representation in the H-module\nRnd ⊗V ∗\nnπ ⊗Symn(W ∗). Then\nTheorem 4.1.2 Analogue of Theorem 4.1.1 holds for ̃sπ,W\nd\n(n).\nFor the purposes of the flip, Theorem 4.1.1 suffices.\nProof: We shall only prove Theorem 4.1.1, the proof of Theorem 4.1.2 be-\ning similar.\nThe proof is an extension of M. Brion’s proof (cf.\n[Dh]) of\nquasi-polynomiality of the stretching function associated with a Littlewood-\nRichardson coefficient of any semisimple Lie algebra.\nClearly (a) follows from (b); cf. [St1].\n66"},{"page":70,"text":"(b) and (c):\nLet Cd be the cyclic group generated by the primitive root ζ of unity of\norder d. It has a natural action on R: x ∈Cd maps z ∈Rk to xkz. Let\nB = RCd = P\nn≥0 Rnd ⊆R be the subring of Cd-invariants. By Boutot\n[Bou], B is a normal C-algebra and spec(B) has rational singularities.\nAssume that H0 is semisimple; extension to the reductive case being easy.\nLet π∗be the dominant weight of H0 such that V ∗\nπ = Vπ∗. By Borel-Weil\n[FH],\nCπ∗= ⊕n≥0V ∗\nnπ = ⊕n≥0Vnπ∗,\nis the homogeneous coordinate ring of the H0-orbit of the point vπ∗∈P(Vπ∗)\ncorresponding to the highest weight vector. This H0-orbit is isomorphic to\nH0/Pπ∗, where Pπ∗⊆H0 is the parabolic stabilizer of vπ∗. Hence Cπ∗is\nnormal and spec(Cπ∗) has rational signularities; cf.\n[Ha, MR, Rm, Sm].\nIt follows that B ⊗Cπ∗is also normal, and spec(B ⊗Cπ∗) has rational\nsingularities. Consider the action of C∗on B ⊗Cπ∗given by:\nx(b ⊗c) = (x · b) ⊗(x−1 · c),\nwhere x ∈C∗maps b ∈Bn to xnb, the action on Cπ∗being similar. Consider\nthe invariant ring\nS = (B ⊗Cπ∗)C∗= ⊕nSn = ⊗n≥0Rnd ⊗V ∗\nnπ.\n(4.2)\nBy Boutot [Bou], it is a normal, and spec(D) has rational singularities.\nSince Vnπ is an H-module, the algebra S has an action of H. Let\nT = T(sπ\nd) = SH = ⊕n≥0Tn\n(4.3)\nbe its subring of H-invariants. By Boutot [Bou], it is normal, and spec(T)\nhas rational singularities–this is the crux of the proof. By Schur’s lemma, the\nmultiplicity of the trivial H-representation in Sn = Rnd⊗V ∗\nnπ is precisely the\nmultiplicity ̃sπ\nd(n) of the H-module Vnπ in Rnd. Hence, the Hilbert function\nof T, i.e., dim(Tn), is precisely ̃sπ\nd(n), and the Hilbert series P\nn≥0 dim(Tn)tn\nis Sπ\nd (t).\nQuasipolynomiality of ̃sπ\nd(n) follows by applying the following\nlemma:\nLemma 4.1.3 (cf. [Dh]) If T = ⊕∞\nn=0Tn is a graded C-algebra, such that\nspec(T) is normal and has rational simgularites, then dim(Tn), the Hilbert\nfunction of T, is a quasi-polynomial function of n.\n67"},{"page":71,"text":"(d) Since spec(T) has rational singularities, T is Cohen-Macaualey.\nLet\nt1, . . . , tu be its homogeneous sequence of parameters (h.s.o.p.), where u =\nk + 1 is the Krull dimension of T. By the theory of Cohen-Macauley rings\n[St2], it follows that its Hilbert series Sπ\nd (t) is of the form\nh0 + h1t + · · · + hktk\nQk+1\ni=1 (1 −tdi)\n,\n(4.4)\nwhere (1) h0 = 1, (2) di is the degree of ti, and (3) hi’s are nonnegative\nintegers. This proves (d). Q.E.D.\nRemark 4.1.4 A careful examination of the proof above shows that ratio-\nnality of Sπ\nd (t), and more strongly, asymptotic quasi-polynomiality of ̃sπ\nd(n)\nas n →∞, can be proved using just Hilbert’s result on finite generation of\nthe algebra of invariants of a reductive-group action. Boutot’s result is nec-\nessary to prove quasi-polynomiality for all n. This is crucial for saturated\nand positive integer programming (Chapter 3).\n4.1.1\nThe minimal positive form and modular index\nThe form (4.4) of Sπ\nd (t) is not unique because it depends on the degrees di’s\nof the paramters ti’s. For future use, let us record the following consequences\nof the proof. Let T be the ring constructed in the proof above.\nCorollary 4.1.5 Suppose T has an h.s.o.p.\nt = (t1, . . . , tu) with di =\ndeg(ti). Then Sπ\nd (T) has a positive rational form (4.4) with di = deg(ti)\ntherein.\nThe proof above is lets us define a minimal positive form of the rational\nfunction Sπ\nd (t) associated with a structural constant s. For this, let us or-\nder h.s.o.p.’s of T lexicographically as per their degree sequences. Here the\ndegree seqeunce of an h.s.o.p. t = (t1, . . . , tu) is defined to be (d1, . . . , du),\nwhere di = deg(ti). The form (4.4) is the same for any h.s.o.p. of lexi-\ncographically minimum degree sequence. We call it the minimal positive\nform of Sπ\nd (t). The modular index of sπ\nd is defined to be max{di}, where\n(d1, . . . , du) is the degree sequence of a lexicographically minimal h.s.o.p.\nSince Problems 1.1.1, 1.1.2,1.1.3, 1.2.1 are special cases of Problem 1.1.4,\nthis defines minimal positive forms of the rational generating functions of the\nstretching quasi-polynomials (cf. Theorem 3.4.1) associated with the struc-\ntural constants in these problems, and also the modular indices of these\nstructural constants.\n68"},{"page":72,"text":"4.1.2\nThe rings associated with a structural constant\nThe preceding proof also associates with the structural constant s a few\nrings which will be important later. Specifically, let S = S(s) and T = T(s)\nbe the rings as in Theorem 4.1.1 (c) associated with the structural constant\ns = sπ\nd.\nLet R = R(s) be the homogeneous coordinate ring of X as in\nTheorem 4.1.1. We call R(s), S(s) and T(s) the rings associated with the\nstructure constant s.\nWhen s = mπ\nλ, as in the subgroup restriction problem (Problem 1.1.3),\nX ∼= G/P as given in eq.(3.10. Then these rings are explicitly as follows:\nR(mπ\nλ)\n=\n⊕n≥0Vnλ(G),\nS(mπ\nλ)\n=\n⊕n≥0Vnλ(G) ⊗Vnπ(H)∗,\nT(mπ\nλ)\n=\n⊕n≥0(Vnλ(G) ⊗Vnπ(H)∗)H.\n(4.5)\nBy specializing the subgroup restriction problem further to the Littlewood-\nRichardson problem (Problem 1.2.1), we get the following rings associated\nby Brion (cf. [Dh]) with the Littlewood-Richardson coefficient cλ\nα,β:\nR(cλ\nα,β)\n=\n⊕n≥0Vnα(H) ⊗Vnβ(H),\nS(cλ\nα,β)\n=\n⊕n≥0Vnα(H) ⊗Vnβ(H) ⊗Vnλ(H)∗,\nT(cλ\nα,β)\n=\n⊕n≥0(Vnα(H) ⊗Vnβ(H) ⊗Vnλ(H)∗)H.\n(4.6)\n4.2\nCanonical models\nThere are several rings other than T(cλ\nα,β) whose Hilbert function coincides\nwith the Littlewood-Richardson stretching quasi-polynomial ̃cλ\nα,β(n).\nFor\nexample, let P = P λ\nα,β be the BZ-polytope [BZ] whose Ehrhart quasi-\npolynomial coincides with ̃cλ\nα,β(n).\nWe can associate with P a ring TP\nas in Stanley [St3] whose Hilbert function coincides with ̃cλ\nα,β(n).\nThere\nare many other choices for P. For example, in type A, we can consider a\nhive polytope or a honeycomb polytope [KT1] instead of the BZ-polytope.\nThe rings TP ’s associated with different P’s will, in general, be different,\nand there is nothing canonical about them. In contrast, the ring T(cλ\nα,β) is\nspecial because:\nProposition 4.2.1 (PH0) The rings R(cλ\nα,β), S(cλ\nα,β), T(cλ\nα,β) have quan-\ntizations Rq(cλ\nα,β), Sq(cλ\nα,β), Tq(cλ\nα,β) endowed with canonical bases in the ter-\nminology of Lusztig [Lu4]. Furthermore, the canonical bases of Rq(cλ\nα,β), Sq(cλ\nα,β)\n69"},{"page":73,"text":"are compatible with the action of the Drinfeld-Jimbo quantum group associ-\nated with H = GLn(C), and the canonical basis of Sq(cλ\nα,β) is an extension\nof the canonical basis of Tq(cλ\nα,β) in a natural way.\nThis follows from the work of Lusztig (cf. [Lu3], Chapter 27 in [Lu4]) and\nKashiwara (cf.\nTheorem2 in [Kas3]).\nSpecializations of these canonical\nbases at q = 1 will be called canonical bases of R(cλ\nα,β), S(cλ\nα,β), T(cλ\nα,β).\nLusztig [Lu4] has conjectured that the structural constants associated with\nthe canonical bases in Proposition 4.2.1 are polynomials in q with nonnega-\ntive integral coefficients as in the case of the canonical basis of the (negative\npart of the) Drinfeld-Jimbo enveloping algebra. We refer to Proposition 4.2.1\nas PH0 in view of this (conjectural) positivity property.\nIn view of this proposition, we call the rings R(cλ\nα,β), S(cλ\nα,β) and T(cλ\nα,β)\nthe canonical rings associated with the Littlewood-Richardson coefficient\ncλ\nα,β, and X = Proj(R(cλ\nα,β)), Y = Proj(S(cλ\nα,β)) and Z = Proj(T(cλ\nα,β)) the\ncanonical models associated with cλ\nα,β.\n4.2.1\nFrom PH0 to PH1,3\nNow we study the relevance of PH0 above in the context of PH1,SH,PH2,\nand PH3 for Littlewood-Richardson coefficients (Section 1.2).\nPH1\nAs already remarked in Section 1.7, PH1 for Littlewood-Richardson coeffi-\ncients is a formal consequence of the properties of Kashiwara’s crystal oper-\nators on the canonical bases in PH0 (Proposition 4.2.1); [Dh, Kas2, Li, Lu4].\nSpecifically, the canonical basis of the ring Rq(cλ\nα,β) also yields a canon-\nical basis for the tensor product Vq,α ⊗Vq,β of the irreducible Hq modules\nwith highest weights α and β. The Littlwood-Richardson rule for arbitrary\ntypes follows from the study of Kashiwara’s crystal operators on this canon-\nical basis for the tensor product; [Lu4]. This rule is equivalent to the one\nin [Li] based on combinatorial interpretation of the crystal operators in the\npath model therein. The article [Dh] derives a convex polyhedral formula\nfor Littlewood-Richardson coefficients (of arbitrary type) using this com-\nbinatorial interpretation. Though the complexity-theoretic issues are not\naddressed in [Dh], it can be verified that the polyhedral formula therein is a\nconvex #P-formula. This yields PH1 for Littlewood-Richardson coefficients\nof arbitrary types using PH0.\n70"},{"page":74,"text":"SH\nNow let us see the relevance of PH0 in the context of SH for Littlewood-\nRichardson coefficients of arbitrary type.\nThe polytope in [Dh], mentioned above, for type A is equivalent to the\nhive polytope in [KT1] in the sense that the number integer points in both\nthe polytopes is the same. Knutson and Tao prove SH for type A by show-\ning that the hive polytope always has in integral vertex. To extend this\nproof to an arbitrary type, one has to convert the polytope in [Dh] into a\npolytope that is guaranteed to contain an integral vertex if the index of the\nstretching quasipolynomial ̃cλ\nα,β(n) is one. The main difficulty here is that\nwe do not have a nice mathematical interpretation for the index. Algorithm\nin Theorem 3.1.1 applied to the polytope in [Dh] computes this index in\npolynomial time. But it does not give a nice interpretation that can be used\nin a proof as above.\nThis index is simply the largest integer dividing the degrees of all ele-\nments in any basis of the canonical ring T(cλ\nα,β)–in particular, the canon-\nical basis. This follows by applying Proposition 3.1.3 to the polytope in\n[Dh]. This leads us to ask: is there an interpretation for the index based on\nLusztig’s topological construction of the canonical basis in Proposition 4.2.1?\nIf so, this may be used to extend the known polyhedral proof for SH in type\nA to arbitrary types. Alternatively, it may be possible to prove SH using\ntopological properties of the canonical basis in the spirit of the topological\n(intersection-theoretic) proof [Bl] of SH in type A.\nPH3\nNow let us see the relevance of PH0 in the context of PH3 for Littlewood-\nRichardson coefficients.\nFirst, let us consider the minimal positive form (Section 4.1.1) associated\nwith a Littlewood-Richardson coefficient cλ\nα,β of type A. Let T = T(cλ\nα,β)\ndenote the ring that arises in this case; cf. eq.(4.6). Now we can ask:\nQuestion 4.2.2 Are all di’s occuring in the minimal positive form (cf.\n(4.4)) one in this special case?\nThis is equivalent to asking if the ring\nT = T(cλ\nα,β) in this case is integral over T1, the degree one component of T.\nIf so, this would provide an explanation for the conjecture of King at al\n[KTT] (cf. eq.(1.3)) in the theory of Cohen-Macauley rings:\n71"},{"page":75,"text":"Proposition 4.2.3 Assuming yes, the conjecture of King et al [KTT] (Hy-\npothesis 1.2.6) holds.\nRemark 4.2.4 In contrast, the ring TP associated with the hive polytope\n(cf. beginning of Section 4.2) need not be integral over its degree one compo-\nnent, in view of the fact that the hive polytope can have nonintegral vertices\n[DM1].\nRemark 4.2.5 T = T(cλ\nα,β) need not be generated by its degree one compo-\nnent T1. If this were always so, the h-vector (hd, · · · , h0) in eq.(1.3) would\nbe an M-vector (Macauley-vector) [St2]. But one can construct α, β and λ\nfor which this does not hold.\nProof: (of the proposition) Since T is integral over T1, it has an h.s.o.p., all of\nwhose elements have degree 1. By Theorem 3.4.1, the singularities of spec(T)\nare rational.\nHence T is Cohen-Macaulay.\nNow the result immediately\nfollows from the theory of Cohen-Macauley rings [St2]. Q.E.D.\nIn view of this Proposition, the conjecture of King et al will follow if all\ncanonical basis elements of T(cλ\nα,β) can be shown to be integral over the basis\nelements of degree one. This requires a further study of the multiplicative\nstructure of this canonical basis. Considerations for PH3 (Hypothesis 1.2.8)\nfor Littlewood-Richardson coefficients of arbitrary type are similar.\nPH2\nSimilarly, the positivity property (PH2) of the stretching quasipolynomial\nassociated with Littlewood-Richardson coefficients may possibly follow from\na deep study of the multiplicative structure of the canonical basis as per\nPH0 (Proposition 4.2.1), just as positivity of the multiplicative structural\ncoefficients of the canonical basis for the (negative part of the) Drinfeld-\nJimbo enveloping algebra follows from a deep study of the multiplicative\nstructure of this basis [Lu4].\n4.2.2\nOn PH0 in general\nThe discussion above indicates that for Littlewood-Richardson coefficients\nPH1,SH,PH3, and plausibly PH2 as well are intimately related to PH0\n(Proposition 4.2.1).\nThis leads us to ask if the rings associated in Sec-\ntion 4.1.2 with other structural constants under consideration in this paper\n72"},{"page":76,"text":"have quantizations which satisfy appropriate forms of PH0. If so, this PH0\nmay be used to derive PH1, SH, PH3, and PH2 (Hypotheses 3.3.1-3.3.4)\nfor these structural constants. Note that SH (a) follows from PH3 (see the\nremark after Hypothesis 3.3.4); PH2 may also follow from PH3. Thus PH1\nand PH3 are the ones to focus on.\nTo formalize this, let s be a structural constant which is either the Kro-\nnecker coefficient as in Problem 1.1.1, or the plethysm constant as in Prob-\nlem 1.1.2, or the multiplicity mπ\nλ in Problem 1.1.3, or the multiplicity sπ\nd, as\nin Problem 1.1.4, when X therein is a class variety. Let R(s), S(s), T(s) be\nthe rings associated with s (Section 4.1.2). Let X(s) = Proj(R(s)), Y (s) =\nProj(S(s)) and Z(s) = Proj(R(s)). We call R = R(s), S = S(s), T = T(s)\nthe canonical rings associated with s, and X(s), Y (s), Z(s) the canonical\nmodels associated with s, because we expect these rings and models to be\nspecial as in the case of the Littlewood-Richardson coefficients.\nLet H be as in Problem 1.1.3 or Problem 1.1.4.\nAssume that H is\nconnected. Let Hq denote the Drifeld-Jimbo quantization of H. Now we\nask:\nQuestion 4.2.6 (PH0??) Are there quatizations Rq, Sq of R, S, with Hq-\naction, and a quantization Tq of T with “canonical” bases (in some appro-\npriate sense) B(Rq), B(Sq), B(Tq), where B(Rq) and B(Sq) are compatible\nwith the Hq-action and B(Sq) is an extension of B(Tq)? Furthermore, do\nthese canonical bases have appropriate positivity properties?\nIn other words, are there quantizations of R, S and T for which PH0\n(Proposition 4.2.1) can be extended in a natural way?\nIf so, this extended PH0 may be used to prove PH1 and SH for s just as\nin the case of Littlewood-Richardson coefficients (of type A).\n4.3\nNonstandard quantum group for the Kronecker\nand the plethysm problems\nWe now consider this question when s is the kronecker or the plethysm\nconstant (cf. Problems 1.1.1 and 1.1.2). PH0 for Littlewood-Richardson\ncoefficients (Proposition 4.2.1) depends critically on the theory of Drinfeld-\nJimbo quantum groups. This is intimately related (in type A) [GrL] to the\nrepresentation theory of Hecke algebras. To extend PH0 in the context of the\nkronecker and the plethysm constants, one needs extensions of these theories\n73"},{"page":77,"text":"in the context of Problems 1.1.1-1.1.2. In this section, we briefly review the\nresults in [GCT4, GCT8, GCT7] in this direction and the theoretical and\nexperimental evidence it provides in support of PH0–that is, affirmative\nanswer to Question 4.2.6–in this context.\nSo let us consider the generalized plethysm problem (Problem 1.1.2).\nAs expected, the representation theory of Drinfeld-Jimbo quantum groups\nand Hecke algebras does not work in the context of this general problem.\nBriefly, the problem is that if H is a connected, reductive group and V its\nrepresentation, then the homomorphism H →G = GL(V ) does not quan-\ntize in the setting of Drinfeld-Jimbo quantum groups. That is, there is no\nquantum group homomorphism from Hq, the Drinfeld-Jimbo quantization\nof H, to Gq, the Drinfeld-Jimbo quantization of G. In [GCT4, GCT7], a new\nnonstandard quantization GH\nq of G– called a nonstandard quantum group–is\nconstructed so that there is a quantum group homomorphism Hq →GH\nq .\nWhen H = G, GH\nq coincides with the Drinfeld-Jimbo quantum group. The\narticle [GCT8] gives a conjectural scheme for constructing a nonstandard\ncanonical basis for the matrix coordinate ring of GH\nq\nthat is akin to the\ncanonical basis for the matrix coordinate ring of the Drinfeld-Jimbo quan-\ntum group [Lu4, Kas3].\nIt is known that the Drinfeld-Jimbo quantum group Gq = GLq(V ) and\nthe Hecke algebra Hn(q) are dually paired: i.e., they have commuting ac-\ntions on V ⊗n\nq\nfrom the left and the right that determine each other, where Vq\ndenotes the standard quantization of V . Furthermore, the Kazhdan-Lusztig\nbasis for Hn(q) is intimately related to the canonical basis for Gq [GrL]. Sim-\nilarly, [GCT7] constructs a nonstandard generalization BH\nn (q) of the Hecke\nalgebra which is (conjecturally) dually paired to GH\nq . The article [GCT8]\ngives a conjectural scheme for constructing a nonstandard canonical basis of\nBH\nn (q) akin to the Kazhdan-Lusztig basis of the Hecke algebra Hn(q).\nThe nonstandard quantum group GH\nq and the nonstandard algebra BH\nn (q)\nturn out to be fundamentally different from the standard Drinfeld-Jimbo\nquantum group Gq and the Hecke algebra Hn(q). For example, the non-\nstandard quantum group GH\nq is a nonflat deformation of G in general. This\nmeans the Poincare series of the matrix coordinate ring of GH\nq is different\nfrom the Poincare series of the matrix coordinate ring of G. Specifically,\nthe terms of the first series can be smaller than the respective terms of the\nsecond series. Similarly, BH\nn (q) is a nonflat deformation of the group algebra\nC[Sn] of the symmetric group Sn; i.e., its dimension can be bigger than that\nof C[Sn].\n74"},{"page":78,"text":"Nonflatness of GH\nq intuitively means that it is “smaller” than G in gen-\neral. Hence, it may seem that there is a loss of information when one goes\nfrom G to GH\nq . Fortunately, there is none, as per the reciprocity conjecture\nin [GCT7]. This roughly says that the information which is lost in the tran-\nsition from G to GH\nq simply gets transfered to BH\nn (q), which is bigger than\nHn(q). In other words, there is no information loss overall. Hence analogues\nof the properties in the standard setting should also hold in the nonstandard\nsetting, though in a far more complex way.\nThat is what seems to happen to positivity. Specifically, experimental ev-\nidence suggests that the conjectural nonstandard canonical bases in [GCT8]\nhave nonstandard positivity properties which are complex versions of the\npositivity properties in the standard setting. See [GCT7, GCT8, GCT10]\nfor a detailed story.\n4.4\nThe cone associated with the subgroup restric-\ntion problem\nIn this section, we prove Theorem 3.4.2, by extending the proof of Brion\nand Knop (cf. [El]) for the Littlewood-Richardson problem. The proof is in\nthe spirit of the proof of quasipolynomiality in Section 4.1.\nLet G be a connected, reductive group, H a connected, reductive sub-\ngroup, and ρ : H →G a homomorphism. Theorem 3.4.2 has the following\nequivalent formulation.\nLet S(H, G) be the set of pairs (μ, λ) such that\nVμ(H) ⊗Vλ(G) has a nonzero H-invariant. Then,\nTheorem 4.4.1 The set S(H, G) is a finitely generated semigroup with re-\nspect to addition.\nWhen G = H × H and the embedding H ⊆G is diagonal, this special-\nizes to the Brion-Knop result mentioned above. The proof follows by an\nextension the technique therein.\nProof: Let B be a Borel subgroup of G, U the unipotent radical of B and\nT the maximal torus in B. Similarly, let B′ be a Borel subgroup of H, U ′\nthe unipotent radical of B′ and T ′ the maximal torus in B′. Without loss\nof generality, we can assume that B′ ⊆B, U ′ ⊆U, T ′ ⊆T. Let A = C[G]U\nbe the algebra of regular functions on G that are invariant with respect to\nthe right multiplication by U. It is known to be finitely generated [El]. The\ngroups G and T act on A via left and right multiplication, respectively. As\n75"},{"page":79,"text":"a G × T-module,\nA = ⊕λVλ(G),\n(4.7)\nwhere the torus T acts on Vλ(G) via multiplication by the highest weight\nλ∗of the dual module. Similarly,\nA′ = C[H]U′ = ⊕λVμ(H),\n(4.8)\nwhere the torus T ′ acts on Vμ(H) via multiplication by the highest weight\nμ∗of the dual module.\nNow A ⊗A′ is finitely generated since A and A′ are. Let X = (A ⊗A′)H\nbe the ring of invariants of H acting diagonally on A⊗A′. The torus T ×T ′\nacts on X from the right. Since H is reductive, X is finitely generated [PV].\nHence, the semigroup of the weights of the right action of T × T ′ on X is\nfinitely generated. We have\nX = (A ⊗A′)H = ((⊕Vλ(G)) ⊗(⊕Vμ(H)))H = ⊕(Vλ(G) ⊗Vμ(H))H,\nand the weights of the algebra X are of the form (λ∗, μ∗) such that Vλ(G) ⊗\nVμ(H) contains a nontrivial H-invariant.\nTherefore these pairs form a\nfinitely generated semigroup. Q.E.D.\nFor the sake of simplicity, assume that G and H are semisimple in what\nfollows.\nLet TR(H, G) denote the polyhedral convex cone in the weight\nspace of H × G generated by T(H, G), as defined in Theorem 3.4.2. This is\na generalization of the Littlewood-Richardson cone (Section 2.2.2).\nThe following generalization of Corollary 3.2.3 is a consequence of The-\norem 3.1.1 and its proof.\nTheorem 4.4.2 Assume that the positivity hypothesis PH1 (Section 3.3)\nholds for the subgroup restriction problem for the pair (H, G), where both H\nand G are classical. Given dominant weights μ, λ of H and G, the polytope\nPμ,λ as in PH1 has a specification of the form\nAx ≤b\n(4.9)\nwhere A depends only on H and G, but not on μ or λ, and b depends\nhomogeneously and linearly on μ, λ. Let n be the total number of columns\nin A.\nThen, there exists a decomposition of TR(H, G) into a set of polyhedral\ncones, which form a cell complex C(H, G), and, for each chamber C in this\n76"},{"page":80,"text":"complex, a set M(C) of O(n) modular equations, each of the form\nX\ni\naiμi +\nX\ni\nbiλi = 0\n(mod d),\nsuch that\n1. Saturation hypothesis SH is equivalent to saying that: (μ, λ) ∈T(H, G)\niff(μ, λ) ∈TR(H, G) and (μ, λ) satisfies the modular equations in the\nset M(Cμ,λ) associated with the smallest cone Cμ,λ ∈C(H, G) contain-\ning (μ, λ).\n2. Given (μ, λ), whether (μ, λ) ∈TR(H, G) can be determined in polyno-\nmial time.\n3. If so, whether (μ, λ) satisfies the modular equations associated with\nthe smallest cone in C(H, G) containing it can also be determined in\npolynomial time.\nProof: Given a point p = (μ′, λ′) in the weight space of H ×G, where μ′ and\nλ′ are arbitrary rational points, let S(p) denote the constraints (half-spaces)\nin the sytem (4.9) whose bounding hyperplanes contain the polytope Pμ′,λ′.\nWe can decompose TR(H, G) into a conical, polyhedral cell complex, so that\ngiven a cone C in this complex, and a point p in its interior, the set S(p)\ndoes not depend on p. We shall denote this set by S(C). Thus the affine\nspan of Pμ,λ, for any (μ, λ) ∈C, is determined by the linear system\nA′x = b′,\nwhere [A′, b′] consists of the rows of [A, b] in (4.9) corresponding to the set\nS(C). By finding the Smith normal form of A′, we can associate with C a set\nof modular equations that the entries of b′ must satisfy for this affine span to\ncontain an integer point; see the proof of Theorem 3.1.1. Since the entries of\nA′ depend only on H and G, these equations depend only on C. If (μ, λ) ∈\nT(H, G), then (μ, λ) is integral, and hence these equations are satisfied.\nConversely, if (μ, λ) ∈TR(H, G) and these equations are satisfied, then the\nsaturation property implies that (μ, λ) ∈T(H, G), as seen by examining\nthe proof of Theorem 3.1.1. Furthermore, given (μ, λ), the algorithm in the\nproof of Theorem 3.1.1 implicitly determines if (μ, λ) ∈TR(H, G) and if\nthese modular equations are satisfied in polynomial time. Q.E.D.\n77"},{"page":81,"text":"4.5\nElementary proof of rationality\nIn this section we give an elementary proof of rationality in Theorem 3.4.1\n(a), when H therein is connected–actually of a slightly stronger statement:\nnamely, the stretching function ̃mπ\nλ(n) is asymptotically a quasipolynomial,\nas n →∞; cf. Remark 4.1.4. But this proof cannot be extended to prove\nquasipolynomiality for all n. The proof here is motivated by the work of\nRassart [Rs], De Loera and McAllister on the stretching function associated\nwith a Littlewood-Richardson coefficient.\nFirst, we recall some standard results that we will need.\nVector partition functions\nGiven an integral s×n matrix B and integral n-vector c, consider the vector\nparitition function φB(c), which is the number of integer solutions to the\ninteger programming problem\nBy = c,\ny ≥0.\n(4.10)\nFor a fixed c, b, let\nφB,c(n) = φB(nc)\nφB,c,b(n) = φB(nc + b).\n(4.11)\nBy Sturmfels [Stm] and Szenes-Vergne residue formula [SV], φB(c) is a\npiecewise quasipolynomial function of c. That is, Rn can be decomposed into\npolyhedral cones, called chambers, so that the restriction of φB(c) to each\nchamber R is a multivariate quasipolynomial function of the coordinates of c.\nThis implies that φB,c(n) is a quasipolynomial function of n. It also implies\nthat the function φB,c,b(n) is asymptotically a quasipolynomial function of\nn, as n →∞, because the points nc + b, as n →∞, lie in just one chamber.\nThe Szenes-Verne residue formula [SV] for vector partition functions also\nimplies that there is a constant d(B), depending only on B, such that the\nperiod of φB,c(n), for any c, divides d(B).\nKlimyk’s formula\nLet H ⊆G and mπ\nλ be as in Theorem 3.4.1 (a), with H connected. Let us\nassume that H is semisimple, the general case being similar. Let H and G\nbe the Lie algebras of H and G respectively. We recall Klimyk’s formula for\nmπ\nλ. Without loss of generality, we can assume that the Cartan subalgebra\n78"},{"page":82,"text":"C ⊆H is a subalgebra of the Cartan subalgebra D ⊆G.\nSo we have a\nrestriction from D∗to C∗, and we assume that the half-spaces determining\npositive roots are compatible. We denote weights of H by symbols such as μ\nand of G by symbols such as ̄μ. To be consistent, we shall use the notation\nmπ ̄λ instead of mπ\nλ in this proof. We write ̄μ ↓μ if the weight ̄μ of G restricts\nto the weight μ of H. We denote a typical element of the Weyl group of\nH by W, and a typical element of the Weyl group of G by ̄W. Given a\ndominant weight π of G and a weight ̄μ of G, let n ̄μ( ̄λ) denote the dimension\nof the weight space for ̄μ in B ̄λ = V ̄λ(G).\nWe assume that:\n(A): For any weight μ of H, the number of ̄μ’s such that ̄μ ↓μ is finite.\nFor example, this is so in the plethysm problem (Problem 1.1.2). We\nshall see later how this assumption can be removed.\nBy Klimyk’s formula (cf. page 428, [FH]),\nmπ ̄λ =\nX\nW\n(−1)W\nX\n ̄μ↓π−ρ−W (ρ)\nn ̄μ(V ̄λ),\n(4.12)\nwhere ρ is half the sum of positive roots of H. We allow ̄μ in the inner sum\nto range over all weights ̄μ of G such that ̄μ ↓π −ρ −W(ρ) by defining\nn ̄μ(V ̄λ) to be zero if ̄μ does not occur in V ̄λ.\nProof of Theorem 3.4.1 (a)\nThe goal is to express ̃mπ ̄λ(n) as a linear combination of vector partition\nfunctions φB,c,b(n)’s, for suitable B, c, b’s, using Klimyk’s formula for mπ ̄λ.\nAfter this, we can deduce asymptotic quasipolynomiality of ̃mπ ̄λ(n) from\nasymptotic quasipolynomiality of φB,c,b(n)’s.\nBy Kostant’s multiplicity formula (cf. page 421 [FH]),\nn ̄μ(V ̄λ) =\nX\n ̄\nW\n(−1)\n ̄\nW P( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ)),\n(4.13)\nwhere P( ̄λ), for a weight ̄λ of G, denotes the Kostant partition function;\ni.e., the number of ways to write ̄λ as a sum of positive roots of G. It is\nimportant for the proof that Kostant’s formula (4.13) holds even if ̄μ is not\na weight that occurs in the representation V ̄λ–in this case, n ̄μ(V ̄λ) = 0, and\nthe right hand side of (4.13) vanishes.\nBy eq.(4.12) and (4.13),\n79"},{"page":83,"text":"mπ ̄λ =\nX\nW\nX\n ̄\nW\n(−1)W (−1)\n ̄\nW\nX\n ̄μ↓π−ρ−W (ρ)\nP( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ)).\n(4.14)\nLet D denote the dominant Weyl chamber in the weight space of G. Let\nC denote the Weyl chamber complex associated with the weight space of G.\nThe cells in this complex are closed polyhedral cones. Each cone is either\nthe chamber ̄W(D), for some Weyl group element ̄W, or a closed face of\n ̄W(D) of any dimension.\nUsing M ̈obius inversion, the inner sum\nX\n ̄μ↓π−ρ−W (ρ)\nP( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ))\nin eq.(4.14) can be written as a linear combination\nX\nC\na(C)\nX\n ̄μ∈C: ̄μ↓π−ρ−W (ρ)\nP( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ)),\nwhere C ranges over chambers in the Weyl chamber complex C, a(C) is an\nappropriate constant for each C.\nHence,\nmπ ̄λ =\nX\nW\nX\n ̄\nW\n(−1)W (−1)\n ̄\nW X\nC\na(C)\nX\n ̄μ∈C: ̄μ↓π−ρ−W (ρ)\nP( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ)).\n(4.15)\nNow think of π and ̄λ as variables. But H and G are fixed, and hence\nalso the quantities such as ρ and ̄ρ.\nClaim 4.5.1 For fixed Weyl group elements W, ̄\nW and a fixed C, the sum\nX\n ̄μ∈C: ̄μ↓π−ρ−W (ρ)\nP( ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ))\n(4.16)\ncan be expressed as a vector partition function associated with an appropriate\nlinear system\nBy = c,\ny ≥0,\n(4.17)\nwhere the matrix\nB = BH,G,C,\n80"},{"page":84,"text":"depends only on C and the root systems of H and G, but not on π and ̄λ,\nand the coordinates of the vector\nc = mW, ̄\nW,C( ̄λ, π, ρ, ̄ρ),\ndepend on W, ̄W, C, ρ, ̄ρ, π, π, and furthermore, their dependence on π, ̄λ, ρ, ̄ρ\nis linear.\nHere assumption (A) is crucial. Without it, the sum (4.16) can diverge. Of\ncourse, without assumption (A), we can still make the sum finite, by requir-\ning that ̄μ lie within the convex hull H ̄λ generated by the points { ̄W( ̄λ)},\nwhere ̄W ranges over all Weyl group elements. This means we have to add\nconstraints to the system (4.17) corresponding to the facets of H ̄λ.\nBut\nthe entries of the resulting B would depend on ̄λ, and the theory of vector\npartition functions will no longer apply.\nProof of the claim: Let ̄μi’s denote the integer coordinates of ̄μ in the basis\nof fundamental weights.\nWe denote the integer vector ( ̄μ1, ̄μ2, · · · ) by ̄μ\nagain. The Kostant partition function P(ν) is a vector partition function\nassociated with an integer programming problem:\nBPv = ν,\nv ≥0,\nwhere the columns of BP correspond to positive roots of G. The sum in\n(4.16) is equal to the number of integral pairs ( ̄μ, v) such that\n1. ̄μ ∈C,\n2. ̄μ ↓π −ρ −W(ρ),\n3. BPv = ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ), v ≥0.\nThe first two condititions here can be expressed in terms of linear con-\nstraints (equalities and inequalities) on the coordinates ̄μi’s. Thus the three\nconditions together can be expressed in terms of linear constraints on ( ̄μ, v).\nBy the finiteness assumption (A), the polytope determined by these con-\nstraints is a bounded polytope.\nThe number of integer points in such a\npolytope can be expressed as a vector partition function (cf. [BBCV]). This\nproves the claim.\nLet us denote the vector partition associated with the integer program-\nming problem (4.17) in the claim by φW, ̄\nW,C(c( ̄λ, π, ρ, ̄ρ)). Then\nmπ ̄λ =\nX\nW\nX\n ̄\nW\n(−1)W (−1)\n ̄\nW X\nC\na(C)φW, ̄\nW,C(c( ̄λ, π, ρ, ̄ρ)).\n(4.18)\n81"},{"page":85,"text":"Hence,\n ̃mπ ̄λ(n) = mn ̄λ\nnπ =\nX\nW\nX\n ̄\nW\n(−1)W (−1)\n ̄\nW X\nC\na(C)φW, ̄\nW,C(c(n ̄λ, nπ, ρ, ̄ρ)).\n(4.19)\nIt follows from Claim 4.5.1 and the standard results on vector partition\nfunctions mentioned in the begining of this section that\ngW, ̄\nW,C(n) = φW, ̄\nW,C(c(n ̄λ, nπ, ρ, ̄ρ)),\nis asymptitically a quasipolynomial function of n.\nHence, ̃mπ ̄λ(n) is also\nasymptotically a quasipolynomial function of n.\nThis implies (cf.\n[St1])\nthat\nMπ ̄λ (t) =\nX\nn≥0\n ̃mπ ̄λ(n)tn\n(4.20)\nis rational function of t.\nThis proves Theorem 3.4.1 (a) under the finiteness assumption (A).\nIt remains to remove the assumption (A). Let G′ ⊇H be the smallest\nLevi subalgebra of G containing H. Then\nmπ ̄λ =\nX\nπ′\nmπ′\n ̄λ mπ\nπ′,\n(4.21)\nwhere π′ ranges over dominant weights of G′, mπ′\n ̄λ denotes the multiplicity of\nVπ′(G′) in V ̄λ(G), and mπ\nπ′ the multiplicity of Vπ(H) in Vπ′(G′). Furthermore,\n1. the finiteness asssumption (A) is now satisfied for the pair (G′, H): i.e.,\nfor any weight μ of H, the number of weights μ′’s of G′ such that μ′ ↓μ\nis finite.\n2. There is a polyhedral expression for mπ′\n ̄λ ; this follows from [Li, Dh].\nBy the first condition and the argument above, we get an expression for\nmπ\nπ′ akin to (4.18). Substituting this expression and the polyhedral expres-\nsion for mπ′\n ̄λ in (4.21), leads to a formula for ̃mπ ̄λ(n) as a linear combination\nof φB,c,b(n)’s for appropriate B, c, b’s. After this, we proceed as before.\nThis proves Theorem 3.4.1 (a). Q.E.D.\nWe also note down the following consequence of the proof.\nProposition 4.5.2 There is a constant D depending only G and H, such\nthat for any ̄λ, π, orders of the poles of Mπ ̄λ (t) (cf. (4.20), as roots of unity,\ndivide D.\n82"},{"page":86,"text":"A bound on D provided by the proof below is very weak: D = O(2O(rank(G))).\nProof: It suffices to to bound the period of the quasipolynomial ̃mπ ̄λ(n). For\nthis, it suffices to let n →∞. For a fixed W, ̄\nW , C, the chamber containing\nc(n ̄λ, nπ, ρ, ̄ρ)) is completely determined by ̄λ and π as n →∞. Under these\nconditions, the degree of φW, ̄\nW,C(c(n ̄λ, nπ, ρ, ̄ρ)) is equal to the dimension of\nthe polytope associated with this vector partition function. This dimension\nis clearly O(rank(G)2).\nBy Szenes-Vergne residue formula [SV], there is a constant D depending\non only G, H, W, ̄\nW , C, such that the period of the quasipolynomial h(n) =\nφW, ̄\nW,C(c(n ̄λ, nπ, 0, 0)) divides D for every ̄λ, π; here we are putting ρ and\n ̄ρ equal to zero, since we are interested in what happens as n →∞. Q.E.D.\n83"},{"page":87,"text":"Chapter 5\nParallel and PSPACE\nalgorithms\nIn this chapter we give PSPACE algorithms (cf. Theorem 3.4.3) for com-\nputing the various structural constants under consideration . We shall only\nprove Theorem 3.4.3, when H is therein is either a complex, semisimple\ngroup, or a symmetric group, or a general linear group over a finite field,\nthe extension to the general case being routine.\nWe recall two standard results in parallel complexity theory [KR], which\nwill be used repeatedly.\nLet NC(t(N), p(N) denote the class of problems that can be solved\nin O(t(N)) parallel time using O(p(N)) processors, where N denotes the\nbitlength of the input. Let\nNC = ∪iNC(logi(N), poly(N)).\nThis is the class of problems having efficient parallel algorithms.\nProposition 5.0.3 [Cs, KR] Let A be an n × n-matrix with entries in a\nring R of characteristic zero. Then the determinant of A, and A−1, if A\nis nonsingular, can be computed in O(log2 n) parallel steps using poly(n)\nprocessors; here each operation in the ring is considered one step. Hence, if\nR = Q, the problems of computing the determinant, the inverse and solving\nlinear systems belong to NC.\nProposition 5.0.4 The class NC(t(N), 2t(N)) ⊆SPACE(O(t(N))).\nIn\nparticular, NC(poly(N), 2O(poly(N))) ⊆PSPACE.\n84"},{"page":88,"text":"5.1\nComplex semisimple Lie group\nIn this section we prove a special case of Theorem 3.4.3 for the general-\nized plethym problem (Problem 1.1.2). Accordingly, let H be a complex,\nsemisimple, simply connected Lie group, G = GL(V ), where V = Vμ(H) is\nan irreducible representation of H with dominant weight μ, ρ : H →G the\nhomomorphism corresponding to the representation, and mπ\nλ the multiplic-\nity of Vπ(H) in Vλ(G), considered as an H-module via ρ; cf. Problem 1.1.3.\nThen:\nTheorem 5.1.1 The multiplicity mπ\nλ can be computed in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, dim(H))\nspace.\nHere it is assumed that the partition λ = λ1 ≥λ2 ≥· · · λr > 0 is rep-\nresented in a compact form by specifying only its nonzero parts λ1, . . . , λr.\nThis is important since dim(G) can be exponential in dim(H) and ⟨μ⟩. A\ncompact representation allows ⟨λ⟩to be small, say poly(dim(H), ⟨μ⟩), in this\ncase.\nWe begin with a simpler special case.\nProposition 5.1.2 If dim(V ) = poly(dim(H)), then mπ\nλ can be computed\nin PSPACE; i.e., in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩, dim(H)) space.\nThis implies that the Kronecker coefficient (Problem 1.1.1) can be computed\nin PSPACE.\nProof: Let us use the notation ̄λ instead of λ to be consistent with the\nnotation used in Klimyk’s formula (4.12). By the latter, mπ ̄λ can be computed\nin PSPACE if n ̄μ(V ̄λ) in that formula can be computed in PSPACE for every\n ̄μ and ̄λ. In type A, this is just the number of Gelfand-Tsetlin tableau with\nthe shape ̄λ and weight ̄μ.\nIf dim(V ) = poly(dim(H)), the size of such\na tableau is O(dim(V )2) = poly(dim(H)).\nSo we can count the number\nof such tableu in PSPACE as follows: Begin with a zero count, and cycle\nthrough all tableaux of shape ̄λ in polynomial space one by one, increasing\nthe count by one everytime the tableau satisfies all constraints for Gelfand-\nTsetlin tableau and has weight ̄μ. In general, the role of Gelfand-Tsetlin\ntableaux is played by Lakshmibai-Seshadri (LS) paths [Li, Dh]. Q.E.D.\nThe argument above does not work if dim(V ) is not poly(dim(H)), as\nin the plethym problem (Problem 1.1.2), where dim(V ) = dim(Vμ) can\nbe exponential in n = dim(H) and the bitlength of μ. In this case, the\n85"},{"page":89,"text":"algorithm cannot even afford to write down a tableau since its size need not\nbe polynomial.\nNext we turn to Theorem 5.1.1.\nFor the sake of simplicity, we shall\nprove it only for H = SLn(C), or rather GLn(C)–i.e., the usual plethysm\nproblem. This illustrates all the basic ideas. The general case is similar. We\nshall prove a slightly stronger result in this case:\nTheorem 5.1.3 The plethysm constant aπ\nλ,μ can be can be computed in\npoly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) space.\nHere the dependence on n = dim(H) is not there. This makes a difference\nif the heights of μ and π are less than n = dim(H)–remember that we are\nusing a compact representation of a partition in which only nonzero parts\nare specified. This is really not a big issue. Because aπ\nλ,μ depends only on\nthe partitions λ, μ, π and not n. Hence, without loss of generality, we can\nassume that n is the maximum of the heights of μ and π. It is possible to\nstrengthen Theorem 5.1.1 similarly.\nTo prove Theorem 5.1.3, we shall give an efficient parallel algorithm to\ncompute ̃aπ\nλ,μ that works in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) parallel time using O(2poly(⟨λ⟩,⟨μ⟩,⟨π⟩))\nprocessors. This will show that the problem of computing ̃aπ\nλ,μ is in the com-\nplexity class NC(poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩), 2poly(⟨λ⟩,⟨μ⟩,⟨π⟩)), which is contained in\nPSPACE by Proposition 5.0.4. The basic idea is to parallelize the classical\ncharacter-based algorithm for computing aπ\nλ,μ by using efficient parallel algo-\nrithm for inverting a matrix and solving a linear system (Proposition 5.0.3).\nWe begin by recalling the standard facts concerning the characters of\nthe general linear group.\nGiven a representation W of GLm(C), let ρ :\nGLm(C) →GL(W) be the representation map.\nLet χρ(x1, . . . , xm) de-\nnote the formal character of this representation W. This is the trace of\nρ(diag(x1, . . . , xm)), where diag(x1, . . . , xn) denotes the generic diagonal ma-\ntrix with variable entries x1, . . . , xm on its diagonal. If W is an irreducible\nrepresentation Vλ(GLm(C)), then χρ(x1, . . . , xm) is the Schur polynomial\nSλ(x1, . . . , xm). By the Weyl character formula,\nSλ =\n|xλi+m−i\nj\n|\n|xm−i\nj\n| ,\n(5.1)\nwhere |ai\nj| denotes the determinant of an m×m-matrix a. The Schur polyno-\nmials form a basis of the ring of symmetric polynomials in x1, . . . , xm. The\n86"},{"page":90,"text":"simplest basis of this ring consists of the complete symmetric polynomials\nMβ(x1, . . . , xm) defined by\nMβ(x1, . . . , xm) =\nX\nγ\ntγ,\nwhere γ ranges over all permutations of β and tγ = Q\ni xγi\ni . Schur polyno-\nmials are related to Mβ by:\nSλ =\nX\nβ\nkβ\nλMβ,\n(5.2)\nwhere kβ\nλ is the Kostka number. This is the number of semistandard tableau\nof shape λ and weight β.\nIf the representation W is reducible, its decomposition into irreducibles\nis given by:\nW =\nX\nπ\nm(π)Vπ(GLn(C)),\n(5.3)\nwhere m(π)’s are the coefficients of the formal character χρ(x1, . . . , xm) in\nthe Schur basis:\nχρ =\nX\nπ\nm(π)Sπ.\nProof of Theorem 5.1.3\nLet λ, μ, π be as in Theorem 5.1.3. Let H = GLn(C), V = Vμ(H), G =\nGL(V ). Let sλ(x1, . . . , xm) be the formal character of the representation\nVλ(G) of G. Here m = dim(Vμ) can be exponential in n and ⟨μ⟩. The basis\nof Vμ(H) is indexed by semistandard tableau of shape μ with entries in [1, n].\nLet us order these tableau, say lexicographically, and let Ti, 1 ≤i ≤m,\ndenote the i-th tableau in this order. With each tableau T, we associate a\nmonomial\nt(T) =\nn\nY\ni=1\ntwi(T)\ni\n,\nwhere wi(T) denotes the number of i’s in T. Given a polynomial f(x1, . . . , xm),\nlet us define fμ = fμ(t1, . . . , tn) to be the polynomial obtained by substi-\ntuting xi = t(Ti) in f(x1, . . . , xm). Then the formal character of Vλ(G),\nconsidered as an H-representation of via the homomorphism H →G =\n87"},{"page":91,"text":"GL(Vμ(H)), is the symmetric polynomial Sλ,μ(t1, . . . , tn) = (Sλ)μ.\nThe\nplethysm constant aπ\nλ,μ is defined by:\nSλ,μ(t1, . . . , tn) =\nX\nπ\naπ\nλ,μSπ(t1, . . . , tn).\n(5.4)\nAn efficient parallel algorithm to compute aπ\nλ,μ is as follows. Here by an\nefficient parallel algorithm, we mean an algorithm that works in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)\ntime using 2poly(⟨λ⟩,⟨μ⟩,⟨π⟩) processors.\nWe will repeatedly use Proposi-\ntion 5.0.3.\nAlgorithm\n(1) Compute Sλ,μ(t1, . . . , tn). By the Weyl character formula (5.1),\nSλ,μ(t1, . . . , tn) = Aλ,μ(t1, . . . , tn)\nBλ,μ(t1, . . . , tn),\nwhere Aλ(x1, . . . , xm) and Bλ(x1, . . . , xm) denote the numerator and denom-\ninator in (5.1), and Aλ,μ = (Aλ)μ, and Bλ,μ = (Bλ)μ. Let R = C[t1, . . . , tn].\nThen\nAλ,μ(t1, . . . , tn) = |t(Tj)λi+m−i|.\nThis is the determinant of an m × m matrix with entries in R, where m =\ndim(V ) can be exponential in n and ⟨μ⟩. It can be evaluated in O(log2 m)\nparallel ring operations using poly(m) processors. Each ring element that\narises in the course of this algorithm is a polynomial in t1, . . . , tn of total\ndegree O(|λ|m), where |λ| denotes the size of λ. The total number of its\ncoefficients is r = O((|λ|m)n). Hence each ring operation can be carried\nout efficiently in O(log2(r)) parallel time using poly(r) processors. Since\nlog m = poly(n, ⟨μ⟩) and log r = poly(n, ⟨λ⟩, ⟨μ⟩), it follows that Aλ,μ can\nbe evaluated in poly(n, ⟨μ⟩, ⟨λ⟩) parallel time using 2poly(n,⟨μ,λ⟩) processors.\nThe determinant Bλ,μ can also be computed efficiently in parallel in a similar\nfashion. To compute Sλ,μ, we have to divide Aλ,μ by Bλ,μ. This can be done\nby solving an r × r linear system, which, again, can be done efficiently in\nparallel. This computation yields representation of Sλ,μ in the monomial\nbasis {Mβ} of the ring of symmetric polynomials in t1, . . . , tn.\n(2) To get the coefficients aπ\nλ,μ, we have to get the representation of Sλ,μ(t)\nin the Schur basis. This change of basis requires inversion of the matrix\nin the linear system (5.2). The entries of the matrix K occuring in this\n88"},{"page":92,"text":"linear system are Kostka numbers. Each Kostka number can be computed\nefficiently in parallel.\nHence, all entries of this matrix can be computed\nefficiently in parallel. After this, the matrix can be inverted efficiently in\nparallel, and the coefficients aπ\nλ,μ’s of Sλ,μ in the Schur basis can be computed\nefficiently in parallel. Finally, we use Proposition 5.0.4 to conclude that aπ\nλ,μ\ncan be computed in PSPACE. Q.E.D.\n5.2\nSymmetric group\nNext we prove Theorem 3.4.3 when H = Sm.\nLet X = Vμ(Sm) be an\nirreducible representation (the Specht module) of Sm corresponding to a\npartition μ of size m.\nLet ρ : H →G = GL(X) be the corresponding\nhomomorphism.\nTheorem 5.2.1 Given partitions λ, μ, π, where μ and π have size m, the\nmultiplicity mπ\nλ,μ of the Specht module Vπ(Sm) in Vλ(G) can be computed in\npoly(m, ⟨λ⟩) space.\nRemark 5.2.2 The bitlengths ⟨μ⟩and ⟨π⟩are not mentioned in the com-\nplexity bound because they are bounded by m.\nFor this, we need three lemmas.\nLemma 5.2.3 The character of a symmetric group can be computed in\nPSPACE.\nSpecifically, given a partition π of size m, and a sequence\ni = (i1, i2, . . .) of nonnegative integers such that P jij = m, the value of\nthe character χπ of Sm on the conjugacy class Ci of permutations indexed\nby i can be computed in poly(m) parallel time using 2poly(m) processors.\nHence it can be computed in poly(m) space (cf. Proposition 5.0.4).\nHere the conjugacy class Ci consists of those permutations that have i1\n1-cycles, i2 2-cycles, and so on.\nProof: Let k be the height of the partition π. Let x = (x1, . . . , xk) be the\ntuple of variables xi’s. Given a formal series f(x) and a tuple (l1, . . . , lk) of\nnonnegative integers, let [f(x)](l1,...,lk) denote the coefficient of xl1\n1 · · · xlk\nk in\nf.\nBy the Frobenius character formula [FH],\nχλ(Ci) = [f(x)](l1,...,lk),\n(5.5)\n89"},{"page":93,"text":"where\nl1 = π1 + k −1, l2 = π2 + k −2, . . . , lk = πk,\nand\nf(x) = ∆(x)\nm\nY\nj=1\nPj(x)ij,\nwith\n∆(x)\n=\nQ\ni<j(xi −xj),\nPj(x)\n=\nxj\n1 + · · · + xj\nk.\n(5.6)\nSince deg(f) = poly(m) and k ≤m, the total number of coefficients of\nf(x) is 2poly(m). Hence, we can evaluate f(x) in PSPACE by setting up\nappropriate recurrence relations.\nAlternatively, we can easily evaluate f(x) in poly(m) parallel time using\n2poly(m) processors, and then extract its required coefficient. After this, the\nresult follows from Proposition 5.0.4. Q.E.D.\nLemma 5.2.4 Suppose φ is a character of Sm whose value on any conju-\ngacy class Ci can be computed in O(s) space, for some parameter s. Then,\nthe multiplicity of the representation Vπ(Sm) in the representation Vφ(Sm)\ncorresponding φ can be computed in O(poly(m) + s) space.\nProof: The multiplicity is given by the inner product\n⟨φ, χπ⟩= 1\nm!\nX\nσ∈Sm\n ̄\nφ(σ)χπ(σ).\n(5.7)\nBy assumption, φ(σ) can be computed in O(s) space, and by Lemma 5.2.3,\nχπ(σ) can be computed in poly(m) space. Hence, the result follows from\nthe preceding formula. Q.E.D.\nGiven an irreducible representation X = Vμ(Sm) and an irreducible rep-\nresentation W = Vλ(G) of G = GL(X)), let ρμ denote the representation\nmap Sm →G, ρλ the representation map G →GL(W), and\nρ : Sm →G →GL(W)\ntheir composition. This is a representation of Sm. Let χρ be the character\nof ρ.\nLemma 5.2.5 For any σ ∈Sm, χρ(σ) can be computed in poly(m, ⟨λ⟩) in\npoly(m, ⟨λ⟩) space.\n90"},{"page":94,"text":"The bitlength ⟨μ⟩is not mentioned in the complexity bound because it\nis bounded by m.\nProof: Let r = dim(X). The formal character of the representation Vλ(G)\nof G = GL(X) is the Schur polynomial Sλ(x1, . . . , xr), r = dim(X). Hence,\nχρ(σ) = Sλ(α)\nwhere α = (α1, . . . , αr) is the tuple of eigenvalues of ρμ(σ). We shall compute\nthe right hand side fast in parallel–i.e., in poly(m, ⟨λ⟩) parallel time using\n2poly(m,⟨λ⟩) processors–and then use Proposition 5.0.4 to conclude the proof.\nThis is done as follows.\n(1) Let χμ denote the character of the representation ρμ. Let pi(α) = αi\n1 +\n· · · + αi\nr denote the i-th power sum of the eigenvalues. For any i,\npi(α) = χμ(σi).\nWe can compute σi, for i ≤|λ|, where |λ| denotes the size of λ, in poly(log i, m) =\npoly(m, ⟨λ⟩) time using repeated squaring. After this χμ(σi) can be com-\nputed fast in parallel in poly(m) time using Lemma 5.2.3. Thus each pi(α)\ncan be computed in poly(m, ⟨λ⟩) time in parallel using 2poly(m,⟨λ⟩) proces-\nsors. We calculate pi(α) in parallel for all i ≤|λ|, and all pγ(α) = Q\nj pγj(α)\nin parallel for all partitions γ of size at most m.\n(2) After this, we calculate the complete symmetric function hi(α), for\neach i ≤|λ|, fast in parallel, by using the relation [Mc]:\nhi =\nX\n|γ|=i\nz−1\nγ pγ,\nwhere zγ = Q\ni≥1 imimi!, and mi = mi(γ) denotes the number of parts of γ\nequal to i. Thus we can calculate hγ(α) = Q\nj hγj(α), for all partitions γ of\nsize m, fast in parallel.\n(3) To compute Sλ(α), we recall that the transition matrix between\nthe Schur basis {Sλ} and the complete symmetric basis {hγ} of the ring\nof symmetric functions is K∗, the transpose inverse of the Kostka matrix\nK = [Kλ,γ], where Kλ,γ denote the Kostka number; cf. [Mc]. As we noted in\nthe proof of Theorem 5.1.3, each Kostka number can be computed in fast in\nparallel. Hence, K can be computed fast in parallel. After this, its inverse\nK−1 can be computed fast in parallel by Proposition 5.0.3–this is the crux\nof the proof–and finally K∗as well. Thus Sλ(α) can be computed fast in\nparallel, since each hγ(α) can be computed fast in parallel. Q.E.D.\nTheorem 5.2.1 follows from Lemma 5.2.3,5.2.4 and 5.2.5. Q.E.D.\n91"},{"page":95,"text":"5.3\nGeneral linear group over a finite field\nIn this section we prove Theorem 3.4.3, when H therein is the general lin-\near group GLn(Fpk) over a finite field Fpk. Irreducible representations of\nH = GLn(Fpk) have been classified by Green [Mc]. They are labelled by\ncertain partition-valued functions. See [Mc] for a precise description of these\nlabelling functions. It is clear from the description therein that each labelling\nfunction has a compact representation of bitlength O(n + k + ⟨p⟩), where\n⟨p⟩= log2 p; we specify a function by giving its partition values at the places\nwhere it is nonzero. Let μ denote any such label. Let X = Vμ(H) be the\ncorresponding irreducible representation of H, and ρ : H →G = GL(X)\nthe corresponding homomorphism.\nTheorem 5.3.1 Given a partition λ and labelling functions μ and π as\nabove, the multiplicity mπ\nλ,μ of the irreducible representation Vπ(H) in Vλ(G)\ncan be computed in poly(n, k, ⟨p⟩, ⟨λ⟩) space.\nThe proof is similar to that of Theorem 5.2.1 for the symmetric group\nwith the following result playing the role of Lemma 5.2.3:\nLemma 5.3.2 Given a label γ of an irreducible character χγ of H = GLn(Fpk)\nand a label δ of a conjugacy class in H, the value χγ(δ) can be computed\nin poly(n, k, ⟨p⟩) parallel time using 2poly(n,k,⟨p⟩) processors, and hence by\nProposition 5.0.4, in poly(n, k, ⟨p⟩) space.\nThe label δ of a conjugacy class in H is also a partition valued function\n[Mc], which admits a compact representation of bitlength poly(n, k, ⟨p⟩).\nProof: We shall parallelize Green’s algorithm [Mc] for computing the charac-\nter values, and then conclude by Proposition 5.0.3. Green shows that χγ(δ)’s\nare entries of a transition matrix between a two polynomial bases: the first\nconstructed using Hall-Littlewood polynomials, and the second using Schur\npolynomials. We have construct this transition matrix fast in parallel. We\nshall only indicate here how the transition matrix between the basis of Hall-\nLittlewood polynomials and the Schur basis for the ring symmetric functions\nover Z[t] can be constructed fast in parallel. This idea can then be easily\nextended to complete the proof.\nFirst, we recall the definition of the Hall-Littlewood polynomial Pπ(x; t) =\nPπ(x1, . . . , xk; t) [Mc].\nThis is a symmetric polynomial in xi’s with co-\nefficients in Z[t].\nIt interpolates between the Schur function sπ(x) and\n92"},{"page":96,"text":"the monomial symmetric function mπ(x) because Pπ(x; 0) = sπ(x) and\nPπ(x; 1) = mπ(x). The formal definition is as follows:\nFor a given partition π, let vπ(t) = Q\ni≥0 vmi(t), where mi is the number\nof parts of π equal to i, and\nvm(t) =\nm\nY\ni=1\n1 −ti\n1 −t .\nThen\nPπ(x; t) = Aπ(x, t)\nBπ(x, t),\n(5.8)\nwhere\nAπ(x, t) = P\nσ∈Sk sgn(σ)σ(xπ1\n1 · · · xπk\nk\nQ\ni<j xi −txj,\nBπ(x, t) = vπ(t) Q\ni<j(xi −xj).\n(5.9)\nHere sgn(σ) denotes the sign of σ.\nLet wπ,α(t)’s be the coeffcients of Pπ(x, t) in the Schur basis:\nPπ(x; t) =\nX\nα\nwπ,α(t)sα(x).\n(5.10)\nWe want to calculate the matrix W = [wπ,α] fast in parallel.\nUsing\nformula (5.9), we calculate Aπ(x; t) fast in parallel; i.e., we calculate the\ncoefficients of Aπ(x; t) in the basis of monomials in x and t. We calculate\nBπ(x; t) similarly.\nAfter this the division in (5.8) can be carried out by\nsolving a an appropriate linear system. This can be done fast in parallel\nby Proposition 5.0.3.\nSince, Pπ(x; t) is symmetric in xi’s, this yields its\ncoefficients in the monomial symmetric basis {mα(x)} with the coefficients\nbeing in Z[t]. The transition matrix [Mc] from the monomial symmetric\nbasis to the Schur basis is given by the inverse of the Kostka matrix. This\ninverse can be computed fast in parallel by Proposition 5.0.3. After this,\nthe coefficients wπ,α’s of Pπ(x; t) in the Schur basis can be computed fast in\nparallel.\nFurthermore, the inverse of W = [Wπ,α] can also be computed fast in\nparallel by Proposition 5.0.3. Q.E.D.\n5.3.1\nTensor product problem\nAnalogue of the Kronecker problem (Problem 1.1.1) for H = GLn(Fpk) is:\n93"},{"page":97,"text":"Problem 5.3.3 Given partition valued functions λ, μ, π, decide if the mul-\ntiplicity bπ\nλ,μ of Vπ(H) in the tensor product Vλ(H) ⊗Vμ(H) is nonzero.\nIn this context:\nTheorem 5.3.4 The multiplicity mπ\nλ,μ can be computed in PSPACE; i.e.,\nin poly(n, k, ⟨p⟩) space.\nProof: This follows from Lemma 5.3.2 and analogues of Lemmas 5.2.4 and\n5.2.5 in this setting. Q.E.D.\nA possible canditate for a stretching function assoociated with bπ\nλ,μ is:\n ̃bπ\nλ,μ(n) = bnπ\nnλ,nμ,\nwhere nλ denotes the stretched partition-valed function obtained by stretch-\ning each partition value of λ by a factor of n. In other words ̃bπ\nλ,μ(n) is\nthe multiplicity of Vnπ(H(n)) in Vnλ(H(n)) ⊗Vnμ(H(n)), where H(n) =\nGLnm(Fpk) is the stretched group. Is it a quasi-polynomial? If so, we can\nalso ask for a good bound on its saturation and positivity indices.\n5.4\nFinite simple groups of Lie type\nThe work of Deligne-Lusztig [DL] and Lusztig[Lu5] yield an algorithm for\ncomputing the character values for finite simple groups of Lie type.\nQuestion 5.4.1 Can this algorithm be parallelized?\nIf so, Lemma 5.3.2, and hence Theorem 5.3.1, can be extended to finite\nsimple groups of Lie type.\n94"},{"page":98,"text":"Chapter 6\nExperimental evidence for\npositivity\nIn this chapter we give experimental evidence for positivity (PH2,3).\n6.1\nLittlewood-Richardson problem\nExperimental evidence for PH2 in the context of the Littlewood-Richardson\nproblem (Problem 1.2.1) has been given in [DM2], and for PH3 in type A\nin [KTT]. We give experimental evidence for PH3 in types B, C, D here.\nLet Cλ\nα,β(t) be as in eq.(1.2). Its reduced positive form for various values\nof α, β, λ is shown in Figure 6.1 for type B, in Figure 6.2 for type C, and\nFigure 6.3 for type D. The rank of the Lie algebra is three in all cases.\nIn these types, the period of the stretching quasipolynomial ̃cλ\nα,β(n) is at\nmost two. Accordingly, the period of every pole of Cλ\nα,β(t) is at most two.\nThe tables were computed from the tables in [DM2] for the stretching quasi-\npolynomial ̃cλ\nα,β(n) in these cases.\n6.2\nKronecker problem, n = 2\nLet kπ\nλ,μ be the Kronecker coefficient in Problem 1.1.1. Let ̃kπ\nλ,μ(n) = ̃knπ\nnλ,nμ\nbe the associated stretching quasi-polynomial, and\nKπ\nλ,μ(t) =\nX\nn≥0\n ̃kπ\nλ,μ(n)tn,\n95"},{"page":99,"text":"the associated rational function. An explicit formula (with alternating signs)\nfor the Kronecker coefficient, when n = 2, has given by Remmel and White-\nhead [RW] and Rosas [Ro], and a positive formula in [GCT9]. This case\nturns out to be nontrivial. For example, the number of chambers (domains)\nof quasi-polynomiality in this case turns out to be more than a million. Their\nexplicit description can be found out using the formula for the Kronecker\ncoefficient in [RW].\nWe implemented Rosas’ formula to check PH2 for the quasipolynomial\n ̃kπ\nλ,μ(n) for a few thousand values of μ, ν and λ with the help of a computer.\nA large number of samples was chosen to ensure that a significant fraction\nof the chambers were sampled. The quasi-polynomial ̃kπ\nλ,μ(n) and a positive\nform of the rational function Cπ\nλ,μ(t) are shown Figures 6.4 and 6.5 for few\nsample values of λ = (λ1, λ2), μ = (μ1, μ2), and π = (π1, π2, π3, π4).\nIt\nmay be noted that ̃kπ\nλ,μ(n) need not be a polynomial; this answers Kirillov’s\nquestion [Ki] in the negative. But its period is at most two for n = 2. This\nfollows from the formula in [RW].\nFor the λ, μ and π that we sampled,\npositivity index of ̃kπ\nλ,μ(n) is always zero.\nBut it turns out [BOR] that\nthere are some λ, μ and π for which the saturation and positivity indices of\n ̃kπ\nλ,μ(n) are nonzero (one), but very small and thus consistent with SH and\nPH2 (Hypothesis1.6.6) in this paper; in the earlier version of this paper, SH\nand PH2 stipulated that the saturation and positivity indices are always\nzero. These (λ, μ, π) escaped our random sampling, because their density is\nextremely small [BOR].\n6.3\nG/P and Schubert varieties\nLet V = Vλ(G) be an irreducible representation of G = SLk(C) correspond-\ning to a partition λ.\nLet vλ be the point in P(V ) corresponding to the\nhighest weight vector, and X = Gvλ ∼= G/Pλ its closed orbit. Let hk,λ(n)\nbe the Hilbert function of the homogeneous coordinate ring R of X. It is\na quasipolynomial since spec(R) has rational singularities. In fact, it is a\npolynomial, since t = 1 is the only pole of the Hilbert series\nHk,λ(t) =\nX\nn≥0\nhk,λ(n)tn.\nFigure 6.6 gives experimental evidence for strict positivity (PH2) of hk,λ(n)\n(as discussed in Section 3.5.5) for a few sample values of k and λ. Figure 6.7\ngives experimental evidence for strict positivity of the Hilbert polynomial\n96"},{"page":100,"text":"of the Schubert subvarieties of the Grassmanian; there Gn,k denotes the\nGrassmannian of k-planes in V = Cn, and Ωa, a = (a(1), . . . , a(d)) its\nSchubert subvariety consisting of the k-subspaces W such that dim(W ∩\nVn−k+i−a(i)) ≥i for all i, where V = Vn ⊃· · · V1 ⊃0 is a complete flag of\nsubspaces in V . The Hilbert polynomials were computed using the explicit\npolyhedral interpretation for them deduced from the theory of algebras with\nstraightening laws (Hodge algebras) [DEP2].\n6.4\nThe ring of symmetric functions\nLet V = Ck, G = GL(V ), H = Sk, with the natural embedding H →\nG.\nLet us consider the spacial case of the subgroup restriction problem\n(Problem 1.1.3), with Vλ(G) = V , and Vπ(H) the trivial representation of\nH.\nThen s = mπ\nλ, the multiplicity of the trivial representation in V , is\none. Though the decision problem (Problem 1.1.3) is trivial in this case, the\ncanonical model associated with s is nontrivial.\nThe canonical rings R = R(s) and S = S(s) associated with s in this\ncase coincide with C[V ] = C[x1, . . . , xk].\nThe ring T = T(s) = SH =\nC[x1, . . . , xk]Sk is the subring of symmetric functions. Its Hilbert function\nh(n) is a quasipolynomial. PH1 and PH3 for Z = Proj(T), as per Defini-\ntion 3.5.2, follow easily, the latter from the well known rational generating\nfunction for the partition function [St1]. But PH2 turns out to be nontriv-\nial. Figures 6.8-6.13 give experimental evidence for strict positivity of h(n)\n(PH2). In these figures, the i-th row of the table for a given k shows hi(n),\nwhere hi(n), 1 ≤i ≤l, are such that h(n) = hi(n), when n = i modulo the\nperiod l of h(n).\n97"},{"page":101,"text":"α\nβ\nλ\nCλ\nα,β(t)\n(0, 15, 5)\n(12, 15, 3)\n(6, 15, 6)\n350 t8+19121 t7+123576 t6+297561 t5+342064 t4+192779 t3+46992 t2+2641 t+1\n(1−t)3(1−t2)3\n(4, 8, 11)\n(3, 15, 10)\n(10, 1, 3)\n1+5 t+6 t2+t3\n(1−t2)3\n(8, 1, 3)\n(11, 13, 3)\n(8, 6, 14)\n2 t8+45 t7+259 t6+591 t5+773 t4+522 t3+198 t2+29 t+1\n(1−t)3(1−t2)4\n(8, 9, 14)\n(8, 4, 5)\n(1, 5, 15)\n136 t9+3422 t8+20204 t7+53608 t6+76076 t5+60986 t4+26674 t3+5568 t2+345 t+1\n(1−t)3(1−t2)4\n(10, 5, 6)\n(5, 4, 10)\n(0, 7, 12)\n219 t8+12135 t7+79231 t6+193003 t5+223919 t4+127907 t3+31704 t2+1870 t+1\n(1−t)6(1+t)3\n \n \nFigure 6.1: Cλ\nα,β(t) for B3\n98"},{"page":102,"text":"α\nβ\nλ\nCλ\nα,β(t)\n(1, 13, 6)\n(14, 15, 5)\n(5, 11, 7)\n18145 t8+267151 t7+1070716 t6+1917716 t5+1735692 t4+778184 t3+144596 t2+5538 t+1\n(1−t)4(1−t2)3\n(4, 15, 14)\n(12, 12, 10)\n(4, 9, 8)\n2280 t9+267658 t8+2746131 t7+9276935 t6+14682332 t5+11903923 t4+4746803 t3+751126 t2+21249 t+1\n(1−t)4(1−t2)3\n(9, 0, 8)\n(8, 12, 9)\n(7, 7, 3)\n3 t2+4 t+1\n(1−t)6\n(10, 2, 7)\n(8, 10, 1)\n(7, 5, 5)\n8984 t8+132826 t7+534183 t6+960491 t5+873227 t4+394045 t3+74067 t2+2941 t+1\n(1−t)4(1−t2)3\n(10, 10, 15)\n(11, 3, 15)\n(10, 7, 15)\n7162 t9+736327 t8+7178960 t7+23540366 t6+36359642 t5+28788904 t4+11166361 t3+1693696 t2+43515 t+1\n(1−t)7(1+t)3\n \n \nFigure 6.2: Cλ\nα,β(t) for C3\n99"},{"page":103,"text":"α\nβ\nλ\nCλ\nα,β(t)\n(0, 15, 5)\n(12, 15, 3)\n(6, 15, 6)\n633 t7+24259 t6+142236 t5+252113 t4+168220 t3+36131 t2+1414 t+1\n(1−t)7(1−t2)\n(4, 8, 11)\n(3, 15, 10)\n(10, 1, 3)\n7962 t8+503679 t7+4525372 t6+11944350 t5+12218255 t4+4879052 t3+586370 t2+10862 t+1\n(1−t)8(1−t2)\n(8, 1, 3)\n(11, 13, 3)\n(8, 6, 14)\n81 t7+19407 t6+211964 t5+513585 t4+426652 t3+110317 t2+4609 t+1\n(1−t)6(1−t2)\n(8, 9, 14)\n(8, 4, 5)\n(1, 5, 15)\n9 t2+8 t+1+2 t3\n(1−t)3\n(10, 5, 6)\n(5, 4, 10)\n(0, 7, 12)\n3647 t7+111208 t6+570739 t5+920201 t4+560336 t3+106748 t2+3435 t+1\n(1−t)8(1+t)\n \n \nFigure 6.3: Cλ\nα,β(t) for D3\n100"},{"page":104,"text":"λ1\nλ2\nμ1\nμ2\nπ1\nπ2\nπ3\nπ4\n ̃kπ\nλ,μ(n); n odd\n ̃kπ\nλ,μ(n); n even\nKπ\nλ,μ(t)\n87\n62\n97\n52\n64\n39\n24\n22\n1/2 + 4 n + 11/2 n2\n1 + 4 n + 11/2 n2\n1+8 t+11 t2+2 t3\n(1−t)2(1−t2)\n104\n95\n149\n50\n95\n78\n15\n11\n1/2 + 13/2 n + 18 n2\n1 + 13/2 n + 18 n2\n1+23 t+36 t2+12 t3\n(1−t)2(1−t2)\n101\n85\n102\n84\n78\n72\n24\n12\n17/2 n + 71\n2 n2\n1 + 17/2 n + 71\n2 n2\n1+42 t+72 t2+27 t3\n(1−t)2(1−t2)\n79\n63\n93\n49\n88\n37\n14\n3\n3/4 + 27\n2 n + 303\n4 n2\n1 + 27\n2 n + 303\n4 n2\n1+88 t+151 t2+63 t3\n(1−t)2(1−t2)\n97\n93\n114\n76\n77\n66\n47\n0\n1/2 + 15/2 n + 21 n2\n1 + 15/2 n + 21 n2\n1+27 t+42 t2+14 t3\n(1−t)2(1−t2)\n88\n56\n113\n31\n99\n35\n7\n3\n1/2 + 11/2 n + 10 n2\n1 + 11/2 n + 10 n2\n1+14 t+20 t2+5 t3\n(1−t)2(1−t2)\n134\n82\n140\n76\n91\n72\n49\n4\n3/4 + 21 n + 669\n4 n2\n1 + 21 n + 669\n4 n2\n1+187 t+334 t2+147 t3\n(1−t)2(1−t2)\n133\n69\n149\n53\n98\n55\n43\n6\n1 + 6 n + 8 n2\n1 + 6 n + 8 n2\n15 t2+13 t+1+3 t3\n(1−t)3\n80\n63\n111\n32\n88\n38\n10\n7\n1\n1\n1+t\n1−t\n118\n69\n151\n36\n95\n63\n20\n9\n1 + 4 n + 4 n2\n1 + 4 n + 4 n2\n7 t2+7 t+1+t3\n(1−t)3\n96\n51\n103\n44\n90\n53\n3\n1\n1/2 + 39\n2 n + 36 n2\n1 + 39\n2 n + 36 n2\n1+54 t+72 t2+17 t3\n(1−t)2(1−t2)\n117\n72\n133\n56\n82\n57\n41\n9\n1 + 9 n + 18 n2\n1 + 9 n + 18 n2\n35 t2+26 t+1+10 t3\n(1−t)3\n72\n63\n77\n58\n49\n38\n28\n20\n1/2 + 7 n + 55\n2 n2\n1 + 7 n + 55\n2 n2\n1+33 t+55 t2+21 t3\n(1−t)2(1−t2)\n48\n37\n49\n36\n34\n24\n16\n11\n1/2 + 6 n + 37\n2 n2\n1 + 6 n + 37\n2 n2\n1+23 t+37 t2+13 t3\n(1−t)2(1−t2)\n108\n56\n113\n51\n73\n50\n29\n12\n1 + 4 n + 4 n2\n1 + 4 n + 4 n2\n7 t2+7 t+1+t3\n(1−t)3\n \n \nFigure 6.4: The quasipolynomial ̃kπ\nλ,μ and the rational function Kπ\nλ,μ(t) for the Kronecker problem, n = 2.\n101"},{"page":105,"text":"λ1\nλ2\nμ1\nμ2\nπ1\nπ2\nπ3\nπ4\n ̃kπ\nλ,μ(n); n odd\n ̃kπ\nλ,μ(n); n even\nKπ\nλ,μ(t)\n77\n40\n78\n39\n58\n29\n24\n6\n1 + 19/2 n + 57\n2 n2\n1 + 19/2 n + 57\n2 n2\n56 t2+37 t+1+20 t3\n(1−t)3\n153\n81\n157\n77\n96\n63\n61\n14\n1 + 3 n + 2 n2\n1 + 3 n + 2 n2\n3 t2+4 t+1\n(1−t)3\n90\n89\n102\n77\n90\n42\n30\n17\n1/2 + 13/2 n + 6 n2\n1 + 13/2 n + 6 n2\n1+11 t+12 t2\n(1−t)2(1−t2)\n145\n102\n160\n87\n96\n84\n39\n28\n1 + 10 n + 25 n2\n1 + 10 n + 25 n2\n49 t2+34 t+1+16 t3\n(1−t)3\n109\n95\n136\n68\n78\n60\n46\n20\n1 + 3 n + 2 n2\n1 + 3 n + 2 n2\n3 t2+4 t+1\n(1−t)3\n100\n42\n104\n38\n85\n27\n27\n3\n1 + 8 n\n1 + 8 n\n8 t+1+7 t2\n(1−t)2\n74\n51\n86\n39\n52\n34\n26\n13\n1\n1\n1+t\n1−t\n98\n90\n124\n64\n92\n67\n22\n7\n1/2 + 23/2 n + 60 n2\n1 + 23/2 n + 60 n2\n1+70 t+120 t2+49 t3\n(1−t)2(1−t2)\n57\n38\n75\n20\n52\n25\n17\n1\n1 + 3 n + 2 n2\n1 + 3 n + 2 n2\n3 t2+4 t+1\n(1−t)3\n159\n140\n170\n129\n89\n82\n73\n55\n1 + 3/2 n + 1/2 n2\n1 + 3/2 n + 1/2 n2\n1+t\n(1−t)3\n144\n122\n157\n109\n88\n86\n74\n18\n3/4 + n + 1/4 n2\n1 + n + 1/4 n2\n1\n(1−t)2(1−t2)\n90\n68\n92\n66\n88\n37\n23\n10\n1/4 + 12 n + 351\n4 n2\n1 + 12 n + 351\n4 n2\n1+98 t+176 t2+76 t3\n(1−t)2(1−t2)\n89\n42\n100\n31\n76\n28\n19\n8\n1 + 6 n + 8 n2\n1 + 6 n + 8 n2\n15 t2+13 t+1+3 t3\n(1−t)3\n88\n56\n107\n37\n71\n39\n20\n14\n1 + 9/2 n + 9/2 n2\n1 + 9/2 n + 9/2 n2\n8 t2+8 t+1+t3\n(1−t)3\n124\n111\n133\n102\n98\n89\n27\n21\n1/2 + 7 n + 53\n2 n2\n1 + 7 n + 53\n2 n2\n1+32 t+53 t2+20 t3\n(1−t)2(1−t2)\n \n \nFigure 6.5: Continuation of Figure 6.4\n102"},{"page":106,"text":"k\nλ\nhk,λ(n)\n3\n(21, 19)\n399 n3 + 35527969472513\n137438953472 n2 + 4329327034365\n137438953472 n + 1\n5\n(21, 19)\n3700378042361\n4194304\nn7 + 575575719967\n524288\nn6 + 2157156441\n4096\nn5 + 266554253\n2048\nn4 + 4643843\n256\nn3 + 1468423\n1024\nn2 + 7619\n128 n + 1\n3\n(21, 9, 6)\n270 n3 + 40819369181185\n274877906944 n2 + 3092376453119\n137438953472 n + 1\n3\n(12, 9, 5)\n42 n3 + 40132174413825\n1099511627776 n2 + 11544872091645\n1099511627776 n + 1\n3\n(21, 9, 6)\n27396522639355\n536870912\nn6 + 463063744509\n8388608\nn5 + 6265700353\n262144\nn4 + 5577375771\n1048576\nn3 + 84246529\n131072 n2 + 20971505\n524288 n + 1048573\n1048576\n3\n(21, 19, 16)\n15 n3 + 81363860455425\n4398046511104 n2 + 8246337208319\n1099511627776 n + 1\n4\n(9, 7, 5)\n7215545057279\n17179869184 n6 + 4183298146289\n4294967296\nn5 + 247765925897\n268435456\nn4 + 1914699777\n4194304\nn3 + 4160749567\n33554432 n2 + 587202553\n33554432 n + 67108863\n67108864\n4\n(21, 12, 9)\n16437913583613\n268435456\nn6 + 132498063359\n2097152\nn5 + 109509083155\n4194304\nn4 + 1462763527\n262144\nn3 + 171442179\n262144\nn2 + 10485755\n262144 n + 524287\n524288\n4\n(21, 9, 5)\n32469952757755\n536870912\nn6 + 129805320191\n2097152\nn5 + 108129157137\n4194304\nn4 + 2926313487\n524288\nn3 + 86638593\n131072 n2 + 10616825\n262144 n + 262143\n262144\n4\n(21, 9, 6)\n27396522639355\n536870912\nn6 + 463063744509\n8388608\nn5 + 6265700353\n262144\nn4 + 5577375771\n1048576\nn3 + 84246529\n131072 n2 + 20971505\n524288 n + 1048573\n1048576\n4\n(31, 19, 5)\n35969680015355\n33554432\nn6 + 1424674346311\n2097152\nn5 + 22705493343\n131072\nn4 + 46973953\n2048\nn3 + 3423915\n2048\nn2 + 65365\n1024 n + 16383\n16384\nFigure 6.6: Hilbert polynomial for G/Pλ, G = SLk(C). There is a slight rounding error caused by interpolation–\ne.g., the constant term of each polynomial should be one.\n103"},{"page":107,"text":"n\nk\na\n7\n3\n(1, 3, 5)\n1/3 n3 + 59373627899905\n39582418599936 n2 + 28587302322173\n13194139533312 n + 1\n7\n3\n(1, 2, 4)\nn + 1\n7\n3\n(1, 4, 6)\n22265110462465\n534362651099136 n5 + 4638564679679\n11132555231232 n4 + 13\n8 n3 + 105942526633\n34359738368 n2 + 146028888073\n51539607552 n + 34359738361\n34359738368\n6\n2\n(1, 4, 5)\n15637498706143\n2251799813685248 n6 + 3665038759245\n35184372088832 n5 + 1389660529559\n2199023255552 n4 + 272014595421\n137438953472 n3\n+ 230973796809\n68719476736 n2 + 100215903571\n34359738368 n + 1\n6\n2\n(1, 4, 6)\n69578470195\n25048249270272 n7 + 1217623228439\n25048249270272 n6 + 372534725887\n1043677052928 n5 + 30953963537\n21743271936 n4 + 12044363351\n3623878656 n3\n+ 683671553\n150994944 n2 + 1335466297\n402653184 n + 268435457\n268435456\n7\n3\n(1, 4, 6)\n23456248059223\n562949953421312 n5 + 7330077518505\n17592186044416 n4 + 1786706395137\n1099511627776 n3 + 423770106525\n137438953472 n2 + 24338148015\n8589934592 n + 17179869169\n17179869184\n6\n3\n(1, 3, 6)\n1/8 n4 + 16126170540715\n17592186044416 n3 + 19\n8 n2 + 710101259605\n274877906944 n + 1\n8\n3\n(1, 3, 6)\n171798691840001\n1374389534720000 n4 + 31496426837333\n34359738368000 n3 + 4080218931199\n1717986918400 n2 + 443813287253\n171798691840 n + 1\nFigure 6.7: Hilbert polynomial of the Schubert subvariety Ωa, a = (a(1), . . . , a(k)), of the Grassmannian Gn,k.\n104"},{"page":108,"text":"k = 2\n\"\n1/2 n + 1/2\n1/2 n + 1\n#\nk = 3\n \n \n1/12 n2 + 1/2 n + 5\n12\n1/12 n2 + 1/2 n + 2/3\n1/12 n2 + 1/2 n + 3/4\n1/12 n2 + 1/2 n + 46912496118443\n70368744177664\n1/12 n2 + 1/2 n + 58640620148053\n140737488355328\n1/12 n2 + 1/2 n + 1\n \n \nk = 4\n \n \n1\n144 n3 + 5\n48 n2 + 61572651155457\n140737488355328 n + 15881834623431\n35184372088832\n1\n144 n3 + 117281240296107\n1125899906842624 n2 + 140737488355325\n281474976710656 n + 19\n36\n1\n144 n3 + 234562480592215\n2251799813685248 n2 + 123145302310909\n281474976710656 n + 19791209299969\n35184372088832\n1\n144 n3 + 234562480592215\n2251799813685248 n2 + 70368744177667\n140737488355328 n + 62549994824587\n70368744177664\n1\n144 n3 + 5\n48 n2 + 61572651155453\n140737488355328 n + 748278746681\n2199023255552\n1\n144 n3 + 117281240296107\n1125899906842624 n2 + 70368744177665\n140737488355328 n + 26388279066621\n35184372088832\n1\n144 n3 + 117281240296107\n1125899906842624 n2 + 7\n16 n + 7940917311717\n17592186044416\n1\n144 n3 + 117281240296107\n1125899906842624 n2 + 35184372088831\n70368744177664 n + 6841405683939\n8796093022208\n1\n144 n3 + 29320310074027\n281474976710656 n2 + 30786325577729\n70368744177664 n + 9\n16\n1\n144 n3 + 58640620148053\n562949953421312 n2 + 35184372088831\n70368744177664 n + 5619726097523\n8796093022208\n1\n144 n3 + 117281240296105\n1125899906842624 n2 + 7\n16 n + 2993114986727\n8796093022208\n1\n144 n3 + 58640620148055\n562949953421312 n2 + 1/2 n + 1\n \n \nFigure 6.8: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk; k = 2, 3, 4.\n105"},{"page":109,"text":"1\n2880 n4 +\n46912496118441\n4503599627370496 n3 + 3787206717893\n35184372088832 n2 + 469583091025\n1099511627776 n + 499743305817\n1099511627776\n1\n2880 n4 +\n46912496118445\n4503599627370496 n3 + 3787206717891\n35184372088832 n2 + 503942829399\n1099511627776 n + 310001195055\n549755813888\n1\n2880 n4 +\n46912496118441\n4503599627370496 n3 + 3787206717897\n35184372088832 n2 + 469583091031\n1099511627776 n + 30279519437\n68719476736\n400319966877379\n1152921504606846976 n4 +\n23456248059221\n2251799813685248 n3 + 3787206717893\n35184372088832 n2 + 503942829403\n1099511627776 n + 47340083975\n68719476736\n400319966877379\n1152921504606846976 n4 +\n5864062014805\n562949953421312 n3 + 3787206717895\n35184372088832 n2 + 117395772755\n274877906944 n + 89955703917\n137438953472\n400319966877379\n1152921504606846976 n4 +\n5864062014805\n562949953421312 n3 + 3787206717893\n35184372088832 n2 + 31496426837\n68719476736 n + 92771293595\n137438953472\n400319966877379\n1152921504606846976 n4 +\n46912496118447\n4503599627370496 n3 + 1893603358947\n17592186044416 n2 + 234791545515\n549755813888 n + 5661005505\n17179869184\n1\n2880 n4 +\n23456248059223\n2251799813685248 n3 + 1893603358949\n17592186044416 n2 + 62992853675\n137438953472 n + 94680167945\n137438953472\n400319966877379\n1152921504606846976 n4 +\n46912496118441\n4503599627370496 n3 + 3787206717891\n35184372088832 n2 + 117395772757\n274877906944 n + 38869454029\n68719476736\n1\n2880 n4 +\n46912496118441\n4503599627370496 n3 + 3787206717897\n35184372088832 n2 + 62992853675\n137438953472 n + 52494044729\n68719476736\n1\n2880 n4 +\n5864062014805\n562949953421312 n3 + 946801679473\n8796093022208 n2 + 234791545515\n549755813888 n + 2830502753\n8589934592\n400319966877379\n1152921504606846976 n4 +\n23456248059219\n2251799813685248 n3 + 1893603358949\n17592186044416 n2 + 251971414695\n549755813888 n + 27487790695\n34359738368\n400319966877379\n1152921504606846976 n4 +\n23456248059223\n2251799813685248 n3 + 1893603358945\n17592186044416 n2 + 234791545509\n549755813888 n + 31233956605\n68719476736\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 251971414699\n549755813888 n + 19375074691\n34359738368\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 29348943189\n68719476736 n + 11005853695\n17179869184\n400319966877379\n1152921504606846976 n4 +\n23456248059219\n2251799813685248 n3 + 473400839737\n4398046511104 n2 + 251971414699\n549755813888 n + 5917510497\n8589934592\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 234791545511\n549755813888 n + 15616978311\n34359738368\n1\n2880 n4 +\n23456248059223\n2251799813685248 n3 + 1893603358947\n17592186044416 n2 + 62992853673\n137438953472 n + 23192823403\n34359738368\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 946801679475\n8796093022208 n2 + 117395772757\n274877906944 n + 2830502755\n8589934592\n400319966877379\n1152921504606846976 n4 +\n11728124029609\n1125899906842624 n3 + 1893603358949\n17592186044416 n2 + 31496426837\n68719476736 n + 30541989663\n34359738368\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 117395772759\n274877906944 n + 9717363505\n17179869184\n1\n2880 n4 +\n2932031007403\n281474976710656 n3 + 1893603358945\n17592186044416 n2 + 125985707353\n274877906944 n + 4843768673\n8589934592\n400319966877379\n1152921504606846976 n4 +\n23456248059223\n2251799813685248 n3 + 59175104967\n549755813888 n2 + 117395772755\n274877906944 n + 2830502755\n8589934592\n400319966877379\n1152921504606846976 n4 +\n23456248059221\n2251799813685248 n3 + 946801679475\n8796093022208 n2 + 62992853675\n137438953472 n + 3435973837\n4294967296\n400319966877379\n1152921504606846976 n4 +\n23456248059223\n2251799813685248 n3 + 1893603358947\n17592186044416 n2 + 58697886379\n137438953472 n + 11244462985\n17179869184\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 946801679475\n8796093022208 n2 + 15748213419\n34359738368 n + 9687537337\n17179869184\n1\n2880 n4 +\n11728124029609\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 117395772753\n274877906944 n + 1892469965\n4294967296\n1\n2880 n4 +\n23456248059221\n2251799813685248 n3 + 946801679473\n8796093022208 n2 + 7874106709\n17179869184 n + 11835020991\n17179869184\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 117395772753\n274877906944 n + 3904244579\n8589934592\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 125985707349\n274877906944 n + 1879048193\n2147483648\n \n \nFigure 6.9: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 5; the\nfirst 30 rows.\n106"},{"page":110,"text":"1\n2880 n4 +\n11728124029609\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 58697886375\n137438953472 n + 88453211\n268435456\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 15748213419\n34359738368 n + 2958755247\n4294967296\n400319966877379\n1152921504606846976 n4 +\n23456248059223\n2251799813685248 n3 + 946801679475\n8796093022208 n2 + 58697886375\n137438953472 n + 4858681755\n8589934592\n400319966877379\n1152921504606846976 n4 +\n11728124029609\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 62992853673\n137438953472 n + 151367771\n268435456\n1\n2880 n4 +\n23456248059221\n2251799813685248 n3 + 946801679473\n8796093022208 n2 + 58697886375\n137438953472 n + 2274244835\n4294967296\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 236700419869\n2199023255552 n2 + 62992853671\n137438953472 n + 858993459\n1073741824\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 946801679473\n8796093022208 n2 + 14674471595\n34359738368 n + 3904244571\n8589934592\n400319966877379\n1152921504606846976 n4 +\n23456248059219\n2251799813685248 n3 + 59175104967\n549755813888 n2 + 7874106709\n17179869184 n + 2421884337\n4294967296\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 59175104967\n549755813888 n2 + 29348943189\n68719476736 n + 3784939927\n8589934592\n400319966877379\n1152921504606846976 n4 +\n23456248059223\n2251799813685248 n3 + 473400839737\n4398046511104 n2 + 62992853673\n137438953472 n + 1908874355\n2147483648\n400319966877379\n1152921504606846976 n4 +\n5864062014805\n562949953421312 n3 + 946801679473\n8796093022208 n2 + 29348943189\n68719476736 n + 1952122291\n4294967296\n400319966877379\n1152921504606846976 n4 +\n11728124029609\n1125899906842624 n3 + 946801679471\n8796093022208 n2 + 62992853675\n137438953472 n + 2899102929\n4294967296\n400319966877379\n1152921504606846976 n4 +\n11728124029611\n1125899906842624 n3 + 473400839735\n4398046511104 n2 + 58697886381\n137438953472 n + 707625689\n2147483648\n400319966877379\n1152921504606846976 n4 +\n11728124029613\n1125899906842624 n3 + 59175104967\n549755813888 n2 + 15748213419\n34359738368 n + 2958755253\n4294967296\n1\n2880 n4 +\n11728124029609\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 58697886377\n137438953472 n + 3288334339\n4294967296\n100079991719345\n288230376151711744 n4 +\n5864062014805\n562949953421312 n3 + 59175104967\n549755813888 n2 + 62992853675\n137438953472 n + 1210942169\n2147483648\n1\n2880 n4 +\n11728124029611\n1125899906842624 n3 + 946801679477\n8796093022208 n2 + 29348943193\n68719476736 n + 1415251375\n4294967296\n1\n2880 n4 +\n5864062014805\n562949953421312 n3 + 59175104967\n549755813888 n2 + 31496426839\n68719476736 n + 3435973835\n4294967296\n100079991719345\n288230376151711744 n4 +\n1466015503701\n140737488355328 n3 + 473400839737\n4398046511104 n2 + 29348943195\n68719476736 n + 976061147\n2147483648\n100079991719345\n288230376151711744 n4 +\n11728124029611\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 31496426839\n68719476736 n + 3280877793\n4294967296\n1\n2880 n4 +\n5864062014805\n562949953421312 n3 + 946801679473\n8796093022208 n2 + 29348943191\n68719476736 n + 946234983\n2147483648\n100079991719345\n288230376151711744 n4 +\n2932031007403\n281474976710656 n3 + 946801679475\n8796093022208 n2 + 15748213419\n34359738368 n + 1479377623\n2147483648\n1\n2880 n4 +\n5864062014805\n562949953421312 n3 + 59175104967\n549755813888 n2 + 29348943195\n68719476736 n + 122007643\n268435456\n100079991719345\n288230376151711744 n4 +\n2932031007403\n281474976710656 n3 + 473400839737\n4398046511104 n2 + 3937053355\n8589934592 n + 1449551461\n2147483648\n100079991719345\n288230376151711744 n4 +\n2932031007403\n281474976710656 n3 + 236700419869\n2199023255552 n2 + 29348943193\n68719476736 n + 71070151\n134217728\n1\n2880 n4 +\n1466015503701\n140737488355328 n3 + 59175104967\n549755813888 n2 + 15748213419\n34359738368 n + 739688813\n1073741824\n100079991719345\n288230376151711744 n4 +\n11728124029611\n1125899906842624 n3 + 473400839739\n4398046511104 n2 + 14674471597\n34359738368 n + 607335219\n1073741824\n1\n2880 n4 +\n2932031007403\n281474976710656 n3 + 236700419869\n2199023255552 n2 + 3937053355\n8589934592 n + 605471085\n1073741824\n100079991719345\n288230376151711744 n4 +\n11728124029611\n1125899906842624 n3 + 473400839737\n4398046511104 n2 + 29348943193\n68719476736 n + 88453211\n268435456\n100079991719345\n288230376151711744 n4 +\n1466015503701\n140737488355328 n3 + 473400839737\n4398046511104 n2 + 3937053355\n8589934592 n + 2147483647\n2147483648\n \n \nFigure 6.10: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 5;\nthe last 30 rows.\n107"},{"page":111,"text":"53375995583651\n4611686018427387904 n5 +\n21892498188609\n36028797018963968 n4 +\n418085902907\n35184372088832 n3 + 115486898397\n1099511627776 n2 + 26847522421\n68719476736 n + 8448724291\n17179869184\n53375995583651\n4611686018427387904 n5 +\n10946249094305\n18014398509481984 n4 +\n836171805815\n70368744177664 n3 + 7396888121\n68719476736 n2 + 15302809397\n34359738368 n + 9853503363\n17179869184\n106751991167299\n9223372036854775808 n5 +\n21892498188599\n36028797018963968 n4 +\n1672343611625\n140737488355328 n3 + 57743449207\n549755813888 n2 + 14060052663\n34359738368 n + 975427339\n2147483648\n13343998895913\n1152921504606846976 n5 +\n10946249094301\n18014398509481984 n4 +\n1672343611641\n140737488355328 n3 + 59175104961\n549755813888 n2 + 15302809423\n34359738368 n + 610309549\n1073741824\n106751991167305\n9223372036854775808 n5 +\n21892498188611\n36028797018963968 n4 +\n1672343611623\n140737488355328 n3 + 115486898411\n1099511627776 n2 + 13423761203\n34359738368 n + 4461910959\n8589934592\n13343998895913\n1152921504606846976 n5 +\n21892498188605\n36028797018963968 n4 +\n1672343611631\n140737488355328 n3 + 29587552483\n274877906944 n2 + 15939100865\n34359738368 n + 1927366573\n2147483648\n213503982334599\n18446744073709551616 n5 +\n10946249094305\n18014398509481984 n4 +\n418085902909\n35184372088832 n3 + 57743449199\n549755813888 n2 + 13423761217\n34359738368 n + 835973461\n2147483648\n106751991167299\n9223372036854775808 n5 +\n10946249094305\n18014398509481984 n4 +\n836171805821\n70368744177664 n3 + 59175104963\n549755813888 n2 + 7651404713\n17179869184 n + 320001575\n536870912\n106751991167299\n9223372036854775808 n5 +\n10946249094299\n18014398509481984 n4 +\n1672343611637\n140737488355328 n3 + 115486898395\n1099511627776 n2 + 14060052667\n34359738368 n + 255936429\n536870912\n213503982334597\n18446744073709551616 n5 +\n2736562273577\n4503599627370496 n4 +\n1672343611629\n140737488355328 n3 + 7396888121\n68719476736 n2 + 7651404703\n17179869184 n + 2859997491\n4294967296\n106751991167299\n9223372036854775808 n5 +\n342070284197\n562949953421312 n4 + 104521475727\n8796093022208 n3 + 57743449199\n549755813888 n2 + 1677970153\n4294967296 n + 1790721319\n4294967296\n1\n86400 n5 +\n10946249094299\n18014398509481984 n4 +\n836171805815\n70368744177664 n3 + 59175104959\n549755813888 n2 + 3984775219\n8589934592 n + 987842475\n1073741824\n1\n86400 n5 +\n5473124547153\n9007199254740992 n4 +\n209042951453\n17592186044416 n3 + 57743449193\n549755813888 n2 + 3355940307\n8589934592 n + 884291849\n2147483648\n106751991167303\n9223372036854775808 n5 +\n10946249094301\n18014398509481984 n4 +\n836171805817\n70368744177664 n3 + 118350209919\n1099511627776 n2 + 1912851173\n4294967296 n + 264972307\n536870912\n213503982334605\n18446744073709551616 n5 +\n2736562273577\n4503599627370496 n4 +\n836171805827\n70368744177664 n3 + 57743449201\n549755813888 n2 + 7030026327\n17179869184 n + 1233125369\n2147483648\n53375995583651\n4611686018427387904 n5 +\n10946249094299\n18014398509481984 n4 +\n836171805819\n70368744177664 n3 + 118350209933\n1099511627776 n2 + 1912851177\n4294967296 n + 1478317113\n2147483648\n106751991167297\n9223372036854775808 n5 +\n1368281136787\n2251799813685248 n4 +\n836171805817\n70368744177664 n3 + 57743449195\n549755813888 n2 + 1677970151\n4294967296 n + 117959881\n268435456\n1667999861989\n144115188075855872 n5 +\n10946249094303\n18014398509481984 n4 + 104521475727\n8796093022208 n3 + 59175104961\n549755813888 n2 + 3984775215\n8589934592 n + 109722991\n134217728\n106751991167297\n9223372036854775808 n5 +\n342070284197\n562949953421312 n4 +\n836171805815\n70368744177664 n3 + 28871724597\n274877906944 n2 + 1677970153\n4294967296 n + 332087389\n1073741824\n213503982334607\n18446744073709551616 n5 +\n10946249094307\n18014398509481984 n4 +\n836171805821\n70368744177664 n3 + 59175104963\n549755813888 n2 + 7651404709\n17179869184 n + 384426083\n536870912\n \n \nFigure 6.11: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 6; the first 20 rows.\n108"},{"page":112,"text":"213503982334615\n18446744073709551616 n5 +\n10946249094307\n18014398509481984 n4 +\n836171805813\n70368744177664 n3 + 28871724597\n274877906944 n2 + 7030026337\n17179869184 n + 640721865\n1073741824\n106751991167309\n9223372036854775808 n5 +\n10946249094297\n18014398509481984 n4 +\n209042951455\n17592186044416 n3 + 29587552479\n274877906944 n2 + 1912851179\n4294967296 n + 157275009\n268435456\n26687997791827\n2305843009213693952 n5 +\n342070284197\n562949953421312 n4 +\n836171805825\n70368744177664 n3 + 57743449199\n549755813888 n2 + 3355940301\n8589934592 n + 90445245\n268435456\n13343998895913\n1152921504606846976 n5 +\n5473124547153\n9007199254740992 n4 + 104521475729\n8796093022208 n3 + 29587552481\n274877906944 n2 + 1992387605\n4294967296 n + 450971557\n536870912\n106751991167303\n9223372036854775808 n5 +\n10946249094307\n18014398509481984 n4 +\n836171805827\n70368744177664 n3 + 28871724597\n274877906944 n2 + 209746269\n536870912 n + 17843591\n33554432\n106751991167303\n9223372036854775808 n5 +\n10946249094297\n18014398509481984 n4 +\n836171805819\n70368744177664 n3 + 924611015\n8589934592 n2 + 239106397\n536870912 n + 5146825\n8388608\n1667999861989\n144115188075855872 n5 +\n5473124547153\n9007199254740992 n4 +\n418085902911\n35184372088832 n3 + 7217931151\n68719476736 n2 + 54922081\n134217728 n + 265331665\n536870912\n1667999861989\n144115188075855872 n5 +\n10946249094291\n18014398509481984 n4 +\n836171805825\n70368744177664 n3 + 59175104963\n549755813888 n2 + 478212795\n1073741824 n + 326629601\n536870912\n1667999861989\n144115188075855872 n5 +\n10946249094297\n18014398509481984 n4 +\n418085902909\n35184372088832 n3 + 14435862303\n137438953472 n2 + 3355940305\n8589934592 n + 24121261\n67108864\n53375995583655\n4611686018427387904 n5 +\n10946249094291\n18014398509481984 n4 +\n209042951453\n17592186044416 n3 + 59175104973\n549755813888 n2 + 3984775217\n8589934592 n + 503316475\n536870912\n26687997791827\n2305843009213693952 n5 +\n10946249094285\n18014398509481984 n4 +\n836171805825\n70368744177664 n3 + 57743449207\n549755813888 n2 + 3355940305\n8589934592 n + 115234099\n268435456\n106751991167303\n9223372036854775808 n5 +\n2736562273573\n4503599627370496 n4 +\n209042951459\n17592186044416 n3 + 7396888121\n68719476736 n2 + 3825702363\n8589934592 n + 42684549\n67108864\n106751991167301\n9223372036854775808 n5 +\n10946249094305\n18014398509481984 n4 +\n209042951453\n17592186044416 n3 + 28871724605\n274877906944 n2 + 1757506587\n4294967296 n + 277411267\n536870912\n106751991167303\n9223372036854775808 n5 +\n10946249094299\n18014398509481984 n4 +\n418085902905\n35184372088832 n3 + 924611015\n8589934592 n2 + 3825702353\n8589934592 n + 33950041\n67108864\n106751991167307\n9223372036854775808 n5 +\n1368281136787\n2251799813685248 n4 +\n52260737865\n4398046511104 n3 + 28871724601\n274877906944 n2 + 209746269\n536870912 n + 61328751\n134217728\n53375995583651\n4611686018427387904 n5 +\n2736562273575\n4503599627370496 n4 + 104521475727\n8796093022208 n3 + 29587552487\n274877906944 n2 + 249048451\n536870912 n + 128849017\n134217728\n106751991167301\n9223372036854775808 n5 +\n5473124547145\n9007199254740992 n4 +\n209042951459\n17592186044416 n3 + 7217931151\n68719476736 n2 + 1677970157\n4294967296 n + 30318473\n67108864\n106751991167303\n9223372036854775808 n5 +\n342070284197\n562949953421312 n4 +\n418085902919\n35184372088832 n3 + 14793776243\n137438953472 n2 + 1912851181\n4294967296 n + 143223581\n268435456\n53375995583651\n4611686018427387904 n5 +\n5473124547153\n9007199254740992 n4 +\n209042951459\n17592186044416 n3 + 1804482787\n17179869184 n2 + 878753295\n2147483648 n + 13898875\n33554432\n106751991167303\n9223372036854775808 n5 +\n5473124547147\n9007199254740992 n4 +\n418085902907\n35184372088832 n3 + 29587552479\n274877906944 n2 + 956425585\n2147483648 n + 195527051\n268435456\n \n \nFigure 6.12: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 6; the middle 20 rows.\n109"},{"page":113,"text":"13343998895913\n1152921504606846976 n5 +\n5473124547145\n9007199254740992 n4 +\n418085902919\n35184372088832 n3 + 7217931149\n68719476736 n2 + 209746269\n536870912 n + 64348647\n134217728\n53375995583651\n4611686018427387904 n5 +\n5473124547143\n9007199254740992 n4 + 104521475729\n8796093022208 n3 + 14793776237\n137438953472 n2 + 498096901\n1073741824 n + 28772927\n33554432\n106751991167301\n9223372036854775808 n5 +\n5473124547141\n9007199254740992 n4 +\n209042951457\n17592186044416 n3 + 3608965575\n34359738368 n2 + 1677970153\n4294967296 n + 11719905\n33554432\n13343998895913\n1152921504606846976 n5 +\n5473124547151\n9007199254740992 n4 +\n418085902905\n35184372088832 n3 + 3698444059\n34359738368 n2 + 478212797\n1073741824 n + 74631683\n134217728\n106751991167311\n9223372036854775808 n5 +\n5473124547147\n9007199254740992 n4 +\n418085902907\n35184372088832 n3 + 28871724591\n274877906944 n2 + 878753299\n2147483648 n + 10682367\n16777216\n106751991167313\n9223372036854775808 n5 +\n5473124547151\n9007199254740992 n4 + 104521475729\n8796093022208 n3 + 14793776237\n137438953472 n2 + 478212797\n1073741824 n + 84006215\n134217728\n106751991167307\n9223372036854775808 n5 +\n2736562273573\n4503599627370496 n4 +\n209042951459\n17592186044416 n3 + 28871724589\n274877906944 n2 + 104873135\n268435456 n + 3161957\n8388608\n53375995583655\n4611686018427387904 n5 +\n5473124547149\n9007199254740992 n4 +\n209042951451\n17592186044416 n3 + 29587552485\n274877906944 n2 + 996193803\n2147483648 n + 118111589\n134217728\n106751991167309\n9223372036854775808 n5 +\n1368281136787\n2251799813685248 n4 +\n418085902907\n35184372088832 n3 + 28871724601\n274877906944 n2 + 419492539\n1073741824 n + 49899533\n134217728\n106751991167305\n9223372036854775808 n5 +\n2736562273573\n4503599627370496 n4 +\n52260737863\n4398046511104 n3 + 29587552479\n274877906944 n2 + 1912851173\n4294967296 n + 87717907\n134217728\n106751991167309\n9223372036854775808 n5 +\n1368281136787\n2251799813685248 n4 + 104521475729\n8796093022208 n3 + 28871724595\n274877906944 n2 + 878753303\n2147483648 n + 8962701\n16777216\n106751991167305\n9223372036854775808 n5 +\n2736562273571\n4503599627370496 n4 +\n209042951453\n17592186044416 n3 + 29587552499\n274877906944 n2 + 956425585\n2147483648 n + 10878263\n16777216\n53375995583651\n4611686018427387904 n5 +\n5473124547157\n9007199254740992 n4 + 104521475727\n8796093022208 n3 + 3608965575\n34359738368 n2 + 838985083\n2147483648 n + 53611225\n134217728\n13343998895913\n1152921504606846976 n5 +\n5473124547145\n9007199254740992 n4 +\n418085902907\n35184372088832 n3 + 14793776249\n137438953472 n2 + 498096903\n1073741824 n + 104354293\n134217728\n106751991167299\n9223372036854775808 n5 +\n2736562273575\n4503599627370496 n4 +\n209042951447\n17592186044416 n3 + 3608965575\n34359738368 n2 + 838985087\n2147483648 n + 7873221\n16777216\n106751991167303\n9223372036854775808 n5 +\n5473124547151\n9007199254740992 n4 +\n418085902899\n35184372088832 n3 + 14793776243\n137438953472 n2 + 956425599\n2147483648 n + 90737805\n134217728\n53375995583655\n4611686018427387904 n5 +\n5473124547155\n9007199254740992 n4 + 104521475727\n8796093022208 n3 + 14435862303\n137438953472 n2 + 878753295\n2147483648 n + 18680385\n33554432\n106751991167305\n9223372036854775808 n5 +\n2736562273573\n4503599627370496 n4 + 104521475725\n8796093022208 n3 + 3698444063\n34359738368 n2 + 478212799\n1073741824 n + 36634403\n67108864\n106751991167311\n9223372036854775808 n5 +\n684140568395\n1125899906842624 n4 +\n209042951463\n17592186044416 n3 + 7217931153\n68719476736 n2 + 52436567\n134217728 n + 19926951\n67108864\n26687997791827\n2305843009213693952 n5 +\n1368281136789\n2251799813685248 n4 +\n6532592233\n549755813888 n3 + 14793776243\n137438953472 n2 + 498096903\n1073741824 n + 67108871\n67108864\n \n \nFigure 6.13: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 6; the last 20 rows.\n110"},{"page":114,"text":"Chapter 7\nOn verification and discovery\nof obstructions\nIn this chapter we give applications of the results and positivity hypotheses\nin this paper to the problem of verifying or discovering an obstruction, i.e.,\na “proof of hardness” [GCT2] in the context of the P vs.\nNP and the\npermanent vs. determinant problems in characteristic zero.\n7.1\nObstruction\nAn obstruction in an abstract setting of Problem 1.1.4 is defined as follows.\nLet X and Y be H-varieties with compact specifications (Section 3.5),\nH a connected reductive group. Let ⟨X⟩and ⟨Y ⟩denote the bit lengths of\ntheir specifications (Section 3.5). Suppose we wish to show that X cannot\nbe embedded as an H-subvariety of Y . Pictorially:\nX ̸֒→Y.\n(7.1)\nFor example, in the context of the P vs. NP problem in characteristic\nzero [GCT1, GCT2], X is a class variety XNP (n, l) associated with the\ncomplexity class NP for the given input size parameter n and the circuit\nsize parameter l. The variety Y is the class variety XP (l) associated with\nthe class P for given l. And H is SLl(C). If NP ⊆P (over C) to the\ncontrary, then it would turn out that\nXNP (n, l) ֒→XP (l)\n111"},{"page":115,"text":"as an H-subvariety, for every l = poly(n). The goal is to show that this\nembedding cannot exist when l = poly(n) and n →∞.\nLet R(X) and R(Y ) be the homogeneous coordinate rings of X and\nY , respectively. Let R(X)d and R(Y )d denote their degree d-compoenents.\nSuppose to the contrary that an H-embedding as in (7.1) exists. Then there\nexists a degre preserving H-equivariant surjection from R(Y )d to R(X)d for\nevery d, and hence, a degree-preserving H-equivariant injection from R(X)∗\nd\nto R(Y )∗\nd. Hence, every irreducible H-module Vλ(G) that occurs in R(X)∗\nd\nalso occurs in R(Y )∗\nd. This leads to:\nDefinition 7.1.1 An irreducible representation Vλ(H) is called an obstruc-\ntion for the pair (X,Y) if it occurs (as an H-submodule) in R(X)∗\nd but not\nin R(Y )∗\nd, for some d. We say that Vλ(H) is an obstruction of degree d.\nRemark 7.1.2 The obstruction as defined here is dual to the obstrution as\ndefined in [GCT2].\nExistence of such an obstruction implies that X cannot be embedded in\nY as an H-subvariety.\nLet us assume that X and Y are H-subvarieties of P(V ), where V is an\nH-module, and that we are given a point y ∈Y ⊆P(V ) that is distiguished\nin the following sense. Let Hy ⊆H be the stabilizer of y. Then Cy, the line\nin V corresponding to y, is invariant under Hy. Let [y] be the set of points\nin P(V ) stabilized by Hy. We say that y is characterized by its stabilizer Hy\nif y = [y]; i.e., y is the only point in P(V ) stabilized by Hy. Let\nH[y] = ∪z∈[y]Hz\nbe the union of the H-orbits of all points in [y]. We say that y is a distin-\nguished point of Y if Y equals the projective closure of H[y] in P(V ). If y\nis characterized by its stabilizer, this means Y is the projective closure of\nthe orbit Hy of y.\nIf Vλ(H) occurs in R(Y )∗\nd, then it can be shown (cf. Proposition 4.2 in\n[GCT2]) that Vλ(H) contains an Hy-submodule isomorphic to (Cy)d, the\nd-th tensor power of Cy.\nThis leads to the following stronger notion of\nobstruction:\nDefinition 7.1.3 [GCT2] We say that Vλ(H) is a strong obstruction for\nthe pair (X, Y ) if, for some d, it occurs in R(X)∗\nd, but it does not contain\nan Hy-module isomorphic to (Cy)d.\n112"},{"page":116,"text":"Existence of a strong obstruction also implies that X cannot be embedded\nin Y as an H-subvariety. The results in [GCT2] suggest that strong obstruc-\ntions exist in the context of the lower bound problems under cosideration.\nThe goal then is to show their existence.\n7.2\nDecision problems\nThe decisions problems that arise in this context are the following. Let sλ\nd\nbe the multiplicity of Vλ(H) in R(X)∗\nd, and md\nλ the multiplicity of the Hy-\nmodule (Cy)d in Vλ(H), considered an Hy-module via the the embedding\nρ : Hy ֒→H. Thus λ is a strong obstruction of degree d iffsλ\nd is nonzero\nand md\nλ is zero.\nProblem 7.2.1 (Decision Problems)\n(a) Given d, λ and the specification of X, decide if sλ\nd is nonzero.\n(b) Given d, λ and the specifications of H, Hy and ρ, decide if md\nλ is nonzero.\n(c) Given d, λ and the specifications of X, H, Hy and ρ, decide if λ is a\nstrong obstruction of degree d.\nThe first is an instance of the decision Problem 1.1.4 in geometric in-\nvariant theory, and the second of the subgroup restriction Problem 1.1.3.\nBy the results in Chapter 3-4, relaxed forms of the decision problems in (a)\nand (b) belong to P assuming appropriate PH1 and SH; this implies that\na relaxed form of the decision problem in (c) also belongs to P assumming\nPH1 and SH. We will only need a weak relaxed form of (c), for which the\nweak form of SH that is implied by PH1 (cf. Theorem 3.3.5) will suffice.\n7.3\nVerification of obstructions\nThe relevant PH1 are as follows.\nAssume that that singularities of spec(R(X)) are rational.\nBy Theo-\nrem 3.5.1, the stretching function ̃sλ\nd(k) = skλ\nkd is a quasipolynomial. Hence\nPH1 for sλ\nd (Hypothesis 3.3.1, or rather its slight variant obtained by replac-\ning R(X)d with R(X)∗\nd) is well defined. It is:\nHypothesis 7.3.1 (PH1):\nThere exists a polytope P λ\nd such that:\n113"},{"page":117,"text":"1. The number of integer points in P λ\nd is equal to sλ\nd.\n2. The Ehrhart quasi-polynomial of P λ\nd coincides with the stretching quasi-\npolynomial ̃sλ\nd(n) (cf. Theorem 3.5.1).\n3. The polytope P λ\nd is given by a separating oracle, as in Section 2.3. Its\nencoding bitlength ⟨P λ\nd ⟩is poly(⟨d⟩, ⟨λ⟩, ⟨X⟩), and the combinatorial\nsize ∥P λ\nd ∥is poly(ht(λ), ∥X∥), where ∥X∥is the combinatorial size of\nX (Section 3.5), and ht(λ) is the height of λ.\nSimilarly by Theorem 3.4.1 (or rather its slight variant which can be\nproved similarly), the stretching function mkd\nkλ is a quasipolynomial. Hence\nPH1 for md\nλ (cf. Hypothesis 3.3.1 and Section 3.4) is also well defined. It is:\nHypothesis 7.3.2 (PH1:)\nThere exists a polytope Qd\nλ such that:\n1. The number of integer points in Qd\nλ is equal to md\nλ.\n2. The Ehrhart quasi-polynomial of Qd\nλ coincides with the stretching quasi-\npolynomial ̃md\nλ(n) (Theorem 3.5.1).\n3. The polytope Qd\nλ is given by a separating oracle. Its encoding bitlength\n⟨Qd\nλ⟩is poly(⟨d⟩, ⟨λ⟩, ⟨ρ⟩, ⟨Hy⟩, ⟨H⟩), and the combinatorial size ∥Qd\nλ∥\nis O(poly(ht(λ), ⟨H⟩, ⟨Hy⟩, ⟨ρ⟩)). Here ⟨H⟩, ⟨Hy⟩and ⟨ρ⟩denote the\nbitlengths of H, Hy and ρ (Section 3.4).\nTheorem 7.3.3 (Weak SH:)\n(a) Assuming PH1 (Hypothesis 7.3.1), the saturation index of ̃sλ\nd(n) is at\nmost apoly(∥P λ\nd ∥), for some explicit constant a > 0.\n(b) Assuming PH1 (Hypothesis 7.3.2), the saturation index of ̃md\nλ(n) is at\nmost bpoly(∥Qd\nλ∥), for some explicit constant b > 0.\nThis follows from Theorem 3.3.5.\nTheorem 7.3.4 Assume PH1 (Hypotheses 7.3.1-7.3.2). Then, given d, λ,\nthe specifications of X, H, Hy and ρ, and a relaxation parameter c greater\nthan the explicit bounds on the saturation indices in Theorem 7.3.3, whether\ncλ is an obstruction of degree d can be decided in\npoly(⟨d⟩, ⟨λ⟩, ⟨X⟩, ⟨H⟩, ⟨Hy⟩, ⟨ρ⟩, ⟨c⟩)\ntime.\n114"},{"page":118,"text":"This follows by applying Theorem 3.1.1 to the polytopes P λ\nd and Qd\nλ with\nthe saturation index estimates in Theorem 7.3.3.\n7.4\nRobust obstruction\nWe now define a notion of obstruction that is well behaved with respect to\nrelaxation.\nDefinition 7.4.1 Assume PH1 for both sλ\nd and md\nλ (Hypotheses 7.3.1-7.3.2).\nWe say that Vλ(H) is a robust obstruction for the pair (X, Y ) if one of the\nfollowing hold:\n1. Qd\nλ is empty, and P λ\nd is nonempty.\n2. Both Qdλ and P λ\nd are nonempty, the affine span of Qd\nλ does not contain\nan integer point and the affine span of P λ\nd contains an integer point.\nIf Vλ(H) is a robust obstruction, so is Vlλ(H), for all or most positive\nintegral l, hence the name robust.\nProposition 7.4.2 Assume PH1 for both sλ\nd and md\nλ as above. If Vλ(H)\nis a robust obstruction for the pair (X, Y ), then for some positive integer\nk–called a relaxation parameter–Vkλ(H) is a strong obstruction for (X, Y ).\nIn fact, this is so for most large enough k.\nProof:\n(1) Suppose Qd\nλ is empty, and P λ\nd is nonempty. Let k be a large enough\npositive integer k such that kP λ\nd = P kλ\ndk contains an integer point. Then skλ\nkd\nis nonzero. But mkd\nkλ is zero since Qkd\nkλ = kQd\nλ is empty. Thus kλ is a strong\nobstruction.\n(2) Suppose both Qdλ and P λ\nd are nonempty, the affine span of Qd\nλ does not\ncontain an integer point and the affine span of P λ\nd contains an integer point.\nWe can choose a positive integer k such that the affine span of kQd\nλ = Qdk\ndλ\ndoes not contain an integer point, but kP λ\nd = P kλ\ndk contains an integer point;\nmost large enough k have this property. This means skλ\nkd is nonzero, but mkd\nkλ\nis zero. Thus kλ is a strong obstruction. Q.E.D.\n115"},{"page":119,"text":"7.5\nVerification of robust obstructions\nTheorem 7.5.1 Assume that the singularities of spec(R(X)) are rational.\nAssume PH1 for both sλ\nd and md\nλ as above. Then, given λ, d and the speci-\nfications of ρ : Hy ֒→H and X, whether Vλ(H) is a robust obstruction can\nbe verified in poly(⟨ρ⟩, ⟨Hy⟩, ⟨H⟩, ⟨X⟩, ⟨d⟩, ⟨λ⟩) time. Furthermore, a positive\nintegral relaxation parameter k such that Vkλ(G) is a strong obstruction can\nalso be found in the same time.\nThe crucial result used implicitly here is the quasipolynomiality theorem\n(Theorem 4.1.1) because of which PH1 for both sλ\nd and md\nλ are well defined.\nProof: By linear programming [GLS], whether Qd\nλ is nonempty or not can\nbe determined in poly(⟨Qd\nλ⟩) = poly(⟨ρ⟩, ⟨Hy⟩, ⟨H⟩, ⟨d⟩, ⟨λ⟩) time.\nIf it\nis nonempty, the linear programming algorithm also gives its affine span.\nWhether this contains an integer point can be determined in polynomial\ntime, using the polynomial time algorithm for computing the Smith normal\nform, as in the proof of Theorem 3.1.1.\nSimilarly, whether P λ\nd is nonempty or not can be determined in poly(⟨P λ\nd ⟩) =\npoly(⟨X⟩, ⟨d⟩, ⟨λ⟩) time. If it is nonempty, whether its affine span contains\nan integer point can be determined in polynomial time similarly. Further-\nmore, the algorithm can also be made to return a vertex v of the polytope\nP λ\nd if it is nonempty.\nUsing these observations, whether Vλ(G) is a robust obstruction can be\ndetermined in polynomial time.\nAs far as the computation of the relaxation parameter k is concerned,\nlet us consider the second case in Definition 7.4.1–when both Qd\nλ and P λ\nd are\nnonempty, the affine span of Qd\nλ does not contain an integer point and the\naffine span of P λ\nd contains an integer point–the first case being simpler. In\nthis case, by examining the Smith normal forms of the defining equations\nof the affine spans of P λ\nd and Qd\nλ and the rational coordinates of a vertex\nv ∈P λ\nd , we can find a large enough k so that the affine span of Qkd\nkλ does not\ncontain an integer point, the affine span of P kλ\nkd contains an integer point,\nand P kλ\nkd contains an integer point that is some multiple of v. Q.E.D.\nThe value of the relaxation parameter k computed above is rather con-\nservative.\nOne may wish to compute as small value of k as possible for\nwhich Vkλ(G) is a strong obstruction (though in our application this is not\nnecessary).\nIf SH for holds for the structural constant sλ\nd (cf.\nHypothe-\nsis 3.3.2 and Section 3.5), then we can let k be the smallest integer larger\n116"},{"page":120,"text":"than the saturation index (estimate) for P λ\nd such that affine span of Qkd\nkλ (if\nnonempty) does not contain an integer point (as can be ensured by looking\nat the Smith normal of the defining equations of the affine span).\n7.6\nArithemetic version of the P #P vs. NC prob-\nlem in characteristric zero\nWe now specialize the discussion in the preceding sections in the context\nof the arithmetic form of the P #P vs. NC problem in characteristric zero\n[V]. In concrete terms, the problem is to show that the permanent of an\nn × n complex matrix X cannot be expressed as a determinant of an m × m\ncomplex matrix, whose entries are (possibly nonhomogeneous) linear com-\nbinations of the entries of X.\n7.6.1\nClass varieties\nThe class varieties in this context are as follows [GCT1]. Let Y be an m×m\nvariable matrix, which can also be thought of as a variable l-vector, l = m2.\nLet X be its, say, principal bottom-right n × n submatrix, n < m, which\ncan be thought of as a variable k-vector, k = n2. Let V = Symm(Y ) be the\nspace of homogeneous forms of degree m in the variable entries of Y . The\nspace V , and hence P(V ), has a natural action of G = GL(Y ) = GLl(C)\ngiven by\n(σf)(Y ) = f(σ−1Y ),\nfor any f ∈V , σ ∈G, and thinking of Y as an l-vector. Let W = Symn(X)\nbe the space of homogeneous forms of degree n in the variable entries of\nX. The space W, and also P(W), has a similar action of K = GL(X) =\nGLk(C). We use any entry y of Y not in X as the homogenizing variable\nfor embedding W in V via the map φ : W →V defined by:\nφ(h)(Y ) = ym−nh(X),\n(7.2)\nfor any h(X) ∈W. We also think of φ as a map from P(W) to P(V ).\nLet g = det(Y ) ∈P(V ) be the determinant form, and f = φ(h), where\nh = perm(X) ∈P(W). Let ∆V [g], ∆V [f] ⊆P(V ) be the projective closures\nof the orbits Gg and Gf, respectively, in P(V ). Let ∆W [h] ⊆P(W) be the\nprojective closure of the K-orbit Kh of h in P(W). Then ∆V [g] is called the\nclass variety associated with NC and ∆V [f] the class variety associated with\n117"},{"page":121,"text":"P #P ; ∆W[h] is called the base class variety associated with P #P . (The base\nclass variety is not used in what follows. Rather its variant, called a reduced\nclass variety defined below, will be used.) These class varieties depend on\nthe lower bound parameters n and m. If we wish to make these explicit, we\nwould write ∆V [f, n, m] and ∆V [g, m] instead of ∆V [f] and ∆V [g].\nThe class varieties ∆V [g] = ∆V [g, m] and ∆V [f] = ∆V [f, n, m] are\nG-subvarieties of P(V ), and their homogeneous coordinate rings RV [g] =\nRV [g, m] and RV [f] = RV [f, n, m] have natural degree-preserving G-action.\nIt is conjectured in [GCT1] that, if m = poly(n) and n →∞, then\nf ̸∈∆V [g]; this is equivalent to saying that the class variety ∆V [f, n, m]\ncannot be embedded in the class variety ∆V [g, m] (as a subvariety). This\nimplies the arithmetic form of the P #P ̸= NC conjecture in characteristic\nzero.\n7.6.2\nObstructions\nThe obstruction in this context is defined as follows. A G-module Vλ(G) is\ncalled an obstruction for the pair (f, g) if it occurs in RV [f, n, m]∗\nd but not\nRV [g, m]∗\nd for some d. It is called a strong obstruction if, for some d, it occurs\nin RV [f, n, m]∗\nd but it does not contain (Cg)d as a Gg-submodule, where\n(Cg) ⊆V denotes the one dimensional line corresponding to g, and Gg ⊆G\nis the stabilizer of g = det(Y ) ∈P(V ). If Vλ(G) is a (strong) obstruction\nof degree d, then the size |λ| = dm; hence d is completely determined by λ\nand m.\nExistence of an obstruction or a strong obstruction implies that the\nclass variety ∆V [f, n, m] cannot be embedded in the class variety ∆V [g, m],\nas sought. The main algebro-geometric results of [GCT1, GCT2] suggest\nthat strong obstructions should indeed exist for all n →∞, assuming m =\npoly(n); cf. Section 4, Conjecture 2.10 and Theorem 2.11 in [GCT2]. The\ngoal then is to prove existence of strong obstructions for all n.\nThe definition of a strong obstruction can be simplified further as follows.\nLet X′ denote the set of variables, which consists of the variable entries in\nX and the homogenizing variable y above. Let W ′ = Symm(X′) ⊆V =\nSymm(Y ) be the space of homogeneous forms of degree m in the variables\nof X′.\nWe have a natural action of H = GL(X′) = GLn2+1(C) on W ′\nand hence on P(W ′).\nWe have a natural map φ′ : W →W ′ given by\nφ′(h)(X′) = ym−nh(X). The map φ in (7.2) is φ′ followed by the inclusion\nfrom W ′ to V . We also think of φ′ as a map from P(W) to P(W ′).\n118"},{"page":122,"text":"Let f ′ = φ′(h), for h = perm(X) ∈P(W). Let ∆W ′[f ′] ⊆P(W ′) be\nthe orbit closure of Hf ′.\nIt is an H-subvariety of P(W ′), and hence its\nhomogeneous coordinate ring RW ′[f ′] has the natural degree preserving H-\naction. We call ∆W ′[f ′] the reduced class variety for P #P . It is known (cf.\nTheorem 8.2 in [GCT2]) that Vλ(G) occurs in RV [f]∗\nd iffVλ(H) occurs in\nRW ′[f ′]∗\nd. Here the dominant weight λ of G is considered a dominant weight\nof H by restriction from G to H.\nHence Vλ(G) is a strong obstruction for the pair (f, g), ifffor some d,\nVλ(H) occurs in RW ′[f ′]∗\nd as an H-submodule and Vλ(G) does not contain\n(Cg)d as a Gg-submodule.\nIn particular, we can assume without loss of\ngenerality that the height of the Young diagram for λ is at most n2 + 1;\notherwise Vλ(H) would be zero.\n7.6.3\nRobust obstructions\nIt is known that the stabilizer Gg of g = det(Y ) ∈P(V ) consists of lin-\near transformations in G of the form Y →AY ∗B−1, thinking of Y as an\nm × m matrix, where Y ∗is either Y or Y T , A, B ∈GLm(C). Thus the con-\nnected component of Gg is essentially GLm(C) × GLm(C) ⊆G = GLl(C) =\nGLm2(C). This means the subgroup restriction problem for the embedding\nρ : Gg ֒→G is essentially the Kronecker problem (Problem 1.1.1).\nAssume PH1 (Hypothesis 7.3.2) for the subgroup restriction ρ : Gg ֒→G;\nwhich is essentially PH1 for the Kronecker problem. It now assumes the\nfollowing concrete form. Let md\nλ denote the multiplicity of the Gg-module\n(Cg)d in Vλ(G). Assume that the height of λ is at most n2+1 for the reasons\ngive above.\nHypothesis 7.6.1 (PH1:)\nThere exists a polytope Qd\nλ such that:\n1. The number of integer points in Qd\nλ is equal to md\nλ.\n2. The Ehrhart quasi-polynomial of Qd\nλ coincides with the stretching quasi-\npolynomial ̃md\nλ(n) (cf. Theorem 3.5.1).\n3. The polytope Qd\nλ is given by a separating oracle, and its encoding\nbitlength ⟨Qd\nλ⟩is poly(n, ⟨m⟩, ⟨d⟩, ⟨λ⟩) time.\nWe have to explain why ⟨Qd\nλ⟩is stipulated to depend polynomially on\nn and ⟨m⟩, rather than m. After all, the bitlengths ⟨G⟩, ⟨Gg⟩and ⟨ρ⟩are\n119"},{"page":123,"text":"O(poly(m2)) as per the definitions in Section 3.4. So, as per PH1 for sub-\ngroup restriction in Section 3.4.3, ⟨Qd\nλ⟩should depend polynomially on m.\nWe are stipulating a stronger condition for the following reason. First, as we\nalready mentioned, the above hypothesis is essentially PH1 for the Kronecker\nproblem, which is obtained by specializing PH1 for the plethysm problem\n(Hypothesis 1.6.4). In Hypothesis 1.6.4, the encoding bitlength of the poly-\ntope depends polynomially on the bitlengths of the various partition param-\neters λ, π, μ of the plethysm constant aπ\nλ,μ, but is independent of the rank of\nthe group G therein. (As explained in the remarks after Hypothesis 1.6.4,\nthis is justified because the bound in Theorem 1.6.3 is also independent of\nthe rank of G). For the same reason, the encoding bitlength of the polytope\nhere should be independent of the rank of G (which is m2), but should de-\npend polynomiallly on the total bit length of the partitions parametrizing\nthe representations Vλ(G) and (Cy)d. This is O(n + ⟨m⟩+ ⟨d⟩+ ⟨λ⟩). (Note\nthat the one dimensional representation (Cy)d of Gg is essentially the d-th\npower of the determinant representation of Gg, since the connected compo-\nnent of Gg is isomorphic to GLm(C)×GLm(C). The Young diagram for the\npartition corresponding to the d-th power of the determinant representation\nof GLm(C) is a rectangle of height m and width d. It can be specified by\nsimply giving m and d–this specification has bit length ⟨m⟩+ ⟨d⟩.)\nNext let us specialize PH1 as per Hypothesis 7.3.1. The class variety\n∆V [f] = ∆[f, n, m] will now play the role of X in Hypothesis 7.3.1. But,\nfor the reasons explained in the proof of Proposition 7.6.4 below, we shall\ninstead specialize Hypothesis 7.3.1 to the (simpler) reduced class variety\nZ = ∆W ′[f ′]. It now assumes that following concrete form. Let sλ\nd denote\nthe multiplicity of Vλ(H) in RW ′[f ′]∗\nd. Putting Z in place of X in Hypothe-\nsis 7.3.1, we get:\nHypothesis 7.6.2 (PH1):\nThere exists a polytope P λ\nd such that:\n1. The number of integer points in P λ\nd is equal to sλ\nd.\n2. The Ehrhart quasi-polynomial of P λ\nd coincides with the stretching quasi-\npolynomial ̃sλ\nd(n) (cf. Theorem 3.5.1).\n3. The polytope P λ\nd is given by a separating oracle, and its encoding\nbitlength ⟨P λ\nd ⟩is\npoly(⟨d⟩, ⟨λ⟩, ⟨Z⟩) = poly(⟨d⟩, ⟨λ⟩, n, ⟨m⟩).\n(7.3)\n120"},{"page":124,"text":"Here (7.3) follows because ⟨Z⟩= n+⟨m⟩. To see why, let us observe that\nZ = ∆W ′[f ′] is completely specified once the point f ′ = ym−nh ∈P(W ′) is\nspecified. To specify f ′, it sufficies to specify m, n and the point h ∈P(W).\nIt is known [GCT2] that the point h = perm(X) ∈P(W) is completely\ncharacterized by its stabilizer Kh ⊆K = GL(X) = GLk(C). Furthermore,\nKh is explicitly known [Mc]. It is generated by the linear transformation in\nK of the form X →λXμ−1, thinking of X as an n × n matrix, where λ and\nμ are either diagonal or permutation matrices. So to specifiy h, it suffices\nto specify Kh, K and the embedding ρ′ : Kh ֒→K. The bit length of this\nspecification is O(n) (cf. Section 3.4). To specify f ′, and hence Z, it suffices\nto specify m, n, K, Kh and ρ′. The total bit length of this specification is\nO(n + ⟨m⟩).\nAssume PH1 for both md\nλ and sλ\nd, i.e., Hypotheses 7.6.1 and 7.6.2.\nDefinition 7.6.3 We say that Vλ(G) is a robust obstruction for the pair\n(f, , g) if one of the following hold:\n1. Qd\nλ is empty, and P λ\nd is nonempty.\n2. Both Qd\nλ and P λ\nd are nonempty, the affine span of Qd\nλ does not contain\nan integer point and the affine span of P λ\nd contains an integer point.\nIf the first condition holds, we say that Vλ(G) is a geometric obstruction.\nIf the second condition holds, it is called a modular obstruction.\nProposition 7.6.4 Assume PH1 for both md\nλ and sλ\nd (Hypotheses 7.6.1 and\n7.6.2). If Vλ(G) is a robust obstruction for the pair (f, g), then for some\npositive integral relaxation parameter k, Vkλ(G) is a strong obstruction for\n(f, g). In fact, this is so for most large enough k.\nProof: This essentially follows from Proposition 7.4.2. It only remains to\nclarify why we can use PH1 for the reduced class variety ∆W ′[f ′]–as we are\ndoing here– in place of PH1 for the class variety ∆V [f]. This is because,\nas already mentioned, Vλ(G) occurs in RV [f]∗\nd iffVλ(H) occurs in RW ′[f ′]∗\nd.\nQ.E.D.\n7.6.4\nVerification of robust obstructions\nTheorem 7.6.5 Assume that the singularities of spec(RW ′[f ′]) are ratio-\nnal. Assume PH1 for both md\nλ and sλ\nd as above (Hypotheses 7.6.1 and 7.6.2).\n121"},{"page":125,"text":"Then, given n, m, λ and d, whether Vλ(H) is a robust obstruction can be\nverified in poly(n, ⟨m⟩, ⟨d⟩, ⟨λ⟩) time. Furthermore, a positive integral relax-\nation parameter k such that kλ is a strong obstruction can also be computed\nin this much time.\nOnce n and m are specified, the various class varieties and K, Kh, ρ′, G, Gg, ρ\nabove are automatically specified implicitly.\nProof: This follows from Theorem 7.5.1; cf. also the remark following its\nproof. Q.E.D.\nTheorem 7.3.4 can be similarly specialized in this context; we leave that\nto the reader.\n7.6.5\nOn explicit construction of obstructions\nTheorem 7.6.6 Assume that m = poly(n) or even 2polylog(n), and:\n1. (RH) [Rationality Hypothesis]: The singularities of spec(RW ′[f ′]) are\nrational.\n2. PH1 for both md\nλ and sλ\nd (Hypotheses 7.6.1 and 7.6.2).\n3. OH [Obstruction Hypothesis]: For every (large enough) n, there exists\nλ of poly(n) bit length such that |λ| is divisible by m and one of the\nfollowing holds (with d = |λ|/m):\n(a) Qd\nλ is empty, and P λ\nd is nonempty.\n(b) Both Qdλ and P λ\nd are nonempty, the affine span of Qd\nλ does not\ncontain an integer point and the affine span of P λ\nd,x contains an\ninteger point.\nThen there exists an explicit family {λn} of robust obstructions.\nHere we say that {λn} is an explicit family of robust obstructions if each\nλn is short and easy to verify. Short means ⟨λn⟩is O(poly(n)). Easy to verify\nmeans whether λn is a robust obstruction can be verified in O(poly(n)) time.\nThe poly(n) bound here and in OH is meant to be independent of m,\nas long as m << 2n; i.e., it should hold even when m = 2polylog(n). In\nother words, the family {λn} should continue to remain an explicit robust\nobstruction family, as we vary m over all values ≤2polylog(n), and perhaps\n122"},{"page":126,"text":"even values ≤2o(n), but will cease to be an obstruction family for some large\nenough m = 2Ω(n). This is an important uniformity condition.\nProof: OH basically says that there exists a short robust obstruction λn for\nevery n. By Theorem 7.6.5, it is easy to verify. Q.E.D.\n7.6.6\nWhy should robust obstructions exist?\nThe main question now is: why should OH hold?\nThat is, why should\n(short) robust obstructions exist?\nAs we already mentioned, the results in [GCT1, GCT2] indicate that\nstrong obstructions should exist for every n, assuming m = poly(n). We\nshall give a heuristic argument for existence of robust obstructions assuming\nthat strong obstructions exist. This will crucially depend on the following SH\nfor md\nλ, which is essentially SH for the Kronecker problem (i.e. specialization\nof Hypothesis 1.6.5 to the Kronecker problem), good experimental evidence\nfor which is provided in [BOR].\nHypothesis 7.6.7 (SH:) (a): The saturation index of ̃md\nλ(k) is bounded by\na polynomial in m. (Observe that the rank of G is poly(m) and the height of\nλ is at most n2 + 1). (b): The quasi-polynomial ̃md\nλ(n) is strictly saturated,\ni.e. the saturation index is zero, for almost all λ (and d).\nIf Vλ(G) is a strong obstruction, sλ\nd is nonzero but md\nλ is zero. Thus,\nassumming PH1, there are three possibilities:\n1. Qd\nλ is empty, and P λ\nd is nonempty and contains an integer point.\n2. Both Qdλ and P λ\nd are nonempty, the affine span of Qd\nλ does not contain\nan integer point and P λ\nd contains an integer point.\n3. Both Qd\nλ and P λ\nd are nonempty. The affine span of Qd\nλ contains an\ninteger point, but Qd\nλ does not. And P λ\nd contains an integer point.\nIn the first two cases, λ is a robust obstruction. As per SH (Hypothe-\nsis 7.6.7), for almost all λ, the Ehrhart quasipolynomial of Qd\nλ is saturated:\nthis means (cf. the proof of Theorem 3.1.1), if the affine span of Qd\nλ contains\nan integer point then Qd\nλ also contains an integer point. And hence, with\na high probability, the third case should not occur. In other words, strong\nobstructions can be expected to be robust with a high probability.\n123"},{"page":127,"text":"Let us call a strong obstruction λ fragile if it is not robust; this means\nthe affine span of Qd\nλ contains an integer point, but Qd\nλ does not. By SH\n(Hypothesis 7.6.7), if λ is fragile, then for some k = poly(m), Qkd\nkλ contains\nan intger point, and hence, kλ is not obstruction. Thus fragile obstructions\nare close to not being obstructions, and furthermore, are expected to be\nrare, as argued above. This is why we are focussing on robust obstructions.\nIt may be remarked that the only SH needed in the argument above\nis the one (Hypothesis 7.6.7) for the structural constant md\nλ.\nThis is a\nspecial case of the SH for the subgroup restriction problem (cf. Section 3.4)\nspecialized to the embedding Gg ֒→G. In particular, we do not need SH\nfor the structural constant sλ\nd; i.e., for the more difficult decision problem in\ngeometric invariant theory (cf. Problem 1.1.4 and Section 3.5).\n7.6.7\nOn discovery of robust obstructions\nIt may be conjectured that not just the verification (cf.\nTheorem 7.6.5)\nbut also the discovery of robust obstructions is easy for the problem under\nconsideration. In this section we shall give an argument in support of this\nconjecture for geometric (robust) obstructions (which may be conjectured to\nexist in the problem under consideration). For this we need to reformulate\nthe notions of strong and robust obstructions (Definition 7.6.3) as follows.\nLet TZ be the set of pairs (d, λ) such that sλ\nd is nonzero and SZ the set\nof pairs (d, λ) such that md\nλ is nonzero.\nProposition 7.6.8 Assuming PH1 above (Hypotheses 7.6.1 and 7.6.2), TZ\nand SZ are finitely generated semigroups with respect to addition.\nThese semi-groups are analogues of the Littlewood-Richardson semigroup\n(Section 2.2.2) in this setting.\nProof: The proof is similar to that for the Littlewood-Richardson semigroup\n[Z].\nFor given d and λ, the polytope P λ\nd in PH1 for sλ\nd (Hypothesis 7.6.2) has\na specification of the form\nAx ≤b\n(7.4)\nwhere A depends only the variety Z = ∆W ′[f ′], but not on d or λ, and\nb depends homogeneously and linearly on d and λ. Let P be the polytope\ndefined by the inequalities (7.4) where both d and λ are treated as variables.\nThen P is a polyhedral cone (through the origin) in the ambient space\n124"},{"page":128,"text":"containing P with the coordinates x, d and λ. Let PZ be the set of integer\npoints in P. It is a finitely generated semigroup since P is a polyhedral cone.\nLet TR be the orthogonal projection of P on the hyperplane corresponding\nto the coordinates d and λ. Now TZ is simply the projection of PZ. Hence\nit is a finitely generated semigroup as well.\nThe proof for SZ is similar, with SR defined similarly. Q.E.D.\nThe polyhedral cones TR and SR here are analogues of the Littlewood-\nRichardson cone (Section 2.2.2) in this setting. Note that (d, λ) ∈TR iffP λ\nd\nis nonempty; similarly for SR.\nA Weyl module Vλ(G) is a strong obstruction for the pair (f, g) of degree\nd iff(d, λ) occurs in TZ but not in SZ. It is a robust obstruction iffit occurs\nin TR but not in SZ. It is a geometric obstruction iffit occurs in TR but not\nin SR. It is a modular obstruction iffit occurs in TR and also in SR but not\nin SZ.\nAssuming PH1 (Hypothesis 7.6.2), whether (d, λ) belongs to TR can be\ndetermined in polynomial time by linear programming, since (d, λ) ∈TR\niffP λ\nd is nonempty. Similarly, assuming PH1 (Hypothesis 7.6.1), whether\n(d, λ) ∈SR can be determined in polynomial time.\nThe following is a stronger complement to PH1.\nHypothesis 7.6.9 (PH1*)\nWhether TR\\SR is nonempty can be determined in polynomial time; i.e.,\npoly(n, ⟨m⟩) time. If so, the algorithm can also output (d, λ) ∈TR \\ SR of\npolynomial bit length.\nProposition 7.6.10 Assuming PH1*, given n and m, the problem of decid-\ning if a geometric obstruction exists for the pair (f, g), and finding one if one\nexists, belongs to the complexity class P; i.e., it can be done in poly(n, ⟨m⟩)\ntime.\nThis immediately follows from Hypothesis 7.6.9 since (d, λ) is a geometric\nobstruction iff(d, λ) ∈TR \\ SR.\nHypothesis 7.6.9 is supported by the following:\nProposition 7.6.11 Assuming PH1 (Hypotheses 7.6.1 and 7.6.2), Hypoth-\nesis 7.6.9 holds if TR and SR have polynomially many explicitly given con-\nstraints with the specification of polynomial bit length; here polynomial means\npoly(n, ⟨m⟩).\n125"},{"page":129,"text":"The proposition holds even if the polytope SR has exponentially many\nconstraints, as long as it is given by a separation oracle that works in poly-\nnomial time.\nProof: It suffices to check if SR satisfies each constraint of TR. This can be\ndone in polynomial time using the linear programming algorithm in [GLS].\nSpecifically, let l(y) ≥0 be a constraint of TR. Then we just need to minimize\nl(y) on SR and check if the minimum exceeds zero. Q.E.D.\nBut this method does not work when the number of constraints of TR is\nexponential, as expected in the context of the lower bound problems under\nconsideration. In fact, no generic black-box-type algorithm, like the one in\n[GLS] based on just a membership or separation oracle for TR, can be used\nto prove (4) when the number of constraints of TR is exponential.\nFortunately, this is not a serious problem. A basic principle in combi-\nnatorial optimization, as illustrated in [GLS], is that a complexity theoretic\nproperty that holds for polytopes with polynomially many constraints will\nalso hold for polytopes with exponentially many constraints, provided these\nconstraints are sufficiently well-behaved.\nFor example, Edmond’s perfect\nmatching polytope for nonbipartite graphs has complexity-theoretic proper-\nties similar to the perfect matching polytope for bipartite graphs, though it\ncan have exponentially many constraints. We have already remarked that\nTR and SR are analogues of the Littlewood-Richardson cones. The facets of\nthe Littlewood-Richardson cone have a very nice explicit description [Kl, Z].\nThe cones TR, SR here are expected to have similar nice explicit descrip-\ntion.\nThis is why Hypothesis 7.6.9 can be expected to hold even if the\nnumber of constraints of TR is exponential, just as it holds even when SR\nhas exponentially many constraints. But a polynomial-time algorithm as in\nHypothesis 7.6.9 would have to depend crucially on the specific nature of\nthe facets (constraints) of TR in the spirit of the linear-programming-based\nalgorithm for the construction of a maximum-weight perfect matching in\nnonbipartite graphs [Ed], where too the number of constraints is exponen-\ntial but the algorithm still works because of the structure theorems based\non the specific nature of the constraints.\n7.7\nArithmetic form of the P vs NP problem in\ncharacteristic zero\nWe turn now to the arithemetic form of the P vs. NP problem in character-\nistic zero. The arguments are essentially verbatim translations of those for\n126"},{"page":130,"text":"the arithmetic form of the P #P vs. NC problem in the preceding section.\nHence we shall be brief.\nIn the preceding section h(X) was perm(X) and g(Y ) was det(Y ). Now\nh(X) and g(Y ) would be explicit (co)-NP-complete and P-complete func-\ntions E(X) and H(Y ) constructed in [GCT1]. They can be thought of as\npoints in suitable W = Symk(X) and V = Syml[Y ], k = O(n2), l = O(m2),\nwith the natural action of GL(X) and G = GL(Y ), where n denotes the\nnumber of input parameters and m denotes the circuit size parameter in\nthe lower bound problem. These functions are extremely special like the\ndeterminant and the permanent in the sense that they are “almost” char-\nacterized by their stabilizers as explained in [GCT1]–and this is enough for\nour purposes.\nWe again have a natural embedding φ : P(W) →P(V ), which lets us\ndefine f = φ(h). The class variety for NP is defined to be ∆V [f] ⊆P(V ),\nthe projective closure of the orbit Gf. The class variety for P is ̃∆V [g] ⊆\nP(V ), which is defined to be the projective closure of G[g], where [g] denotes\nthe set of points in P(V ) that are stabilized by Gg ⊆G, the stabilizer of g.\nAn explicit description of Gg is given in [GCT1]; cf. Section 7 therein. To\nshow P ̸= NP in characteristic zero, it suffices to show that ∆V [f] is not a\nsubvariety of ̃∆V [g] for all large enough n, if m = poly(n) (cf. Conjecture\n7.4. in [GCT1]). For this, in turn, it suffices to show existence of strong\nobstructions, defined very much as in Section 7.6, for all n, assumming\nm = poly(n).\nWe can then formulate PH1 for the new h(X) and g(Y ) just as in Hy-\npotheses 7.6.1 and 7.6.2, and the notion of a robust obstruction as in Defi-\nnition 7.6.3. We then have:\nTheorem 7.7.1 (Verification of obstructions)\nAnalogues of Theorems 7.6.5 and 7.6.6 holds for h(X) = E(X) and\ng(Y ) = H(Y ).\nFurthermore, even discovery of robust obstructions can be conjectured\nto be easy (poly-time)–this would follow from the obvious analogue of Hy-\npothesis 7.6.9 here.\nHeuristic argument for existence of robust obstructions is very similar\nto the one in Section 7.6.6. It needs SH for the special case of the subgroup\nrestriction problem for the embedding Gg ֒→G. The group Gg, as described\nin [GCT1], is a product of some copies of the algebraic torus and the sym-\n127"},{"page":131,"text":"metric group. The subgroup restriction problem in this case is akin to but\nharder than the plethysm problem.\n128"},{"page":132,"text":"Bibliography\n[BGS]\nT. Baker, J. Gill, R. Soloway, Relativization of the P =?NP ques-\ntion, SIAM J. Comput. 4, 431-442, 1975.\n[BBCV] M. Baldoni, M. Beck, C. Cochet, M. Vergne, Volume computation\nfor polytopes and partition functions for classical root systems,\nmath.CO/0504231, Apr, 2005.\n[Bar]\nA. Barvinok, A polynomial time algorithm for counting integral\npoints in polyhedra when the dimension is fixed. Math. Oper. Res.,\n19 (4): 769-779, 1994.\n[BDR]\nThe Frobenius problem, rational polytopes, and Fourier-Dedekind\nsums, Journal of number theory, vol. 96, issue 1, 2002.\n[BBD]\nA.\nBeilinson,\nJ.\nBernstein,\nP.\nDeligne,\nFaisceaux\npervers,\nAst ́erisque 100, (1982), Soc. Math. France.\n[Bl]\nP. 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Harris, Representation theory, A first course,\nSpringer, 1991.\n[KM]\nM. Kapovich, J. Millson, Structure of the tensor product semi-\ngroup, math.RT/0508186.\n[GCTabs] K. Mulmuley, M. Sohoni, Geometric complexity theory, P vs. NP\nand explicit obstructions, in “Advances in Algebra and Geometry”,\nEdited by C. Musili, the proceedings of the International Confer-\nence on Algebra and Geometry, Hyderabad, 2001.\n[GCTflip] K. Mulmuley, On P vs. NP, geometric complexity theory, and the\nflip I: a high level view, Technical report TR-2007-16, computer\nscience department, The university of Chicago, September 2007.\nAvailable at http://ramakrishnadas.cs.uchicago.edu.\n[GCT1] K. Mulmuley, M. Sohoni, Geometric complexity theory I: an ap-\nproach to the P vs. NP and related problems, SIAM J. Comput.,\nvol 31, no 2, pp 496-526, 2001.\n[GCT2] K. Mulmuley, M. Sohoni, Geometric complexity theory II: towards\nexplicit obstructions for embeddings among class varieties, SIAM\nJ. 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Valiant, The complexity of computing the permanent, Theoret-\nical Computer Science 8, pp 189-201, 1979.\n[Z]\nA.\nZelevinsky,\nLittlewood-Richardson\nsemigroups,\narXiv:math.CO/9704228 v1 30 Apr 1997.\n136"}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"of the P ̸= NP conjecture in characteristic zero is intimately linked to the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"Kronecker problem, n = 2 . . . . . . . . . . . . . . . . . . . .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"complexity theory, such as the P ̸=?NP problem in characteristic zero, to","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"ity of the P ̸= NP conjecture in characteristic zero is intimately linked","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"Equivalently [FH], let H = GLn(C) × GLn(C) and ρ : H →G =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"GL(Cn ⊗Cn) = GLn2(C) the natural embedding. Then kπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"G = GLn2(C).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"λ,μ. This is the multiplicity of the irreducible representation Vπ(H) of H =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"GLn(C) in the irreducible representation Vλ(G) of G = GL(Vμ), where Vμ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"module via the representation map ρ : H →G = GL(Vμ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"action. Let R = ⊕dRd be the homogeneous coordinate ring of X. Assume","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"is given in the form of a triple x = (λ, μ, π). Here the partition λ is specified","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"(a) Let h = dim G + htλ + htπ, where dim(G) is the dimension of the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"puting any of them, assuming P ̸= NP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"concerns the following special case of Problem 1.1.3, with G = H × H, the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"α,β(n) = cnλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"that f(n) = fj(n) if n = j mod l. Here l is supposed to be the smallest such","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"such that f(n) = fj(n) if n = j mod l. The index of f, index(f), is defined","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"If f(n) is identically zero, we let index(f) = 0. If f(1) ̸= 0, then clearly","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"index(f) = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"α,β(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"(and h0 = 1).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"Let F(t) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"F(t) = hdtd + · · · + h0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"i=0(1 −tai)di ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"where (1) h0 = 1, and hi’s are nonnegative integers, (2) ai’s and di’s are","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"i di = d + 1, where d = max deg(fj(n)) is the degree","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"If F(t) has a positive form with a0 = 1, then f(n) is strictly saturated","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"form, with a0 = 1, of modular index bounded by a constant depending only","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"forms of Problems 1.1.1, 1.1.2, 1.1.3, and 1.1.4, with X = G/P or a class","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"efficient (type A), then sie(P) = 0, by the saturation theorem [KT1], and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"pie(P) = 0, by PH2 (Hypothesis 1.2.3). For the polytopes P that would","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"in Problems 1.1.1, 1.1.2, 1.1.3 and 1.1.4, with X = G/P or a class variety.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"lem 1.1.4, with X = G/P or a class variety. As for Littlewood-Richardson","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"λ,μ(n) = anπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"λ,μ(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"(c) There exist graded, normal C-algebras S = S(aπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"λ,μ) = ⊕nSn, and T =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"λ,μ) = ⊕nTn such that:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"2. T = SH, the subring of H-invariants in S, where H = GLn(C) as in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"λ,μ(t) = h0 + h1t + · · · + hdtd","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"j d(j) = d + 1, where d is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"λ,μ(n), h0 = 1, and hi’s are nonnegative","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"λ,μ) = Proj(S(aπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"irreducible representation Vπ(H) of H = GLn(C) occurs in the irreducible","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"representation Vλ(G) of G = GL(Vμ(H)) with nonzero multiplicity. Then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"For every (λ, μ, π) there exists a polytope P = P π","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"λ,μ = φ(P),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"(n = 2) of the Kronecker problem analysed in [BOR], the saturation index","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"where h = dim G + htλ + htπ.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"λ,μ is at t = 1–i.e. if aμ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"the Kronecker problem when n = 2; the earlier known formulae [RW, Ro]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"indices of the Kronecker coefficient, for n = 2, are very small, and almost","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"constants in Problems 1.1.3, and 1.1.4, with X = G/P or a class variety.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"and positivity hypotheses for Problems 1.1.3 and 1.1.4 (with X = G/P or","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"λ. The meaning depends on H. Thus if H = GLn(C), λ is a partition and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"Vλ(H) the Weyl module indexed by λ, if H = Sm, then λ is a partition of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"size |λ| = m, and Vλ(H) the Specht module indexed by λ, and so on.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"An algorithm is called strongly polynomial [GLS], if given an input x =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"but does not depend ⟨x⟩, where ⟨x⟩= P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"f(x) = f(x1, · · · , xk) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"and y runs over all tuples such that ⟨y⟩= poly(⟨x⟩). The formula (2.1) is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"and Littlewood-Richardson function c(α, β, λ) = cλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"α,β, assuming P ̸= NP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq65","equation_number":null,"raw_text":"function of P: i.e., χP (y) = 1, if y ∈P, and zero otherwise. We say that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq66","equation_number":null,"raw_text":"f = f(x) = f(x1, . . . , xk) has a convex #P-formula if, for every x ∈Zk,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq67","equation_number":null,"raw_text":"f(x) = φ(Px),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq68","equation_number":null,"raw_text":"f(x) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq69","equation_number":null,"raw_text":"α,β = φ(P λ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq70","equation_number":null,"raw_text":"c(α, β, λ) = cλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq71","equation_number":null,"raw_text":"When G = GLn(C), the facets of LRR(G) have an explicit description by","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq72","equation_number":null,"raw_text":"(α, β, λ) ∈Rr such that α = λ(A + B), β = λ(A), λ = λ(B) for some","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq73","equation_number":null,"raw_text":"Littlewood-Richardson semisgroup LRr = LR(GLr(C)): HEr ∩P 3","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq74","equation_number":null,"raw_text":"r = LRr,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq75","equation_number":null,"raw_text":"Given that the answer is yes for the closely related LRr = LR(GLr(C))","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq76","equation_number":null,"raw_text":"polynomial time computable. If χP(y) = 0 for some y ∈Rr, it is natural to","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq77","equation_number":null,"raw_text":"assume that the hyperplane is given in the form l = 0, where a linear function","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq78","equation_number":null,"raw_text":"a separation oracle that works in polynomial time with ⟨Px⟩= poly(⟨x⟩).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq79","equation_number":null,"raw_text":"Problem 1.1.3 and Problem 1.1.4, with X = G/P or a class variety therein.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq80","equation_number":null,"raw_text":"Let fP(n) = φ(nP) be the Ehrhart","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq81","equation_number":null,"raw_text":"fP (n) = fi,P(n) if n = i modulo l(P). Let FP (t) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq82","equation_number":null,"raw_text":"oracle works in polynomial time. Furthermore, if index(fP) ̸= 0 (i.e. if P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq83","equation_number":null,"raw_text":"O(poly(⟨x⟩)) time, where ⟨x⟩is the bitlength of x, and ⟨Px⟩= O(poly(⟨x⟩)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq84","equation_number":null,"raw_text":"C and an integral vector d such that span(P) is defined by Cx = d. This","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq85","equation_number":null,"raw_text":"If P is empty, index(fP) = 0. So assume that it is nonempty. Let ̄C be","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq86","equation_number":null,"raw_text":"the Smith normal form of C; i.e., ̄C = ACB for some unimodular matrices","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq87","equation_number":null,"raw_text":"z = B−1x, span(P) is specified by the linear system ̄Cz = ̄d = Ad. The","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq88","equation_number":null,"raw_text":"̄cizi = ̄di,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq89","equation_number":null,"raw_text":"Claim 3.1.2 index(fP ) = ̃c and fi,P(n) is not an identically zero polyno-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq90","equation_number":null,"raw_text":"Proof of the claim: Indeed, nP = {nz | z ∈P} contains no integer point","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq91","equation_number":null,"raw_text":"Hence, it is easy to see that FP (t) = F ̄P (t ̃c), where","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq92","equation_number":null,"raw_text":"F ̄P (x) is the Ehrhart series of the dilated polytope ̄P = ̃cP. By eq.(3.1),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq93","equation_number":null,"raw_text":"zi = ̄di( ̃c/ ̄ci),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq94","equation_number":null,"raw_text":"Clearly, ̃c divides the least period l(P) of fP, and l( ̄P) = l(P)/ ̃c is the period","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq95","equation_number":null,"raw_text":"with zi = ai/b, for some integers ai’s and b such that b = j modulo l( ̄P).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq96","equation_number":null,"raw_text":"(b): Let s = sie(P) be the given saturation index estimate. This means","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq97","equation_number":null,"raw_text":"is divisible by index(fP ) (by letting n = c −s). This can be checked in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq98","equation_number":null,"raw_text":"almost always, by the argument in (b) with s = 0 and c = 1, “Probably","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq99","equation_number":null,"raw_text":"Proposition 3.1.3 The rational function FP (t) = F ̄P (t ̃c), where F ̄P (x) is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq100","equation_number":null,"raw_text":"the Ehrhart series of the dilated polytope ̄P = ̃cP, and ̃c is the index of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq101","equation_number":null,"raw_text":"where A is an m × n matrix, b an m vector and m = poly(n), then the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq102","equation_number":null,"raw_text":"where A is an m × n matrix, m possibly exponential. Let ∥P∥= n + ψ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq103","equation_number":null,"raw_text":"fP(n) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq104","equation_number":null,"raw_text":"FP (t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq105","equation_number":null,"raw_text":"j=0(1 −taj),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq106","equation_number":null,"raw_text":"where h0 = 1, hi’s are nonnegative, and aj is the least positive integer","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq107","equation_number":null,"raw_text":"the following lemma to FP (t), since ⟨aj⟩= O(poly(∥P∥)). Q.E.D.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq108","equation_number":null,"raw_text":"F(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq109","equation_number":null,"raw_text":"j=0(1 −taj),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq110","equation_number":null,"raw_text":"where h0 = 1, hi’s are nonnegative, and aj’s are positive and relatively","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq111","equation_number":null,"raw_text":"Let a = max{aj}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq112","equation_number":null,"raw_text":"Proof: Let g(n) be the quasi-polynomial whose generating function G(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq113","equation_number":null,"raw_text":"j=0(1 −taj). It is known that this is the Ehrhart quasipoly-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq114","equation_number":null,"raw_text":"ajxj = 1, xj > 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq115","equation_number":null,"raw_text":"ajxj = m","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq116","equation_number":null,"raw_text":"a0a1a2(a0 + a1 + a2) = O(poly(a)),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq117","equation_number":null,"raw_text":"assumming that a0 ≤a1 . . .. Hence, s(g) = O(poly(a)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq118","equation_number":null,"raw_text":"j=0(1 −taj). Then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq119","equation_number":null,"raw_text":"s(gi) ≤i + s(g) = O(poly(a, n)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq120","equation_number":null,"raw_text":"Since, hi’s in (3.6) are nonnegative, s(f) = max s(gi). The result follows.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq121","equation_number":null,"raw_text":"where we have allowed strict inequalities. The function FP (n) = φ(nP), the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq122","equation_number":null,"raw_text":"If s(P) = sie(P), then this guarantee holds, as can be seen from the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq123","equation_number":null,"raw_text":"G = GLn(C), by the saturation theorem, a triple (α, β, λ) of dominant","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq124","equation_number":null,"raw_text":"ciλi = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq125","equation_number":null,"raw_text":"When G = SLn(C), the modular complex is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq126","equation_number":null,"raw_text":"Corollary 3.2.3 says that, given x = (α, β, λ), whether x ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq127","equation_number":null,"raw_text":"f(x) = f(α, β, λ) = cλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq128","equation_number":null,"raw_text":"α, β, λ, ∥x∥is the total number of coordinates of α, β and λ, and rank(x) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq129","equation_number":null,"raw_text":"5. (PH1?): Is there a polytope Px, for every x, with ⟨Px⟩= O(poly(⟨x⟩))","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq130","equation_number":null,"raw_text":"and ∥Px∥= O(poly(∥x∥)), such that ̃f(x, n) = fPx(n)?","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq131","equation_number":null,"raw_text":"In the rest of this paper, we study these questions when f = f(x) is a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq132","equation_number":null,"raw_text":"Hypothesis 3.3.1 (PH1) Let f = f(x) be the function associated with a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq133","equation_number":null,"raw_text":"f(x) = φ(Px),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq134","equation_number":null,"raw_text":"2. ⟨P⟩= O(poly(P)) and ∥P∥= O(poly(∥x∥)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq135","equation_number":null,"raw_text":"F(x, t) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq136","equation_number":null,"raw_text":"combinatorial bit lengths ∥λ∥, ∥π∥. We let ∥H∥= ⟨H⟩and ∥ρ∥= ⟨ρ⟩, since","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq137","equation_number":null,"raw_text":"simple components. With this terminology, we let f(x) = mπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq138","equation_number":null,"raw_text":"H = GLn(C). Its bitlength ⟨H⟩is n. The specification [ρ] of the repre-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq139","equation_number":null,"raw_text":"sentation map ρ : H →G = GL(Vμ(H)) consists of just the partition μ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq140","equation_number":null,"raw_text":"specified in terms of its nonzero parts. Its bitlength ⟨ρ⟩= ⟨μ⟩. The ranks","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq141","equation_number":null,"raw_text":"ρ : H →G = GL(V ) is an explicit polynomial homomorphism. The iden-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq142","equation_number":null,"raw_text":"V its rational representation, and G = GL(V ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq143","equation_number":null,"raw_text":"Let Vλ(G) = Symd(V ),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq144","equation_number":null,"raw_text":"the bitlength of this specification. Thus, if H = SLn(C), then ⟨H⟩= O(n).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq145","equation_number":null,"raw_text":"(3) If H is the alternating group An, we only specify n. Let ⟨H⟩= n.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq146","equation_number":null,"raw_text":"In this case the specification of ρ : H →G = GL(V ) is a pair [ρ] = ([H], [V ])","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq147","equation_number":null,"raw_text":"(1) If H is a semisimple, simply connected Lie group, and V = Vμ(H) its","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq148","equation_number":null,"raw_text":"Thus [V ] = μ, and its bitlength ⟨V ⟩is the total bitlength of all","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq149","equation_number":null,"raw_text":"(2) If H = Sn, and V = Sγ its irreducible representation (Specht module),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq150","equation_number":null,"raw_text":"be the bitlength of this partition, and ∥V ∥= ⟨V ⟩.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq151","equation_number":null,"raw_text":"of this specification; it is O(poly(n, k, ⟨p⟩)). We let ∥V ∥= ⟨V ⟩. More gener-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq152","equation_number":null,"raw_text":"of G. When G = GLm(C), λ is a partition, which we specify by only giving","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq153","equation_number":null,"raw_text":"Let f(x) = mπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq154","equation_number":null,"raw_text":"λ as above, with x = ([H], [ρ], [λ], [π]). Here λ is the dominant","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq155","equation_number":null,"raw_text":"̃f(x, n) = ̃mπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq156","equation_number":null,"raw_text":"λ(n) = mnπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq157","equation_number":null,"raw_text":"λ (t) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq158","equation_number":null,"raw_text":"(c) There exist graded, normal C-algebras S = S(mπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq159","equation_number":null,"raw_text":"λ) = ⊕nSn and T =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq160","equation_number":null,"raw_text":"λ) = ⊕nTn such that:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq161","equation_number":null,"raw_text":"2. T = SH, the subring of H-invariants in S.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq162","equation_number":null,"raw_text":"λ (t) = h0 + h1t + · · · + hdtd","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq163","equation_number":null,"raw_text":"j d(j) = d+1, where d is the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq164","equation_number":null,"raw_text":"degree of the quasi-polynomial, h0 = 1, and hi’s are nonnegative integers.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq165","equation_number":null,"raw_text":"λ) = Proj(S(mπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq166","equation_number":null,"raw_text":"f(x) = mπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq167","equation_number":null,"raw_text":"X = Gvλ ∼= G/Pλ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq168","equation_number":null,"raw_text":"where the P = Pλ = Gvλ is the parabolic stabilizer of vλ. We have a natural","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq169","equation_number":null,"raw_text":"tuple ([X], d, [π]) = ([H], [ρ], [λ], d, [π]), where [π] is the specification of π as","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq170","equation_number":null,"raw_text":"phism as in Section 3.4. Let V = Vλ(G) be an irreducible representation of G","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq171","equation_number":null,"raw_text":"be connected. We associate with f(x) = sπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq172","equation_number":null,"raw_text":"̃f(x, n) = ̃sπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq173","equation_number":null,"raw_text":"d(n) = snπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq174","equation_number":null,"raw_text":"degree nd. Let S(t) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq175","equation_number":null,"raw_text":"(c) There exist graded, normal C-algebras S = S(sπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq176","equation_number":null,"raw_text":"d) = ⊕nSn and T =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq177","equation_number":null,"raw_text":"d) = ⊕nTn such that:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq178","equation_number":null,"raw_text":"2. T = SH, the subring of H-invariants in S.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq179","equation_number":null,"raw_text":"d (t) = h0 + h1t + · · · + hktk","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq180","equation_number":null,"raw_text":"j k(j) = k + 1, where k is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq181","equation_number":null,"raw_text":"d(n), h0 = 1, and hi’s are nonnegative","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq182","equation_number":null,"raw_text":"Let T = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq183","equation_number":null,"raw_text":"Let Z = Proj(T).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq184","equation_number":null,"raw_text":"Let hT (n) = dim(Tn) be its Hilbert function,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq185","equation_number":null,"raw_text":"polynomial hT (n) is convex. This means there exists a polytope P = PT","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq186","equation_number":null,"raw_text":"Let us illustrate this definition with an example. Let X ∼= G/Pλ be as in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq187","equation_number":null,"raw_text":"that it has a compact specification: namely [X] = λ. Since singularities","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq188","equation_number":null,"raw_text":"with h0 = 1 and hi’s nonnegative. This is so because R is Cohen-Macauley","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq189","equation_number":null,"raw_text":"Now let s = sπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq190","equation_number":null,"raw_text":"compact specification [X] as above. Let T = T(s) be the ring associated","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq191","equation_number":null,"raw_text":"Let Z = Z(s) = Proj(T).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq192","equation_number":null,"raw_text":"specification [Z] = ([X], d, π). Let ⟨Z⟩be its bitlength.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq193","equation_number":null,"raw_text":"problem, s = sπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq194","equation_number":null,"raw_text":"1 = mπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq195","equation_number":null,"raw_text":"FP (t) = h(d)td + · · · + h(0)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq196","equation_number":null,"raw_text":"1. the value of a Kazhdan-Lusztig polynomial at q = 1, [KL1];","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq197","equation_number":null,"raw_text":"2. the values at q = 1 of the well behaved special cases of the parabolic","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq198","equation_number":null,"raw_text":"Let R = ⊕kRd be a normal graded C-algebra with an action of a reduc-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq199","equation_number":null,"raw_text":"the connected component of H containing the identity. Let HD = H/H0 be","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq200","equation_number":null,"raw_text":"module Vπ = Vπ(H0), an H-module with trivial action of HD. Let sπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq201","equation_number":null,"raw_text":"d (t) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq202","equation_number":null,"raw_text":"(c) There exist graded, normal C-algebras S = S(sπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq203","equation_number":null,"raw_text":"d) = ⊕nSn and T =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq204","equation_number":null,"raw_text":"d) = ⊕nTn such that:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq205","equation_number":null,"raw_text":"2. T = SH, the subring of H-invariants in S.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq206","equation_number":null,"raw_text":"d (t) = h0 + h1t + · · · + hktk","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq207","equation_number":null,"raw_text":"j k(j) = k + 1, where k is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq208","equation_number":null,"raw_text":"d(n), h0 = 1, and hi’s are nonnegative","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq209","equation_number":null,"raw_text":"B = RCd = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq210","equation_number":null,"raw_text":"π = Vπ∗. By Borel-Weil","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq211","equation_number":null,"raw_text":"Cπ∗= ⊕n≥0V ∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq212","equation_number":null,"raw_text":"nπ = ⊕n≥0Vnπ∗,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq213","equation_number":null,"raw_text":"x(b ⊗c) = (x · b) ⊗(x−1 · c),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq214","equation_number":null,"raw_text":"S = (B ⊗Cπ∗)C∗= ⊕nSn = ⊗n≥0Rnd ⊗V ∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq215","equation_number":null,"raw_text":"T = T(sπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq216","equation_number":null,"raw_text":"d) = SH = ⊕n≥0Tn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq217","equation_number":null,"raw_text":"multiplicity of the trivial H-representation in Sn = Rnd⊗V ∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq218","equation_number":null,"raw_text":"Lemma 4.1.3 (cf. [Dh]) If T = ⊕∞","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq219","equation_number":null,"raw_text":"n=0Tn is a graded C-algebra, such that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq220","equation_number":null,"raw_text":"t1, . . . , tu be its homogeneous sequence of parameters (h.s.o.p.), where u =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq221","equation_number":null,"raw_text":"i=1 (1 −tdi)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq222","equation_number":null,"raw_text":"where (1) h0 = 1, (2) di is the degree of ti, and (3) hi’s are nonnegative","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq223","equation_number":null,"raw_text":"t = (t1, . . . , tu) with di =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq224","equation_number":null,"raw_text":"d (T) has a positive rational form (4.4) with di = deg(ti)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq225","equation_number":null,"raw_text":"degree seqeunce of an h.s.o.p. t = (t1, . . . , tu) is defined to be (d1, . . . , du),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq226","equation_number":null,"raw_text":"where di = deg(ti). The form (4.4) is the same for any h.s.o.p. of lexi-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq227","equation_number":null,"raw_text":"rings which will be important later. Specifically, let S = S(s) and T = T(s)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq228","equation_number":null,"raw_text":"s = sπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq229","equation_number":null,"raw_text":"Let R = R(s) be the homogeneous coordinate ring of X as in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq230","equation_number":null,"raw_text":"When s = mπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq231","equation_number":null,"raw_text":"X ∼= G/P as given in eq.(3.10. Then these rings are explicitly as follows:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq232","equation_number":null,"raw_text":"example, let P = P λ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq233","equation_number":null,"raw_text":"ated with H = GLn(C), and the canonical basis of Sq(cλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq234","equation_number":null,"raw_text":"bases at q = 1 will be called canonical bases of R(cλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq235","equation_number":null,"raw_text":"α,β, and X = Proj(R(cλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq236","equation_number":null,"raw_text":"α,β)), Y = Proj(S(cλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq237","equation_number":null,"raw_text":"α,β)) and Z = Proj(T(cλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq238","equation_number":null,"raw_text":"α,β of type A. Let T = T(cλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq239","equation_number":null,"raw_text":"T = T(cλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq240","equation_number":null,"raw_text":"Remark 4.2.5 T = T(cλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq241","equation_number":null,"raw_text":"the rings associated with s (Section 4.1.2). Let X(s) = Proj(R(s)), Y (s) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq242","equation_number":null,"raw_text":"Proj(S(s)) and Z(s) = Proj(R(s)). We call R = R(s), S = S(s), T = T(s)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq243","equation_number":null,"raw_text":"representation, then the homomorphism H →G = GL(V ) does not quan-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq244","equation_number":null,"raw_text":"When H = G, GH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq245","equation_number":null,"raw_text":"It is known that the Drinfeld-Jimbo quantum group Gq = GLq(V ) and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq246","equation_number":null,"raw_text":"When G = H × H and the embedding H ⊆G is diagonal, this special-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq247","equation_number":null,"raw_text":"of generality, we can assume that B′ ⊆B, U ′ ⊆U, T ′ ⊆T. Let A = C[G]U","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq248","equation_number":null,"raw_text":"A = ⊕λVλ(G),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq249","equation_number":null,"raw_text":"A′ = C[H]U′ = ⊕λVμ(H),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq250","equation_number":null,"raw_text":"Now A ⊗A′ is finitely generated since A and A′ are. Let X = (A ⊗A′)H","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq251","equation_number":null,"raw_text":"X = (A ⊗A′)H = ((⊕Vλ(G)) ⊗(⊕Vμ(H)))H = ⊕(Vλ(G) ⊗Vμ(H))H,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq252","equation_number":null,"raw_text":"biλi = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq253","equation_number":null,"raw_text":"Proof: Given a point p = (μ′, λ′) in the weight space of H ×G, where μ′ and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq254","equation_number":null,"raw_text":"A′x = b′,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq255","equation_number":null,"raw_text":"By = c,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq256","equation_number":null,"raw_text":"φB,c(n) = φB(nc)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq257","equation_number":null,"raw_text":"φB,c,b(n) = φB(nc + b).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq258","equation_number":null,"raw_text":"of the weight space for ̄μ in B ̄λ = V ̄λ(G).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq259","equation_number":null,"raw_text":"mπ ̄λ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq260","equation_number":null,"raw_text":"n ̄μ(V ̄λ) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq261","equation_number":null,"raw_text":"a weight that occurs in the representation V ̄λ–in this case, n ̄μ(V ̄λ) = 0, and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq262","equation_number":null,"raw_text":"mπ ̄λ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq263","equation_number":null,"raw_text":"mπ ̄λ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq264","equation_number":null,"raw_text":"By = c,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq265","equation_number":null,"raw_text":"B = BH,G,C,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq266","equation_number":null,"raw_text":"c = mW, ̄","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq267","equation_number":null,"raw_text":"BPv = ν,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq268","equation_number":null,"raw_text":"3. BPv = ̄W( ̄λ + ̄ρ) −( ̄μ + ̄ρ), v ≥0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq269","equation_number":null,"raw_text":"mπ ̄λ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq270","equation_number":null,"raw_text":"̃mπ ̄λ(n) = mn ̄λ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq271","equation_number":null,"raw_text":"nπ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq272","equation_number":null,"raw_text":"W,C(n) = φW, ̄","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq273","equation_number":null,"raw_text":"Mπ ̄λ (t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq274","equation_number":null,"raw_text":"mπ ̄λ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq275","equation_number":null,"raw_text":"A bound on D provided by the proof below is very weak: D = O(2O(rank(G))).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq276","equation_number":null,"raw_text":"W , C, such that the period of the quasipolynomial h(n) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq277","equation_number":null,"raw_text":"NC = ∪iNC(logi(N), poly(N)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq278","equation_number":null,"raw_text":"R = Q, the problems of computing the determinant, the inverse and solving","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq279","equation_number":null,"raw_text":"semisimple, simply connected Lie group, G = GL(V ), where V = Vμ(H) is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq280","equation_number":null,"raw_text":"Here it is assumed that the partition λ = λ1 ≥λ2 ≥· · · λr > 0 is rep-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq281","equation_number":null,"raw_text":"Proposition 5.1.2 If dim(V ) = poly(dim(H)), then mπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq282","equation_number":null,"raw_text":"If dim(V ) = poly(dim(H)), the size of such","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq283","equation_number":null,"raw_text":"a tableau is O(dim(V )2) = poly(dim(H)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq284","equation_number":null,"raw_text":"in the plethym problem (Problem 1.1.2), where dim(V ) = dim(Vμ) can","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq285","equation_number":null,"raw_text":"be exponential in n = dim(H) and the bitlength of μ. In this case, the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq286","equation_number":null,"raw_text":"prove it only for H = SLn(C), or rather GLn(C)–i.e., the usual plethysm","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq287","equation_number":null,"raw_text":"Here the dependence on n = dim(H) is not there. This makes a difference","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq288","equation_number":null,"raw_text":"if the heights of μ and π are less than n = dim(H)–remember that we are","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq289","equation_number":null,"raw_text":"Sλ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq290","equation_number":null,"raw_text":"Mβ(x1, . . . , xm) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq291","equation_number":null,"raw_text":"where γ ranges over all permutations of β and tγ = Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq292","equation_number":null,"raw_text":"Sλ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq293","equation_number":null,"raw_text":"W =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq294","equation_number":null,"raw_text":"Let λ, μ, π be as in Theorem 5.1.3. Let H = GLn(C), V = Vμ(H), G =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq295","equation_number":null,"raw_text":"Vλ(G) of G. Here m = dim(Vμ) can be exponential in n and ⟨μ⟩. The basis","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq296","equation_number":null,"raw_text":"t(T) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq297","equation_number":null,"raw_text":"i=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq298","equation_number":null,"raw_text":"let us define fμ = fμ(t1, . . . , tn) to be the polynomial obtained by substi-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq299","equation_number":null,"raw_text":"tuting xi = t(Ti) in f(x1, . . . , xm). Then the formal character of Vλ(G),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq300","equation_number":null,"raw_text":"considered as an H-representation of via the homomorphism H →G =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq301","equation_number":null,"raw_text":"GL(Vμ(H)), is the symmetric polynomial Sλ,μ(t1, . . . , tn) = (Sλ)μ.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq302","equation_number":null,"raw_text":"Sλ,μ(t1, . . . , tn) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq303","equation_number":null,"raw_text":"Sλ,μ(t1, . . . , tn) = Aλ,μ(t1, . . . , tn)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq304","equation_number":null,"raw_text":"inator in (5.1), and Aλ,μ = (Aλ)μ, and Bλ,μ = (Bλ)μ. Let R = C[t1, . . . , tn].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq305","equation_number":null,"raw_text":"Aλ,μ(t1, . . . , tn) = |t(Tj)λi+m−i|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq306","equation_number":null,"raw_text":"This is the determinant of an m × m matrix with entries in R, where m =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq307","equation_number":null,"raw_text":"coefficients is r = O((|λ|m)n). Hence each ring operation can be carried","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq308","equation_number":null,"raw_text":"log m = poly(n, ⟨μ⟩) and log r = poly(n, ⟨λ⟩, ⟨μ⟩), it follows that Aλ,μ can","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq309","equation_number":null,"raw_text":"Next we prove Theorem 3.4.3 when H = Sm.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq310","equation_number":null,"raw_text":"Let X = Vμ(Sm) be an","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq311","equation_number":null,"raw_text":"Let ρ : H →G = GL(X) be the corresponding","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq312","equation_number":null,"raw_text":"i = (i1, i2, . . .) of nonnegative integers such that P jij = m, the value of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq313","equation_number":null,"raw_text":"Proof: Let k be the height of the partition π. Let x = (x1, . . . , xk) be the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq314","equation_number":null,"raw_text":"χλ(Ci) = [f(x)](l1,...,lk),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq315","equation_number":null,"raw_text":"l1 = π1 + k −1, l2 = π2 + k −2, . . . , lk = πk,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq316","equation_number":null,"raw_text":"f(x) = ∆(x)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq317","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq318","equation_number":null,"raw_text":"Since deg(f) = poly(m) and k ≤m, the total number of coefficients of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq319","equation_number":null,"raw_text":"Given an irreducible representation X = Vμ(Sm) and an irreducible rep-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq320","equation_number":null,"raw_text":"resentation W = Vλ(G) of G = GL(X)), let ρμ denote the representation","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq321","equation_number":null,"raw_text":"Proof: Let r = dim(X). The formal character of the representation Vλ(G)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq322","equation_number":null,"raw_text":"of G = GL(X) is the Schur polynomial Sλ(x1, . . . , xr), r = dim(X). Hence,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq323","equation_number":null,"raw_text":"χρ(σ) = Sλ(α)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq324","equation_number":null,"raw_text":"where α = (α1, . . . , αr) is the tuple of eigenvalues of ρμ(σ). We shall compute","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq325","equation_number":null,"raw_text":"(1) Let χμ denote the character of the representation ρμ. Let pi(α) = αi","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq326","equation_number":null,"raw_text":"pi(α) = χμ(σi).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq327","equation_number":null,"raw_text":"We can compute σi, for i ≤|λ|, where |λ| denotes the size of λ, in poly(log i, m) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq328","equation_number":null,"raw_text":"sors. We calculate pi(α) in parallel for all i ≤|λ|, and all pγ(α) = Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq329","equation_number":null,"raw_text":"hi =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq330","equation_number":null,"raw_text":"|γ|=i","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq331","equation_number":null,"raw_text":"where zγ = Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq332","equation_number":null,"raw_text":"i≥1 imimi!, and mi = mi(γ) denotes the number of parts of γ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq333","equation_number":null,"raw_text":"equal to i. Thus we can calculate hγ(α) = Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq334","equation_number":null,"raw_text":"K = [Kλ,γ], where Kλ,γ denote the Kostka number; cf. [Mc]. As we noted in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq335","equation_number":null,"raw_text":"H = GLn(Fpk) have been classified by Green [Mc]. They are labelled by","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq336","equation_number":null,"raw_text":"⟨p⟩= log2 p; we specify a function by giving its partition values at the places","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq337","equation_number":null,"raw_text":"where it is nonzero. Let μ denote any such label. Let X = Vμ(H) be the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq338","equation_number":null,"raw_text":"corresponding irreducible representation of H, and ρ : H →G = GL(X)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq339","equation_number":null,"raw_text":"Lemma 5.3.2 Given a label γ of an irreducible character χγ of H = GLn(Fpk)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq340","equation_number":null,"raw_text":"First, we recall the definition of the Hall-Littlewood polynomial Pπ(x; t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq341","equation_number":null,"raw_text":"the monomial symmetric function mπ(x) because Pπ(x; 0) = sπ(x) and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq342","equation_number":null,"raw_text":"Pπ(x; 1) = mπ(x). The formal definition is as follows:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq343","equation_number":null,"raw_text":"For a given partition π, let vπ(t) = Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq344","equation_number":null,"raw_text":"vm(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq345","equation_number":null,"raw_text":"i=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq346","equation_number":null,"raw_text":"Pπ(x; t) = Aπ(x, t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq347","equation_number":null,"raw_text":"Aπ(x, t) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq348","equation_number":null,"raw_text":"Bπ(x, t) = vπ(t) Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq349","equation_number":null,"raw_text":"Pπ(x; t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq350","equation_number":null,"raw_text":"We want to calculate the matrix W = [wπ,α] fast in parallel.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq351","equation_number":null,"raw_text":"Furthermore, the inverse of W = [Wπ,α] can also be computed fast in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq352","equation_number":null,"raw_text":"Analogue of the Kronecker problem (Problem 1.1.1) for H = GLn(Fpk) is:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq353","equation_number":null,"raw_text":"λ,μ(n) = bnπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq354","equation_number":null,"raw_text":"the multiplicity of Vnπ(H(n)) in Vnλ(H(n)) ⊗Vnμ(H(n)), where H(n) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq355","equation_number":null,"raw_text":"Kronecker problem, n = 2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq356","equation_number":null,"raw_text":"λ,μ(n) = ̃knπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq357","equation_number":null,"raw_text":"λ,μ(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq358","equation_number":null,"raw_text":"for the Kronecker coefficient, when n = 2, has given by Remmel and White-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq359","equation_number":null,"raw_text":"sample values of λ = (λ1, λ2), μ = (μ1, μ2), and π = (π1, π2, π3, π4).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq360","equation_number":null,"raw_text":"question [Ki] in the negative. But its period is at most two for n = 2. This","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq361","equation_number":null,"raw_text":"Let V = Vλ(G) be an irreducible representation of G = SLk(C) correspond-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq362","equation_number":null,"raw_text":"highest weight vector, and X = Gvλ ∼= G/Pλ its closed orbit. Let hk,λ(n)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq363","equation_number":null,"raw_text":"polynomial, since t = 1 is the only pole of the Hilbert series","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq364","equation_number":null,"raw_text":"Hk,λ(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq365","equation_number":null,"raw_text":"Grassmannian of k-planes in V = Cn, and Ωa, a = (a(1), . . . , a(d)) its","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq366","equation_number":null,"raw_text":"Vn−k+i−a(i)) ≥i for all i, where V = Vn ⊃· · · V1 ⊃0 is a complete flag of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq367","equation_number":null,"raw_text":"Let V = Ck, G = GL(V ), H = Sk, with the natural embedding H →","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq368","equation_number":null,"raw_text":"(Problem 1.1.3), with Vλ(G) = V , and Vπ(H) the trivial representation of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq369","equation_number":null,"raw_text":"Then s = mπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq370","equation_number":null,"raw_text":"The canonical rings R = R(s) and S = S(s) associated with s in this","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq371","equation_number":null,"raw_text":"case coincide with C[V ] = C[x1, . . . , xk].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq372","equation_number":null,"raw_text":"The ring T = T(s) = SH =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq373","equation_number":null,"raw_text":"h(n) is a quasipolynomial. PH1 and PH3 for Z = Proj(T), as per Defini-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq374","equation_number":null,"raw_text":"where hi(n), 1 ≤i ≤l, are such that h(n) = hi(n), when n = i modulo the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq375","equation_number":null,"raw_text":"λ,μ(t) for the Kronecker problem, n = 2.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq376","equation_number":null,"raw_text":"Figure 6.6: Hilbert polynomial for G/Pλ, G = SLk(C). There is a slight rounding error caused by interpolation–","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq377","equation_number":null,"raw_text":"Figure 6.7: Hilbert polynomial of the Schubert subvariety Ωa, a = (a(1), . . . , a(k)), of the Grassmannian Gn,k.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq378","equation_number":null,"raw_text":"k = 2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq379","equation_number":null,"raw_text":"k = 3","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq380","equation_number":null,"raw_text":"k = 4","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq381","equation_number":null,"raw_text":"Figure 6.8: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk; k = 2, 3, 4.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq382","equation_number":null,"raw_text":"Figure 6.9: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 5; the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq383","equation_number":null,"raw_text":"Figure 6.10: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 5;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq384","equation_number":null,"raw_text":"Figure 6.11: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 6; the first 20 rows.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq385","equation_number":null,"raw_text":"Figure 6.12: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 6; the middle 20 rows.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq386","equation_number":null,"raw_text":"Figure 6.13: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]Sk, k = 6; the last 20 rows.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq387","equation_number":null,"raw_text":"as an H-subvariety, for every l = poly(n). The goal is to show that this","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq388","equation_number":null,"raw_text":"embedding cannot exist when l = poly(n) and n →∞.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq389","equation_number":null,"raw_text":"if y = [y]; i.e., y is the only point in P(V ) stabilized by Hy. Let","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq390","equation_number":null,"raw_text":"H[y] = ∪z∈[y]Hz","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq391","equation_number":null,"raw_text":"d(k) = skλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq392","equation_number":null,"raw_text":"d = P kλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq393","equation_number":null,"raw_text":"kλ = kQd","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq394","equation_number":null,"raw_text":"λ = Qdk","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq395","equation_number":null,"raw_text":"d = P kλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq396","equation_number":null,"raw_text":"λ⟩) = poly(⟨ρ⟩, ⟨Hy⟩, ⟨H⟩, ⟨d⟩, ⟨λ⟩) time.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq397","equation_number":null,"raw_text":"d ⟩) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq398","equation_number":null,"raw_text":"variable matrix, which can also be thought of as a variable l-vector, l = m2.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq399","equation_number":null,"raw_text":"can be thought of as a variable k-vector, k = n2. Let V = Symm(Y ) be the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq400","equation_number":null,"raw_text":"space V , and hence P(V ), has a natural action of G = GL(Y ) = GLl(C)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq401","equation_number":null,"raw_text":"(σf)(Y ) = f(σ−1Y ),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq402","equation_number":null,"raw_text":"for any f ∈V , σ ∈G, and thinking of Y as an l-vector. Let W = Symn(X)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq403","equation_number":null,"raw_text":"X. The space W, and also P(W), has a similar action of K = GL(X) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq404","equation_number":null,"raw_text":"φ(h)(Y ) = ym−nh(X),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq405","equation_number":null,"raw_text":"Let g = det(Y ) ∈P(V ) be the determinant form, and f = φ(h), where","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq406","equation_number":null,"raw_text":"h = perm(X) ∈P(W). Let ∆V [g], ∆V [f] ⊆P(V ) be the projective closures","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq407","equation_number":null,"raw_text":"The class varieties ∆V [g] = ∆V [g, m] and ∆V [f] = ∆V [f, n, m] are","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq408","equation_number":null,"raw_text":"G-subvarieties of P(V ), and their homogeneous coordinate rings RV [g] =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq409","equation_number":null,"raw_text":"RV [g, m] and RV [f] = RV [f, n, m] have natural degree-preserving G-action.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq410","equation_number":null,"raw_text":"It is conjectured in [GCT1] that, if m = poly(n) and n →∞, then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq411","equation_number":null,"raw_text":"implies the arithmetic form of the P #P ̸= NC conjecture in characteristic","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq412","equation_number":null,"raw_text":"is the stabilizer of g = det(Y ) ∈P(V ). If Vλ(G) is a (strong) obstruction","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq413","equation_number":null,"raw_text":"of degree d, then the size |λ| = dm; hence d is completely determined by λ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq414","equation_number":null,"raw_text":"that strong obstructions should indeed exist for all n →∞, assuming m =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq415","equation_number":null,"raw_text":"X and the homogenizing variable y above. Let W ′ = Symm(X′) ⊆V =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq416","equation_number":null,"raw_text":"We have a natural action of H = GL(X′) = GLn2+1(C) on W ′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq417","equation_number":null,"raw_text":"φ′(h)(X′) = ym−nh(X). The map φ in (7.2) is φ′ followed by the inclusion","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq418","equation_number":null,"raw_text":"Let f ′ = φ′(h), for h = perm(X) ∈P(W). Let ∆W ′[f ′] ⊆P(W ′) be","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq419","equation_number":null,"raw_text":"It is known that the stabilizer Gg of g = det(Y ) ∈P(V ) consists of lin-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq420","equation_number":null,"raw_text":"nected component of Gg is essentially GLm(C) × GLm(C) ⊆G = GLl(C) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq421","equation_number":null,"raw_text":"∆V [f] = ∆[f, n, m] will now play the role of X in Hypothesis 7.3.1. But,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq422","equation_number":null,"raw_text":"Z = ∆W ′[f ′]. It now assumes that following concrete form. Let sλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq423","equation_number":null,"raw_text":"poly(⟨d⟩, ⟨λ⟩, ⟨Z⟩) = poly(⟨d⟩, ⟨λ⟩, n, ⟨m⟩).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq424","equation_number":null,"raw_text":"Here (7.3) follows because ⟨Z⟩= n+⟨m⟩. To see why, let us observe that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq425","equation_number":null,"raw_text":"Z = ∆W ′[f ′] is completely specified once the point f ′ = ym−nh ∈P(W ′) is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq426","equation_number":null,"raw_text":"It is known [GCT2] that the point h = perm(X) ∈P(W) is completely","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq427","equation_number":null,"raw_text":"characterized by its stabilizer Kh ⊆K = GL(X) = GLk(C). Furthermore,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq428","equation_number":null,"raw_text":"Theorem 7.6.6 Assume that m = poly(n) or even 2polylog(n), and:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq429","equation_number":null,"raw_text":"following holds (with d = |λ|/m):","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq430","equation_number":null,"raw_text":"as long as m << 2n; i.e., it should hold even when m = 2polylog(n). In","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq431","equation_number":null,"raw_text":"enough m = 2Ω(n). This is an important uniformity condition.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq432","equation_number":null,"raw_text":"strong obstructions should exist for every n, assuming m = poly(n). We","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq433","equation_number":null,"raw_text":"(Hypothesis 7.6.7), if λ is fragile, then for some k = poly(m), Qkd","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq434","equation_number":null,"raw_text":"where A depends only the variety Z = ∆W ′[f ′], but not on d or λ, and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq435","equation_number":null,"raw_text":"points in suitable W = Symk(X) and V = Syml[Y ], k = O(n2), l = O(m2),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq436","equation_number":null,"raw_text":"with the natural action of GL(X) and G = GL(Y ), where n denotes the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq437","equation_number":null,"raw_text":"define f = φ(h). The class variety for NP is defined to be ∆V [f] ⊆P(V ),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq438","equation_number":null,"raw_text":"show P ̸= NP in characteristic zero, it suffices to show that ∆V [f] is not a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq439","equation_number":null,"raw_text":"subvariety of ̃∆V [g] for all large enough n, if m = poly(n) (cf. Conjecture","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq440","equation_number":null,"raw_text":"m = poly(n).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq441","equation_number":null,"raw_text":"Analogues of Theorems 7.6.5 and 7.6.6 holds for h(X) = E(X) and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq442","equation_number":null,"raw_text":"g(Y ) = H(Y ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq443","equation_number":null,"raw_text":"T. Baker, J. Gill, R. Soloway, Relativization of the P =?NP ques-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":268300,"parse_confidence":0.5,"equation_parse_rate_proxy":1.0,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}