{"paper_meta":{"paper_id":"arxiv:0709.0746","title":"0709.0746","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:0709.0746v1 [cs.CC] 5 Sep 2007\nGeometric Complexity Theory: Introduction\nDedicated to Sri Ramakrishna\nKetan D. Mulmuley 1\nThe University of Chicago\nMilind Sohoni\nI.I.T., Mumbai\nTechnical Report TR-2007-16\nComputer Science Department\nThe University of Chicago\nSeptember 2007\nAugust 2, 2014\n1Part of the work on GCT was done while the first author was visiting I.I.T.\nMumbai to which he is grateful for its hospitality\n\nForeword\nThese are lectures notes for the introductory graduate courses on geo-\nmetric complexity theory (GCT) in the computer science department, the\nuniversity of Chicago. Part I consists of the lecture notes for the course\ngiven by the first author in the spring quarter, 2007. It gives introduction\nto the basic structure of GCT. Part II consists of the lecture notes for the\ncourse given by the second author in the spring quarter, 2003. It gives in-\ntroduction to invariant theory with a view towards GCT. No background\nin algebraic geometry or representation theory is assumed. These lecture\nnotes in conjunction with the article [GCTflip1], which describes in detail\nthe basic plan of GCT based on the principle called the flip, should provide\na high level picture of GCT assuming familiarity with only basic notions\nof algebra, such as groups, rings, fields etc. Many of the theorems in these\nlecture notes are stated without proofs, but after giving enough motivation\nso that they can be taken on faith. For the readers interested in further\nstudy, Figure 1 shows logical dependence among the various papers of GCT\nand a suggested reading sequence.\nThe first author is grateful to Paolo Codenotti, Joshua Grochow, Sourav\nChakraborty and Hari Narayanan for taking notes for his lectures.\n1\n\nGCTabs\n|\n↓\nGCTflip1\n|\n↓\nThese lecture notes\n−−→\nGCT3\n|\n|\n↓\n|\nGCT1\n|\n|\n|\n↓\n|\nGCT2\n|\n|\n|\n↓\n↓\nGCT6\n←−−\nGCT5\n|\n↓\nGCT4\n−−→\nGCT9\n|\n↓\nGCT7\n|\n↓\nGCT8\n|\n↓\nGCT10\n|\n↓\nGCT11\n|\n↓\nGCTflip2\nFigure 1: Logical dependence among the GCT papers\n2\n\nContents\nI\nThe basic structure of GCT\nBy Ketan D. Mulmuley\n8\n1\nOverview\n9\n1.1\nOutline\n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n9\n1.2\nThe G ̈odelian Flip\n. . . . . . . . . . . . . . . . . . . . . . . .\n12\n1.3\nMore details of the GCT approach . . . . . . . . . . . . . . .\n13\n2\nRepresentation theory of reductive groups\n16\n2.1\nBasics of Representation Theory\n. . . . . . . . . . . . . . . .\n16\n2.1.1\nDefinitions\n. . . . . . . . . . . . . . . . . . . . . . . .\n16\n2.1.2\nNew representations from old . . . . . . . . . . . . . .\n17\n2.2\nReductivity of finite groups . . . . . . . . . . . . . . . . . . .\n19\n2.3\nCompact Groups and GLn(C) are reductive . . . . . . . . . .\n20\n2.3.1\nCompact groups\n. . . . . . . . . . . . . . . . . . . . .\n20\n2.3.2\nWeyl’s unitary trick and GLn(C) . . . . . . . . . . . .\n21\n3\nRepresentation theory of reductive groups (cont)\n22\n3.1\nProjection Formula . . . . . . . . . . . . . . . . . . . . . . . .\n23\n3.2\nThe characters of irreducible representations form a basis\n. .\n25\n3.3\nExtending to Infinite Compact Groups . . . . . . . . . . . . .\n27\n4\nRepresentations of the symmetric group\n29\n4.1\nRepresentations and characters of Sn . . . . . . . . . . . . . .\n30\n4.1.1\nFirst Construction . . . . . . . . . . . . . . . . . . . .\n30\n4.1.2\nSecond Construction . . . . . . . . . . . . . . . . . . .\n31\n4.1.3\nThird Construction . . . . . . . . . . . . . . . . . . . .\n32\n4.1.4\nCharacter of Sλ [Frobenius character formula] . . . . .\n32\n4.2\nThe first decision problem in GCT . . . . . . . . . . . . . . .\n33\n3\n\n5\nRepresentations of GLn(C)\n35\n5.1\nFirst Approach [Deruyts]\n. . . . . . . . . . . . . . . . . . . .\n35\n5.1.1\nHighest weight vectors . . . . . . . . . . . . . . . . . .\n38\n5.2\nSecond Approach [Weyl] . . . . . . . . . . . . . . . . . . . . .\n39\n6\nDeciding nonvanishing of Littlewood-Richardson coefficients 41\n6.1\nLittlewood-Richardson coefficients\n. . . . . . . . . . . . . . .\n41\n7\nLittlewood-Richardson coefficients (cont)\n46\n7.1\nThe stretching function\n. . . . . . . . . . . . . . . . . . . . .\n47\n7.2\nOn(C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n48\n8\nDeciding nonvanishing of Littlewood-Richardson coefficients\nfor On(C)\n52\n9\nThe plethysm problem\n56\n9.1\nLittlewood-Richardson Problem [GCT 3,5] . . . . . . . . . . .\n57\n9.2\nKronecker Problem [GCT 4,6] . . . . . . . . . . . . . . . . . .\n57\n9.3\nPlethysm Problem [GCT 6,7] . . . . . . . . . . . . . . . . . .\n58\n10 Saturated and positive integer programming\n61\n10.1 Saturated, positive integer programming . . . . . . . . . . . .\n61\n10.2 Application to the plethysm problem . . . . . . . . . . . . . .\n63\n11 Basic algebraic geometry\n64\n11.1 Algebraic geometry definitions\n. . . . . . . . . . . . . . . . .\n64\n11.2 Orbit closures . . . . . . . . . . . . . . . . . . . . . . . . . . .\n66\n11.3 Grassmanians . . . . . . . . . . . . . . . . . . . . . . . . . . .\n67\n12 The class varieties\n70\n12.1 Class Varieties in GCT . . . . . . . . . . . . . . . . . . . . . .\n70\n13 Obstructions\n73\n13.1 Obstructions\n. . . . . . . . . . . . . . . . . . . . . . . . . . .\n74\n13.1.1 Why are the class varieties exceptional? . . . . . . . .\n75\n14 Group theoretic varieties\n78\n14.1 Representation theoretic data . . . . . . . . . . . . . . . . . .\n79\n14.2 The second fundamental theorem . . . . . . . . . . . . . . . .\n80\n14.3 Why should obstructions exist? . . . . . . . . . . . . . . . . .\n81\n4\n\n15 The flip\n82\n15.1 The flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n83\n16 The Grassmanian\n86\n16.1 The second fundamental theorem . . . . . . . . . . . . . . . .\n87\n16.2 The Borel-Weil theorem . . . . . . . . . . . . . . . . . . . . .\n88\n17 Quantum group: basic definitions\n90\n17.1 Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . .\n90\n18 Standard quantum group\n97\n19 Quantum unitary group\n103\n19.1 A q-analogue of the unitary group\n. . . . . . . . . . . . . . . 103\n19.2 Properties of Uq . . . . . . . . . . . . . . . . . . . . . . . . . . 105\n19.3 Irreducible Representations of Gq . . . . . . . . . . . . . . . . 106\n19.4 Gelfand-Tsetlin basis . . . . . . . . . . . . . . . . . . . . . . . 106\n20 Towards positivity hypotheses via quantum groups\n108\n20.1 Littlewood-Richardson rule via standard quantum groups\n. . 108\n20.1.1 An embedding of the Weyl module . . . . . . . . . . . 109\n20.1.2 Crystal operators and crystal bases . . . . . . . . . . . 110\n20.2 Explicit decomposition of the tensor product\n. . . . . . . . . 112\n20.3 Towards nonstandard quantum groups for the Kronecker and\nplethysm problems . . . . . . . . . . . . . . . . . . . . . . . . 113\nII\nInvariant theory with a view towards GCT\nBy Milind Sohoni\n116\n21 Finite Groups\n117\n21.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117\n21.2 The finite group action . . . . . . . . . . . . . . . . . . . . . . 118\n21.3 The Symmetric Group . . . . . . . . . . . . . . . . . . . . . . 122\n22 The Group SLn\n124\n22.1 The Canonical Representation . . . . . . . . . . . . . . . . . . 124\n22.2 The Diagonal Representation . . . . . . . . . . . . . . . . . . 125\n22.3 Other Representations . . . . . . . . . . . . . . . . . . . . . . 128\n22.4 Full Reducibility\n. . . . . . . . . . . . . . . . . . . . . . . . . 130\n5\n\n23 Invariant Theory\n131\n23.1 Algebraic Groups and affine actions\n. . . . . . . . . . . . . . 131\n23.2 Orbits and Invariants . . . . . . . . . . . . . . . . . . . . . . . 132\n23.3 The Nagata Hypothesis\n. . . . . . . . . . . . . . . . . . . . . 136\n24 Orbit-closures\n139\n25 Tori in SLn\n142\n26 The Null-cone and the Destabilizing flag\n147\n26.1 Characters and the half-space criterion . . . . . . . . . . . . . 147\n26.2 The destabilizing flag . . . . . . . . . . . . . . . . . . . . . . . 149\n27 Stability\n154\n6\n\n7\n\nPart I\nThe basic structure of GCT\nBy Ketan D. Mulmuley\n8\n\nChapter 1\nOverview\nScribe: Joshua A. Grochow\nGoal: An overview of GCT.\nThe purpose of this course is to give an introduction to Geometric Com-\nplexity Theory (GCT), which is an approach to proving P ̸= NP via al-\ngebraic geometry and representation theory. A basic plan of this approach\nis described in [GCTflip1, GCTflip2]. It is partially implemented in a se-\nries of articles [GCT1]-[GCT11]. The paper [GCTconf] is a conference an-\nnouncement of GCT. The paper [Ml] gives an unconditional lower bound\nin a PRAM model without bit operations based on elementary algebraic\ngeometry, and was a starting point for the GCT investigation via algebraic\ngeometry.\nThe only mathematical prerequisites for this course are a basic knowl-\nedge of abstract algebra (groups, ring, fields, etc.)\nand a knowledge of\ncomputational complexity. In the first month we plan to cover the represen-\ntation theory of finite groups, the symmetric group Sn, and GLn(C), and\nenough algebraic geometry so that in the remaining lectures we can cover\nbasic GCT. Most of the background results will only be sketched or omitted.\nThis lecture uses slightly more algebraic geometry and representation\ntheory than the reader is assumed to know in order to give a more complete\npicture of GCT. As the course continues, we will cover this material.\n1.1\nOutline\nHere is an outline of the GCT approach. Consider the P vs. NP question\nin characteristic 0; i.e., over integers. So bit operations are not allowed, and\n9\n\nbasic operations on integers are considered to take constant time. For a sim-\nilar approach in nonzero characteristic (characteristic 2 being the classical\ncase from a computational complexity point of view), see GCT 11.\nThe basic principle of GCT is the called the flip [GCTflip1]. It “reduces”\n(in essence, not formally) the lower bound problems such as P vs. NP in\ncharacteristic 0 to upper bound problems: showing that certain decision\nproblems in algebraic geometry and representation theory belong to P. Each\nof these decision problems is of the form: is a given (nonnegative) structural\nconstant associated to some algebro-geometric or representation theoretic\nobject nonzero? This is akin to the decision problem: given a matrix, is\nits permanent nonzero? (We know how to solve this particular problem in\npolynomial time via reduction to the perfect matching problem.)\nNext, the preceding upper bound problems are reduced to purely math-\nematical positivity hypotheses [GCT6]. The goal is to show that these and\nother auxilliary structural constants have positive formulae. By a positive\nformula we mean a formula that does not involve any alternating signs like\nthe usual positive formula for the permanent; in contrast the usual formula\nfor the determinant involves alternating signs.\nFinally, these positivity hypotheses are “reduced” to conjectures in the\ntheory of quantum groups [GCT6, GCT7, GCT8, GCT10] intimately related\nto the Riemann hypothesis over finite fields proved in [Dl2], and the related\nworks [BBD, KL2, Lu2].\nA pictorial summary of the GCT approach is\nshown in Figure 1.1, where the arrows represent reductions, rather than\nimplications.\nTo recap: we move from a negative hypothesis in complexity theory\n(that there does not exist a polynomial time algorithm for an NP-complete\nproblem) to a positive hypotheses in complexity theory (that there exist\npolynomial-time algorithms for certain decision problems) to positive hy-\npotheses in mathematics (that certain structural constants have positive\nformulae) to conjectures on quantum groups related to the Riemann hy-\npothesis over finite fields, the related works and their possible extensions.\nThe first reduction here is the flip: we reduce a question about lower bounds,\nwhich are notoriously difficult, to the one about upper bounds, which we\nhave a much better handle on. This flip from negative to positive is already\npresent in G ̈odel’s work: to show something is impossible it suffices to show\nthat something else is possible. This was one of the motivations for the GCT\napproach. The G ̈odelian flip would not work for the P vs. NP problem be-\ncause it relativizes. We can think of GCT as a form of nonrelativizable (and\nnon-naturalizable, if reader knows what that means) diagonalization.\nIn summary, this approach very roughly “reduces” the lower bound prob-\n10\n\nP vs. NP\nchar. 0\nFlip\n=⇒\nDecision problems\nin alg. geom.\n& rep. thy.\n=⇒\nShow certain\nconstants in alg.\ngeom. and repr.\ntheory have\npositive formulae\nLower bounds\n(Neg. hypothesis\nin complexity thy.)\nUpper bounds\n(Pos. hypotheses\nin complexity thy.)\nPos. hypotheses\nin mathematics\n=⇒\nConjectures on\nquantum groups\nrelated to RH over\nfinite fields\nFigure 1.1: The basic approach of GCT\n11\n\nlems such as P vs. NP in characteristic zero to as-yet-unproved quantum-\ngroup-conjectures related to the Riemann Hypothesis over finite fields. As\nwith the classical RH, there is experimental evidence to suggest these con-\njectures hold – which indirectly suggests that certain generalizations of the\nRiemann hypothesis over finite fields also hold – and there are hints on how\nthe problem might be attacked. See [GCTflip1, GCT6, GCT7, GCT8] for a\nmore detailed exposition.\n1.2\nThe G ̈odelian Flip\nWe now re-visit G ̈odel’s original flip in modern language to get the flavor of\nthe GCT flip.\nG ̈odel set out to answer the question:\nQ: Is truth provable?\nBut what “truth” and “provable” means here is not so obvious a priori. We\nstart by setting the stage: in any mathematical theory, we have the syntax\n(i.e. the language used) and the semantics (the domain of discussion). In\nthis case, we have:\nSyntax (language)\nSemantics (domain)\nFirst order logic\n(∀, ∃, ¬, ∨, ∧, . . . )\nConstants\n0,1\nVariables\nx, y, z, . . .\nBasic Predicates\n>, <, =\nFunctions\n+,−,×,exponentiation\nAxioms\nAxioms of the natural numbers N\nUniverse: N\nA sentence is a valid formula with all variables quantified, and by a\ntruth we mean a sentence that is true in the domain. By a proof we mean a\nvalid deduction based on standard rules of inference and the axioms of the\ndomain, whose final result is the desired statement.\nHilbert’s program asked for an algorithm that, given a sentence in num-\nber theory, decides whether it is true or false.\nA special case of this is\nHilbert’s 10th problem, which asked for an algorithm to decide whether a\nDiophantine equation (equation with only integer coefficients) has a nonzero\ninteger solution.\nG ̈odel showed that Hilbert’s general program was not\n12\n\nachievable. The tenth problem remained unresolved until 1970, at which\npoint Matiyasevich showed its impossibility as well.\nHere is the main idea of G ̈odel’s proof, re-cast in modern language.\nFor a Turing Machine M, whether the empty string ε is in the language\nL(M) recognized by M is undecidable. The idea is to reduce a question\nof the form ε ∈L(M) to a question in number theory. If there were an\nalgorithm for deciding the truth of number-theoretic statements, it would\ngive an algorithm for the above Turing machine problem, which we know\ndoes not exist.\nThe basic idea of the reduction is similar to the one in Cook’s proof that\nSAT is NP-complete. Namely, ε ∈L(M) iffthere is a valid computation\nof M which accepts ε. Using Cook’s idea, we can use this to get a Boolean\nformula:\n∃m∃a valid computation of M with configurations of size ≤m\ns.t. the computation accepts ε.\nThen we use G ̈odel numbering – which assigns a unique number to each\nsentence in number theory – to translate this formula to a sentence in number\ntheory. The details of this should be familiar.\nThe key point here is: to show that truth is undecidable in number theory\n(a negative statement), we show that there exists a computable reduction\nfrom ε\n?∈L(M) to number theory (a positive statement). This is the essence\nof the G ̈odelian flip, which is analogous to – and in fact was the original\nmotivation for – the GCT flip.\n1.3\nMore details of the GCT approach\nTo begin with, GCT associates to each complexity class such as P and NP\na projective algebraic variety χP, χNP , etc. [GCT1]. In fact, it associates\na family of varieties χNP (n, m): one for each input length n and circuit\nsize m, but for simplicity we suppress this here. The languages L in the\nassociated complexity class will be points on these varieties, and the set\nof such points is dense in the variety. These varieties are thus called class\nvarieties. To show that NP ⊈P in characteristic zero, it suffices to show\nthat χNP cannot be imbedded in χP .\nThese class varieties are in fact G-varieties. That is, they have an action\nof the group G = GLn(C) on them. This action induces an action on the\nhomogeneous coordinate ring of the variety, given by (σf)(x) = f(σ−1x)\nfor all σ ∈G. Thus the coordinate rings RP and RNP of χP and χNP are\n13\n\nG-algebras, i.e., algebras with G-action. Their degree d-components RP (d)\nand RNP (d) are thus finite dimensional G-representations.\nFor the sake of contradiction, suppose NP ⊆P in characteristic 0. Then\nthere must be an embedding of χNP into χP as a G-subvariety, which\nin turn gives rise (by standard algebraic geometry arguments) to a sur-\njection RP ։ RNP of the coordinate rings.\nThis implies (by standard\nrepresentation-theoretic arguments) that RNP (d) can be embedded as a G-\nsub-representation of RP (d). The following diagram summarizes the impli-\ncations.\ncomplexity\nclasses\nclass\nvarieties\ncoordinate\nrings\nrepresentations\nof GLn(C)\nNP _\n \n//o/o/o/o/o/o/o/o/o/o\nχNP _\n \n//o/o/o/o/o/o/o/o/o/o\nRNP\n//o/o/o/o/o/o/o/o/o\nRNP (d)\n _\n \nP\n//o/o/o/o/o/o/o/o/o/o/o\nχP\n//o/o/o/o/o/o/o/o/o/o/o\nRP\nOO\n//o/o/o/o/o/o/o/o/o/o\nRP(d)\nWeyl’s theorem–that all finite-dimensional representations of G = GLn(C)\nare completely reducible, i.e. can be written as a direct sum of irreducible\nrepresentations–implies that both RNP (d) and RP (d) can be written as di-\nrect sums of irreducible G-representations. An obstruction [GCT2] of degree\nd is defined to be an irreducible G-representation occuring (as a subrepre-\nsentation) in RNP (d) but not in RP (d). Its existence implies that RNP (d)\ncannot be embedded as a subrepresentation of RP (d), and hence, χNP can-\nnot be embedded in χP as a G-subvariety; a contradiction.\nWe actually have a family of varieties χNP (n, m): one for each input\nlength n and circuit size m. Thus if an obstruction of some degree exists for\nall n →∞, assuming m = nlog n (say), then NP ̸= P in characteristic zero.\nConjecture 1.1. [GCTflip1] There is a polynomial-time algorithm for con-\nstructing such obstructions.\nThis is the GCT flip: to show that no polynomial-time algorithm exists\nfor an NP-complete problem, we hope to show that there is a polynomial\ntime algorithm for finding obstructions. This task then is further reduced to\nfinding polynomial time algorithms for other decision problems in algebraic\ngeometry and representation theory.\nMere existence of an obstruction for all n would actually suffice here. For\nthis, it suffices to show that there is an algorithm which, given n, outputs\n14\n\nan obstruction showing that χNP (n, m) cannot be imbedded in χP(n, m),\nwhen m = nlog n. But the conjecture is not just that there is an algorithm,\nbut that there is a polynomial-time algorithm.\nThe basic principle here is that the complexity of the proof of existence\nof an object (in this case, an obstruction) is very closed tied to the computa-\ntional complexity of finding that object, and hence, techniques underneath\nan easy (i.e. polynomial time) time algorithm for deciding existence may\nyield an easy (i.e. feasible) proof of existence. This is supported by much\nanecdotal evidence:\n• An obstruction to planar embedding (a forbidden Kurotowski minor)\ncan be found in polynomial, in fact, linear time by variants of the\nusual planarity testing algorithms, and the underlying techniques, in\nretrospect, yield an algorithmic proof of Kurotowski’s theorem that\nevery nonplanar graph contains a forbidden minor.\n• Hall’s marriage theorem, which characterizes the existence of per-\nfect matchings, in retrospect, follows from the techniques underlying\npolynomial-time algorithms for finding perfect matchings.\n• The proof that a graph is Eulerian iffall vertices have even degree is,\nessentially, a polynomial-time algorithm for finding an Eulerian circuit.\n• In contrast, we know of no Hall-type theorem for Hamiltonians paths,\nessentially, because finding such a path is computationally difficult\n(NP-complete).\nAnalogously the goal is to find a polynomial time algorithm for deciding\nif there exists an obstruction for given n and m, and then use the underlying\ntechniques to show that an obstruction always exists for every large enough\nn if m = nlog n. The main mathematical work in GCT takes steps towards\nthis goal.\n15\n\nChapter 2\nRepresentation theory of\nreductive groups\nScribe: Paolo Codenotti\nGoal: Basic notions in representation theory.\nReferences: [FH, F]\nIn this lecture we review the basic representation theory of reductive\ngroups as needed in this course. Most of the proofs will be omitted, or just\nsketched.\nFor complete proofs, see the books by Fulton and Harris, and\nFulton [FH, F]. The underlying field throughout this course is C.\n2.1\nBasics of Representation Theory\n2.1.1\nDefinitions\nDefinition 2.1. A representation of a group G, also called a G-module, is\na vector space V with an associated homomorphism ρ : G →GL(V ). We\nwill refer to a representation by V .\nThe map ρ induces a natural action of G on V , defined by g·v = (ρ(g))(v).\nDefinition 2.2. A map φ : V →W is G-equivariant if the following dia-\ngram commutes:\nV\nφ\n−−−−→W\n yg\n yg\nV\nφ\n−−−−→W\n16\n\nThat is, if φ(g·v) = g·φ(v). A G-equivariant map is also called G-invariant\nor a G-homomorphism.\nDefinition 2.3. A subspace W ⊆V is said to be a subrepresentation, or a\nG-submodule of a representation V over a group G if W is G-equivariant,\nthat is if g · w ∈W for all w ∈W.\nDefinition 2.4. A representation V of a group G is said to be irreducible\nif it has no proper non-zero G-subrepresentations.\nDefinition 2.5. A group G is called reductive if every finite dimensional\nrepresentation V of G is a direct sum of irreducible representation.\nHere are some examples of reductive groups:\n• finite groups;\n• the n-dimensional torus (C∗)n;\n• linear groups:\n– the general linear group GLn(C),\n– the special linear group SLn(C),\n– the orthogonal group On(C) (linear transformations that preserve\na symmetric form),\n– and the symplectic group Spn(C) (linear transformations that\npreserve a skew symmetric form);\n• Exceptional Lie Groups\nTheir reductivity is a nontrivial fact.\nIt will be proved later in this\nlecture for finite groups, and the general and special linear groups. In some\nsense, the list above is complete: all reductive groups can be constructed\nby basic operations from the components which are either in this list or are\nrelated to them in a simple way.\n2.1.2\nNew representations from old\nGiven representations V and W of a group G, we can construct new repre-\nsentations in several ways, some of which are described below.\n• Tensor product: V ⊗W. g · (v ⊗w) = (g · v) ⊗(g · w).\n• Direct sum: V ⊕W.\n17\n\n• Symmetric tensor representation: The subspace Symn(V ) ⊂V ⊗· · ·⊗\nV spanned by elements of the form\nX\nσ\n(v1 ⊗· · · ⊗vn) · σ =\nX\nσ\nvσ(1) ⊗· · · vσ(n),\nwhere σ ranges over all permutations in the symmetric group Sn.\n• Exterior tensor representation: The subspace Λn(V ) ⊂V ⊗· · · ⊗V\nspanned by elements of the form\nX\nσ\nsgn(σ)(v1 ⊗· · · ⊗vn) · σ =\nX\nσ\nsgn(σ)vσ(1) ⊗· · · vσ(n).\n• Let V and W be representations, then Hom(V, W) is also a represen-\ntation, where g · φ is defined so that the following diagram commutes:\nV\nφ\n−−−−→W\n yg\n yg\nV\ng·φ\n−−−−→W\nMore precisely,\n(g · φ)(v) = g · (φ(g−1 · v)).\n• In particular, V ∗: V →C is a representation, and is called the dual\nrepresentation.\n• Let G be a finite group. Let S be a finite G-set (that is, a finite set\nwith an associated action of G on its elements). We construct a vector\nspace over any field K (we will be mostly concerned with the case\nK = C), with a basis vector associated to each element in S. More\nspecifically, consider the set K[S] of formal sums P\ns∈S αses, where\nαs ∈K, and es is a vector associated with S ∈s. Note that this set\nhas a vector space structure over K, and there is a natural induced\naction of G on K[S], defined by:\ng ·\nX\ns∈S\nαses =\nX\ns∈S\nαseg·s.\nThis action gives rise to a representation of G.\n• In particular, G is a G-set under the action of left multiplication. The\nrepresentation we obtain in the manner described above from this G-\nset is called the regular representation.\n18\n\n2.2\nReductivity of finite groups\nProposition 2.1. Let G be a finite group. If W is a subrepresentation of a\nrepresentation V , then there exists a representation W ⊥s.t. V = W ⊕W ⊥.\nProof. Choose any Hermitian form H0 of V , and construct a new Hermitian\nform H defined as:\nH(v, w) =\nX\ng∈G\nHo(g · v, g · w).\nAveraging is a useful trick that is used very often in representation theory,\nbecause it ensures G-invariance. In fact, H is G-invariant, that is,\nH(v, w) =\nX\ng∈G\nHo(g · v, g · w) = H(h · v, h · w)\nLet W ⊥be the perpendicular complement to W with respect to the Her-\nmitian form H.\nThen W ⊥is also G-invariant, and therefore it is a G-\nsubmodule.\nCorollary 2.1. Every representation of a finite group is a direct sum of\nirreducible representations.\nLemma 2.1. (Schur) If V and W are irreducible representations over C,\nand φ : V →W is a homomorphism (i.e. a G-invariant map), then:\n1. Either φ is an isomorphism or φ = 0.\n2. If V = W, φ = λI for some λ ∈C.\nProof.\n1. Since Ker(φ), and Imφ are G-submodules, either Im(φ) = V\nor Im(φ) = 0.\n2. Let φ : V →V . Since C algebraically closed, there exists an eigenvalue\nλ of φ. Look at the map φ−λI : V →V . By (1), φ−λI = 0 (it can’t\nbe an isomorphism because something maps to 0). So φ = λI.\nCorollary 2.2. Every representation is a unique direct sum of irreducible\nrepresentations. More precisely, given two decompositions into irreducible\nrepresentations,\nV =\nM\nV ai\ni\n=\nM\nW bj\nj ,\nthere is a one to one correspondence between the Vi’s and Wj’s, and the\nmultiplicities correspond.\nProof. exercise (follows from Schur’s lemma).\n19\n\ngu\nu\nR\ngR\nFigure 2.1: Example of a left Haar measure for the circle (U1(C)). Left action by\na group element g on a small region R around u does not change the area.\n2.3\nCompact Groups and GLn(C) are reductive\nNow we prove reductivity of compact groups.\n2.3.1\nCompact groups\nExamples of compact groups:\n• Un(C) ⊆GLn(C), the unitary groups (all rows are normal and orthog-\nonal).\n• SUn(C) ⊆SLn(C), the special unitary group.\nGiven a compact group, a left-invariant Haar measure is a measure that\nis invariant under the left action of the group. In other words, multiplication\nby a group element does not change the area of a small region (i.e., the group\naction is an isometry, see figure 2.1).\nTheorem 2.1. Compact groups are reductive\nProof. We use the averaging trick again. In fact the proof is the same as\nin the case of finite groups, using integration instead of summation for the\naveraging trick. Let H0 be any Hermitian form on V. Then define H as:\nH(v, w) =\nZ\nG\nH(gv, gw)dG\n20\n\nwhere dG is a left-invariant Haar measure.\nNote that H is G-invariant.\nLet W ⊥be the perpendicular complement to W. Then W ⊥is G-invariant.\nHence it is a G-submodule.\nThe same proof as before then gives us Schur’s lemma for compact\ngroups, from which follows:\nTheorem 2.2. If G is compact, then every finite dimensional representation\nof G is a unique direct sum of irreducible representations.\n2.3.2\nWeyl’s unitary trick and GLn(C)\nTheorem 2.3. (Weyl) GLn(C) is reductive\nProof. (general idea)\nLet V be a representation of GLn(C). Then GLn(C) acts on V :\nGLn(C) ֒→V.\nBut Un(C) is a subgroup of GLn(C). Therefore we have an induced action\nof Un(C) on V , and we can look at V as a representation of Un(C). As a\nrepresentation of Un(C), V breaks into irreducible representations of Un(C)\nby the theorem above. To summarize, we have:\nUn(C) ⊆GLn(C) ֒→V = ⊕iVi,\nwhere the Vi’s are irreducible representations of Un(C). Weyl’s unitary trick\nuses Lie algebra to show that every finite dimensional representation of\nUn(C) is also a representation of GLn(C), and irreducible representations of\nUn(C) correspond to irreducible representations of GLn(C). Hence each Vi\nabove is an irreducible representation of GLn(C).\nOnce we know these groups are reductive, the goal is to construct and\nclassify their irreducible finite dimensional representations.\nThis will be\ndone in the next lectures: Specht modules for Sn, and Weyl modules for\nGLn(C).\n21\n\nChapter 3\nRepresentation theory of\nreductive groups (cont)\nScribe: Paolo Codenotti\nGoal: Basic representation theory, continued from the last lecture.\nIn this lecture we continue our introduction to representation theory.\nAgain we refer the reader to the book by Fulton and Harris for full details\n[FH]. Let G be a finite group, and V a finite-dimensional G-representation\ngiven by a homomorphism ρ : G →GL(V ). We define the character of the\nrepresentation V (denoted χV ) by χV (g) = Tr(ρ(g)).\nSince Tr(A−1BA) = Tr(B), χV (hgh−1) = χV (g). This means charac-\nters are constant on conjugacy classes (sets of the form {hgh−1|h ∈G}, for\nany g ∈G). We call such functions class functions.\nOur goal for this lecture is to prove the following two facts:\nGoal 1 A finite dimensional representation is completely determined by its\ncharacter.\nGoal 2 The space of class functions is spanned by the characters of irreducible\nrepresentations. In fact, these characters form an orthonormal basis\nof this space.\nFirst, we prove some useful lemmas about characters.\nLemma 3.1. χV ⊕W = χV + χW\nProof. Let g ∈G, and let ρ, σ be homomorphisms from G into V and W,\nrespectively. Let λ1, . . . , λr be the eigenvalues of ρ(g), and μ1, . . . , μs the\n22\n\neigenvalues of σ(g). Then (ρ ⊕σ)(g) = (ρ(g), σ(g)), so the eigenvalues of\n(ρ ⊕σ)(g) are just the eigenvalues of ρ(g) together with the eigenvalues of\nσ(g).\nThen χV (g) = P\ni λi, χW (g) = P\ni μi, and χV ⊕W = P\ni λi + P\ni μi.\nLemma 3.2. χV ⊗W = χV χW\nProof. Let g ∈G, and let ρ, σ be homomorphisms into V and W, respec-\ntively. Let λ1, . . . , λr be the eigenvalues of ρ(g), and μ1, . . . , μs the eigenval-\nues of σ(g). Then (ρ ⊗σ)(g) is the Kronecker product of the matrices ρ(g)\nand σ(g). So its eigenvalues are all λiμj where 1 ≤i ≤r, 1 ≤j ≤s.\nThen, Tr((ρ ⊗σ)(g)) = P\ni,j λiμj = (P\ni λi)\n P\nj μj\n \n, which is equal to\nTr(ρ(g))Tr(σ(g)).\n3.1\nProjection Formula\nIn this section, we derive a projection formula needed for Goal 1 that allows\nus to determine the multiplicity of an irreducible representation in another\nrepresentation. Given a G-module V , let V G = {v|∀g ∈G, g · v = v}. We\nwill call these elements G-invariant. Let\nφ =\n1\n|G|\nX\ng∈G\ng ∈End(V ),\n(3.1)\nwhere each g, via ρ is considered an element of End(V ).\nLemma 3.3. The map φ : V\n→V is a G-homomorphism; i.e., φ ∈\nHomG(V, V ) = (Hom(V, V ))G.\nProof. The set End(V ) is a G-module, as we saw in last class, via the fol-\nlowing commutative diagram: for any π ∈End(V ), and h ∈G:\nV\nπ\n−−−−→V\n yh\n yh\nV\nh·π\n−−−−→V.\nTherefore π ∈HomG(V, V ) (i.e., π is a G-equivariant morphism) iff\nh · π = π for all h ∈G.\nWhen φ is defined as in equation (3.1) above,\nh · φ =\n1\n|G|\nX\ng\nhgh−1 =\n1\n|G|\nX\ng\ng = φ.\n23\n\nThus\nh · φ = φ, ∀h ∈G,\nand φ : V →V is a G-equivariant morphism, i.e. φ ∈HomG(V, V ).\nLemma 3.4. The map φ is a G-equivariant projection of V onto V G\nProof. For every w ∈W, let\nv = φ(w) =\n1\n|G|\nX\ng∈G\ng · w.\nThen\nh · v = h · φ(w) =\n1\n|G|\nX\ng∈G\nhg · w = v, for any h ∈G.\nSo v ∈V G. That is, Im(φ) ⊆V G. But if v ∈V G, then\nφ(v) =\n1\n|G|\nX\ng∈G\ng · v =\n1\n|G||G|v = v.\nSo V G ⊆Im(φ), and φ is the identity on V G. This means that φ is the\nprojection onto V G.\nLemma 3.5.\ndim(V G) =\n1\n|G|\nX\ng∈G\nχV (g).\nProof. We have: dim(V G) = Tr(φ), because φ is a projection (φ = φ|V G ⊕\nφ|Ker(φ)). Also,\nTr(φ) =\n1\n|G|\nX\ng∈G\nTrV (g) =\n1\n|G|\nX\ng∈G\nχV (g).\nThis gives us a formula for the multiplicity of the trivial representation\n(i.e., dim(V G)) inside V .\nLemma 3.6. Let V, W be G-representations. If V is irreducible, dim(HomG(V, W))\nis the multiplicity of V inside W. If W is irreducible, dim(HomG(V, W)) is\nthe multiplicity of W inside V .\nProof. By Schur’s Lemma.\n24\n\nLet Cclass(G) be the space of class functions on (G), and let (α, β) =\n1\n|G|\nP\ng α(g)β(g) be the Hermitian form on Cclass\nLemma 3.7. If V and W are irreducible G representations, then\n(χV , χW ) =\n1\n|G|\nX\ng∈G\nχV (g)χW (g) =\n(\n1\nif V ∼= W\n0\nif V ≇W.\n(3.2)\nProof. Since Hom(V, W) ∼= V ∗⊗W, χHom(V,W ) = χV ∗χW = χV χW . Now\nthe result follows from Lemmas 3.5 and 3.6.\nLemma 3.8. The characters of the irreducible representations form an or-\nthonormal set.\nProof. Follows from Lemma 3.7.\nIf V ,W are irreducible, then ⟨χV , χW ⟩is 0 if V ̸= W and 1 otherwise.\nThis implies that:\nTheorem 3.1 (Goal 1). A representation is determined completely by its\ncharacter.\nProof. Let V = L\ni V ⊕ai\ni\n. So χV = P\ni aiχVi, and ai = (χV , χVi). This\ngives us a formula for the multiplicity of an irreducible representation in\nanother representation, solely in terms of their characters.\nTherefore, a\nrepresentation is completely determined by its character.\n3.2\nThe characters of irreducible representations\nform a basis\nIn this section, we address Goal 2.\nLet R be the regular representation of G, V an irreducible representation\nof G.\nLemma 3.9.\nR =\nM\nV\nEnd(V, V ),\nwhere V ranges over all irreducible representations of G.\n25\n\nProof. χR(g) is 0 if g is not the identity and |G| otherwise.\n(χR, χV ) =\n1\n|G|\nX\ng∈G\nχR(g)χV (g) =\n1\n|G||G|χV (e) = χV (e) = dim(V )\nLet α : G →C. For any G-module V , let φα,V = P\ng α(g)g : V →V\nExercise 3.1. φα,V is G equivariant (i.e. a G-homomorphism) iffα is a\nclass function.\nProposition 3.1. Suppose α : G →C is a class function, and (α, χV ) = 0\nfor all irreducible representations V . Then α is identically 0.\nProof. If V is irreducible, then, by Schur’s lemma, since φα,V is a G-homomorphism,\nand V is irreducible, φα,V = λId, where λ = 1\nnTr(φα,V ), n = dim(V ). We\nhave:\nλ = 1\nn\nX\ng\nα(g)χV (g) = 1\nn|G|(α, χV ∗).\nNow V is irreducible iffV ∗is irreducible. So λ = 1\nn|G|0 = 0. Therefore,\nφα,V = 0 for any irreducible representation, and hence for any representa-\ntion.\nNow let V be the regular representation. Since g as endomorphisms of\nV are linearly independent, φα,V = 0 implies that α(g) = 0.\nTheorem 3.2. Characters form an orthonormal basis for the space of class\nfunctions.\nProof. Follows from Proposition 3.1, and Lemma 3.8\nIf V = L\ni V ⊕ai\ni\n, and πi : V →V ⊕ai\ni\nis the projection operator. We have\na formula π =\n1\n|G|\nP\ng g for the trivial representation. Analogously:\nExercise 3.2. πi = dimVi\n|G|\nP\ng χVi(g)g.\n26\n\n3.3\nExtending to Infinite Compact Groups\nIn this section, we extend the preceding results to infinite compact groups.\nWe must take some facts as given, since these theorems are much more\ncomplicated than those for finite groups.\nConsider compact G, specifically Un(C), the unitary subgroup of G(C).\nU1(C) is the circle group. Since U1(C) is abelian, all its representations are\none-dimensional.\nSince the group G is infinite, we can no longer sum over it. The idea\nis to replace the sum\n1\n|G|\nP\ng f(g) in the previous setting with\nR\nG f(g)dμ,\nwhere μ is a left-invariant Haar measure on G.\nIn this fashion, we can\nderive analogues of the preceding results for compact groups. We need to\nnormalize, so we set\nR\nG dμ = 1.\nLet ρ : G →GL(V ), where V is a finite dimensional G-representation.\nLet χV (g) = Tr(ρ(g)). Let V = L\ni V ai\ni\nbe the complete decomposition of\nV into irreducible representations.\nWe can again create a projection operator π : V →V G, by letting\nπ =\nR\nG ρ(g)dμ.\nLemma 3.10. We have:\ndim(V G) =\nZ\nG\nχV (g)dμ.\nProof. This result is analogous to Lemma 3.5 for finite groups.\nFor class functions α, β, define an inner product\n(α, β) =\nZ\nG\nα(g)β(g)dμ.\nLemma 3.10 applied to HomG(V, W) gives\n(χV , χW ) =\nZ\nG\nχV χW dμ = dim(HomG(V, W)).\nLemma 3.11. If V, W are irreducible, (χV , χW) = 1 if V and W are iso-\nmorphic, and (χV , χW ) = 0 otherwise.\nProof. This result is analogous to Lemma 3.7 for finite groups.\nLemma 3.12. The irreducible representations are orthonormal, just as in\nLemma 3.8 in the case of finite groups.\n27\n\nIf V is reducible, V = L\ni V ⊕ai\ni\n, then\nai = (χV , χVi) =\nZ\nG\nχV χVidμ.\nHence\nTheorem 3.3. A finite dimensional representation is completely determined\nby its character.\nThis achieves Goal 1 for compact groups. Goal 2 is much harder:\nTheorem 3.4 (Peter-Weyl Theorem). (1) The characters of the irreducible\nrepresentations of G span a dense subset of the space of continuous class\nfunctions.\n(2) The coordinate functions of all irreducible matrix representations of\nG span a dense subset of all continuous functions on G.\nBy a coordinate function of a representation ρ : G →GL(V ), we mean\nthe function on G corresponding to a fixed entry of the matrix form of\nρ(g). For G = U1(C), (2) gives the Fourier series expansion on the circle.\nHence, the Peter-Weyl theorem constitutes a far reaching generalization of\nthe harmonic analyis from the circle to general Un(C).\n28\n\nChapter 4\nRepresentations of the\nsymmetric group\nScribe: Sourav Chakraborty\nGoal:\nTo determine the irreducible representations of the Symmetric\ngroup Sn and their characters.\nReference:\n[FH, F]\nRecall\nLet G be a reductive group. Then\n1. Every finite dimensional representation of G is completely reducible,\nthat is, can be written as a direct sum of irreducible representations.\n2. Every irreducible representation is determined by its character.\nExamples of reductive groups:\n• Continuous: algebraic torus (C∗)m, general linear group GLn(C), spe-\ncial linear group Sln(C), symplectic group Spn(C), orthogonal group\nOn(C).\n• Finite: alternating group An, symmetric group Sn, Gln(Fp), simple lie\ngroups of finite type.\n29\n\n4.1\nRepresentations and characters of Sn\nThe number of irreducible representations of Sn is the same as the the\nnumber of conjugacy classes in Sn since the irreducible characters form a\nbasis of the space of class functions.\nEach permutation can be written\nuniquely as a product of disjoint cycles. The collection of lengths of the\ncycles in a permutation is called the cycle type of the permutation. So a\ncycle type of a permutation on n elements is a partition of n. And in Sn\neach conjugacy class is determined by the cycle type, which, in turn, is\ndetermined by the partition of n. So the number of conjugacy class is same\nas the number of partitions of n. Hence:\nNumber of irreducible representations of Sn = Number of partitions of n\n(4.1)\nLet λ = {λ1 ≥λ2 ≥. . . } be a partition of n; i.e., the size |λ| = P λi is\nn. The Young diagram corresponding to λ is a table shown in Figure 1. It\nis like an inverted staircase. The top row has λ1 boxes, the second row has\nλ2 boxes and so on. There are exactly n boxes.\nrow 1\nrow 5\nrow 6\nrow 2\nrow 3\nrow 4\nFigure 4.1: Row i has λi number of boxes\nFor a given partition λ, we want to construct an irreducible representa-\ntion Sλ, called the Specht-module of Sn for the partition λ, and calculate\nthe character of Sλ. We shall give three constructions of Sλ.\n4.1.1\nFirst Construction\nA numbering T of a Young diagram is a filling of the boxes in its table\nwith distinct numbers from 1, . . . , n. A numbering of a Young diagram is\n30\n\nalso called a tableau. It is called a standard tableaux if the numbers are\nstrictly increasing in each row and column. By Tij we mean the value in the\ntableaux at i-th row and j-th column. We associate with each tableaux T a\npolynomial in C[X1, X2, . . . , Xn]:\nfT = ΠjΠi<i′(XTij −XTi′j).\nLet Sλ be the subspace of C[X1, X2, . . . , Xn] spanned by fT ’s, where T\nranges over all tableaux of shape λ. It is a representation of Sn. Here Sn\nacts on C[X1, X2, . . . , Xn] as:\n(σ.f)(X1, X2, . . . , Xn) = f(Xσ(1), Xσ(2), . . . , Xσ(n))\nTheorem 4.1.\n1. Sλ is irreducible.\n2. Sλ ̸≈Sλ′ if λ ̸= λ′\n3. The set {fT }, where T ranges over standard tableau of shape λ, is a\nbasis of Sλ.\n4.1.2\nSecond Construction\nLet T be a numbering of a Young diagram with distinct numbers from\n{1, . . . , n}. An element σ in Sn acts on T in the usual way by permuting\nthe numbers. Let R(T), C(T) ⊂Sn be the sets of permutations that fix\nthe rows and columns of T, respectively. We have R(σT) = σR(T)σ−1 and\nC(σT) = σR(T)σ−1. We say T ≡T ′ if the rows of T and T ′ are the same\nup to ordering. The equivalence class of T, called the tabloid, is denoted by\n{T}. Its orbit is isomorphic to Sn/R(T).\nLet C[Sn] be the group algebra of Sn. Representations of Sn are the same\nas the representations of C[Sn]. The element aT = P\np∈R(T) p in C[Sn] is\ncalled the row symmetrizer, bT = P\nq∈C(T) sign(q)q the column symmetrizer,\nand cT = aT bT the Young symmetrizer.\nLet\nVT = bT .{T} =\nX\nq∈C(T)\nsign(q){qT}.\nThen σ.VT = VσT . Let Sλ be the span of all VT ’s, where T ranges over\nall numberings of shape λ.\nTheorem 4.2.\n1. Sλ is irreducible\n31\n\n2. Sλ ̸≈Sλ′ if λ ̸= λ′.\n3. The set {VT |T standard} forms a basis for Sλ.\n4.1.3\nThird Construction\nLet T be a canonical numbering of shape λ = (λ1, . . . , λk). By this, we mean\nthe first row is numbered by 1, . . . , λ1, the second row by λ1 + 1, . . . λ1 + λ2,\nand so on, and the rows are increasing. Let aλ = aT , bλ = bT , and cλ =\ncT = aT .bT .\nThen Sλ = C[Sn].cλ is a representation of Sn from the left.\nTheorem 4.3.\n1. Sλ is irreducible\n2. Sλ ̸∼= Sλ′ if λ ̸= λ′.\n3. The basis: an exercise.\n4.1.4\nCharacter of Sλ [Frobenius character formula]\nLet i = (i1, i2, . . . , ik) be such that P\nj jij = n. Let Ci be the conjugacy\nclass consisting of permutations with ij cycles of length j. Let χλ be the\ncharacter of Sλ. The goal is to find χλ(Ci).\nLet λ : λ1 ≥· · · ≥λk be a partition of length k. Given k variables\nX1, X2, . . . , Xk, let\nPj(X) =\nX\nXj\ni ,\nbe the power sum, and\n∆(X) = Πi<j(Xj −Xi)\nthe discriminant. Let f(X) be a formal power series on Xi’s. Let [f(X)]l1,l2,...,lk\ndenote the coefficient of Xl1\n1 Xl2\n2 . . . Xlk\nk in f(X). Let li = λi + k −i.\nTheorem 4.4 (Frobenius Character Formula).\nχλ(Ci) =\n \n∆(X) · ΠjPj(X)ij \nl1,l2,...,lk\n32\n\n4.2\nThe first decision problem in GCT\nNow we can state the first hard decision problem in representation theory\nthat arises in the context of the flip. Let Sα and Sβ be two Specht modules\nof Sn. Since Sn is reductive, Sα ⊗Sβ decomposes as\nSα ⊗Sβ =\nM\nkλ\nαβSλ\nHere kλ\nαβ is called the Kronecker coefficient.\nProblem 4.1. (Kronecker problem) Given λ, α and β decide if kλ\nαβ > 0.\nConjecture 4.1 (GCT6). This can be done in polynomial time; i.e. in time\npolynomial in the bit lengths of the inputs λ, α and β.\n33\n\n55\n34\n\nChapter 5\nRepresentations of GLn(C)\nScribe: Joshua A. Grochow\nGoal: To determine the irreducible representations of GLn(C) and their\ncharacters.\nReferences: [FH, F]\nThe goal of today’s lecture is to classify all irreducible representations\nof GLn(C) and compute their characters. We will go over two approaches,\nthe first due to Deruyts and the second due to Weyl.\nA polynomial representation of GLn(C) is a representation ρ : GLn(C) →\nGL(V ) such that each entry in the matrix ρ(g) is a polynomial in the entries\nof the matrix g ∈GLn(C).\nThe main result is that the polynomial irreducible representations of\nGLn(C) are in bijective correspondence with Young diagrams λ of height\nat most n, i.e. λ1 ≥λ2 ≥· · · ≥λn ≥0. Because of the importance of\nWeyl’s construction (similar constructions can be used on many other Lie\ngroups besides GLn(C)), the irreducible representation corresponding to λ\nis known as the Weyl module Vλ.\n5.1\nFirst Approach [Deruyts]\nLet X = (xij) be a generic n × n matrix with variable entries xij. Consider\nthe polynomial ring C[X] = C[x11, x12, . . . , xnn].\nThen GLn(C) acts on\nC[X] by (A ◦f)(X) = f(ATX) (it is easily checked that this is in fact a left\naction).\nLet T be a tableau of shape λ. To each column C of T of length r, we\nassociate an r × r minor of X as follows: if C has the entries i1, . . . , ir, then\n35\n\ntake from the first r columns of X the rows i1, . . . , ir. Visually:\nC =\n \n \n \ni1\n...\nir\n \n \n −→eC =\n1\n· · ·\nr\n↓\n↓\ni1 →\ni2 →\n...\nir →\n \n \n \n \n \n \n \n \n \n \n \nxi1,1\n· · ·\nxi1,r\n· · ·\nxi1,n\nxi2,1\n· · ·\nxi2,r\n· · ·\nxi2,n\n...\n...\n...\nxir,1\n· · ·\nxir,r\n· · ·\nxir,n\n \n \n \n \n \n \n \n \n \n \n \n(Thus if there is a repeated number in the column C, eC = 0, since\nthe same row will get chosen twice.) Using these monomials eC for each\ncolumn C of the tableau T, we associate a monomial to the entire tableau,\neT = Q\nC eC. (Thus, if in any column of T there is a repeated number,\neT = 0. Furthermore, the numbers must all come from {1, . . . , n} if they\nare to specify rows of an n × n matrix.\nSo we restrict our attention to\nnumberings of T from {1, . . . , n} in which the numbers in any given column\nare all distinct.)\nLet Vλ be the vector space generated by the set {eT }, where T ranges over\nall such numberings of shape λ. Then GLn(C) acts on Vλ: for g ∈GLn(C),\neach row of gX is a linear combination of the rows of X, and since eC is a\nminor of X, g · eC is a linear combination of minors of X of the same size,\ni.e. g(eC) = P\nD ag\nC,DeD (this follows from standard linear algebra). Then\ng(eT )\n=\ng(eC1eC2 · · · eCk)\n=\n X\nD\nag\nC1,DeD\n!\n· · ·\n X\nD\nag\nCk,DeD\n!\nIf we expand this product out, we find that each term is in fact eT ′ for some\nT ′ of the appropriate shape. We then have the following theorem:\nTheorem 5.1.\n1. Vλ is an irreducible representation of GLn(C).\n2. The set {eT |T is a semistandard tableau of shape λ} is a basis for Vλ.\n(Recall that a semistandard tableau is one whose numbering is weakly\nincreasing across each row and strictly increasing down each column.)\n3. Every polynomial irreducible representation of GLn(C) of degree d is\nisomorphic to Vλ for some partition λ of d of height at most n.\n36\n\n4. Every rational irreducible representation of GLn(C) (each entry of ρ(g)\nis a rational function in the entries of g ∈GLn(C)) is isomorphic to\nVλ⊗detk for some partition λ of height at most n and for some integer\nk (where det is the determinant representation).\n5. (Weyl’s character formula) Define the character χλ of Vλ by χλ(g) =\nTr(ρ(g)), where ρ : GLn(C) →GL(Vλ) is the representation map.\nThen, for g ∈GLn(C) with eigenvalues x1, . . . , xn,\nχλ(g) = Sλ(x1, . . . , xn) :=\n xλi+n−i\nj\n \n xn−i\nj\n \n(where |yi\nj| is the determinant of the n × n matrix whose entries are\nyij = yi\nj, so, e.g., the determinant in the denominator is the usual van\nder Monde determinant, which is equal to Q\ni<j(xi −xj)). Here Sλ is\na polynomial, called the Schur polynomial.\nNB: It turns out that all holomorphic representations of GLn(C) are\nrational, and, by part (4) of the theorem, the Weyl modules classify all such\nrepresentations up to scalar multiplication by powers of the determinant.\nWe’ll give here a very brief introduction to the Schur polynomial in-\ntroduced in the above theorem, and explain why the Schur polynomial Sλ\nassociated to λ gives the character of Vλ.\nLet λ be a partition, and T a semistandard tableau of shape λ. Define\nx(T) = Q\ni xμi(T)\ni\n∈C[x1, . . . , xn], where μi(T) is the number of times i\nappears in T. Then it can be shown [F] that\nSλ(x1, . . . , xn) =\nX\nT\nx(T),\nwhere the sum is taken over all semistandard tableau of shape λ.\nProposition 5.1. Sλ(x1, . . . , xn) is the character of Vλ, where x1, . . . , xn\ndenote the eigenvalues of an element of GLn(C).\nProof. It suffices to show this diagonalizable g ∈GLn(C), since the diago-\nnalizable matrices are dense in GLn(C).\nSo let g ∈GLn(C) be diagonalizable with eigenvalues x1, . . . , xn. We\ncan assume that g is diagonal. If not, let A be a matrix that diagonalizes\ng.\nSo AgA−1 is diagonal with x1, . . . , xn as its diagonal entries.\nIf ρ :\nGLn(C) →GL(Vλ) is the representation corresponding to the module Vλ,\n37\n\nthen conjugate ρ by A to get ρ′ : GLn(C) →GL(Vλ) defined by ρ′(h) =\nAρ(h)A−1. In particular, since trace is invariant under conjugation, ρ and\nρ′ have the same character. The module corresponding to ρ′ is simply A·Vλ,\nwhich is clearly isomorphic to Vλ since A is invertible. Thus to compute the\ncharacter χλ(g), it suffices to compute the character of g under ρ′, i.e., when\ng is diagonal, as we shall assume now.\nWe will show that eT is an eigenvector of g with eigenvalue x(T), i.e.\ng(eT ) = x(T)eT . Then since {eT |T is a semistandard tableau of shape λ} is\na basis for Vλ, the trace of g on Vλ will just be P\nT x(T), where the sum is\nover semistandard T of shape λ; this is exactly Sλ(x1, . . . , xn).\nWe reduce to the case where T is a single column. Suppose the claim\nis true for all columns C. Then since eT is a product of eC where C is\na column, the corresponding eigenvalue of eT will be Q\nC x(C) (where the\nproduct is taken over the columns C of T), which is exactly x(T).\nSo assume T is a single column, say with entries i1, . . . , ir. Then eT is\nsimply the above-mentioned r×r minor of the generic n×n matrix X = (xij)\n(do not confuse the double-indexed entries of the matrix X with the single-\nindexed eigenvalues of g). Since g is diagonal, gt = g. So (g ◦eT )(X) =\neT (gtX) = eT (gX). Thus g multiplies the ij-th column by xij. Thus its\neffect on eT is simply to multiply it by Qr\nj=1 xij, which is exactly x(T).\n5.1.1\nHighest weight vectors\nThe subgroup B ⊂GLn(C) of lower triangular invertible matrices, called\nthe Borel subgroup, is solvable.\nSo every irreducible representation of B\nis one-dimensional. A weight vector for GLn(C) is a vector v which is an\neigenvector for every matrix b ∈B.\nIn other words, there is a function\nλ : B →C such that b · v = λ(b)v for all b ∈B. The restriction of λ to the\nsubgroup of diagonal matrices in B is known as the weight of v.\nAs we showed in the proof of the above proposition,\n \n \n \nx1\n...\nxn\n \n \n eT = x(T)eT = xλ1\n1 · · · xλn\nn eT .\nSo eT is a weight vector with weight x(T). Thus Theorem 5.1 (2) gives a\nbasis consisting entirely of weight vectors. We abbreviate the weight x(T)\nby the sequence of exponents (λ1, . . . , λn). We say eT is a highest weight\nvector if its weight is the highest in the lexicographic ordering (using the\nabove sequence notation for the weight).\n38\n\nEach Vλ has a unique (up to scalars) B-invariant vector, which turns\nout to be the highest weight vector: namely eT , where T is canonical. For\nexample, for λ = (5, 3, 2, 2, 1), the canonical T is:\nT =\n1 1 1 1 1\n2 2 2\n3 3\n4 4\n5\nNote that the weight of such eT is (λ1, . . . , λn), so that the highest weight\nvector uniquely determines λ, and thus the entire representation Vλ. (This\nis a general feature of highest weight vectors in the representation theory of\nLie algebras and Lie groups.) Thus the irreducible representations GLn(C)\nare in bijective correspondence wih the highest weights of GLn(C), i.e. the\nsequences of exponents of the eigenvalues of the B-invariant eigenvectors.\n5.2\nSecond Approach [Weyl]\nLet V = Cn and consider the d-th tensor power V ⊗d. The group GLn(C)\nacts on V ⊗d on the left by the diagonal action\ng(v1 ⊗· · · ⊗vd) = gv1 ⊗· · · ⊗gvd (g ∈GL(V ))\nwhile the symmetric group Sd acts on the right by\n(v1 ⊗· · · ⊗vd)τ = v1τ ⊗· · · ⊗vdτ (τ ∈Sd) .\nThese two actions commute, so V ⊗d is a representation of H = GLn(C) ×\nSd. Every irreducible representation of H is of the form U ⊗W for some\nirreducible representation U of GLn(C) and some irreducible representation\nW of Sd. Since both GLn(C) and Sd are reductive (every finite-dimensional\nrepresentation is a direct sum of irreducible representations), their product\nH is reductive as well. So there are some partitions α and β and integers\nmαβ such that\nV ⊗d =\nM\n(Vα ⊗Sβ)mαβ,\nwhere Vα are Weyl modules and Sβ are Specht modules (irreducible repre-\nsentations of the symmetric group Sd).\nTheorem 5.2. V ⊗d = L\nλ Vλ⊗Sλ, where the sum is taken over partitions λ\nof d of height at most n. (Note that each summand appears with multiplicity\none, so this is a “multiplicity-free” decomposition.)\n39\n\nNow, let T be any standard tableau of shape λ, and recall the Young\nsymmetrizer cT from our discussion of the irreducible representations of Sd.\nThen V ⊗dcT is a representation of GLn(C) from the left (since cT ∈C[Sd]\nacts on the right, and the left action of GLn(C) and the right action of Sd\ncommute.)\nTheorem 5.3. V ⊗dcT ∼= Vλ\nThus\nV ⊗d =\nM\nλ:|λ|=d\nM\nstd. tableau\nT of shape λ\nV ⊗dcT ,\nwhere |λ| = P λi denotes the size of λ. In particular, Vλ occurs in V ⊗d with\nmultiplicity dim(Sλ).\nFinally, we construct a basis for V ⊗dcT . A bitableau of shape λ is a pair\n(U, T) where U is a semistandard tableau of shape λ and T is a standard\ntableau of shape λ. (Recall that the the semistandard tableau of shape λ\nare in natural bijective correspondence with a basis for the Weyl module Vλ,\nwhile the standard tableau of shape λ are in natural bijective correspondence\nwith a basis for the Specht module Sλ.)\nTo each bitableau we associate a vector e(U,T) = ei1 ⊗· · · ⊗eid where\nij is defined as follows. Each number 1, . . . , n appears in T exactly once.\nThe number ij is the entry of U in the same location as the number j in T;\npictorially:\nU\nT\nij\nj\nThen:\nTheorem 5.4. The set {e(U,T)cT } is a basis for V ⊗dcT .\n40\n\nChapter 6\nDeciding nonvanishing of\nLittlewood-Richardson\ncoefficients\nScribe: Hariharan Narayanan\nGoal:\nTo show that nonvanishing of Littlewood-Richardson coefficients\ncan be decided in polynomial time.\nReferences: [DM2, GCT3, KT2]\n6.1\nLittlewood-Richardson coefficients\nFirst we define Littlewood-Richardson coefficients, which are basic quanti-\nties encountered in representation theory. Recall that the irreducible rep-\nresentations Vλ of GLn(C), the Weyl modules, are indexed by partitions λ,\nand:\nTheorem 6.1 (Weyl). Every finite dimensional representation of GLn(C)\nis completely reducible.\nLet G = GLn(C). Consider the diagonal embedding of G ֒→G × G.\nThis is a group homomorphism. Any G × G module, in particular, Vα ⊗Vβ\ncan also be viewed as a G module via this homomorphism. It then splits\ninto irreducible G-submodules:\n41\n\nVα ⊗Vβ = ⊕γcγ\nαβVλ.\n(6.1)\nHere cγ\nαβ is the multiplicity of Vγ in Vα ⊗Vβ and is known as the Littlewood-\nRichardson coefficient.\nThe character of Vλ is the Schur polynomial Sλ. Hence, it follows from\n(6.1) that the Schur polynomials satisfy the following relation:\nSαSβ = ⊕γcγ\nαβSλ.\n(6.2)\nTheorem 6.2. cγ\nαβ is in PSPACE.\nProof: This easily follows from eq.(6.2) and the definition of Schur poly-\nnomials.\nAs a matter of fact, a stronger result holds:\nTheorem 6.3. cγ\nαβ is in #P.\nRecall that #P ⫅PSPACE.\nProof:\nThis is an immediate consequence of the following Littlewood-\nRichardson rule (formula) for cγ\nαβ. To state it, we need a few definitions.\nGiven partitions γ and α, a skew Young diagram of shape γ/α is the dif-\nference between the Young diagrams for γ and α, with their top-left corners\naligned; cf. Figure 6.2. A skew tableau of shape γ/α is a numbering of the\nboxes in this diagram. It is called semi-standard (SST) if the entries in each\ncolumn are strictly increasing top to bottom and the entries in each row\nare weakly increasing left to right; see Figures 6.1 and 6.2. The row word\nrow(T) of a skew-tableau T is the sequence of numbers obtained by reading\nT left to right, bottom to top; e.g. row(T) for Figure 6.2 is 13312211. It is\ncalled a reverse lattice word, if when read right to left, for each i, the number\nof i’s encountered at any point is at least the number of i + 1’s encountered\ntill that point; thus the row word for Figure 6.2 is a reverse lattice word.\nWe say that T is an LR tableau for given α, β, γ of shape γ/α and content\nβ if\n1. T is an SST,\n2. row(T) is a reverse lattice word,\n3. T has shape γ/α, and\n4. the content of T is β, i. e. the number of i’s in T is βi.\n42\n\n1 1 2 5\n2 2 3\n4\nFigure 6.1: semi-standard Young tableau\n:\n:\n:\n:\n:\n: 1 1\n:\n:\n: 1 2 2\n:\n: 3\n1 3\nFigure 6.2: Littlewood–Richardson skew tableau\nFor example, Figure 6.2 shows an LR tableau with α = (6, 3, 2), β =\n(4, 2, 2) and γ = (8, 6, 3, 2).\nThe Littlewood-Richardson rule [F, FH]: cγ\nαβ is equal to the number\nof LR skew tableaux of shape γ/α and content β.\nRemark: It may be noticed that the Littlewood-Richardson rule depends\nonly on the partitions α, β and γ and not on n, the rank of GLn(C) (as long\nas it is greater than or equal to the maximum height of α, β or γ). For this\nreason, we can assume without loss of generalitity that n is the maximum\nof the heights of α, β and γ, as we shall henceforth.\nNow we express cγ\nαβ as the number of integer points in some polytope\nP γ\nαβ using the Littlewood-Richardson rule:\nLemma 6.1. There exists a polytope P = P γ\nαβ of dimension polynomial in\nn such that the number of integer points in it is cγ\nαβ.\nProof: Let ri\nj(T), i ≤n, j ≤n, denote the number of j’s in the i-th row of\nT. If T is an LR-tableau of shape γ/α with content β then these integers\nsatisfy the following constraints:\n1. Nonnegativity: ri\nj ≥0.\n2. Shape constraints: For i ≤n,\nαi +\nX\nj\nri\nj = γi.\n43\n\n3. Content constraints: For j ≤n:\nX\ni\nri\nj = βj.\n4. Tableau constraints:\nαi+1 +\nX\nk≤j\nri+1\nk\n≤αi +\nX\nk′<j\nri\nk′.\n5. Reverse lattice word constraints: ri\nj = 0 for i < j, and for i ≤n,\n1 < j ≤n:\nX\ni′≤i\nri′\nj ≤\nX\ni′<i\nri′\nj−1.\nLet P γ\nαβ be the polytope defined by these constraints. Then cγ\nαβ is the\nnumber of integer points in this polytope. This proves the lemma.\n□\nThe membership function of the polytope P γ\nαβ is clearly computable in\ntime that is polynomial in the bitlengths of α, β and γ. Hence cγ\nαβ belongs\nto #P. This proves the theorem.\n□\nThe complexity-theoretic content of the Littlewood-Richardson rule is\nthat it puts a quantity, which is a priori only in PSPACE, in #P. We also\nhave:\nTheorem 6.4 ([H]). cγ\nαβ is #P-complete.\nFinally, the main complexity-theoretic result that we are interested in:\nTheorem 6.5 (GCT3, Knutson-Tao, De Loera-McAllister). The problem\nof deciding nonvanishing of cγ\nαβ is in P, i. e. , it can be solved in time that is\npolynomial in the bitlengths of α, β and γ. In fact, it can solved in strongly\npolynomial time [GCT3].\nHere, by a strongly polynomial time algorithm, we mean that the num-\nber of arithmetic steps +, −, ∗, ≤, . . . in the algorithm is polynomial in the\nnumber of parts of α, β and γ regardless of their bitlengths, and the bit-\nlength of each intermediate operand is polynomial in the bitlengths of α, β\nand γ.\nProof: Let P = P γ\nαβ be the polytope as in Lemma 6.1.\nAll vertices of\nP have rational coefficients. Hence, for some positive integer q, the scaled\npolytope qP has an integer point. It follows that, for this q, cqγ\nqα,qβ is positive.\nThe saturation Theorem [KT] says that, in this case, cγ\nα,β is positive. Hence,\nP contains an integer point. This implies:\n44\n\nLemma 6.2. If P ̸= ∅then cγ\nαβ > 0.\nBy this lemma, to decide if cγ\nαβ > 0, it suffices to test if P is nonempty.\nThe polytope P is given by Ax ≤b where the entries of A are 0 or 1–\nsuch linear programs are called combinatorial. Hence, this can be done in\nstrongly polynomial time using Tardos’ algorithm [GLS] for combinatorial\nlinear programming. This proves the theorem.\n□\nThe integer programming problem is NP-complete, in general. However,\nlinear programming works for the specific integer programming problem here\nbecause of the saturation property [KT].\nProblem: Find a genuinely combinatorial poly-time algorithm for deciding\nnon-vanishing of cγ\nαβ.\n45\n\nChapter 7\nLittlewood-Richardson\ncoefficients (cont)\nScribe: Paolo Codenotti\nGoal: We continue our study of Littlewood-Richardson coefficients and\ndefine Littlewood-Richardson coefficients for the orthogonal group On(C).\nReferences: [FH, F]\nRecall\nLet us first recall some definitions and results from the last class. Let cγ\nα,β\ndenote the Littlewood-Richardson coefficient for GLn(C).\nTheorem 7.1 (last class). Non-vanishing of cγ\nα,β can be decided in poly(⟨α⟩, ⟨β⟩, ⟨γ⟩)\ntime, where ⟨⟩denotes the bit length.\nThe positivity hypotheses which hold here are:\n• cγ\nα,β ∈#P, and more strongly,\n• Positivity Hypothesis 1 (PH1): There exists a polytope P γ\nα,β of\ndimension polynomial in the heights of α, β and γ such that cγ\nα,β =\nφ(P γ\nα,β), where φ indicates the number of integer points.\n• Saturation Hypothesis (SH): If ckγ\nkα,kβ ̸= 0 for some k ≥1, then\ncγ\nα,β ̸= 0 [Saturation Theorem].\nProof. (of theorem)\nPH1 + SH + Linear programming.\n46\n\nThis is the general form of algorithms in GCT. The main principle is\nthat linear programming works for integer programming when PH1 and SH\nhold.\n7.1\nThe stretching function\nWe define ecγ\nα,β(k) = ckγ\nkα,kβ.\nTheorem 7.2 (Kirillov, Derkesen Weyman [Der, Ki]). ecγ\nα,β(k) is a polyno-\nmial in k.\nHere we prove a weaker result. For its statement, we will quickly review\nthe theory of Ehrhart quasipolynomials (cf. Stanley [S]).\nDefinition 7.1. (Quasipolynomial) A function f(k) is called a quasipoly-\nnomial if there exist polynomials fi, 1 ≤i ≤l, for some lsuch that\nf(k) = fi(k) if k ≡i mod l.\nWe denote such a quasipolynomial f by f = (fi). Here lis called the period\nof f(k) (we can assume it is the smallest such period).\nThe degree of a\nquasipolynomial f is the max of the degrees of the fi’s.\nNow let P ⊆Rm be a polytope given by Ax ≤b. Let φ(P) be the number\nof integer points inside P. We define the stretching function fP(k) = φ(kP),\nwhere kP is the dilated polytope defined by Ax ≤kb.\nTheorem 7.3. (Ehrhart) The stretching function fP(k) is a quasipolyno-\nmial. Furthermore, fP (k) is a polynomial if P is an integral polytope (i.e.\nall vertices of P are integral).\nIn view of this result, fP(k) is called the Ehrhart quasi-polynomial of\nP. Now ecγ\nα,β(k) is just the Ehrhart quasipolynomial of P γ\nα,β, and cγ\nα,β =\nφ(P γ\nα,β), the number of integer points in P γ\nα,β. Moreover P γ\nα,β is defined by\nthe inequality Ax ≤b, where A is constant, and b is a homogeneous linear\nform in the coefficients of α, β, and γ.\nHowever, P γ\nα,β need not be integral. Therefore Theorem (7.2) does not\nfollow from Ehrhart’s result. Its proof needs representation theory.\nDefinition 7.2. A quasipolynomial f(k) is said to be positive if all the\ncoefficients of fi(k) are nonnegative. In particular, if f(k) is a polynomial,\nthen it’s positive if all its coefficients are nonnegative.\n47\n\nThe Ehrhart quasipolynomial of a polytope is positive only in exceptional\ncases. In this context:\nPH2 (positivity hypothesis 2) [KTT]: The polynomial ecγ\nα,β(k) is positive.\nThere is considerable computer evidence for this.\nProposition 7.1. PH2 implies SH.\nProof. Look at:\nc(k) = ecγ\nα,β(k) =\nX\naiki.\nIf all the coefficients ai are nonnegative (by PH2), and c(k) ̸= 0, then c(1) ̸=\n0.\nSH has a proof involving algebraic geometry [B]. Therefore we suspect\nthat the stronger PH2 is a deep phenomenon related to algebraic geometry.\n7.2\nOn(C)\nSo far we have talked about GLn(C).\nNow we move on to the orthogo-\nnal group On(C). Fix Q, a symmetric bilinear form on Cn; for example,\nQ(V, W) = V T W.\nDefinition 7.3. The orthogonal group On(C) ⊆GLn(C) is the group con-\nsisting of all A ∈GLn(C) s.t.\nQ(AV, AW) = Q(V, W) for all V and\nW ∈Cn. The subgroup SOn(C) ⊆SLn(C), where SLn(C) is the set of\nmatrices with determinant 1, is defined similarly.\nTheorem 7.4 (Weyl). The group On(C) is reductive\nProof. The proof is similar to the reductivity of GLn(C), based on Weyl’s\nunitary trick.\nThe next step is to classify all irreducible polynomial representations of\nOn(C).\nFix a partition λ = (λ1 ≥λ2 ≥. . . ) of length at most n.\nLet\n|λ| = d = P λi be its size. Let V = Cn, V ⊗d = V ⊗· · · ⊗V d times, and\nembed the Weyl module Vλ of GLn(C) in V ⊗d as per Theorem 5.3. Define\na contraction map\nφp,q : V ⊗d →V ⊗(d−2)\nfor 1 ≤p ≤q ≤d by:\nφp,q(vi1 ⊗· · · ⊗vid) = Q(vip, viq)(vi1 ⊗· · · ⊗c\nvip ⊗· · · ⊗c\nviq ⊗· · · ⊗vid),\n48\n\nλ\nFigure 7.1: The first two columns of the partition λ are highlighted.\nwhere c\nvip means omit vip.\nIt is On(C)-equivariant, i.e. the following diagram commutes:\nV ⊗d\nφp,q\n−−−−→V ⊗d−2\n yσ∈On(C)\n yσ∈On(C)\nV ⊗d\nφp,q\n−−−−→V ⊗d−2\nLet\nV [d] =\n\\\npq\nker(φp,q).\nBecause the maps are equivariant, each kernel is an On(C)-module, and V [d]\nis an On(C)-module. Let V[λ] = V [d] T Vλ, where Vλ ⊆V ⊗d is the embedded\nWeyl module as above. Then V[λ] is an On(C)-module.\nTheorem 7.5 (Weyl). V[λ] is an irreducible representation of On(C). More-\nover, the following two conditions hold:\n1. If n is odd, then V[λ] is non-zero if and only if the sum of the lengths\nof the first two columns of λ is ≤n (see figure 7.1).\n2. If n is odd, then each polynomial irreducible representation is isomor-\nphic to V[λ] for some λ.\nLet\nV[λ] ⊗V[μ] = ⊗γdγ\nλ,μV[γ]\n49\n\nbe the decomposition of V[λ]⊗V[μ] into irreducibles. Here dγ\nλ,μ is called the\nLittlewood-Richardson coefficient of type B. The types of various connected\nreductive groups are defined as follows:\n• GLn(C): type A\n• On(C), n odd: type B\n• Spn(C): type C\n• On(C), n even: type D\nThe Littlewood-Richardson coefficient can be defined for any type in a sim-\nilar fashion.\nTheorem 7.6 (Generalized Littlewood-Richardson rule). The Littlewood-\nRichardson coefficient dγ\nλ,μ ∈#P. This also holds for any type.\nProof. The most transparent proof of this theorem comes through the theory\nof quantum groups [K]; cf. Chapter 20.\nAs in type A this leads to:\nHypothesis 7.1 (PH1). There exists a polytope P γ\nλ,μ of dimension polyno-\nmial in the heights of λ, μ and γ such that:\n1. dγ\nλ,μ = φ(P γ\nλ,μ), the number of integer points in P γ\nλ,μ, and\n2. edγ\nλ,μ(k) = dkγ\nkλ,kμ is the Ehrhart quasipolynomial of P γ\nλ,μ.\nThere are several choices for such polytopes; e.g. the BZ-polytope [BZ].\nTheorem 7.7 (De Loera, McAllister [DM2]). The stretching function edγ\nλ,μ(k)\nis a quasipolynomial of degree at most 2; so also for types C and D.\nA verbatim translation of the saturation property fails here [Z]): there\nexist λ, μ and γ such that d2γ\n2λ,2μ ̸= 0 but dγ\nλ,μ = 0. Therefore we change the\ndefinition of saturation:\nDefinition 7.4. Given a quasipolynomial f(k) = (fi), index(f) is the\nsmallest i such that fi(k) is not an identically zero polynomial.\nIf f(k)\nis identically zero, index(f) = 0.\nDefinition 7.5. A quasipolynomial f(k) is saturated if f(index(f)) ̸= 0.\nIn particular, if index(f) = 1, then f(k) is saturated if f(1) ̸= 0.\n50\n\nA positive quasi-polynomial is clearly saturated.\nPositivity Hypothesis 2 (PH2) [DM2]: The stretching quasipolyomial\nedγ\nλ,μ(k) is positive.\nThere is considerable evidence for this.\nSaturation Hypothesis (SH): The stretching quasipolynomial edγ\nλ,μ(k) is\nsaturated.\nPH2 implies SH.\nTheorem 7.8. [GCT5] Assuming SH (or PH2), positivity of the Littlewood-\nRichardson coefficient dγ\nλ,μ of type B can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨γ⟩)\ntime.\nThis is also true for all types.\nProof. next class.\n51\n\nChapter 8\nDeciding nonvanishing of\nLittlewood-Richardson\ncoefficients for On(C)\nScribe: Hariharan Narayanan\nGoal: A polynomial time algorithm for deciding nonvanishing of Littlewood-\nRichardson coefficients for the orthogonal group assuming SH.\nReference: [GCT5]\nLet dν\nλ,μ denote the Littlewood-Richardson coefficient of type B (i.e. for\nthe orthogonal group On(C), n odd) as defined in the earlier lecture. In this\nlecture we describe a polynomial time algorithm for deciding nonvanishing\nof dν\nλ,μ assuming the following positivity hypothesis PH2.\nSimilar result\nalso holds for all types, though we shall only concentrate on type B in this\nlecture.\nLet ̃dν\nλ,μ(k) = dkν\nkλ,kμ denote the associated stretching function.\nIt is\nknown to be a quasi-polynomial of period at most two [DM2]. This means\nthere are polynomials f1(k) and f2(k) such that\ndkν\nkλ,kμ =\n f1(k),\nif k is odd;\nf2(k),\nif k is even.\nPositivity Hypothesis (PH2) [DM2]: The stretching quasi-polynomial\n ̃dν\nλ,μ(k) is positive.\nThis means the coefficients of f1 and f2 are all non-\nnegative.\nThe main result in this lecture is:\n52\n\nTheorem 8.1. [GCT5] If PH2 holds, then the problem of deciding the posi-\ntivity (nonvanishing) of dν\nλμ belongs to P. That is, this problem can be solved\nin time polynomial in the bitlengths of λ, μ and ν.\nWe need a few lemmas for the proof.\nLemma 8.1. If PH2 holds, the following are equivalent:\n(1) dν\nλμ ≥1.\n(2) There exists an odd integer k such that dkν\nkλ kμ ≥1.\nProof: Clearly (1) implies (2). By PH2, there exists a polynomial f1\nwith non-negative coefficients such that\n∀odd k, f1(k) = dkν\nkλ kμ.\nSuppose that for some odd k, dkν\nkλ kμ ≥1. Then f1(k) ≥1. Therefore f1 has\nat least one non-zero coefficient. Since all coefficients of f1 are nonnegative,\ndν\nλμ = f1(1) > 0. Since dν\nλμ is an integer, (1) follows.\n□\nDefinition 8.1. Let Z<2> be the subring of Q obtained by localizing Z at\n2:\nZ<2> :=\n p\nq | p, q −1\n2\n∈Z\n \n.\nThis ring consists of all fractions whose denominators are odd.\nLemma 8.2. Let P ∈Rd be a convex polytope specified by Ax ≤B, xi ≥0\nfor all i, where A and B are integral. Let Aff(P) denote its affine span. The\nfollowing are equivalent:\n(1) P contains a point in Zd\n<2>.\n(2) Aff(P) contains a point in Zd\n<2>.\nProof: Since P ⊆Aff(P), (1) implies (2). Now suppose (2) holds. We\nhave to show (1). Let z ∈Zd\n<2> ∩Aff(P).\nFirst, consider the case when Aff(P) is one dimensional. In this case, P\nis the line segment joining two points x and y in Qd. The point z can be\nexpressed as an affine linear combination, z = ax + (1−a)y for some a ∈Q.\nThere exists q ∈Z such that qx ∈Zd\n<2> and qy ∈Zd\n<2>. Note that\n{z + λ(qx −qy) | λ ∈Z<2>} ⊆Aff(P) ∩Zd\n<2>.\n53\n\nSince Z<2> is a dense subset of Q, the l.h.s. and hence the r.h.s. is a dense\nsubset of Aff(P). Consequently, P ∩Zd\n<2> ̸= ∅.\nNow consider the general case. Let u be any point in the interior of P\nwith rational coordinates, and L the line through u and z. By restricting to\nL, the lemma reduces to the preceding one dimensional case.\n□\nLemma 8.3. Let\nP = {x | Ax ≤B, (∀i)xi ≥0} ⊆Rd\nbe a convex polytope where A and B are integral. Then, it is possible to\ndetermine in polynomial time whether or not Aff(P) ∩Zd\n<2> = ∅.\nProof: Using Linear Programming [Kha79, Kar84], a presentation of\nthe form Cx = D can be obtained for Aff(P) in polynomial time, where C\nis an integer matrix and D is a vector with integer coordinates. We may\nassume that C is square since this can be achieved by padding it with 0’s\nif necessary, and extending D. The Smith Normal Form over Z of C is a\nmatrix S such that C = USV where U and V are unimodular and S has\nthe form\n \n \n \n \n \ns11\n0\n. . .\n0\n0\ns22\n. . .\n0\n...\n...\n...\n0\n0\n0\n. . .\nsdd\n \n \n \n \n \nwhere for 1 ≤i ≤d−1, sii divides si+1 i+1. It can be computed in polynomial\ntime [KB79].\nThe question now reduces to whether USV x = D has a\nsolution x ∈Zd\n<2>. Since V is unimodular, its inverse has integer entries\ntoo, and y := V x ∈Zd\n<2> ⇔x ∈Zd\n<2>.\nThis is equivalent to whether\nSy = U−1D has a solution y ∈Zd\n<2>.\nSince S is diagonal, this can be\nanswered in polynomial time simply by checking each coordinate.\n□\nProof of Theorem 8.1: By [BZ], there exists a polytope P = P ν\nλ,μ\nsuch that the Littlewood-Richardson coefficient dν\nλμ is equal to the number\nof integer points in P. This polytope is such that the number of integer\npoints in the dilated polytope kP is dkν\nkλ kμ. Assuming PH2, we know from\nLemma 8.1 that\nP ∩Zd ̸= ∅⇔(∃odd k), kP ∩Zd ̸= ∅.\nThe latter is equivalent to\nP ∩Zd\n<2> ̸= ∅.\n54\n\nThe theorem now follows from Lemma 8.2 and Lemma 8.3.\n□\nIn combinatorial optimization, LP works if the polytope is integral. In\nour setting, this is not necessarily the case [DM1]: the denominators of the\ncoordinates of the vertices of P can be Ω(l), where l is the total height of\nλ, μ and ν. LP works here nevertheless because of PH2; it can be checked\nthat SH is also sufficient.\n55\n\nChapter 9\nThe plethysm problem\nScribe: Joshua A. Grochow\nGoal: In this lecture we describe the general plethysm problem, state anal-\nogous positivity and saturation hypotheses for it, and state the results from\nGCT 6 which imply a polynomial time algorithm for deciding positivity of\na plethysm constant assuming these hypotheses.\nReference: [GCT6]\nRecall\nRecall that a function f(k) is quasipolynomial if there are functions fi(k)\nfor i = 1, . . . , lsuch that f(k) = fi(k) whenever k ≡i mod l. The number\nlis then the period of f. The index of f is the least i such that fi(k) is\nnot identically zero.\nIf f is identically zero, then the index of f is zero\nby convention. We say f is positive if all the coefficients of each fi(k) are\nnonnegative. We say f is saturated if f(index(f)) ̸= 0. If f is positive, then\nit is saturated.\nGiven any function f(k), we associate to it the rational series F(t) =\nP\nk≥0 f(k)tk.\nProposition 9.1. [S] The following are equivalent:\n1. f(k) is a quasipolynomial of period l.\n2. F(t) is a rational function of the form A(t)\nB(t) where deg A < deg B and\nevery root of B(t) is an l-th root of unity.\n56\n\n9.1\nLittlewood-Richardson Problem [GCT 3,5]\nLet G = GLn(C) and cγ\nα,β the Littlewood-Richardson coefficient – i.e. the\nmultiplicity of the Weyl module Vγ in Vα ⊗Vβ. We saw that the positivity\nof cγ\nα,β can be decided in poly(⟨α⟩, ⟨β⟩, ⟨γ⟩) time, where ⟨·⟩denotes the bit-\nlength. Furthermore, we saw that the stretching function ecγ\nα,β(k) = ckγ\nkα,kβ\nis a polynomial, and the analogous stretching function for type B is a\nquasipolynomial of period at most 2.\n9.2\nKronecker Problem [GCT 4,6]\nNow we study the analogous problem for the representations of the symmet-\nric group (the Specht modules), called the Kronecker problem.\nLet Sα be the Specht module of the symmetric group Sm associated to\nthe partition α. Define the Kronecker coefficient κπ\nλ,μ to be the multiplicity\nof Sπ in Sλ ⊗Sμ (considered as an Sm-module via the diagonal action). In\nother words, write Sλ ⊗Sμ = L\nπ κπ\nλ,μSπ. We have κπ\nλ,μ = (χλχμ, χπ), where\nχλ denotes the character of Sλ. By the Frobenius character formula, this\ncan be computed in PSPACE. More strongly, analogous to the Littlewood-\nRichardson problem:\nConjecture 9.1. [GCT4, GCT6] The Kronecker coefficient κπ\nλ,μ ∈# P.\nIn other words, there is a positive #P-formula for κπ\nλ,μ.\nThis is a fundamental problem in representation theory. More concretely,\nit can be phrased as asking for a set of combinatorial objects I and a char-\nacteristic function χ : {I} →{0, 1} such that χ ∈FP and κπ\nλ,μ = P\nI χ(I).\nContinuing our analogy:\nConjecture 9.2. [GCT6] The problem of deciding positivity of κπ\nλ,μ belongs\nto P.\nTheorem 9.1. [GCT6] The stretching function eκπ\nλ,μ(k) = κkπ\nkλ,kμ is a quasipoly-\nnomial.\nNote that κkπ\nkλ,kμ is a Kronecker coefficient for Skm.\nThere is also a dual definition of the Kronecker coefficients. Namely,\nconsider the embedding\nH = GLn(C) × GLn(C) ֒→G = GL(Cn ⊗Cn),\nwhere (g, h)(v ⊗w) = (gv ⊗hw). Then\n57\n\nProposition 9.2. [FH] The Kronecker coefficient κπ\nλ,μ is the multiplicity of\nthe tensor product of Weyl modules Vλ(GLn(C)) ⊗Vμ(GLn(C)) (this is an\nirreducible H-module) in the Weyl module Vπ(G) considered as an H-module\nvia the embedding above.\n9.3\nPlethysm Problem [GCT 6,7]\nNext we consider the more general plethysm problem.\nLet H = GLn(C), V = Vμ(H) the Weyl module of H corresponding to\na partition μ, and ρ : H →G = GL(V ) the corresponding representation\nmap. Then the Weyl module Vλ(G) of G for a given partition λ can be\nconsidered an H-module via the map ρ. By complete reducibility, we may\ndecompose this H-representation as\nVλ(G) =\nM\nπ\naπ\nλ,μVπ(H).\nThe coefficients aπ\nλ,μ are known as plethsym constants (this definition can\neasily be generalized to any reductive group H). The Kronecker coefficient\nis a special case of the plethsym constant [Ki].\nTheorem 9.2 (GCT 6). The plethysm constant aπ\nλ,μ ∈PSPACE.\nThis is based on a parallel algorithm to compute the plethysm constant\nusing Weyl’s character formula. Continuing in our previous trend:\nConjecture 9.3. [GCT6] aπ\nλ,μ ∈# P and the problem of deciding positivity\nof aπ\nλ,μ belongs to P.\nFor the stretching function, we need to be a bit careful. Define eaπ\nλ,μ =\nakπ\nkλ,μ. Here the subscript μ is not stretched, since that would change G,\nwhile stretching λ and π only alters the representations of G.\nAs in the beginning of the lecture, we can associate a function Aπ\nλ,μ(t) =\nP\nk≥0 eaπ\nλ,μ(k)tk to the plethysm constant. Kirillov conjectured that Aπ\nλ,μ(t) is\nrational. In view of Proposition 9.1, this follows from the following stronger\nresult:\nTheorem 9.3 (GCT 6). The stretching function eaπ\nλ,μ(k) is a quasipolyno-\nmial.\nThis is the main result of GCT 6, which in some sense allows GCT to\ngo forward.\nWithout it, there would be little hope for proving that the\n58\n\npositivity of plethysm constants can be decided in polynomial time.\nIts\nproof is essentially algebro-geometric. The basic idea is to show that the\nstretching function is the Hilbert function of some algebraic variety with nice\n(i.e. “rational”) singularities. Similar results are shown for the stretching\nfunctions in the algebro-geometric problems arising in GCT.\nThe main complexity-theoretic result in [GCT6] shows that, under the\nfollowing positivity and saturation hypotheses (for which there is much ex-\nperimental evidence), the positivity of the plethysm constants can indeed\nbe decided in polynomial time (cf. Conjecture 9.3).\nThe first positivity hypothesis is suggested by Theorem 9.3: since the\nstretching function is a quasipolynomial, we may suspect that it is captured\nby some polytope:\nPositivity Hypothesis 1 (PH1). There exists a polytope P = P π\nλ,μ\nsuch that:\n1. aπ\nλ,μ = φ(P), where φ denotes the number of integer points inside the\npolytope,\n2. The stretching quasipolynomial (cf. Thm. 9.3) eaπ\nλ,μ(k) is equal to the\nEhrhart quasipolynomial fP(k) of P,\n3. The dimension of P is polynomial in ⟨λ⟩, ⟨μ⟩, and ⟨π⟩,\n4. the membership in P π\nλ,μ can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) time, and\nthere is a polynomial time separation oracle [GLS] for P.\nHere (4) does not imply that the polytope P has only polynomially\nmany constraints. In fact, in the plethysm problem there may be a super-\npolynomial number of constraints.\nPositivity Hypothesis 2 (PH2).\nThe stretching quasipolynomial\neaπ\nλ,μ(k) is positive.\nThis implies:\nSaturation Hypothesis (SH). The stretching quasipolynomial is sat-\nurated.\nTheorem 9.3 is essential to state these hypotheses, since positivity and\nsaturation are properties that only apply to quasipolynomials. Evidence for\nPH1, PH2, and SH can be found in GCT 6.\nTheorem 9.4. [GCT6] Assuming PH1 and SH (or PH2), positivity of the\nplethysm constant aπ\nλ,μ can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) time.\n59\n\nThis follows from the polynomial time algorithm for saturated integer\nprogramming described in the next class. As with Theorem 9.3, this also\nholds for more general problems in algebraic geometry.\n60\n\nChapter 10\nSaturated and positive\ninteger programming\nScribe: Sourav Chakraborty\nGoal :\nA polynomial time algorithm for saturated integer programming\nand its application to the plethysm problem.\nReference: [GCT6]\nNotation :\nIn this class we denote by ⟨a⟩the bit-length of the a.\n10.1\nSaturated, positive integer programming\nLet Ax ≤b be a set of inequalities.\nThe number of constraints can be\nexponential. Let P ⊂Rn be the polytope defined by these inequalities. The\nbit length of P is defined to be ⟨P⟩= n + ψ, where ψ is the maximum\nbit-length of a constraint in the set of inequalities. We assume that P is\ngiven by a separating oracle. This means membership in P can be decided\nin poly(⟨P⟩) time, and if x ̸∈P then a separating hyperplane is given as a\nproof as in [GLS].\nLet fP(k) be the Ehrhart quasi-polynomial of P. Quasi-polynomiality\nmeans there exist polynomials fi(k), 1 ≤i ≤l, l the period, so that fP(k) =\nfi(k) if k = i modulo l. Then\nIndex(fP) = min{i|fi(k)not identically 0 as a polynomial}\nThe integer programming problem is called positive if fP(k) is positive\nwhenever P is non-empty, and saturated if fP(k) is saturated whenever P is\nnon-empty.\n61\n\nTheorem 10.1 (GCT6).\n1. Index(fP ) can be computed in time polyno-\nmial in the bit length ⟨P⟩of P assuming that the separation oracle\nworks in poly-⟨P⟩-time.\n2. Saturated and hence positive integer programming problem can be solved\nin poly-⟨P⟩-time.\nThe second statement follow from the first.\nProof. Let Aff(P) denote the affine span of P. By [GLS] we can compute\nthe specifications Cx = d, C and d integral, of Aff(P) in poly(⟨P⟩) time.\nWithout loss of generality, by padding, we can assume that C is square. By\n[KB79] we find the Smith-normal form of C in polynomial time. Let it be\n ̄C. So,\n ̄C = ACB\nwhere A and B are unimodular, and ̄C is a diagonal matrix, where the\ndiagonal entries c1, c2, . . . are such that with ci|ci+1.\nClearly Cx = d iff ̄Cz = ̄d where z = B−1x and ̄d = Ad.\nSo all equations here are of form\n ̄cizi = di\n(10.1)\nWithout loss of generality we can assume that ci and di are relatively\nprime. Let ̃c = lcm(ci).\nClaim 10.1. Index(fP ) = ̃c.\nFrom this claim the theorem clearly follows.\nProof of the claim. Let fP(t) = P\nk≥0 fP(k)tk be the Ehrhart Series of P.\nNow kP will not have an integer point unless ̃c divides k because of\n(10.1).\nHence fP(t) = f ̄P(t ̃c) where ̄P is the stretched polytope ̃cP and f ̄P(s) is\nthe Ehrhart series of ̄P. From this it follows that\nIndex(fP ) = ̃cIndex(f ̄P )\nNow we show that Index(f ̄P) = 1.\nThe equations of ̄P are of the form\nzi = ̃c\nci\ndi\n62\n\nwhere each\n ̃c\nci is an integer.\nTherefore without loss of generality we can\nignore these equations and assume the ̄P is full dimensional.\nThen it suffices to show that ̄P contains a rational point whose denomi-\nnators are all 1 modulo l( ̄P), the period of the quasi-polynomial f ̄P (s).\nThis follows from a simple density argument that we saw earlier (cf. the\nproof of Lemma 8.2).\nFrom this the claim follows.\n10.2\nApplication to the plethysm problem\nNow we can prove the result stated in the last class:\nTheorem 10.2. Assuming PH1 and SH, positivity of the plethysm constant\naπ\nλ,μ can be decided in time polynomial in ⟨λ⟩, ⟨μ⟩and ⟨π⟩.\nProof. Let P = P π\nλ,μ be the polytope as in PH1 such that aπ\nλ,μ is the number\nof integer points in P. The goal is to decide if P contains an integer point.\nThis integer programming problem is saturated because of SH. Hence the\nresult follows from Theorem 10.1.\n63\n\nChapter 11\nBasic algebraic geometry\nScribe: Paolo Codenotti\nGoal: So far we have focussed on purely representation-theoretic aspects of\nGCT. Now we have to bring in algebraic geometry. In this lecture we review\nthe basic definitions and results in algebraic geometry that will be needed\nfor this purpose. The proofs will be omitted or only sketched. For details,\nsee the books by Mumford [Mm] and Fulton [F].\n11.1\nAlgebraic geometry definitions\nLet V = Cn, and v1, . . . , vn the coordinates of V .\nDefinition 11.1.\n• Y is an affine algebraic set in V if Y is the set of\nsimultaneous zeros of a set of polynomials in vi’s.\n• An algebraic set that cannot be written as the union of two proper\nalgebraic sets Y1 and Y2 is called irreducible.\n• An irreducible affine algebraic set is called an affine variety.\n• The ideal of an affine algebraic set Y is I(Y ), the set of all polynomials\nthat vanish on Y .\nFor example, Y = (v1 −v2\n2 + v3, v2\n3 −v2 + 4v1) is an irreducible affine\nalgebraic set (and therefore an affine variety).\nTheorem 11.1 (Hilbert). I(Y ) is finitely generated, i.e. there exist poly-\nnomials g1, . . . , gk that generate I(Y ). This means every f ∈I(Y ) can be\nwritten as f = P figi for some polynomials fi.\n64\n\nLet C[V ], the coordinate ring of V , be the ring of polynomials over\nthe variables v1, . . . , vn. The coordinate ring of Y is defined to be C[Y ] =\nC[V ]/I(Y ). It is the set of polynomial functions over Y .\nDefinition 11.2.\n• P(V ) is the projective space associated with V , i.e.\nthe set of lines through the origin in V .\n• V is called the cone of P(V ).\n• C[V ] is called the homogeneous coordinate ring of P(V ).\n• Y ⊆P(V ) is a projective algebraic set if it is the set of simultaneous\nzeros of a set of homogeneous forms (polynomials) in the variables\nv1, . . . , vn. It is necessary that the polynomials be homogeneous because\na point in P(V ) is a line in V .\n• A projective algebraic set Y is irreducible if it can not be expressed as\nthe union of two proper algebraic sets in P(V ).\n• An irreducible projective algebraic set is called a projective variety.\nLet Y ⊆P(V ) be a projective variety, and define I(Y ), the ideal of Y\nto be the set of all homogeneous forms that vanish on Y . Hilbert’s result\nimplies that I(Y ) is finitely generated.\nDefinition 11.3. The cone C(Y ) ⊆V of a projective variety Y ⊆P(V ) is\ndefined to be the set of all points on the lines in Y .\nDefinition 11.4. We define the homogeneous coordinate ring of Y as\nR(Y ) = C[V ]/I(Y ), the set of homogeneous polynomial forms on the cone\nof Y .\nDefinition 11.5. A Zariski open subset of Y is the complement of a pro-\njective algebraic subset of Y . It is called a quasi-projective variety.\nLet G = GLn(C), and V a finite dimensional representation of G. Then\nC[V ] is a G-module, with the action of σ ∈G defined by:\n(σ · f)(v) = f(σ−1v), v ∈V.\nDefinition 11.6. Let Y ⊆P(V ) be a projective variety with ideal I(Y ). We\nsay that Y is a G-variety if I(Y ) is a G-module, i.e., I(Y ) is a G-submodule\nof C[V ].\n65\n\nIf Y is a projective variety, then R(Y ) = C[V ]/I(Y ) is also a G-module.\nTherefore Y is G-invariant, i.e.\ny ∈Y ⇒σy ∈Y, ∀σ ∈G.\nThe algebraic geometry of G-varieties is called geometric invariant theory\n(GIT).\n11.2\nOrbit closures\nWe now define special classes of G-varieties called orbit closures. Let v ∈\nP(V ) be a point, and Gv the orbit of v:\nGv = {gv|g ∈G}.\nLet the stabilizer of v be\nH = Gv = {g ∈G|gv = v}.\nThe orbit Gv is isomorphic to the space G/H of cosets, called the ho-\nmogeneous space. This is a very special kind of algebraic variety.\nDefinition 11.7. The orbit closure of v is defined by:\n∆V [v] = Gv ⊆P(V ).\nHere Gv is the closure of the orbit Gv in the complex topology on P(V ) (see\nfigure 11.1).\nA basic fact of algebraic geometry:\nTheorem 11.2. The orbit closure ∆V [v] is a projective G-variety\nIt is also called an almost homogeneous space.\nLet IV [v] be the ideal of ∆V [v], and RV [v] the homogeneous coordinate\nring of ∆V [v]. The algebraic geometry of general orbit closures is hopeless,\nsince the closures can be horrendous (see figure 11.1). Fortunately we shall\nonly be interested in very special kinds of orbit closures with good algebraic\ngeometry.\nWe now define the simplest kind of orbit closure, which is obtained when\nthe orbit itself is closed. Let Vλ be an irreducible Weyl module of GLn(C),\nwhere λ = (λ1 ≥λ2 ≥· · · ≥λn ≥0) is a partition. Let vλ be the highest\nweight point in P(Vλ), i.e., the point corresponding to the highest weight\n66\n\nv\nGv\n∆V [v]\nLimit points of Gv\nFigure 11.1: The limit points of Gv in ∆V [v] can be horrendous.\nvector in Vλ. This means bvλ = vλ for all b ∈B, where B ⊆GLn(C) is the\nBorel subgroup of lower triangular matrices. Recall that the highest weight\nvector is unique.\nConsider the orbit Gvλ of vλ. Basic fact:\nProposition 11.1. The orbit Gvλ is already closed in P(V ).\nIt can be shown that the stabilizer Pλ = Gvλ is a group of block lower\ntriangular matrices, where the block lengths only depend on λ (see figure\n11.2). Such subgroups of GLn(C) are called parabolic subgroups, and will be\ndenoted by P. Clearly Gvλ ∼= G/Pλ = G/P.\n11.3\nGrassmanians\nThe simplest examples of G/P are Grassmanians.\nDefinition 11.8. Let G = Gln(C), and V = Cn. The Grassmanian Grn\nd is\nthe space of d-dimensional subspaces (containing the origin) of V .\nExamples:\n1. Gr2\n1 is the set of lines in C2 (see figure 11.3).\n2. More generally, P(V ) = Grn\n1 .\nProposition 11.2. The Grassmanian Grn\nd is a projective variety (just like\nP(V ) = Grn\n1 ).\n67\n\n*\nA1\nA2\nA3\nA4\nA5\nm1\nm2\nm3\nm4\nm5\nFigure 11.2: The parabolic subgroup of block lower triangular matrices. The sizes\nmi only depend on λ.\nFigure 11.3: Gr2\n1 is the set of lines in C2.\n68\n\nIt is easy to see that Grn\nd is closed (since the limit of a sequence of d-\ndimensional subspaces of V is a d-dimensional subspace). Hence this follows\nfrom:\nProposition 11.3. Let λ = (1, . . . , 1) be the partition of d, whose all parts\nare 1. Then Grn\nd ∼= Gvλ ⊆P(Vλ).\nProof. For the given λ, Vλ can be identified with the dth wedge product\nΛd(V ) = span{(vi1 ∧· · ·∧vid)|i1, . . . , id are distinct} ⊆V ⊗· · ·⊗V (d times),\nwhere\n(vi1 ∧· · · ∧vid) = 1\nd!\nX\nσ∈Sd\nsgn(σ)(vσ(i1) ⊗· · · ⊗vσ(id)).\nLet Z be a variable d × n matrix.\nThen C[Z] is a G-module: given\nf ∈C[Z] and σ ∈GLn(C), we define the action of σ by\n(σ · f)(Z) = f(Zσ).\nNow Λd(V ), as a G-module, is isomorphic to the span in C[Z] of all d × d\nminors of Z.\nLet A ∈Grn\nd be a d-dimensional subspace of V . Take any basis {v1, . . . , vd}\nof A. The point v1 ∧· · · ∧vd ∈Λd(V ) depends only on the subspace A, and\nnot on the basis, since the change of basis does not change the wedge prod-\nuct. Let ZA be the d × n complex matrix whose rows are the basis vectors\nv1, . . . , vd of A. The Plucker map associates with A the tuple of all d × d\nminors Aj1,...,jd of ZA, where Aj1,...,jd denotes the minor of ZA formed by\nthe columns j1, . . . , jd. This depends only on A, and not on the choice of\nbasis for A.\nThe proposition follows from:\nClaim 11.1. The Plucker map is a G-equivariant map from Grn\nd to Gvλ ⊆\nP(Vλ) and Grn\nd ≈Gvλ ⊆P(Vλ).\nProof. Exercise. Hint: take the usual basis, and note that the highest weight\npoint vλ corresponds to v1 ∧· · · ∧vd.\n69\n\nChapter 12\nThe class varieties\nScribe: Hariharan Narayanan\nGoal: Associate class varieties with the complexity classes #P and NC\nand reduce the NC ̸= P #P conjecture over C to a conjecture that the class\nvariety for #P cannot be embedded in the class variety for NC.\nreference: [GCT1]\nThe NC ̸= P #P conjecture over C says that the permanent of an n × n\ncomplex matrix X cannot be expressed as a determinant of an m × m com-\nplex matrix Y , m = poly(n), whose entries are (possibly nonhomogeneous)\nlinear forms in the entries of X. This obviously implies the NC ̸= P #P\nconjecture over Z, since multivariate polynomials over Cn are determined\nby the values that they take over the subset Zn. The conjecture over Z is\nimplied by the usual NC ̸= P #P conjecture over a finite field Fp, p ̸= 2,\nand hence, has to be proved first anyway.\nFor this reason, we concentrate on the NC ̸= P #P conjecture over C in\nthis lecture. The goal is to reduce this conjecture to a statement in geometric\ninvariant theory.\n12.1\nClass Varieties in GCT\nTowards that end, we associate with the complexity classes #P and NC\ncertain projective algebraic varieties, which we call class varieties. For this,\nwe need a few definitions.\nLet G = GLl(C), V a finite dimensional representation of G. Let P(V )\nbe the associated projective space, which inherits the group action. Given a\npoint v ∈P(V ), let ∆V [v] = Gv ⊆P(V ) be its orbit closure. Here Gv is the\n70\n\nclosure of the orbit Gv in the complex topology on P(v). It is a projective\nG-variety; i.e., a projective variety with the action of G.\nAll class varieties in GCT are orbit closures (or their slight generaliza-\ntions), where v ∈P(V ) corresponds to a complete function for the class in\nquestion. The choice of the complete function is crucial, since it determines\nthe algebraic geometry of ∆V [v].\nWe now associate a class variety with NC. Let g = det(Y ), Y an m × m\nvariable matrix. This is a complete function for NC. Let V = symm(Y )\nbe the space of homogeneous forms in the entries of Y of degree m. It is a\nG-module, G = GLm2(C), with the action of σ ∈G given by:\nσ : f(Y ) 7−→f(σ−1Y ).\nHere σ−1Y is defined thinking of Y as an m2-vector.\nLet ∆V [g] = ∆V [g, m] = Gg, where we think of g as an element of\nP(V ).\nThis is the class variety associated with NC.\nIf g is a different\nfunction instead of det(Y ), the algebraic geometry of ∆V [g] would have been\nunmanageable. The main point is that the algebraic geometry of ∆V [g] is\nnice, because of the very special nature of the determinant function.\nWe next associate a class variety with #P. Let h = perm(X), X an\nn × n variable matrix. Let W = symn(X). It is similarly an H-module,\nH = GLk(C), k = n2. Think of h as an element of P(W), and let ∆W[h] =\nHh be its orbit closure. It is called the class variety associated with #P.\nNow assume that m > n, and think of X as a submatrix of Y , say the\nlower principal submatrix. Fix a variable entry y of Y outside of X. Define\nthe map φ : W →V which takes w(x) ∈W to ym−nw(x) ∈V . This induces\na map from P(V ) to P(W) which we call φ as well. Let φ(h) = f ∈P(V )\nand ∆V [f, m, n] = Gf its orbit closure. It is called the extended class variety\nassociated with #P.\nProposition 12.1 (GCT 1).\n1. If h(X) ∈W can be computed by a cir-\ncuit (over C) of depth ≤logc(n), c a constant, then f = φ(h) ∈\n∆V [g, m], for m = O(2logc n).\n2. Conversely if f ∈∆V [g, m] for m = 2logc n, then h(X) can be approx-\nimated infinitesimally closely by a circuit of depth log2c m. That is,\n∀ǫ > 0, there exists a function ̃h(X) that can be computed by a circuit\nof depth ≤log2c m such that ∥ ̃h −h∥< ǫ in the usual norm on P(V ).\nIf the permanent h(X) can be approximated infinitesimally closely by\nsmall depth circuits, then every function in #P can be approximated in-\nfinitesimally closely by small depth circuits. This is not expected. Hence:\n71\n\nConjecture 12.1 (GCT 1). Let h(X) = perm(X), X an n × n variable\nmatrix. Then f = φ(h) ̸∈∆V [g; m] if m = 2polylog(n) and n is sufficiently\nlarge.\nThis is equivalent to:\nConjecture 12.2 (GCT 1). The G-variety ∆V [f; m, n] cannot be embedded\nas a G-subvariety of ∆V [g, m], symbolically\n∆V [f; m, n] ̸֒→∆V [g, m],\nif m = 2polylog(n) and n →∞.\nThis is the statement in geometric invariant theory (GIT) that we sought.\n72\n\nChapter 13\nObstructions\nScribe: Paolo Codenotti\nGoal: Define an obstruction to the embedding of the #P-class variety in\nthe NC-class-variety and describe why it should exist.\nReferences: [GCT1, GCT2]\nRecall\nLet us first recall some definitions and results from the last class. Let Y be\na generic m × m variable matrix, and X an n × n minor of Y (see figure\n13.1).\nLet g = det(Y ), h = perm(X), f = φ(h) = ym−nperm(X), and\nV = Symm[Y ] the set of homogeneous forms of degree m in the entries of Y .\nThen V is a G-module for G = GL(Y ) = GLl(C), l = m2, with the action\nY\nX\nn\nn\nm\nm\nFigure 13.1: Here Y is a generic m by m matrix, and X is an n by n minor.\n73\n\nof σ ∈G given by\nσ : f(Y ) →f(σ−1Y ),\nwhere Y is thought of as an l-vector, and P(V ) a G-variety. Let\n∆V [f; m, n] = Gf ⊆P(V ),\nand\n∆V [g; m] = Gg ⊆P(V )\nbe the class varieties associated with #P and NC.\n13.1\nObstructions\nConjecture 13.1. [GCT1] There does not exist an embedding ∆V [f; m, n] ֒→\n∆V [g; m] with m = 2polylog(n), n →∞.\nThis implies Valiant’s conjecture that the permanent cannot be com-\nputed by circuits of polylog depth. Now we discuss how to go about proving\nthe conjecture.\nSuppose to the contrary,\n∆[f; m, n] ֒→∆V [g; m].\n(13.1)\nWe denote ∆V [f; m, n] by ∆V [f], and ∆V [g; m] by ∆V [g]. Let RV [g] be\nthe homogeneous coordinate ring of ∆V [g]. The embedding (13.1) implies\nexistence of a surjection:\nRV [f] եւ RV [g]\n(13.2)\nThis is a basic fact from algebraic geometry. The reason is that RV [g] is the\nset of homogeneous polynomial functions on the cone C of ∆V [g], and any\nsuch function τ can be restricted to ∆V [f] (see figure 13.2). Conversely, any\npolynomial function on ∆V [f] can be extended to a homogeneous polynomial\nfunction on the cone C.\nLet RV [f]d and RV [g]d be the degree d components of RV [f] and RV [g].\nThese are G-modules since ∆V [f] and ∆V [g] are G-varieties. The surjection\n(13.2) is degree preserving. So there is a surjection\nRV [f]d եւ RV [g]d\n(13.3)\nfor every d. Since G is reductive, both RV [f]d and RV [g]d are direct sums\nof irreducible G-modules. Hence the surjection (13.3) implies that RV [f]d\ncan be embedded as a G submodule of RV [g]d.\n74\n\nC\nτ\n∆V [f]\nFigure 13.2: C denotes the cone of ∆V [g].\nDefinition 13.1. We say that a Weyl-module S = Vλ(G) is an obstruction\nfor the embedding (13.1) (or, equivalently, for the pair (f, g)) if Vλ(G) occurs\nin RV [f; m, n]d, but not in RV [g; m]d, for some d. Here occurs means the\nmultiplicity of Vλ(G) in the decomposition of RV [f; m, n]d is nonzero.\nIf an obstruction exists for given m, n, then the embedding (13.1) does\nnot exist.\nConjecture 13.2 (GCT2). An obstruction exists for the pair (f, g) for all\nlarge enough n if m = 2polylog(n).\nThis implies Conjecture 13.1. In essence, this turns a nonexistence prob-\nlem (of polylog depth circuit for the permanent) into an existence problem\n(of an obstruction).\nIf we replace the determinant here by any other complete function in\nNC, an obstruction need not exist. Because, as we shall see in the next\nlecture, the existence of an obstruction crucially depends on the exceptional\nnature of the class variety constructed from the determinant.\nThe main\ngoals of GCT in this context are:\n1. understand the exceptional nature of the class varieties for NC and\n#P, and\n2. use it to prove the existence of obstructions.\n13.1.1\nWhy are the class varieties exceptional?\nWe now elaborate on the exceptional nature of the class varieties. Its signif-\nicance for the existence of obstructions will be discussed in the next lecture.\nLet V be a G-module, G = GLn(C). Let P(V ) be a projective variety\nover V . Let v ∈P(V ), and recall ∆V [v] = Gv. Let H = Gv be the stabilizer\nof v, that is, Gv = {σ ∈G|σv = v}.\n75\n\nDefinition 13.2. We say that v is characterized by its stabilizer H = Gv\nif v is the only point in P(V ) such that hv = v, ∀h ∈H.\nIf v is characterized by its stabilizer, then ∆V [v] is completely determined\nby the group triple H ֒→G ֒→K = GL(V ).\nDefinition 13.3. The orbit closure ∆V [v], when v is characterized by its\nstabilizer, is called a group-theoretic variety.\nProposition 13.1. [GCT1]\n1. The determinant g = det(Y ) ∈P(V ) is characterized by its stabilizer.\nTherefore ∆V [g] is group theoretic.\n2. The permanent h = perm(X) ∈P(W), where W = Symn(X), is also\ncharacterized by its stabilizer. Therefore ∆W[h] is also group theoretic.\n3. Finally, f = φ(h) ∈P(V ) is also characterized by its stabilizer. Hence\n∆V [f] is also group theoretic.\nProof. (1) It is a fact in classical representation theory that the stabilizer\nof det(Y ) in G = GL(Y ) = GLm2(C) is the subgroup Gdet that consists of\nlinear transformations of the form Y →AY ∗B, where Y ∗= Y or Y t, for any\nA, B ∈GLm(C). It is clear that linear transformation of this form stabilize\nthe determinant since:\n1. det(AY B) = det(A)det(B)det(Y ) = c det(Y ), where c = det(A) det(B).\nNote that the constant c doesn’t matter because we get the same point\nin the projective space.\n2. det(Y ∗) = det(Y ).\nIt is a basic fact in classical invariant theory that det(Y ) is the only point in\nP(V ) stabilized by Gdet. Furthermore, the stabilizer Gdet is reductive, since\nits connected part is (Gdet)◦≈GLm × GLm with the natural embedding\n(Gdet)◦= GLm × GLm ֒→GL(Cm ⊗Cm) = GLm2(C) = G.\n(2) The stabilizer of perm(x) is the subgroup Gperm ⊆GL(X) = GLn2(C)\ngenerated by linear transformations of the form X →λX∗μ, where X∗=\nXorXt, and λ and μ are diagonal (which change the permanent by a con-\nstant factor) or permutation matrices (which do not change the permanent).\nFinally, the discrete component of Gperm is isomorphic to S2 ⋊Sn ×Sn,\nwhere ⋊denotes semidirect product. The continuous part is (C∗)n × (C∗)n.\nSo Gperm is reductive.\n76\n\n(3) Similar.\nThe main significance of this proposition is the following.\nBecause\n∆V [g], ∆V [f], and ∆W[h] are group theoretic, the algebraic geometric prob-\nlems concerning these varieties can be “reduced” to problems in the theory\nof quantum groups. So the plan is:\n1. Use the theory of quantum groups to understand the structure of the\ngroup triple associated with the algebraic variety.\n2. Translate this understanding to the structure of the algebraic variety.\n3. Use this to show the existence of obstructions.\n77\n\nChapter 14\nGroup theoretic varieties\nScribe: Joshua A. Grochow\nGoal: In this lecture we continue our discussion of group-theoretic varieties.\nWe describe why obstructions should exist, and why the exceptional group-\ntheoretic nature of the class varieties is crucial for this existence.\nRecall\nLet G = GLn(C), V a G-module, and P(V ) the associated projective space.\nLet v ∈P(V ) be a point characterized by its stabilizer H = Gv ⊂G. In\nother words, v is the only point in P(V ) stabilized by H. Then ∆V [v] = Gv\nis called a group-theoretic variety because it is completely determined by the\ngroup triple\nH ֒→G ֒→GL(V ).\nThe simplest example of a group-theoretic variety is a variety of the form\nG/P that we described in the earlier lecture. Let V = Vλ(G) be a Weyl\nmodule of G and vλ ∈P(V ) the highest weight point (recall: the unique\npoint stabilized by the Borel subgroup B ⊂G of lower triangular matrices).\nThen the stabilizer of vλ consists of block-upper triangular matrices, where\n78\n\nthe block sizes are determined by λ:\nPλ := Gvλ =\n \n \n \n \n \n \n \n \n \n \n \n \n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n0\n0\n0\n∗\n∗\n∗\n∗\n∗\n0\n0\n0\n∗\n∗\n∗\n∗\n∗\n0\n0\n0\n0\n0\n∗\n∗\n∗\n0\n0\n0\n0\n0\n∗\n∗\n∗\n0\n0\n0\n0\n0\n∗\n∗\n∗\n \n \n \n \n \n \n \n \n \n \n \n \nThe orbit ∆V [vλ] = Gvλ ∼= G/Pλ is a group-theoretic variety determined\nentirely by the triple\nPλ = Gvλ ֒→G ֒→K = GL(V ).\nThe group-theoretic varieties of main interest in GCT are the class va-\nrieties associated with the various complexity classes.\n14.1\nRepresentation theoretic data\nThe main principle guiding GCT is that the algebraic geometry of a group-\ntheoretic variety ought to be completely determined by the representation\ntheory of the corresponding group triple.\nThis is a natural extension of\nwork already pursued in mathematics by Deligne and Milne on Tannakien\ncategories [DeM], showing that an algebraic group is completely determined\nby its representation theory. So the goal is to associate to a group-theoretic\nvariety some representation-theoretic data that will analogously capture the\ninformation in the variety completely. We shall now illustrate this for the\nclass variety for NC. First a few definitions.\nLet v ∈P(V ) be the point as above characterized by its stabilizer Gv.\nThis means the line Cv ⊆V corresponding to v is a one-dimensional repre-\nsentation of Gv. Thus (Cv)⊗d is a one-dimensional degree d representation,\ni.e. the representation ρ : G →GL(Cv) ∼= C∗is polynomial of degree d\nin the entries of the matrix of an element in G. Recall that C[V ] is the\ncoordinate ring of V , and C[V ]d is its degree d homogeneous component, so\n(Cv)⊗d ⊆C[V ]d.\nTo each v ∈P(V ) that is characterized by its stabilizer, we associate a\nrepresentation-theoretic data, which is the set of G-modules\nΠv =\n[\nd\nΠv(d),\n79\n\nwhere Πv(d) is the set of all irreducible G-submodules S of C[V ]d whose\nduals S∗do not contain a Gv-submodule isomorphic to (Cv)⊗d∗(the dual of\n(Cv)⊗d). The following proposition elucidates the importance of this data:\nProposition 14.1. [GCT2] Πv ⊆IV [v] (where IV [v] is the ideal of the\nprojective variety ∆V [v] ⊆P(V )).\nProof. Fix S ∈Πv(d). Suppose, for the sake of contradiction, that S ⊈\nIV [v]. Since S ⊆C[V ], S consists of “functions” on the variety P(V ) (ac-\ntually homogeneous polynomials on V ). The coordinate ring of ∆V [v] is\nC[V ]/IV [v], and since S ⊈IV [v], S must not vanish identically on ∆V [v].\nSince the orbit Gv is dense in ∆V [v], S must not vanish identically on this\nsingle orbit Gv. Since S is a G-module, if S were to vanish identically on the\nline Cv, then it would vanish on the entire orbit Gv, so S does not vanish\nidentically on Cv.\nNow S consists of functions of degree d. Restrict them to the line Cv.\nThe dual of this restriction gives an injection of (Cv)⊗d∗as a Gv-submodule\nof S∗, contradicting the definition of Πv(d).\n14.2\nThe second fundamental theorem\nWe now ask essentially the reverse question: when does the representation\ntheoretic data Πv generate the ideal IV [v]? For if Πv generates IV [v], then Πv\ncompletely captures the coordinate ring C[V ]/IV [v], and hence the variety\n∆V [v].\nTheorem 14.1 (Second fundamental theorem of invariant theory for G/P).\nThe G-modules in Πvλ(2) generate the ideal IV [vλ] of the orbit Gvλ ∼= G/Pλ,\nwhen V = Vλ(G).\nThis theorem justifies the main principle for G/P, so we can hope that\nsimilar results hold for the class varieties in GCT (though not always exactly\nin the same form).\nNow, let ∆V [g] be the class variety for NC (in other words, take g =\ndet(Y ) for a matrix Y of indeterminates).\nBased on the main principle,\nwe have the following conjecture, which essentially generalizes the second\nfundamental theorem of invariant theory for G/P to the class variety for\nNC:\nConjecture 14.1 (GCT 2). ∆V [g] = X(Πg) where X(Πg) is the zero-set\nof all forms in the G-modules contained in Πg.\n80\n\nTheorem 14.2 (GCT 2). A weaker version of the above conjecture holds.\nSpecifically, assuming that the Kronecker coefficients satisfy a certain sep-\naration property, there exists a G-invariant (Zariski) open neighbourhood\nU ⊆P(V ) of the orbit Gg such that X(Πg) ∩U = ∆V [g] ∩U.\nThere is a notion of algebro-geometric complexity called Luna-Vust com-\nplexity which quantifies the gap between G/P and class varieties. The Luna-\nVust complexity of G/P is 0. The Luna-Vust complexity of the NC class\nvariety is Ω(dim(Y )). This is analogous to the difference between circuits\nof constant depth and circuits of superpolynomial depth. This is why the\nprevious conjecture and theorem turn out to be far harder than the corre-\nsponding facts for G/P.\n14.3\nWhy should obstructions exist?\nThe following proposition explains why obstructions should exist to separate\nNC from P #P.\nProposition 14.2 (GCT 2). Let g = det(Y ), h = perm(X), f = φ(h),\nn = dim(X), m = dim(Y ). If Conjecture 14.1 holds and the permanent can-\nnot be approximated arbitrarily closely by circuits of poly-logarithmic depth\n(hardness assumption), then an obstruction for the pair (f, g) exists for all\nlarge enough n, when m = 2logc n for some constant c. Hence, under these\nconditions, NC ̸= P #P over C.\nThis proposition may seem a bit circular at first, since it relies on a hard-\nness assumption. But we do not plan to prove the existence of obstructions\nby proving the assumptions of this proposition. Rather, this proposition\nshould be taken as evidence that obstructions exist (since we expect the\nhardness assumption therein to hold, given that the permanent is # P-\ncomplete), and we will develop other methods to prove their existence.\nProof. The hardness assumption implies that f /∈∆V [g] if m = 2logc n [GCT\n1].\nConjecture 14.1 says that X(Πg) = ∆V [g]. So there exists an irreducible\nG-module S ∈Πg such that S does not vanish on f. So S occurs in RV [f]\nas a G-submodule.\nOn the other hand, since S ∈Πg, S ⊆IV [g] by Proposition 14.1. So\nS does not occur in RV [g] = C[V ]/IV [g]. Thus S is not a G-submodule of\nRV [g], but it is a G-submodule of RV [f], i.e., S is an obstruction.\n81\n\nChapter 15\nThe flip\nScribe: Hariharan Narayanan\nGoal: Describe the basic principle of GCT, called the flip, in the context of\nthe NC vs. P #P problem over C.\nreferences: [GCTflip1, GCT1, GCT2, GCT6]\nRecall\nAs in the previous lectures, let g = det(Y ) ∈P(V ), Y an m × m vari-\nable matrix, G = GLm2(C), and ∆V (g) = ∆V [g; m] = Gg ⊆P(V ) the\nclass variety for NC. Let h = perm(X), X an n × n variable matrix,\nf = φ(h) = ym−nh ∈P(V ), and ∆V (f) = ∆V [f; m, n] = Gf ⊆P(V ) the\nclass variety for P #P. Let RV [f; m, n] denote the homogeneous coordinate\nring of ∆V [f; m, n], RV [g; m] the homogeneous coordinate ring of ∆V [g; m],\nand RV [f; m, n]d and RV [g; m]d their degree d-components. A Weyl module\nS = Vλ(G) of G is an obstruction of degree d for the pair (f, g) if Vλ occurs\nin RV [f; m, n]d but not RV [g; m]d.\nConjecture 15.1. [GCT2] An obstruction (of degree polynomial in m) ex-\nists if m = 2polylog(n) as n →∞.\nThis implies NC ̸= P #P over C.\n82\n\n15.1\nThe flip\nIn this lecture we describe an approach to prove the existence of such ob-\nstructions. It is based on the following complexity theoretic positivity hy-\npothesis:\nPHflip [GCTflip1]:\n1. Given n, m and d, whether an obstruction of degree d for m and n\nexists can be decided in poly(n, m, ⟨d⟩) time, and if it exists, the label\nλ of such an obstruction can be constructed in poly(n, m, ⟨d⟩) time.\nHere ⟨d⟩denotes the bitlength of d.\n2.\n(a) Whether Vλ occurs in RV [f; m, n]d can be decided in poly(n, m, ⟨d⟩, ⟨λ⟩)\ntime.\n(b) Whether Vλ occurs in RV [g; m]d can be decided in poly(n, m, ⟨d⟩, ⟨λ⟩)\ntime.\nThis suggests the following approach for proving Conjecture 15.1:\n1. Find polynomial time algorithms sought in PHflip-2 for the basic de-\ncision problems (a) and (b) therein.\n2. Using these find a polynomial time algorithm sought in PHflip-1 for\ndeciding if an obstruction exists.\n3. Transform (the techniques underlying) this “easy” (polynomial time)\nalgorithm for deciding if an obstruction exists for given n and m into\nan “easy” (i.e., feasible) proof of existence of an obstruction for every\nn →∞when d is large enough and m = 2polylog(n).\nThe first step here is the crux of the matter. The main results of [GCT6]\nsay that the polynomial time algorithms for the basic decision problems as\nsought in PHflip-2 indeed exist assuming natural analogues of PH1 and SH\n(PH2) that we have seen earlier in the context of the plethysm problem. To\nstate them, we need some definitions.\nLet Sλ\nd [f] = Sλ\nd [f; m, n] be the multiplicity of Vλ = Vλ(G) in RV [f; m, n].\nThe stretching function ̃Sd[f] = ̃Sλ\nd [f; m, n] is defined by\n ̃Sλ\nd [f](k) := Skλ\nkd[f].\nThe stretching function for g, ̃Sλ\nd [g] = ̃Sλ\nd [g; m], is defined analogously.\nThe main mathematical result of [GCT6] is:\n83\n\nTheorem 15.1. [GCT6] The stretching functions ̃Sλ\nd [g] and ̃Sλ\nd [f] are quasipoly-\nnomials assuming that the singularities of ∆V [f; m, n] and ∆V [g; m] are ra-\ntional.\nHere rational means “nice”; we shall not worry about the exact defini-\ntion.\nThe main complexity-theoretic result is:\nTheorem 15.2. [GCT6] Assuming the following mathematical positivity\nhypothesis PH1 and the saturation hypothesis SH (or the stronger positivity\nhypothesis PH2), PHflip-2 holds.\nPH1: There exists a polytope P = P λ\nd [f] such that\n1. The Ehrhart quasi-polynomial of P, fP(k), is ̃Sλ\nd [f](k).\n2. dim(P) = poly(n, m, ⟨d⟩).\n3. Membership in P can be answered in polynomial time.\n4. There is a polynomial time separation oracle [GLS] for P.\nSimilarly, there exists a polytope Q = Qλ\nd[g] such that\n1. The Ehrhart quasi-polynomial of Q, fQ(k), is ̃Sλ\nd [g](k).\n2. dim(Q) = poly(m, ⟨d⟩).\n3. Membership in Q can be answered in polynomial time.\n4. There is a polynomial time separation oracle for Q.\nPH2: The quasi-polynomials ̃Sλ\nd [g] and ̃Sλ\nd [f] are positive.\nThis implies:\nSH: The quasi-polynomials ̃Sλ\nd [g] and ̃Sλ\nd [f] are saturated.\nPH1 and SH imply that the decision problems in PHflip-2 can be trans-\nformed into saturated positive integer programming problems. Hence The-\norem 15.2 follows from the polynomial time algorithm for saturated linear\nprogramming that we described in an earlier class.\nThe decision problems in PHflip-2 are “hyped” up versions of the plethysm\nproblem discussed earlier. The article [GCT6] provides evidence for PH1\n84\n\nand PH2 for the plethysm problem. This constitutes the main evidence for\nPH1 and PH2 for the class varieties in view of their group-theoretic nature;\ncf. [GCTflip1].\nThe following problem is important in the context of PHflip-2:\nProblem 15.1. Understand the G-module structure of the homogeneous\ncoordinate rings RV [f]d and RV [g]d.\nThis is an instance of the following abstract:\nProblem 15.2. Let X be a projective group-theoretic G-variety. Let R =\nL∞\nd=0 Rd be its homogeneous coordinate ring.\nUnderstand the G-module\nstructure of Rd.\nThe simplest group-theoretic variety is G/P. For it, a solution to this\nabstract problem is given by the following results:\n1. The Borel-Weil theorem.\n2. The Second Fundamental theorem of invariant theory [SFT].\nThese will be covered in the next class for the simplest case of G/P, the\nGrassmanian.\n85\n\nChapter 16\nThe Grassmanian\nScribe: Hariharan Narayanan\nGoal: The Borel-Weil and the second fundamental theorem of invariant\ntheory for the Grassmanian.\nReference: [F]\nRecall\nLet V = Vλ(G) be a Weyl module of G = GLn(C) and vλ ∈P(V ) the point\ncorresponding to its highest weight vector. The orbit ∆V [vλ] := Gvλ, which\nis already closed, is of the form G/P, where P is the parabolic stabilizer of\nvλ. When λ is a single column, it is called the Grassmannian.\nAn alternative description of the Grassmanian is as follows. Assume that\nλ is a single column of length d. Let Z be a d × n matrix of variables zij.\nThen V = Vλ(G) can be identified with the span of d × d minors of Z with\nthe action of σ ∈G given by:\nσ : f(z) 7→f(zσ).\nLet Grn\nd be the space of all d-dimensional subspaces of Cn. Let W be a\nd-dimensional subspace of Cn. Let B = B(W) be a basis of W. Construct\nthe d × n matrix zB, whose rows are vectors in B. Consider the Pl ̈ucker\nmap from Grn\nd to P(V ) which maps any W ∈Grn\nd to the tuple of d × d\nminors of ZB. Here the choice of B = B(W) does not matter, since any\nchoice gives the same point in P(V ). Then the image of Grn\nd is precisely the\nGrassmanian Gvλ ⊆P(V ).\n86\n\n16.1\nThe second fundamental theorem\nNow we ask:\nQuestion 16.1. What is the ideal of Grn\nd ≈Gvλ ⊆P(V )?\nThe homogeneous coordinate ring of P(V ) is C[V ].\nWe want an ex-\nplicit set of generators of this ideal in C[V ]. This is given by the second\nfundamental theorem of invariant theory, which we describe next.\nThe coordinates of P(V ) are in one-to-one correspondence with the d×d\nminors of the matrix Z. Let each minor of Z be indexed by its columns. Thus\nfor 1 ≤i1 < · · · < id ≤n, Zi1,...,id is a coordinate of P(V ) corresponding\nto the minor of Z formed by the columns i1, i2, . . ..\nLet Λ(n, d) be the\nset of ordered d-tuples of {1, . . . , n}. The tuple [i1, . . . , id] in this set will\nbe identified with the coordinate Zi1,...,id of P(V ).\nThere is a bijection\nbetween the elements of Λ(n, d) and of Λ(n, n −d) obtained by associating\ncomplementary sets:\nΛ(n, d) ∋λ ↭λ∗∈Λ(n, n −d).\nWe define sgn(λ, λ∗) to be the sign of the permutation that takes [1, . . . , n]\nto [λ1, . . . , λd, λ∗\n1, . . . , λ∗\nn−d].\nGiven s ∈{1. . . . , d}, α ∈Λ(n, s−1), β ∈Λ(n, d+1), and γ ∈Λ(n, d−s),\nwe now define the Van der Waerden Syzygy [[α, β, γ]], which is an element\nof the degree two component C[V ]2 of C[V ], as follows:\n[[α, β, γ]] =\nP\nτ∈Λ(d+1,s) sgn(τ, τ ∗)[α1, . . . , αs−1, βτ ∗\n1 , . . . , βτ ∗\nd+1−s][βτ1, . . . , βτs, γ1, . . . , γd−s].\nIt is easy to show that this syzygy vanishes on the Grassmanian Grn\nd :\nbecause it is an alternating (d+1)-multilinear-form, and hence has to vanish\non any d-dimensional space W ∈Grn\nd . Thus it belongs to the ideal of the\nGrassmanian. Moreover:\nTheorem 16.1 (Second fundamental theorem). The ideal of the Grassma-\nnian Grn\nd is generated by the Van-der-Waerden syzygies.\nAn alternative formulation of this result is as follows. Let Pλ ⊆G be the\nstabilizer of vλ. Let Πvλ(2) be the set of irreducible G-submodules of C[V ]2\nwhose duals do not contain a Pλ-submodule isomorphic to Cv⊗2∗\nλ\n(the dual\nof Cv⊗2\nλ ). Here Cvλ denotes the line in P(V ) corresponding to vλ, which is\na one-dimensional representation of Pλ since it stabilizes vλ ∈P(V ). It can\n87\n\nbe shown that the span of the G-modules in Πvλ(2) is equal to the span of\nthe Van-der-Waerden syzygies. Hence, Theorem 16.1 is equivalent to:\nTheorem 16.2 (Second Fundamental Theorem(SFT)). The G-modules in\nΠvλ(2) generate the ideal of Grn\nd .\nThis formulation of SFT for the Grassmanian looks very similar to the\ngeneralized conjectural SFT for the NC-class variety described in the ear-\nlier class. This indicates that the class varieties in GCT are “qualitatively\nsimilar” to G/P.\n16.2\nThe Borel-Weil theorem\nWe now describe the G-module structure of the homogeneous coordinate\nring R of the Grassmannian Gvλ ⊆P(V ), where λ is a single column of\nheight d. The goal is to give an explicit basis for R. Let Rs be the degree\ns component of R. Corresponding to any numbering T of the shape sλ,\nwhich is a d × s rectangle, whose columns have strictly increasing elements\ntop to bottom, we have a monomial mT = Q\nc Zc ∈C[V ]s, were Zc is the\ncoordinate of P(V ) indexed by the d-tuple c, and c ranges over the s columns\nof T. We say that mT is (semi)-standard if the rows of T are nondecreasing,\nwhen read left to right. It is called nonstandard otherwise.\nLemma 16.1 (Straightening Lemma). Each non-standard mT can be straight-\nened to a normal form, as a linear combination of standard monomials, by\nusing Van der Waerden Syzygies as straightening relations (rewriting rules).\nFor any numbering T as above, express mT in a normal form as per the\nlemma:\nmT =\nX\n(Semi)-Standard Tableau S\nα(S, T), mS\nwhere α(S, T) ∈C.\nTheorem 16.3 (Borel-Weil Theorem for Grassmannians). Standard mono-\nmials {mT } form a basis of Rs, where T ranges over all semi-standard\ntableaux of rectangular shape sλ. Hence, Rs ∼= V ∗\nsλ, the dual of the Weyl\nmodule Vsλ.\nThis gives the G-module structure of R completely. It follows that the\nproblem of deciding if Vβ(G) occurs in Rs can be solved in polynomial time:\nthis is so if and only if (sλ)∗= β, where (sλ)∗denotes the dual partition,\nwhose description is left as an exercise.\n88\n\nThe second fundamental theorem as well as the Borel-Weil theorem easily\nfollow from the straightening lemma and linear independence of the standard\nmonomials (as functions on the Grassmanian).\n89\n\nChapter 17\nQuantum group: basic\ndefinitions\nScribe: Paolo Codenotti\nGoal: The basic plan to implement the flip in [GCT6] is to prove PH1 and\nSH via the theory of quantum groups. We introduce the basic concepts in\nthis theory in this and the next two lectures, and briefly show their relevance\nin the context of PH1 in the final lecture.\nReference: [KS]\n17.1\nHopf Algebras\nLet G be a group, and K[G] the ring of functions on G with values in the\nfield K, which will be C in our applications. The group G is defined by the\nfollowing operations:\n• multiplication: G × G →G,\n• identity e: e →G,\n• inverse: G →G.\nIn order for G to be a group, the following properties have to hold:\n• eg = ge = g,\n• g1(g2g3) = (g1g2)g3,\n• g−1g = gg−1 = e.\n90\n\nWe now want to translate these properties to properties of K[G]. This\nshould be possible since K[G] contains all the information that G has. In\nother words, we want to translate the notion of a group in terms of K[G].\nThis translate is called a Hopf algebra. Thus if G is a group, K[G] is a Hopf\nalgebra. Let us first define the dual operations.\n• Multiplication is a map:\n· : G × G →G.\nSo co-multiplication ∆will be a map as follows:\nK[G × G] = K[G] ⊗K[G] ←K[G].\nWe want ∆to be the pullback of multiplication. So for a given f ∈\nK[G] we define ∆(f) ∈K[G] ⊗K[G] by:\n∆(f)(g1, g2) = f(g1g2).\nPictorially:\nG × G\n·\n−−−−→G\n∆(f)\n y\n yf\nk\nk\n• The unit is a map:\ne →G.\nTherefore we want the co-unit ǫ to be a map:\nK ǫ←−K[G],\ndefined by: for f ∈K[G], ǫ(f) = f(e).\n• Inverse is a map:\n( )−1 : G →G.\nWe want the dual antipode S to be the map:\nK[G] ←K[G]\ndefined by: for f ∈K[G], S(f)(g) = f(g−1).\nThe following are the abstract axioms satisfied by ∆, ǫ and S.\n91\n\n1. ∆and ǫ are algebra homomorphisms.\n∆: K[G] →K[G] ⊗K[G]\nǫ : K[G] →K.\n2. co-associativity: Associativity is defined so that the following diagram\ncommutes:\nG × G×G\nG×G × G\n·\n y\nid\n y\n yid\n y·\nG\n×G\nG×\nG\n·\n y\n y·\nG\nG\nSimilarly, we define co-associativity so that the following dual diagram\ncommutes:\nK[G] ⊗K[G] ⊗K[G]\nK[G] ⊗K[G] ⊗K[G]\n∆\nx \nid\nx \nx id\nx ∆\nK[G]\n⊗K[G]\nK[G] ⊗\nK[G]\n∆\nx \nx ∆\nK[G]\nK[G]\nTherefore co-associativity says:\n(∆⊗id) ◦∆= (id ⊗∆) ◦∆.\n3. The property ge = g is defined so that the following diagram com-\nmutes:\ne ×G\nG\ne\n y\n yid\n y\nG×G\nid\n y·\n y\nG\nG\n92\n\nWe define the co of this property so that the following diagram com-\nmutes:\nK\n× K[G]\nK[G]\nǫ\nx \nx id\nx \nK[G] × K[G]\nid\nx ∆\nx \nK[G]\nK[G]\nThat is, id = (ǫ⊗id)◦∆. Similarly, ge = g translates to: id = (id⊗ǫ)◦∆.\nTherefore we get\nid = (ǫ ⊗id) ◦∆= (id ⊗ǫ) ◦∆.\n4. The last property is gg−1 = e = g−1g. The first equality is equivalent\nto requiring that the following diagram commute:\nG\nG\ndiag\n y\n y\nG×G\n y\n()−1\n y\n yid\ne\nG×G\n y\n y·\n y\nG\nG\nWhere diag : G →G× G is the diagonal embedding. The co of diag is\nm : K[G] ←K[G] ⊗K[G] defined by m(f1, f2)(g) = f1(g) · f2(g). So\nthe co of this property will hold when the following diagram commutes:\n93\n\nK[G]\nK[G]\nm\nx \nx \nK[G] ⊗\nk[G]\nν\nx \nS\nx \nx id\nK\nK[G] ⊗K[G]\nx \nx ∆\nǫ\nx \nK[G]\nK[G]\nWhere ν is the embedding of K into K[G]. Therefore the last property\nwe want to be satisfied is:\nm ◦(S ⊗id) ◦∆= ν ◦ǫ.\nFor e = g−1g, we similarly get:\nm ◦(id ⊗S) ◦∆= ν ◦ǫ.\nDefinition 17.1 (Hopf algebra). A K-algebra A is called a Hopf algebra if\nthere exist homomorphisms ∆: A ⊗A →A, S : A →A, ǫ : A →K, and\nν : A →K that satisfy (1) −(4) above, with A in place of K[G].\nWe have shown that if G is a group, the ring K[G] of functions on\nG is a (commutative) Hopf algebra, which is non-co-commutative if G is\nnon-commutative. Thus for every usual group, we get a commutative Hopf\nalgebra. However, in general, Hopf algebras may be non-commutative.\nDefinition 17.2. A quantum group is a (non-commutative and non-co-\ncommutative) Hopf algebra.\nA nontrivial example of a quantum group will be constructed in the next\nlecture.\nNext we want to look at what happens to group theoretic notions such\nas representations, actions, and homomorphisms, in the context of Hopf\nalgebras. These will correspond to co-representations, co-actions, and co-\nhomomorphisms.\nLet us look closely at the notion of co-representation. A representation\nis a map · : G × V →V , such that\n• (h1h2) · v = h1 · (h2 · v), and\n94\n\n• e · v = v.\nTherefore a (right) co-representation of A will be a linear mapping φ : V →\nV ⊗A, where V is a K-vector space, and φ satisfies the following:\n• The following diagram commutes:\nV ⊗A ⊗A\nid⊗∆\n←−−−−V ⊗A\nφ⊗id\nx \nx φ\nV ⊗A\n←−−−−\nφ\nV\nThat is, the following equality holds:\n(φ ⊗id) ◦φ = (id ⊗∆) ◦φ.\n• The following diagram commutes:\nV ⊗K\nid\n←−−−−V ⊗K\nid⊗ǫ\nx \n\nV ⊗A ←−−−−\nφ\nV\nThat is, the following equality holds:\n(id ⊗ǫ) ◦φ = id\nIn fact all usual group theoretic notions can be “Hopfified” in this sense\n[exercise].\nLet us look now at an example. Let\nG = GLn(C) = GL(Cn) = GL(V ),\nwhere V = Cn. Let Mn be the matrix space of n×n C-matrices, and O(Mn)\nthe coordinate ring of Mn,\nO(Mn) = C[U] = C[{ui\nj}],\nwhere U is an n × n variable matrix with entries ui\nj. Let C[G] = O(G) be\nthe coordinate ring of G obtained by adjoining det(U)−1 to O(Mn). That\nis, C[G] = O(G) = C[U][det(U)−1], which is the C algebra generated by ui\nj’s\nand det(U)−1.\n95\n\nProposition 17.1. C[G] is a Hopf algebra, with ∆, ǫ, and S as follows.\n• Recall that the axioms of a Hopf algebra require that\n∆: C[G] →C[G] ⊗C[G],\n∆(f)(g1, g2) = f(g1g2).\nTherefore we define\n∆(ui\nj) =\nX\nk\nui\nk ⊗uk\nj,\nwhere U denotes the generic matrix in Mn as above.\n• Again, it is required that\nǫ(f) = f(e).\nTherefore we define\nǫ(ui\nj) = δij,\nwhere δij is the Kronecker delta function.\n• Finally, the antipode is required to satisfy S(f)(g) = f(g−1). Let eU\nbe the cofactor matrix of U, U−1 =\n1\ndet(U) eU, and eui\nj the entries of eU.\nThen we define S by:\nS(ui\nj) =\n1\ndet(U) eui\nj = (U−1)i\nj.\n96\n\nChapter 18\nStandard quantum group\nScribe: Paolo Codenotti\nGoal: In this lecture we construct the standard (Drinfeld-Jimbo) quantum\ngroup, which is a q-deformation of the general linear group GLn(C) with\nremarkable properties.\nReference: [KS]\nLet G = GL(V ) = GL(Cn), and V = Cn. In the earlier lecture, we\nconstructed the commutative and non co-commutative Hopf algebra C[G].\nIn this lecture we quantize C[G] to get a non-commutative and non-co-\ncommutative Hopf algebra Cq[G], and then define the standard quantum\ngroup Gq = GLq(V ) = GLq(n) as the virtual object whose coordinate ring\nis Cq[G].\nWe start by defining GLq(2) and SLq(2), for n = 2.\nThen we will\ngeneralize this construction to arbitrary n. Let O(M2) be the coordinate\nring of M2, the set of 2 × 2 complex matrices, C[V ] the coordinate ring of\nV generated by the coordinates x1 and x2 of V which satisfy x1x2 = x2x1.\nLet\nU =\n a\nb\nc\nd\n \nbe the generic (variable) matrix in M2. It acts on V = C2 from the left and\nfrom the right. Let\nx =\n x1\nx2\n \n.\nThe left action is defined by\nx →x′ := Ux.\n97\n\nLet\nx′ =\n x′\n1\nx′\n2\n \n.\nSimilarly, the right action is defined by\nxT →(x′′)T := xT U.\nLet\nx′′ =\n x′′\n1\nx′′\n2\n \n.\nThe action of M2 on V satisfies\nx′\n1x′\n2 = x′\n2x′\n1, and\nx′′\n1x′′\n2 = x′′\n2x′′\n1.\nNow instead of V , we take its q-deformation Vq, a quantum space, whose\ncoordinates x1 and x2 satisfy\nx1x2 = qx2x1,\n(18.1)\nwhere q ∈C is a parameter. Intuitively, in quantum physics if x1 and x2\nare position and momentum, then q = eiħwhen ħis Planck’s constant. Let\nCq[V ] be the ring generated by x1 and x2 with the relation (18.1). That is,\nCq[V ] = C[x1, x2]/ < x1x2 −qx2x1 > .\nIt is the coordinate ring of the quantum space Vq. Now we want to quantize\nM(2) to get Mq(2), the space of quantum 2 × 2 matrices, and GL(2) to\nGLq(2), the space of quantum 2×2 nonsingular matrices. Intuitively, Mq(2)\nis the space of linear transformations of the quantum space Vq which preserve\nthe equation (18.1) under the left and right actions, and similarly, GLq(2) is\nthe space of non-singular linear transformation that preserve the equation\n(18.1) under the left and right actions. We now formalize this intuition.\nLet U =\n a\nb\nc\nd\n \nbe a quantum matrix whose coordinates do not\ncommute. The left and right actions of U must preserve 18.1.\n[Left action:] Let the left action be φL : x →Ux, and Ux = x′. Then we\nmust have:\n a\nb\nc\nd\n x1\nx2\n \n=\n ax1 + bx2\ncx1 + dx2\n \n=\n x′\n1\nx′\n2\n \n.\n98\n\n[Right action:] Let the right action be φR : xT →xT U, and let x′′ =\n(xT U)T = UT x. Then we must have:\n x1\nx2\n a\nb\nc\nd\n \n=\n ax1 + cx2\nbx1 + dx2\n \n=\n x′′\n1\nx′′\n2\n \n.\nThe preservation of x1x2 = qx2x1 under left multiplication means\nx′\n1x′\n2 = qx′\n2x′\n1.\nThat is,\n(ax1 + bx2)(cx1 + dx2) = q(cx1 + dx2)(ax1 + bx2).\n(18.2)\nThe left hand side of (18.2) is\nacx2\n1 + bcx2x1 + adx1x2 + bdx2\n2 = acx2\n1 + (bc + adq)x2x1 + bdx2\n2.\nSimilarly, the right hand side of (18.2) is\nq(cax2\n1 + (da + cbq)x2x1 + bdx2\n2).\nTherefore equation (18.2) implies:\nac = qca\nbd = qdb\nbc + adq = da + qcb.\nThat is,\nac = qca\nbd = qdb\nad −da −qcb + q−1bc = 0.\nSimilarly, since x′′\n1x′′\n2 = qx′′\n2x′′\n1, we get:\nab = qba\ncd = qdc\nad −da −qbc + q−1cb = 0.\nThe last equations from each of these sets imply bc = cb.\nSo we define O(Mq(2)), the coordinate ring of the space of 2×2 quantum\nmatrices Mq(2), to be the C-algebra with generators a, b, c, and d, satisfying\nthe relations:\nab = qba,\nac = qca,\nbd = qdb,\ncd = qdc,\nbc = cb,\nad −da = (q −q−1)bc.\n99\n\nLet\nU =\n a\nb\nc\nd\n \n=\n u1\n1\nu1\n2\nu2\n1\nu2\n2\n \n.\nDefine the quantum determinant of U to be\nDq = det(U) = ad −qbc = da −q−1bc.\nDefine Cq[G] = O(GLq(2)), the coordinate ring of the virtual quantum group\nGLq(2) of invertible 2 × 2 quantum matrices, to be\nO(GLq(2)) = O(Mq(2))[D−1\nq ],\nwhere the square brackets indicate adjoining.\nProposition 18.1. The coordinate ring O(GLq(2)) is a Hopf algebra, with\n∆(ui\nj) =\nX\nk\nui\nk ⊗uk\nj ,\nS(ui\nj) = 1\nDq\neui\nj = (U−1)i\nj,\nǫ(ui\nj) = δij,\nwhere eU = [ ̃ui\nj] is the cofactor matrix\neU =\n d\n−q−1b\n−qc\na\n \n.\n(defined so that U eU = DqI) and U−1 = ̃U/Dq is the inverse of U.\nThis is a non-commutative and non-co-commutative Hopf algebra.\nNow we go to the general n. Let Vq be the n-dimensional quantum space,\nthe q-deformation of V , with coordinates xi’s which satisfy\nxixj = qxjxi\n∀i < j.\n(18.3)\nLet Cq[V ] be the coordinate ring of Vq defined by\nCq[V ] = C[x1, . . . , xn]/ < xixj −qxjxi > .\n100\n\nLet Mq(n) be the space of quantum n × n matrices, that is the set of linear\ntransformations on Vq which preserve (18.3) under the left as well as the\nright action. The left action is given by:\n \n \nx1\n...\nxn\n \n = x →Ux = x′,\nwhere U is the n × n generic quantum matrix. Similarly, the right action is\ngiven by:\nxT →xT U = (x′′)T .\nPreservation of (18.3) under the left and right actions means:\nx′\niy′\nj = qx′\njx′\ni,\nfor\ni < j\nx′′\ni y′′\nj = qx′′\njx′′\ni ,\nfor\ni < j.\nAfter straightforward calculations, these yield the following relations on\nthe entries uij = ui\nj of U:\nujkuik = q−1uikujk\n(i < j)\nukjuki = q−1ukiukj\n(i < j)\nujkuil= uilujk\n(i < j, k < l)\nujluik = uikujl−(q −q−1)ujkuil\n(i < j, k < l).\n(18.4)\nThe quantum determinant is defined as\nDq =\nX\nσ∈Sn\n(−q)l(σ)u\niσ(1)\nj1\n. . . u\niσ(n)\njn\n,\nwhere l(σ) denotes the length of the permutation σ, that is, the number of\ninversions in σ. This determinant formula is the same as the usual formula\nsubstituting (−q) for (−1).\nWe define the coordinate ring of the space Mq(n) of quantum n × n\nmatrices by\nO(Mq(n)) = C[U]/ < (18.4) >, and\nand the coordinate ring of the virtual quantum group GLq(n) by\nCq[G] = O(GLq(n)) = O(Mq(n))[D−1\nq ].\nWe define the quantum minors and, using these, the quantum co-factor\nmatrix eU and the quantum inverse matrix U−1 = eU/Dq in a straightforward\nfashion (these constructions are left as exercises).\n101\n\nTheorem 18.1. The algebra O(GLq(n)) is a Hopf algebra, with\n∆(ui\nj) =\nX\nk\nui\nk ⊗uk\nj\nǫ(ui\nj) = δij\nS(ui\nj) = 1\nDq\neui\nj = (U−1)i\nj\nS(D−1\nq ) = Dq.\nWe also denote the quantum group GLq(n) by Gq, GLq(Cn) or GLq(V ).\nIt has to be emphasized that this is only a virtual object. Only its coordinate\nring Cq[G] is real. Henceforth, whenever we say representation or action of\nGq, we actually mean corepresentation or coaction of Cq[G], and so forth.\n102\n\nChapter 19\nQuantum unitary group\nScribe: Joshua A. Grochow\nGoal:\nDefine the quantum unitary subgroup of the standard quantum\ngroup.\nReference: [KS]\nRecall\nLet V = Cn, G = GLn(C) = GL(V ) = GL(Cn), and O(G) the coordinate\nring of G. The quantum group Gq = GLq(V ) is the virtual object whose\ncoordinate ring is\nO(Gq) = C[U]/⟨relations ⟩,\nwhere U is the generic n × n matrix of indeterminates, and the relations\nare the quadratic relations on the coordinates uj\ni defined in the last class so\nas to preserve the non-commuting relations among the coordinates of the\nquantum vector space Vq on which Gq acts. This coordinate ring is a Hopf\nalgebra.\n19.1\nA q-analogue of the unitary group\nIn this lecture we define a q-analogue of the unitary subgroup U = Un(C) =\nU(V ) ⊆GLn(C) = GL(V ) = G. This is a q-deformation Uq = Uq(V ) ⊆Gq\nof U(V ). Since Gq is only a virtual object, Uq will also be virtual. To define\nUq, we must determine how to capture the notion of unitarity in the setting\nof Hopf algebras. As we shall see, it is captured by the notion of a Hopf\n∗-algebra.\n103\n\nDefinition 19.1. A ∗-vector space is a vector space V with an involution\n∗: V →V satisfying\n(αv + βw)∗= αv∗+ βw∗\n(v∗)∗= v\nfor all v, w ∈V , and α, β ∈C.\nWe think of ∗as a generalization of complex conjugation; and in fact\nevery complex vector space is a ∗-vector space, where ∗is exactly complex\nconjugation.\nDefinition 19.2. A Hopf ∗-algebra is a Hopf algebra (A, ∆, ǫ, S) with an\ninvolution ∗: A →A such that (A, ∗) is a ∗-vector space, and:\n1. (ab)∗= b∗a∗, 1∗= 1\n2. ∆(a∗) = ∆(a)∗(where ∗acts diagonally on the tensor product A ⊗A:\n(v ⊗w)∗= (v∗⊗w∗))\n3. ǫ(a∗) = ǫ(a)\nThere is no explicit condition here on how ∗interacts with the antipode\nS.\nLet O(G) = C[G] be the coordinate ring of G as defined earlier.\nProposition 19.1. Then O(G) is a Hopf ∗-algebra.\nProof. We think of the elements in O(G) as C-valued functions on G and\ndefine ∗: O(G) →O(G) so that it satisfies the three conditions for a Hopf\n∗-algebra, and\n(4) For all f ∈O(G) and g ∈U ⊆G, f ∗(g) = f(g)\nLet uj\ni be the coordinate functions which, together with D−1, D = det(U),\ngenerate O(G). Because of the first condition on a Hopf ∗-algebra (relating\nthe involution ∗to multiplication), specifying (uj\ni)∗and D∗suffices to define\n∗completely. We define\n(uj\ni)∗= S(ui\nj) = (U−1)i\nj\nand D∗= D−1. We can check that this satifies (1)-(4). Here we will only\ncheck (4), and leave the remaining verification as an exercise. Let g be an\nelement of the unitary group U. Then (uj\ni)∗(g) = S(ui\nj)(g) = (g−1)i\nj = (g)j\ni,\nwhere the last equality follows from the fact that g is unitary (i.e. g−1 = g†,\nwhere † denotes conjugate transpose).\n104\n\nThus, we have defined a map f 7→f ∗purely algebraically in such a way\nthat the restriction of f ∗to the unitary group U is the same as taking the\ncomplex conjugate f on U.\nProposition 19.2. The coordinate ring Cq[G] = O(Gq) of the quantum\ngroup Gq = GLq(V ) is also a Hopf ∗-algebra.\nProof. The proof is syntactically identical to the proof for O(G), except\nthat the coordinate function uj\ni now lives in O(Gq) and the determinant D\nbecomes the q-determinant Dq. The definition of ∗is: (uj\ni)∗= S(ui\nj) and\nD∗\nq = D−1\nq , essentially the same as in the classical case.\nIntuitively, the “quantum subgroup” Uq of Gq is the virtual object such\nthat the restriction to Uq of the involution ∗just defined coincides with the\ncomplex conjugate.\n19.2\nProperties of Uq\nWe would like the nice properties of the classical unitary group to transfer\nover to the quantum unitary group, and this is indeed the case. Some of the\nnice properties of U are:\n1. It is compact, so we can integrate over U.\n2. we can do harmonic analysis on U (viz.\nthe Peter-Weyl Theorem,\nwhich is an analogue for U of the Fourier analysis on the circle U1).\n3. Every finite dimensional representation of U has a G-invariant Hermi-\ntian form, and thus a unitary basis – we say that every finite dimen-\nsional representation of U is unitarizable.\n4. Every finite dimensional representation X of U is completely reducible;\nthis follows from (3), since any subrepresentation W ⊆X has a per-\npendicular subrepresentation W ⊥under the G-invariant Hermitian\nform.\nCompactness is in some sense the key here. The question is how to define\nit in the quantum setting. Following Woronowicz, we define compactness to\nmean that every finite dimensional representation of Uq is unitarizable. Let\nus see what this means formally.\nLet A be a Hopf ∗-algebra, and W a corepresentation of A. Let ρ : W →\nW ⊗A be the corepresentation map. Let {bi} be a basis of W. Then, under\n105\n\nρ, bi 7→P\nj bj ⊗mj\ni for some mj\ni ∈A. We can thus define the matrix of\nthe (co)representation M = (mj\ni) in the basis {bi}. We define M∗such that\n(M∗)j\ni = (Mi\nj)∗. Thus, in the classical case (i.e. when q = 1), M∗= M†.\nWe say that the corepresentation W is unitarizable if it has a basis B =\n{bi} such that the corresponding matrix MB of corepresentation satisfies the\nunitarity condition: MBM∗\nB = I. In this case, we say B is a unitary basis\nof the corepresentation W.\nDefinition 19.3. A Hopf ∗-algebra A is compact if every finite dimensional\ncorepresentation of A is unitarizable.\nTheorem 19.1 (Woronowicz). The coordinte ring Cq[G] = O(Gq) is a\ncompact Hopf ∗-algebra. This implies that every finite dimensional repre-\nsentation of Gq, by which we mean a finite dimensional coorepresentation\nof Cq[G], is completely reducible.\nWoronowicz goes further to show that we can q-integrate on Uq, and that\nwe can do quantized harmonic analysis on Uq; i.e., a quantum analogue of\nthe Peter-Weyl theorem holds.\nNow that we know the finite dimensional representations of Gq are com-\npletely reducible, we can ask what the irreducible representations are.\n19.3\nIrreducible Representations of Gq\nWe proceed by analogy with the Weyl modules Vλ(G) for G. Recall that\nevery polynomial irreducible representation of G = GLn(C) is of this form.\nTheorem 19.2.\n1. For all partitions λ of length at most n, there exists\na q-Weyl module Vq,λ(Gq) which is an irreducible representation of Gq\nsuch that\nlim\nq→1 Vq,λ(Gq) = Vλ(G).\n2. The q-Weyl modules give all polynomial irreducible representations of\nGq.\n19.4\nGelfand-Tsetlin basis\nTo understand the q-Weyl modules better, we wish to get an explicit basis for\neach module Vq,λ. We begin by defining a very useful basis – the Gel’fand-\nTestlin basis – in the classical case for Vλ(G), and then describe the q-\nanalogue of this basis.\n106\n\nBy Pieri’s rule [FH]\nVλ(GLn(C)) =\nM\nλ′\nVλ′(GLn−1(C))\nwhere the sum is taken over all λ′ obtained from λ by removing any number\nof boxes (in a legal way) such that no two removed boxes come from the\nsame column. This is an orthogonal decomposition (relative to the GLn(C)-\ninvariant Hermitian form on Vλ) and it is also multiplicity-free, i.e., each Vλ′\nappears only once.\nFix a G-invariant Hermitian form on Vλ. Then the Gel’fand-Tsetlin basis\nfor Vλ(GLn(C)), denoted GT n\nλ , is the unique orthonormal basis for Vλ such\nthat\nGT n\nλ =\n[\nλ′\nGT n−1\nλ′\n,\nwhere the disjoint union is over the λ′ as in Pieri’s rule, and GT n−1\nλ′\nis defined\nrecursively, the case n = 1 being trivial.\nThe dimension of Vλ is the number of semistandard tableau of shape λ.\nWith any tableau T of this shape, one can also explicitly associate a basis\nelement GT(T) ∈GT n\nλ ; we shall not worry about how.\nWe can define the Gel’fand-Tsetlin basis GT n\nq,λ for Vq,λ(Gq(Cn)) analo-\ngously. We have the q-analogue of Pieri’s rule:\nVq,λ(Gq(Cn)) =\nM\nλ′\nVq,λ′(Gq(Cn−1))\nwhere the decomposition is orthogonal and multiplicity-free, and the sum\nranges over the same λ′ as above. So we can define GT n\nq,λ to be the unique\nunitary basis of Vq,λ such that\nGT n\nq,λ =\n[\nλ′\nGT n−1\nq,λ′ .\nWith any semistandard tableau T, one can also explicitly associate a basis\nelement GTq(T) ∈GT n\nq,λ′; details omitted.\n107\n\nChapter 20\nTowards positivity\nhypotheses via quantum\ngroups\nScribe: Joshua A. Grochow\nGoal: In this final brisk lecture, we indicate the role of quantum groups in\nthe context of the positivity hypothesis PH1. Specifically, we sketch how the\nLittlewood-Richardson rule – the gist of PH1 in the Littlewood-Richardson\nproblem – follows from the theory of standard quantum groups. We then\nbriefly mention analogous (nonstandard) quantum groups for the Kronecker\nand plethysm problems defined in [GCT4, GCT7], and the theorems and\nconjectures for them that would imply PH1 for these problems.\nReferences: [KS, K, Lu2, GCT4, GCT6, GCT7, GCT8]\nLet V = Cn, G = GLn(C) = GL(V ), Vλ = Vλ(G) a Weyl module of\nG, Gq = GLq(V ) the standard quantum group, Vq the q-deformation of V\non which GLq(V ) acts, Vq,λ = Vq,λ(Gq) the q-deformation of Vλ(G), and\nGTq,λ = GT n\nq,λ the Gel’fand-Tsetlin basis for Vq,λ.\n20.1\nLittlewood-Richardson rule via standard quan-\ntum groups\nWe now sketch how the Littlewood-Richardson rule falls out of the standard\nquantum group machinery, specifically the properties of the Gelfand-Tsetlin\nbasis.\n108\n\n20.1.1\nAn embedding of the Weyl module\nFor this, we have to embed the q-Weyl module Vq,λ in V ⊗d\nq\n, where d = |λ| =\nP λi is the size of λ. We first describe how to embed the Weyl module Vλ\nof G in V ⊗d in a standard way that can be quantized.\nIf d = 1, then Vλ(G) = V = V ⊗1. Otherwise, obtain a Young diagram\nμ from λ by removing its top-rightmost box that can be removed to get a\nvalid Young diagram, e.g.:\nx\n⇝\nλ\nμ\nIn the following, the box must be removed from the second row, since\nremoving from the first row would result in an illegal Young diagram:\nx\n⇝\nλ\nμ\nBy induction on d, we have a standard embedding Vμ(G) ֒→V ⊗d−1. This\ngives us an embedding Vμ(G) ⊗V ֒→V ⊗d. By Pieri’s rule [FH]\nVμ(G) ⊗V =\nM\nβ\nVβ(G),\nwhere the sum is over all β obtained from μ by adding one box in a legal way.\nIn particular, Vλ(G) ⊂Vμ(G) ⊗V . By restricting the above embedding, we\nget a standard embedding Vλ(G) ֒→V ⊗d.\nNow Pieri’s rule also holds in a quantized setting:\nVq,μ ⊗Vq =\nM\nβ\nVq,β(G),\nwhere β is as above.\nHence, the standard embedding Vλ ֒→V ⊗d above\ncan be quantized in a straightforward fashion to get a standard embedding\nVq,λ ֒→V ⊗d\nq\n. We shall denote it by ρ. Here the tensor product is meant to\nbe over Q(q). Actually, Q(q) doesn’t quite work. We have to allow square\nroots of elements of Q(q), but we won’t worry about this. For a semistandard\ntableau b of shape λ, we denote the image of a Gelfand-Tsetlin basis element\nGTq,λ(b) ∈GTq,λ under ρ by GT ρ\nq,λ(b) = ρ(GTq,λ(b)) ∈V ⊗d\nq\n.\n109\n\n20.1.2\nCrystal operators and crystal bases\nTheorem 20.1 (Crystallization). [DJM] The Gelfand-Tsetlin basis ele-\nments crystallize at q = 0. This means:\nlim\nq→0 GT ρ\nq,λ(b) = vi1(b) ⊗· · · ⊗vid(b),\n(20.1)\nfor some integer functions i1(b), . . . , id(b), and\nlim\nq→∞GT ρ\nq,λ(b) = vj1(b) ⊗· · · ⊗vjd(b),\n(20.2)\nfor some integer functions j1(b), . . . , jd(b).\nThe phenomenon that these limits consists of monomials, i.e., simple\ntensors is known as crystallization. It is related to the physical phenomenon\nof crystallization, hence the name. The maps b 7→i(b) = (i1(b), . . . , id(b))\nand b 7→j(b) = (j1(b), . . . , jd(b)) are computable in poly(⟨b⟩) time (where\n⟨b⟩is the bit-length of b).\nNow we want to define a special crystal basis of Vq,λ based on this phe-\nnomenon of crystallization. Towards that end, consider the following family\nof n × n matrices:\nEi =\n \n \n \n \n \n \n \n \n \n \n0\n0\n· · ·\n0\n...\n...\n...\n0\n1\n· · ·\n0\n0\n· · ·\n0\n...\n...\n0\n \n \n \n \n \n \n \n \n \n \n,\nwhere the only nonzero entry is a 1 in the i-th row and (i + 1)-st column.\nLet Fi = ET\ni . Corresponding to Ei and Fi, Kashiwara associates certain\noperators ˆEi and ˆFi on Vq,λ(Gq). We shall not worry about their actual\nconstruction here (for the readers familiar with Lie algebras: these are closely\nrelated to the usual operators in the Lie algebra of G associated with Ei and\nFi).\nIf we let ˆEi act on GT ρ\nq,λ(b), we get some linear combination\nˆEi(GT ρ\nq,λ(b)) =\nX\nb′\nab\nb′(q)GT ρ\nq,λ(b′),\nwhere ab\nb′(q) ∈Q(q) (actually an algebraic extension of Q(q) as mentioned\nabove). Essentially because of crystallization (Theorem 20.1), it turns out\n110\n\nthat limq→0 ab\nb′(q) is always either 0 or 1, and for a given b, this limit is 1 for\nat most one b′, if any. A similar result holds for ˆFi(GT ρ\nq,λ(b)). This allows\nus to define the crystal operators (due to Kashiwara):\neei · b =\n b′\nif limq→0 ab\nb′(q) = 1,\n0\nif no such b′ exists,\nand similarly for efi. Although these operators are defined according to a\nparticular embedding Vq,λ ֒→V ⊗d\nq\nand a basis, they can be defined intrinsi-\ncally, i.e., without reference to the embedding or the Gel’fand-Tsetlin basis.\nNow, let W be a finite-dimensional representation of Gq, and R the\nsubring of functions in Q(q) regular at q = 0 (i.e. without a pole at q = 0).\nA lattice within W is an R-submodule of W such that Q(q) ⊗R L = W.\n(Intuition behind this definition: R ⊂Q(q) is analogous to Z ⊂Q. A lattice\nin Rn is a Z-submodule L of Rn such that R ⊗Z L = Rn.)\nDefinition 20.1. An (upper) crystal basis of a representation W of Gq is\na pair (L, B) such that\n• L is a lattice in W preserved by the Kashiwara operators ˆEi and ˆFi,\ni.e. ˆEi(L) ⊆L and ˆFi(L) ⊆L.\n• B is a basis of L/qL preserved by the crystal operators eei and efi, i.e.,\neei(B) ⊆B ∪{0} and efi(B) ⊆B ∪{0}.\n• The crystal operators eei and efi are inverse to each other wherever\npossible, i.e., for all b, b′ ∈B, if eei(b) = b′ ̸= 0 then efi(b′) = b, and\nsimilarly, if efi(b) = b′ ̸= 0 then eei(b′) = b.\nIt can be shown that if W = Vq,λ(Gq), then there exists a unique b ∈B\nsuch that eei(b) = 0 for all i; this corresponds to the highest weight vector\nof Vq,λ (the weight vectors in Vq,λ are analogous to the weight vectors in Vλ;\nwe do not give their exact definition here). By the work of Kashiwara and\nDate et al [K, DJM] above, the Gel’fand-Tsetlin basis (after appropriate\nrescaling) is in fact a crystal basis: just let\nL = LGT\n=\nthe R-module generated by GTq,λ, and\nBGT\n=\nGTq,λ(b),\nwhere GTq,λ(b) is the image under the projection L 7→L/qL of the set of\nbasis vectors in GTq,λ(b).\n111\n\nTheorem 20.2 (Kashiwara).\n1. Every finite-dimensional Gq-module has\na unique crystal basis (up to isomorphism).\n2. Let (Lλ, Bλ) be the unique crystal basis corresponding to Vq,λ. Then\n(Lα, Bα) ⊗(Lβ, Bβ) = (Lα ⊗Lβ, Bα ⊗Bβ) is the unique crystal basis\nof Vq,α ⊗Vq,β, where Bα ⊗Bβ denotes {ba ⊗bb|ba ∈Bα, bb ∈Bβ}.\nIt can be shown that every b ∈Bλ has a weight; i.e., it is the image of a\nweight vector in Lλ under the projection Lλ →Lλ/qLλ.\nNow let us see how the Littlewood-Richardson rule falls out of the prop-\nerties of the crystal bases. Recall that the specialization of Vq,α at q = 1 is\nthe Weyl module Vα of G = GLn(C), and\nVα ⊗Vβ =\nM\nγ\ncγ\nα,βVγ\n(20.3)\nwhere cγ\nα,β are the Littlewood-Richardson coefficients.\nThe Littlewood-\nRichardson rule now follows from the following fact:\ncγ\nα,β = #{b ⊗b′ ∈Bα ⊗Bβ|∀i, eei(b ⊗b′) = 0 and b ⊗b′ has weight γ}.\nIntuitively, b ⊗b′ here correspond to the highest weight vectors of the G-\nsubmodules of Vα ⊗Vβ isomorphic to Vγ.\n20.2\nExplicit decomposition of the tensor product\nThe decomposition (20.3) is only an abstract decomposition of Vα ⊗Vβ as a\nG-module. Next we consider the explicit decomposition problem. The goal\nis to find an explicit basis B = Bα⊗β of Vα ⊗Vβ that is compatible with this\nabstract decomposition. Specifically, we want to construct an explicit basis\nB of Vα ⊗Vβ in terms of suitable explicit bases of Vα and Vβ such that B\nhas a filtration\nB = B0 ⊇B1 ⊇· · · ⊇∅\nwhere each ⟨Bi⟩/⟨Bi+1⟩is an irreducible representation of G and ⟨Bi⟩denotes\nthe linear span of Bi.\nFurthermore, each element b ∈B should have a\nsufficiently explicit representation in terms of the basis Bα ⊗Bβ of Vα ⊗Vβ.\nThe explicit decomposition problem for the q-analogue Vq,α ⊗Vq,β is similar.\nFor example, we have already constructed explicit Gelfand-Tsetlin bases\nof Weyl modules. But it is not known how to construct an explicit basis B\n112\n\nwith filtration as above in terms of the Gelfand-Tsetlin bases of Vα and Vβ\n(except when the Young diagram of either α or β is a single row).\nKashiwara and Lusztig [K, Lu2] construct certain canonical bases Bq,α\nand Bq,β of Vq,α and Vq,β, and Lusztig furthermore constructs a canonical\nbasis Bq = Bq,α⊗β of Vq,α ⊗Vq,β such that:\n1. Bq has a filtration as above,\n2. Each b ∈Bq has an expansion of the form\nb =\nX\nbα∈Bq,α,bβ∈Bq,β\nabα,bβ\nb\nbα ⊗bβ,\nwhere each abα,bβ\nb\nis a polynomial in q and q−1 with nonnegative inte-\ngral coefficients,\n3. Crystallization: For each b, as q →0, exactly one coefficient abα,bβ\nb\n→1,\nand the remaining all vanish.\nThe proof of nonnegativity of the coefficients of abα,bβ\nb\nis based on the Rie-\nmann hypothesis (theorem) over finite fields [Dl2], and explicit formulae for\nthese coefficients are known in terms of perverse sheaves [BBD] (which are\ncertain types of algebro-geometric objects).\nThis then provides a satisfactory solution to the explicit decomposition\nproblem, which is far harder and deeper than the abstract decomposition\nprovided by the Littlewood-Richardson rule. By specializing at q = 1, we\nalso get a solution to the explicit decomposition problem for Vα ⊗Vβ. This\n(i.e. via quantum groups) is the only known solution to the explicit decom-\nposition problem even at q = 1. This may give some idea of the power of\nthe quantum group machinery.\n20.3\nTowards nonstandard quantum groups for the\nKronecker and plethysm problems\nNow the goal is to construct quantum groups which can be used to de-\nrive PH1 and explicit decomposition for the Kronecker and plethysm prob-\nlems just as the standard quantum group can be used for the same in the\nLittlewood-Richardson problem.\nIn the Kronecker problem, we let H = GL(Cn) and G = GL(Cn ⊗Cn).\nThe Kronecker coefficient κγ\nα,β is the multiplicity of Vα(H)⊗Vβ(H) in Vγ(G):\nVγ(G) =\nM\nα,β\nκγ\nα,βVα(H) ⊗Vβ(H).\n113\n\nThe goal is to get a positive # P-formula for κγ\nα,β; this is the gist of PH1\nfor the Kronecker problem.\nIn the plethysm problem, we let H = GL(Cn) and G = GL(Vμ(H)).\nThe plethysm constant aπ\nλ,μ is the multiplicity of Vπ(H) in Vλ(G):\nVλ(G) =\nM\nπ\naπ\nλ,μVπ(H).\nAgain, the goal is to get a positive # P-formula for the plethysm constant;\nthis is the gist of PH1 for the plethysm problem.\nTo apply the quantum group approach, we need a q-analogue of the\nembedding H ֒→G. Unfortunately, there is no such q-analogue in the theory\nof standard quantum groups. Because there is no nontrivial quantum group\nhomomorphism from the standard quantum group Hq = GLq(Cn) and to\nthe standard quantum group Gq.\nTheorem 20.3. (1) [GCT4]: Let H and G be as in the Kronecker problem.\nThen there exists a quantum group ˆGq such that the homomorphism H →G\ncan be quantized in the form Hq ֒→ˆGq. Furthermore, ˆGq has a unitary\nquantum subgroup ˆUq which corresponds to the maximal unitary subgroup\nU ⊆G, and a q-analogue of the Peter-Weyl theorem holds for ˆGq. The\nlatter implies that every finite dimensional representation of ˆGq is completely\ndecomposible into irreducibles.\n(2) [GCT7] There is an analogous (possibly singular) quantum group ˆGq\nwhen H and G are as in the plethysm problem. This also holds for general\nconnected reductive (classical) H.\nSince the Kronecker problem is a special case of the (generalized) plethysm\nproblem, the quantum group in GCT 4 is a special case of the quantum group\nin GCT 7. The quantum group in the plethysm problem can be singular, i.e.,\nits determinant can vanish and hence the antipode need not exist. We still\ncall it a quantum group because its properties are very similar to those of the\nstandard quantum group; e.g. q-analogue of the Peter-Weyl theorem, which\nallows q-harmonic analysis on these groups. We call the quantum group\nˆGq nonstandard, because though it is qualitatively similar to the standard\n(Drinfeld-Jimbo) quantum group Gq, it is also, as expected, fundamentally\ndifferent.\nThe article [GCT8] gives a conjecturally correct algorithm to construct a\ncanonical basis of an irreducible polynomial representation of ˆGq which gen-\neralizes the canonical basis for a polynomial representation of the standard\nquantum group as per Kashiwara and Lusztig. It also gives a conjecturally\n114\n\ncorrect algorithm to construct a canonical basis of a certain q-deformation of\nthe symmetric group algebra C[Sr] which generalizes the Kazhdan-Lusztig\nbasis [KL] of the Hecke algebra (a standard q-deformation of C[Sr]). It is\nshown in [GCT7, GCT8] that PH1 for the Kronecker and plethysm problems\nfollows assuming that these canonical bases in the nonstandard setting have\nproperties akin to the ones in the standard setting. For a discussion on SH,\nsee [GCT6].\n115\n\nPart II\nInvariant theory with a view\ntowards GCT\nBy Milind Sohoni\n116\n\nChapter 21\nFinite Groups\nReferences: [FH, N]\n21.1\nGeneralities\nLet V be a vector space over C, and let GL(V ) denote the group of all\nisomorphisms on V . For a fixed basis of V , GL(V ) is isomorphic to the\ngroup GLn(C), the group of all n × n invertible matrices.\nLet G be a group and ρ : G →GL(V ) be a representation. We also\ndenote this by the tuple (ρ, V ) or say that V is a G-module. Let Z ⊆V be\na subspace such that ρ(g)(Z) ⊆Z for all g ∈G. Then, we say that Z is\nan invariant subspace. We say that (ρ, V ) is irreducible if there is no\nproper subspace W ⊂V such that ρ(g)(W) ⊆W for all g ∈G. We say that\n(ρ, V ) is indecomposable is there is no expression V = W1 ⊕W2 such that\nρ(g)(Wi) ⊆Wi, for all g ∈G.\nFor a point v ∈V , the orbit O(v), and the stabilizer Stab(v) are\ndefined as:\nO(v)\n=\n{v′ ∈V |∃g ∈G with ρ(g)(v) = v′}\nStab(v)\n=\n{g ∈G|ρ(g)(v) = v}\nOne may also define v ∼v′ if there is a g ∈G such that ρ(g)(v) = v′. It is\nthen easy to show that [v]∼= O(v).\nLet V ∗be the dual-space of V . The representation (ρ, V ) induces the\ndual representation (ρ∗, V ∗) defined as ρ∗(v∗)(v) = v∗(ρ(g−1)(v)). It will\nbe convenient for ρ∗to act on the right, i.e., ((v∗)(ρ∗))(v) = v∗(ρ(g−1)(v)).\nWhen ρ is fixed, we abbrieviate ρ(g)(v) as just g · v. Along with this,\nthere are the standard constructions of the tensor T d(V ), the symmetric\npower Symd(V ) and the alternating power ∧d(V ) representations.\n117\n\nOf special significance is Symd(V ∗), the space of homogeneous polyno-\nmial functions on V of degree d. Let dim(V ) = n and let X1, . . . , Xn be a\nbasis of V ∗. We define as follows:\nR = C[X1, . . . , Xn] = ⊕∞\nd=0Rd = ⊕∞\nd=0Symd(V ∗)\nThus R is the ring of all polynomial functions on V and is isomorphic\nto the algebra (over C) of n indeterminates. Since G acts on the domain V ,\nG also acts on all functions f : V →C as follows:\n(f · g)(v) = f(g−1 · v)\nThis action of G on all functions extends the action of G on polynomial\nfunctions above.\nIndeed, for any g ∈G, the map tg : R →R given by\nf →f · g is an algebra isomorphism. This is called the translation map.\nFor an f ∈R, we say that f is an invariant if f · g = f for all g ∈G.\nThe following are equivalent:\n• f ∈R is an invariant.\n• Stab(f) = G.\n• f(g · v) = f(v) for all g ∈G and v ∈V .\n• For all v, v′ such that v′ ∈Orbit(v), we have f(v) = f(v′).\nIf W1 and W2 are two modules of G and φ : W1 →W2 is a linear map\nsuch that g · φ(w1) = φ(g · w1) for all g ∈G and w1 ∈W1 then we say that\nφ is G-equivariant or that φ is a morphism of G-modules.\n21.2\nThe finite group action\nLet G be a finite group and (μ, W) be a representation.\nRecall that a complex inner product on W is a map h : W × W →C\nsuch that:\n• h(αw + βw′, w′′) = αh(w, w′′) + βh(w′, w′′) for all α, β ∈C and all\nw, w′, w′′ ∈W.\n• h(w′′, αw + βw′) = αh(w′′, w) + βh(w′′, w′) for all α, β ∈C and all\nw, w′, w′′ ∈W.\n• h(w, w) > 0 for all w ̸= 0.\n118\n\nAlso recall that if Z ⊆W is a subspace, then Z⊥is defined as:\nZ⊥= {w ∈W|h(w, z) = 0 ∀z ∈Z}\nAlso recall that W = Z ⊕Z⊥.\nWe say that an inner product h is G-invariant if h(g·w, g·w′) = h(w, w′)\nfor all w, w′ ∈W and g ∈G.\nProposition 21.1. Let W be as above, and Z be an invariant subspace of\nW. Then Z⊥is also an invariant subspace. Thus every reducible represen-\ntation of G is also decomposable.\nProof: Let x ∈Z⊥, z ∈Z and let us examine (g · x, z). Applying g−1 to\nboth sides, we see that:\nh(g · x, z) = h(g−1 · g · x, g−1 · z) = h(x, g−1 · z) = 0\nThus, G preserves Z⊥as claimed. □\nLet h be a complex inner product on W. We define the inner product\nhG as follows:\nhG(w, w′) =\n1\n|G|\nX\ng′∈G\nh(g′ · w, g′ · w′)\nLemma 21.1. hG is a G-invariant inner product.\nProof: First we see that\nhG(w, w) =\n1\n|G|\nX\ng′∈G\nh(w′, w′)\nwhere w′ = g′ · w. Thus hG(w, w) > 0 unless w = 0. Secondly, by the\nlinearity of the action of G, we see that hG is indeed an inner product.\nFinally, we see that:\nhG(g · w, g · w′) =\n1\n|G|\nX\ng′∈G\nh(g′ · g · w, g′ · g · w′)\nSince as g′ ranges over G, so will g′ · g for any fixed g, we have that hG is\nG-invariant. □\nTheorem 1.\n• Let G be a finite group and (ρ, V ) be an indecomposable\nrepresentation, then it is also irreducible.\n119\n\n• Every representation (ρ, V ) may be decomposed into irreducible repre-\nsentations Vi. Thus V = ⊕iVi, where (ρi, Vi) is an irreducible repre-\nsentation.\nProof: Suppose that Z ⊆V is an invariant subspace, then V = Z ⊕Z⊥is\na non-trivial decomposition of V contradicting the hypothesis. The second\npart is proved by applying the first, recursively. □\nWe have seen the operation of averaging over the group in going\nfrom the inner product h to the G-invariant inner product hG. A similar\napproach may be used for constructing invariant polynomials functions. So\nlet p(X) ∈R = C[X1, . . . , Xn] be a polynomial function. We define the\nfunction pG : V →C as:\npG(v) =\n1\n|G|\nX\ng∈G\np(g · v)\nThe transition from p to pG is called the Reynold’s operator.\nProposition 21.2. Let p ∈R be of degree atmost d, then pG is also a\npolynomial function of degree atmost d. Next, pG is an invariant.\nLet RG denote the set of all invariant polynomial functions on the space\nV . It is easy to see that RG ⊆R is actually a subring of R.\nLet Z ⊆V be an arbitrary subset of V . We say that Z is G-closed if\ng ·z ∈Z for all g ∈G and z ∈Z. Thus Z is a union of orbits of points in V .\nLemma 21.2. Let p ∈RG be an invariant and let Z = V (p) be the variety\nof p. Then Z is G-closed.\nWe have already seen that O(v), the orbit of v arises from the equivalence\nclass ∼on V . Since the group is finite, |O(v)| ≤|G| for any v. Let O1 and\nO2 be disjoint orbits.\nIt is essential to determine if elements of RG can\nseparate O1 and O2.\nLemma 21.3. Let O1 and O2 be as above, and I1 and I2 be their ideals in\nR. Then there are p1 ∈I1 and p2 ∈I2 so that p1 + p2 = 1.\nProof: This follows from the Hilbert Nullstellensatz. Since the point sets\nare finite, there is an explicit construction based on Lagrange interpolation.\n□\nLet G be a finite group and (ρ, V ) be a representation as above. We\nhave see that this induces an action on C[X1, . . . , Xn]. Also note that this\naction is homogeneous: for a g ∈G and p ∈Rd, we have that p · g ∈Rd\n120\n\nas well. Thus RG, the ring of invariants, is a homogeneous subring of R. In\nother words:\nRG = ⊕∞\nd=0RG\nd\nwhere RG\nd are invariants which are homogeneous of degree d. The existence\nof the above decomposition implies that every invariant is a sum of homo-\ngeneous invariants. Now, since RG\nd ⊆Rd as a vector space over C. Thus\ndimC(RG\nd ) ≤dimC(Rd) ≤\n n + d −1\nn −1\n \nWe define the hilbert function h(RG) of RG (or for that matter, of\nany homogeneous ring) as:\nh(RG) =\n∞\nX\nd=0\ndimC(RG\nd )zd\nWe will see now that h(RG) is actually a rational function which is easily\ncomputed. We need a lemma.\nLet (ρ, W) be a representation of the finite group G. Let\nW G = {w ∈W|g · w = w}\nbe the set of all vector invariants in W, and this is a subspace of W. .\nLemma 21.4. Let (ρ, W) be as above. We have:\ndimC(W G) =\n1\n|G|\nX\ng∈G\ntrace(ρ(g))\nProof: Define P =\n1\n|G|\nP\ng∈G ρ(g), as the average of the representation\nmatrices. We see that ρ(g) · P = P · ρ(g) and that P 2 = P. Thus P is\ndiagonalizable and the eigenvalues of P are in the set {1, 0}. Let W1 and\nW0 be the corresponding eigen-spaces. It is clear that W G ⊆W1 and that\nW1 is fixed by each g ∈G. We now argue that every w ∈W1 is actually an\ninvariant. For that, let wg = g · w. We then have that Pw = w implies that\nw =\n1\n|G|\nX\ng∈G\nwg\nNote that a change-of-basis does not affect the hypothesis nor the assertion.\nWe may thus assume that each ρ(g) is unitary, we have that wg = w for all\ng ∈G. Now, the claim follows by computing trace(P). □\nWe are now ready to state Molien’s Theorem:\n121\n\nTheorem 2. Let (ρ, W) be as above. We have:\nh(RG) =\n1\n|G|\nX\ng∈G\n1\ndet(I −zρ(g))\nProof: Let dimC(W) = n and let {X1, . . . , Xn} be a basis of W ∗. Since\nRG = P\nd RG\nd and each RG\nd ⊆C[X1, . . . , Xn]d. Note that each C[X1, . . . , Xn]d\nis also a representation ρd of G.\nFurthermore, it is easy to see that if\n{λ1, . . . , λn} are the eigenvalues of ρ(g), then the eigen-values of the matrix\nρd(g) are precisely (including multiplicity)\n{\nY\ni\nλdi\ni |\nX\ni\ndi = d}\nThus\ntrace(ρd(g)) =\nX\nd:|d|=d\nY\ni\nλdi\ni\nWe then have:\nh(RG)\n=\nP\nd zddimC(RG\nd )\n=\nP\nd zd[ 1\n|G|\nP\ng trace(ρd(g))]\n=\n1\n|G|\nP\ng\n1\n(1−λ1(g)z)...(1−λn(g)z)\n=\n1\n|G|\nP\ng\n1\ndet(I−zρ(g))\nThis proves the theorem. □\n21.3\nThe Symmetric Group\nSn will denote the symmetric group of all bijections on the set [n]. The\nstandard representation of Sn is obviously on V = Cn with\nσ · (v1, . . . , vn) = (vσ(1), . . . , vσ(n))\nThus, regarding V as column vectors, and Sn as the group of n × n-\npermutation matrices, we see that the action of permutation P on vector v\nis given by the matrix multiplication P · v.\nLet X1, . . . , Xn be a basis of V ∗.\nSn acts on R = C[X1, . . . , Xn] by\nXi · σ = Xσ(i). The orbit of any point v = (v1, . . . , vn) is the collection of\nall permutation of the entries of the vector v and thus the size of the orbit\nis bounded by n!.\n122\n\nThe invariants for this action are given by the elementary symmetric\npolynomials ek(X), for k = 1, . . . , n, where\nek(X) =\nX\ni1<i2<...<ik\nXi1Xi2 . . . Xik\nGiven two vector v and w, if w ̸∈O(v), then there is a k such that\nek(v) ̸= ek(w). This follows from the theory of equations in one variable.\nThe ring RG equals C[e1, . . . , en] and has no algebraic dependencies. The\nhilbert function of RG may then be expressed as:\nh(RG) =\n1\n(1 −z)(1 −z2) . . . (1 −zn)\nIt is an exercise to verify that Molien’s expression agrees with the above.\nA related action of Sn is the diagonal action: Let X = {X1, . . . , Xn},\nY = {Y1, . . . , Yn}, and so on upto W = {W1, . . . , Wn} be a family of r\n(disjoint) variables. Let B = C[X, Y, . . . , W] be the ring of polynomials in\nthe variables of the disjoint union.\nWe define the action of Sn on X ∪Y ∪. . . ∪W as Xi · σ = Xσ(i),\nYi · σ = Yσ(i), and so on.\nThe matrix equivalence of this action is the action of the permutation\nmatrices on n × r matrices A, where the action of P on A is given by P · A.\nThe invariants BG is obtained from the r = 1 case by a curious operation:\nLet DXY denote the operator:\nDXY = Y1\n∂\n∂X1\n+ . . . + Yn\n∂\n∂Xn\nThe ring BG is obtained from RG by applying the operators DXY , DXW , DW X\nand so on, to elements of RG. As an example, we have\ne2(X) = X1X2 + X1X3 + . . . + Xn−1Xn\nWe have DXY (e2) as:\nDXY (e2(X)) =\nX\ni̸=j\nXiYj\nThis is clearly an element of BG.\n123\n\nChapter 22\nThe Group SLn\nReferences: [FH, N]\n22.1\nThe Canonical Representation\nLet V be a vector space of dimension n, and let x1, . . . , xn be a basis for V .\nLet X1, . . . , Xn be the dual basis of V ∗.\nSL(V ) will denote the group of all unimodular linear transformations on\nV . In the above basis, this group is isomorphic to that of all n × n matrices\nof determinant 1, or in other words SLn(C).\nThe standard representation of SL(V ) is obviously V itself: Given φ ∈\nSL(V ) and v ∈V , we have φ · v = φ(v) is the action of φ on v.\nIn terms of the basis x above we may write v = [x1, . . . , xn][α1, . . . , αn]T\nand thus φ·v as [φ·x1, . . . , φ·xn][α1, . . . , αn]T . If φ·xi = [x1, . . . , xn][a1i, . . . , ani]T ,\nthen we have\nφ · v = [x1, . . . , xn]\n \n \na11\n. . .\na1n\n...\n...\nan1\n. . .\nann\n \n \n \n \nα1\n...\nαn\n \n \nWe denote this matrix as Aφ.\nWe may now work with SLn(C) or simply SLn. Given a vector a =\n[α1, . . . , αn]T , we see that the matrix multiplication A · a is the action of A\non the column vector a.\nLet us now understand the orbits of typical elements in the column space\nCn. The typical v ∈Cn is a non-zero column vector. For any non-zero vector\nw, we see that there is an element A ∈SLn such that w = Av. Furthermore,\n124\n\nfor any B ∈SLn, clearly Bv ̸= 0. Thus we see that Cn has exactly two\norbits:\nO0 = {0} O1{v ∈Cn|v ̸= 0}\nNote that O1 is dense in V = Cn and its closure includes the orbit O0 and\ntherefore, the whole of V .\nLet R = C[X1, . . . Xn] be the ring of polynomial functions on Cn. We\nexamine the action of SLn on C[X]. Recall that the action of A on X should\nbe such that the evaluations Xi(xj) = δij must be conserved. Thus if the\ncolumn vectors xi = [0, . . . , 0, 1, 0 . . . , 0]T are the basis vectors of V and the\nrow vectors Xi = [0, . . . , 0, 1, 0 . . . , 0] that of V ∗, then a matrix A ∈SLn\ntransforms xi to Axi and Xj to XjA−1. Thus Xj(xi) = Xj/cdotxi goes to\nXjA−1Axi = Xj · xi.\nNext, we examine R = C[X] for invariants. First note that the action of\nSLn is homogeneous and thus we may assume that an invariant p is actually\nhomogeneous of degree d. Next we see that p must be constant on O1 and\nO0. In this case, if p(O1) = α then by the density of O1 in V , we see that p\nis actually constant on V . Thus RG = R0 = C.\n22.2\nThe Diagonal Representation\nLet us now consider the diagonal representation of the above representation.\nIn other words, let V r be the space of all complex n × r matrices x. The\naction of A on x is obvious A · x. Let X be the r × n-matrix dual to x. It\ntransforms according to X →XA−1.\nLet us examine the case when r < n and compute the orbits in V r. Let\nx ∈V r be a matrix and y be a column vector such that x · y = 0. We see\nthat (A · x) · y = 0 as well. Thus if ann(x) = {y ∈Cr|x · y = 0} is the\nannihilator space of x, then we see that ann(x) = ann(A · x).\nWe show that the converse is also true:\nProposition 22.1. Let r < n and x, x′ ∈V r be such that ann(x) = ann(x′).\nThen there is an A ∈SLn such that x = A · x′.\nProof: Use the row-echelon form construction. Make the pivots as 1 using\nappropriate diagonal matrices. Since r < n these can be chosen from SLn.\n□\nThis decides the orbit structure of V r for r < n. The largest orbit Or is\nof course when ann(x) = 0. Thus, when x ∈Vr is a mtrix of rank r, we see\nthat ann(x) = 0 is trivial. The generic element of V r is of this form. Thus\nOr is dense in V r. For this reason, there are no non-constant invariants.\n125\n\nAnother calculation is the computation of the closure of orbits. Let O, O′\nbe two arbitrary orbits of V r. We have:\nProposition 22.2. O′ lies in the closure of O if and only if ann(O′) ⊇\nann(O).\nProof: One direction is clear. We prove the other direction when ann(O) ⊆\nann(O′) and dim(ann(O)) = dim(ann(O′)) −1.\nThus, we may assume\nthat O = [x] and O′ = [x′] with both x and x′ such that rowspace(x) ⊇\nrowspace(x′) with rank(x′) = rank(x)−1. Then, upto SLn, we may assume\nthat the rows x′[1], . . . , x′[k] match the first k rows of x, and that x′[k+1] =\nx′[k + 2] = . . . = x′[n] = 0. Note that x[k + 1] is non-zero and x[k + 2] exists\nand is zero. We then construct the matrix A(t) as follows:\nA(t) =\n \n \nIk\n0\n0\n0\n0\nt\n0\n0\n0\n0\nt−1\n0\n0\n0\n0\nIn−k−2\n \n \nWe see that A(t) ∈SLn for all t ̸= 0. Next, if we let x(t) = A(t) · x, then\nwe see that:\nlim\nt→0 x(t) = x′\nThis shows that x′ lies in the closure of O. □\nWe now look at the case when r ≥n. We have:\nProposition 22.3. Let r ≥n and x, x′ ∈V r be such that ann(x) = ann(x′).\nIf (i) rank(x) < n then there is an A ∈SLn such that x′ = Ax, (ii) if\nrank(x) = n there is a unique A ∈SLn and a λ ∈C∗such that if z = A · x,\nthen the first n −1 rows of z equal those of x′ and z[n] = λx′[n].\nThe proof is easy. Let x be matrix in V r of rank n, and C be a subset\nof [r] = {1, 2, . . . , r} such that det(x[C]) ̸= 0.\nProposition 22.4. Let x be as above. Then O(x), the orbit of x, equals\nall points x′ ∈V r such that (i) ann(x) = ann(x′) and (ii) det(X′[C]) =\ndet(x[C]). The set of all rank n points in V r is dense in V r.\nThe proof is easy.\nProposition 22.5. The orbit O(x) as above is closed.\n126\n\nProof: Notice that if A ∈SLn and z = Ax then det(x[C]) = det(z[C]).\nWe may rewrite condition (i) above as ann(x′) ⊇ann(x). Condition (ii)\nensures that rank(x′) = n and that condition (i) holds with equality. Thus\nis y1, . . . , yr−n are column vectors generating ann(x), then the equations\nx′ys = 0 and det(x′[C]) = det(x[C]) determines the orbit O(x). Thus O(x)\nis the zero-set of some algebraic equations and thus is closed. □\nProposition 22.6. Let x ∈V r be such that rank(x) < n. Then (i) O(x)\nequals those x′ such that ann(x) = ann(x′), (ii) O(x) is not closed and its\nclosure O(x) equals points x′ such that ann(x′) ⊇ann(x).\nWe now move to the computation of invariants. The space C[V r] equals\nthe space of all polynomials in the variable matrix X = (Xij) where i =\n1, . . . , n and j = 1, . . . , r. It is clear that for any set C of n columns of X,\nwe see that det(X[C]) is an invariant. We will denote C as C = c1 < c2 <\n. . . < cn and det(X[C]) as pC, the C-th Plucker coordinate. We aim to\nshow now that these are the only invariants.\nLet C0 = {1 < 2 < . . . < n} and W ⊆V r be the space of all matrices\nx ∈V r such that x[C0] = diag(1, . . . , 1, λ). Let W ′ be those elements of W\nfor which λ ̸= 0.\nLemma 22.1. Let W ′ be as above. (i) If x ∈W ′, then O(x) ∩W ′ = x, (ii)\nfor any x ∈V r such that det(x[C0]) ̸= 0, there is a unique A ∈SLn such\nthat Ax ∈W ′.\nLet us call Z′ ⊆V r as those x such that det(x[C0]) ̸= 0. We then have\nthe projection map\nπ : Z′ →W ′\ngiven by the above lemma. Note that Z′ is SLn-invariant: if x ∈Z′ and\nA ∈SLn then A · x ∈Z′ as well.\nThe ring C[Z′] of regular functions on Z′ is precisely C[X]det([X[C0]), the\nlocalization of C[X] at det(X[C0]).\nWe may parametrize W ′ as:\nW =\n \n \n1\n. . .\n0\nw1,n+1\n. . .\nw1,r\n...\n...\n...\n...\n0\n. . .\nwn,n\nwn,n+1\n. . .\nwn,r\n \n \nLet\nW = {Wi,j|i = 1, . . . , n, j = n + 1, . . . , r} ∪{Wn,n}\nbe a set of (n −r) · r + 1 variables. The ring C[W ′] of regular function on\nW ′ is precisely C[W]Wn,n.\n127\n\nLet x ∈Z′ be an arbitrary point and A = x[C0]. Let Ci,j be the set\nC0 −i + j. The map π is given by:\nπ(x)n,n\n=\ndet(A)\nπ(x)i,j\n=\n det(x[Ci,j])/det(A)\nfor i ̸= n\ndet(x[Ci,j])\notherwise\nThe map π causes the map:\nπ∗: C[W ′] →C[Z′]\ngiven by:\nπ∗(Wn,n)\n=\ndet(X[C0])\nπ∗(Wi,j)\n=\n det(X[Ci,j])/det(X[C0])\nfor i ̸= n\ndet(X[Ci,j])\notherwise\nNow we note that Z′ is dense in V r. Let p ∈C[X]SLn be an invariant.\nClearly p restricted to W ′ defines an element pW ′ of C[W ′].\nThis then\nextends to Z′ via π∗. Clearly π∗(pW ′) must match p on Z′. Thus we have\nthat p is a polynomial in det(X[Ci,j]) possibly localized at det(X[C0]).\nNote that for a general C, det(X[C]) is already expressible in det(X[C0])\nand det(X[Ci,j]).\n22.3\nOther Representations\nWe discuss two other representations:\nThe Conjugate Action: Let M be the space of all n × n matrices with\ncomplex entries. we define the action of A ∈SLn on M as follows. For an\nM ∈M, A acts on it by conjugation.\nA · M = AMA−1\nNote that dimC(M) = n2. Let X = {Xij|1 ≤i, j ≤n} be the dual space\nto M. The invariants for this action are Tr(X), . . . , Tr(Xi), . . . , Tr(Xn).\nThat these are invariants ic clear, for Tr(AMiA−1) = Tr(M) for any A, M.\nAlso note that Tr(X) = X11 + . . . + Xnn is linear in the X’s.\nThis\nindicates an invariant hyperplane in M.\nThis is precisely the trace zero\nmatrices. Thus M = M0 ⊕M1 where M0 are all matrices M such that\nTr(M) = 0. The one-dimensional complementary space M1 is composed of\nmultiples of the identity matrix.\n128\n\nThe orbits are parametrized by the Jordan canonical form JCF(M).\nIn other words, if JCF(M) = JCF(M′) them M′ ∈O(M). Furthermore,\nif JCF(M) is diagonal then the orbit of M is closed in M.\nThe Space of Forms: Let V be a complex vector space of dimension n\nandV ∗its dual. Let X1, . . . , Xn be the dual basis. The space Symd(V ∗)\nconsists of degree d forms are formal linear combinations of the monomials\nXi1\n1 . . . Xin\nn with i1 + . . . + in = d. The typical form may be written as:\nf(X) =\nX\ni1+...+in=d\nBi1...inXi1\n1 . . . Xin\nn\nLet A ∈SLn be such that\nA−1 =\n \n \na11\n. . .\nan1\n...\n...\na1n\n. . .\nann\n \n \nThe space Symd(V ∗) is the formal linear combination of the generaic coef-\nficients {Bi||i| = d}. The action of A is obtained by substituting\nXi →\nX\nj\naijXi\nin f(X) and recoomputing the coefficients. we illustrate this for n = 2 and\nd = 2. Then, the generic form is given by:\nf(X1, X2) = B20X2\n1 + B11X1\n1X1\n2 + B02X2\n2\nUpon substitution, we get f(a11X1 + a12X2, a21X1 + a22X2): Thus the new\ncoefficients are:\nB20\n→\nB20a2\n11+\nB11a11a21+\nB02a2\n21\nB11\n→\nB202a11a12+\nB11(a11a22 + a12a21)+\nB022a21a22\nB02\n→\nB20a2\n12+\nB11a12a22+\nB02a2\n22\nWe thus see that the variables B move linearly, with coefficients as homo-\ngeneous polynomials of degree 2 in the coefficients of the group elements. In\nthe above case, we know that the discriminant B2\n11−4B20B02 is an invariant,\nThese spaces have been the subject of intense analysis and their study by\nGordan, Hilbert and other heralded the beginning of commutative algebra\nand invariant theory.\n129\n\n22.4\nFull Reducibility\nLet W be a representation of SLn and let Z ⊆W be an invariant subspace.\nThe reducibility question is whether there exists a complement Z′ such that\nW = Z ⊕Z′ and Z′ is SLn-invariant as well.\nThe above result is indeed true although we will not prove it here. There\nare many proofs known, each with a specific objective in a specific situation,\nand each extremely instructive.\nThe simplest is possibly through the Weyl Unitary trick.\nIn this, a\nsuitable compact subgroup U ⊆SLn is chosen. The theory of compact\ngroups is much like that of finite groups and a complement Z′ may easily\nbe found. It is then shown that Z′ is SLn-invariant.\nThe second attack is through showing the fill reducibility of the module\n⊗dV , the d-th tensor product representation of V . This goes through the\nconstruction of the commutator of the same module regarded as a right\nSd-module, with the symmetric group permutating the contents of the d\npositions. The full reducibility then follows from Maschke’s theorem and\nthat of the finite group Sd.\nThe oldest approach was through the construction of a symbolic reynold’s\noperator, which is the Cayley Ω-process. Let C[W] be the ring of polyno-\nmial functions on the space W. The operator Ωis a map:\nΩ: C[W] →C[W]SLn\nsuch that if p is an invariant then Ω(p · p′) = p · Ω(p′). The definition of Ω\nis nothing but curious.\n130\n\nChapter 23\nInvariant Theory\nReferences: [FH, N]\n23.1\nAlgebraic Groups and affine actions\nAn algebraic group (over C) is an affine variety G equipped with (i) the\ngroup product, i.e., a morphism of algebraic varieties · : G×G →G which is\nassociative, i.e. (g1·g2)·g3 = g1·(g2·g3) for all g1, g2, g3 ∈G, (ii) and algebraic\ninverse i : G →G, i.e, g · i(g) = i(g) · g = 1G, where 1G is (iii) a special\nelement 1G ∈G, which functions as the identity, i.e., 1G · g = g · 1G = g for\nall g ∈G. Let C[G] be the ring of regular functions on G. The requirements\n(i)-(iii) above are via morphisms of algebraic varities, and thus the group\nproduct and the inverse must be defined algebraically. Thus the product\n· : G × G →G results in a morphism of C-algebras ·∗: C[G] →C[G] ⊗C[G]\nand the inverse into another morphism i∗: C[G] →C[G].\nThe essential example is obviously SLn. Clearly for G = SLn, we have\nC[G] = C[X]/(det(X)−1), where X is the n×n indeterminate matrix. The\nmorphism ·∗and i∗are clearly:\n·∗(Xij)\n=\nP\nk Xik ⊗Xkj\ni∗(Xij)\n=\ndet(Mji)\nwhere Mji is the corresponding minor of X.\nNext, let Z be another affine variety with C[Z] as its ring of regular\nfunctions. We say that Z is a G-variety if there is a morphism μ : G×Z →Z\nwhich is a group action. Thus, not only must G act on Z, it must do so\nalgebraically. This μ induces the map μ∗:\nμ∗: C[Z] →C[G] ⊗C[Z]\n131\n\nThus every function f ∈C[Z] goes to a finite sum:\nμ∗(f) =\nk\nX\ni=1\nhi ⊗fi\nwhere fi ∈C[Z] and hi ∈C[G] for all i.\nTo continue with our example, consider SL2 and Z = Symd(V ∗). C[Z] =\nC[B20, B11, B02] and C[G] = C[A11, A12, A21, A22]/(A11A22 −A21A12 −1).\nWe have for example:\nμ∗(B20) = A2\n11 ⊗B20 + A11A21 ⊗B11 + A2\n21 ⊗B02\nLet μ : G × Z →Z and μ′ : G × Z′ →Z′ be two G-varities and let\nφ : Z →Z′ be a morphism. We say that φ is G-equivariant if φ(μ(g, z)) =\nμ′(g, φ(z)) for all g ∈G and z ∈Z. Thus φ commutes with the action of G.\n23.2\nOrbits and Invariants\nEvery g ∈G induces an algebraic bijection on Z by restricting the map\nμ : G × Z →Z to g. We denote this map by μ(g) and call it translation\nby g. The map:\nμ(g) : Z →Z\ninduces the isomorphism of C-algebras:\nμ(g)∗: C[Z] →C[Z]\nGiven any function f ∈C[Z], μ(g)∗(f) ∈C[Z] is the translated function\nand denotes the action of G on f. This makes C[Z] into a G-module. If\nφ : Z →Z′ is a G-equivariant map, then the map φ∗: C[Z′] →C[Z] is also\nG-equivariant, for the G action on C[Z] and C[Z′] as above.\nWe next examine the equation:\nμ∗(f) =\nk\nX\ni=1\nhi ⊗fi\nwhere fi ∈C[Z] and hi ∈C[G] for all i. For a fixed g, we see that\nμ(g)∗(f) =\nk\nX\ni=1\nhi(g) ⊗fi\n132\n\nThus we see that every translate of f lies in the k-dimensional vector space\nC · {f1, . . . , fk}. Let\nM(f) = C · {μ(g)∗(f)|g ∈G} ⊆C · {f1, . . . , fk}\nClearly, M(f) is a G-invariant subspace of C[Z]. This may be generalized:\nProposition 23.1. Let S = {s1, . . . , sm} be a finite subset of C[Z]. Then\nthere is a finite-dimensional G-invariant subspace M(S) of C[Z] containing\ns1, . . . , sm.\nNext, let us consider μ : G × Z →Z and fix a z ∈Z. We get the map\nμz : G →X. Since G is an affine variety, we see that the image μz(G) is a\nconstructible set, whose closure is an affine variety. The image is precisely\nthe orbit O(z) ⊆Z. The closure of O(z) will be denoted by ∆(z).\nIf O(z) = ∆(z), then we have the G-equivariant embedding iz : O(z) →\nZ. This gives us the map i∗\nz : C[Z] →C[O(z)] which is a surjection. Thus\nthere is an ideal I(z) = ker(i∗\nz) such that C[O(z)] ∼= C[Z]/I(z). Since the\nmap iz is G-equivariant, I(z) is a G-submodule of C[Z] and is the ideal of\ndefinition of the orbit O(z).\nIn general, if I ⊆C[Z] is an ideal which is G-invariant, then the variety\nof I is also G-invariant and is the union of orbits.\nThe second construction that we make is that of the quotient Z/G.\nSince C[Z] is a G-module, we examine C[Z]G, the subring of G-invariant\nfunctions in C[Z]. We define Z/G as the spectrum Spec(C[Z]G). The inclu-\nsion C[Z]G →C[Z] gives us the quotient map:\nπ : Z →Z/G\nExercise 23.1. Let us consider Z = Sym2(V ∗) ∼= C3 where V is the stan-\ndard representation of G = SL2. As we have seen, C[Z] = C[B20, B11, B02].\nThere is only one invariant δ = B2\n11 −4B02B20. Thus C[Z]G = C[δ]. Thus\nZ/G is precisely Spec(C[δ]) = C, the complex plane. The map π is executed\nas follows: given a form aX2\n1 + bX1X2 + cX2\n2 ≡(a, b, c) ∈Z, we eveluate\nthe invariant δ at the point (a, b, c). Thus π : C3 →C is given by:\nπ(a, b, c) = b2 −4ac\nClearly, if f, f ′ ∈Z such that f = g · f ′ for g ∈SL2 then δ(f) = δ(f ′). We\nlook at the converse: if f, f ′ are such that δ(f) = δ(f ′) then is it that f ∈\nO(f ′)? We begin with f = aX2\n1 +bX1X2 +cX2\n2. We assume for the moment\nthat a ̸= 0. If that is the case, we make the substitution X1 →X1 −αX2\n133\n\nand X2 →X2. Note that this transformation is unimodular for all α. This\ntransforms f to:\na(X1 −αX2)2 + b(X1 −αX2)X2 + cX2\n2\nThe coefficient of X1X2 is −2aα+b. Thus by choosing α as b/2a we see that\nf is transformed into a′′X2\n1 −c′′X2\n2 for some a′′, c′′. By a similar token, even\nif a = 0 one may do a similar transform. Thus in general, if f is not the\nzero form, there is a point in O(f) which is of the form aX2\n1 −cX2\n2. Thus,\nwe may assume that both f and f ′ are in this form. We can simplify the\nform further by a diagonal element of SL2 to put both f and f ′ as X2\n1 −cX2\n2\nand X2\n1 −c′X2\n2. It is now clear that δ(f) = δ(f ′) implies that c = c′. Thus\nthe general answer is that if f, f ′ ̸= 0 and δ(f) = δ(f ′) then f ∈O(f ′).\nNext, let us examine the form 0. we see that δ(0) = 0. Thus we see that\n(i) for any point d ∈C, if d ̸= 0, then π−1(d) consists of a single orbit, (ii)\nπ−1(0) consists of two orbits, O(0) and O(X2\n1), the perfect square, (iii) the\norbits O(f) are closed when δ(f) ̸= 0, (iv) O(X2\n1) is not closed. Its closure\nincludes the closed orbit 0.\nThe above example illustrates the utility of contructing the quotient Z/G\nas a variety parametrizing closed orbits albeit with some deviant points. In\nthe example above, the discrepancy was at the point 0, wherein the pre-\nimage is closed but decomposes into two orbits.\nWe state the all-important theorem linking a space and its quotient in\nthe restricted case when G is finite and Z is a finite-dimensional G-module.\nThus C[Z] is a polynomial ring and the action of G is homogeneous.\nTheorem 3. Let G be a finite group and act on the space Z. Let R = C[Z]\nand RG = C[Z]G be the ring of invariants. Let π : Z →Z/G be the quotient\nmap. Then\n(i) For any ideal J ⊆RG, we have (J · R) ∩RG = J.\n(ii) The map π is surjective. Further, for any x ∈Z/G, π−1(x) is a single\norbit in Z.\nProof: Let f1, . . . , fk be elements of RG and f ∈RG be such that:\nf = r1f1 + . . . + rkfk\nwhere ri are elements of R. Since G is finite, we apply the Reynolds operator\n134\n\np →pG. In other words, we have:\nf\n=\n1\n|G|\nX\ng∈G\nf\n=\n1\n|G|\nX\ng∈G\nX\ni\nrifi\n=\n1\n|G|\nX\ni\nfi\nX\ng∈G\nri\n=\nX\ni\nfi\n1\n|G|\nX\ng∈G\nri\n=\nX\ni\nfirG\ni\nNote that we have used the fact that if h ∈RG and p ∈R then (h · p)G =\nh · pG. Thus we see that any element f of RG which is expressible as an\nR-linear combination of elements fi’s of RG is already expressible as an\nRG-linear combination of the same elements. This proves (i) above.\nNow we prove (ii). Firstly, let J be a maximal ideal of RG. By part (i)\nabove, J · R is a proper ideal of R and thus π is surjective. Let x ∈Z/G\nand Jx ⊆RG be the maximal ideal for the point x. Let z be such that\nπ(z) = x and let IO(x) ⊆R be the ideal of all functions in R vanishing at\nall points of the orbit O(z) of z. We show that rad(Jx · R) = IO(z) which\nproves (ii). Towards that, it is clear that (a) JxR ⊆IO(z) and (b) the variety\nof JxR is G-invariant. Let O(z′) is another orbit in the variety of Jx · R.\nIf O(z′) ̸= O(z) then we already have that there is a p ∈RG such that\np(O(z)) = 0 and p(O(z′)) = 1. Since this p ∈Jx · R, the variety of Jx · R\nexcludes the orbit O(z′) proving our claim. □\nTheorem 4. Let Z be a finite-dimesional G-module for a finite group G.\nThen C[Z]G, the ring of G-invariants, is finitely generated as a C-algebra.\nProof: Since the action of G is homogeneous, every invariant f ∈C[Z]G is\nthe sum of homogeneous invariants. Let IG ⊆C[Z] be the ideal generated\nby the positive degree invariants.\nBy the Hilbert Basis theorem, IG =\n(f1, . . . , fk)·C[Z], where each fi is itself an invariant. we claim that C[Z]G =\nC[f1, . . . , fk] ⊆C[Z], or in other words, every invariant is a polynomial over\nC in the terms f1, . . . , fk. To this end, let f be a homogeneous invariant.\nWe prove this by induction over the degree d of f. Since f ∈IG, we have\nf =\nX\ni\nrifi where for all i, ri ∈C[Z]\n135\n\nBy applying the reynolds operator, we have:\nf =\nX\ni\nhifi where for all i, hi ∈C[Z]G\nNow since each hi has degree less than d, by our induction hypothesis, each\nis an element of C[f1, . . . , fk]. This then shows that f too is an element of\nC[f1, . . . , fk]. □\n23.3\nThe Nagata Hypothesis\nWe now generalize the above two theorems to the case of more general\ngroups. This generlization is possible if the group G satisfies what we call\nthe Nagata Hypothesis.\nDefinition 23.1. Let G be an algebraic group. We say that G satisfies the\nNagata hypothesis if for every finite-dimensional module M over C and a\nG-invariant hyperplane N ⊆M, there is a decomposition M = H ⊕P as\nG-modules, where H is a hyperplane and P is an invariant (i.e., P is the\ntrivial representation.\nThe group SLn satisfies the hypothesis, and so do the so-called reductive\ngroups over C.\nTheorem 5. Let G satisfy the Nagata hypothesis and let Z be an affine\nG-variety. The ring C[Z]G is finitely generated as a C-algebra.\nThe proof will go through several steps.\nLet f ∈C[Z] be an arbitrary element. We define two modules M(f) and\nN(f).\nM(f)\n=\nC · {s · f|s ∈G}\nN(f)\n=\nC · {s · f −t · f|s, t ∈G}\nThus M(f) ⊇N(f) are finite-dimensional submodules of C[Z].\nLemma 23.1. Let f ∈C[Z] be an arbitrary element. Then there exists an\nf ∗∈C[Z]G ∩M(f) such that f −f ∗∈N(f).\nProof: We prove this by induction over dim(M(f)).\nNote that s · f =\nf+(s·f−1·f) and thus M(f) = N(f)⊕C·f as vector spaces. Thus N(f) is a\nG-invariant hyperplane of M(f). By the Nagata hypothesis, M(f) = f ′⊕H,\nwhere f ′ is an invariant. If f ′ ̸∈N(f) then M(f) = f ′⊕N(f) and the lemma\nfollows. However, if f ′ ∈N(f), then f = f ′ + h where h ∈H. Since H is\n136\n\na G-module, we see that M(h) ⊆H which is of dimension less than that of\nM(f). Thus there is an invariant h∗such that h −h∗∈N(h). Note that\nf −f ′ = h with f ′ invariant, implies that s · h −t · h = s · f −t · f. Thus\nN(h) ⊆N(f). We take f ∗= h∗. Examining f −f ∗, we see that\nf −h∗= f ′ + h −h∗∈N(f)\nThis proves the lemma. □\nLemma 23.2. Let f1, . . . , fr ∈C[Z]G.\nThen C[Z]G ∩(P\ni C[Z] · fi) =\nP\ni C[Z]G · fi. Thus if an invariant f is a C[Z]linear combination of invari-\nants, then it is already a C[Z]G linear combination.\nProof: This is proved by induction on r. Say f = Pr\ni=1 hifi with hi ∈C[Z].\nApplying the above lemma to hr, there is an h′′ ∈C[Z]G and an h′ ∈N(hk)\nsuch that hr = h′ + h′′. We tackle h′ as follows. Since f is an invariant, we\nhave for all s, t ∈G:\nr\nX\ni=1\n(s · hi −t · hi)fi = s · f −t · f = 0\nHence:\n(s · hr −t · hr)fr =\nr−1\nX\ni=1\n(s · hi −t · hi)fi\nIt follows from this that h′fr = Pr−1\ni=1 h′\nifi for some h′\ni ∈C[Z]. Substituting\nthis in the expression\nf =\nr−1\nX\ni=1\nhifi + (h′ + h′′)fr\nwe get:\nf −h′′fr =\nr−1\nX\ni=1\n(hi + h′\ni)fi\nThus the invariant f −h′′fr is C[Z]-linear combination of r −1 invariants,\nand thus the induction hypothesis applies. This then results in an expression\nfor f as a C[Z]G-linear combination for f. □\nThe above lemma proves part (i) of Theorem 3 for groups G for which\nthe Nagata hypothesis holds. The route to Theorem 4 is now a straight-\nforward adaptation of its proof for finite groups. In the case when Z is a\nG-module, C[Z]G would then be homogeneous and the proof of Theorem 4\nholds. For general Z here is a trick which converts it to the homogeneous\ncase:\n137\n\nProposition 23.2. Let Z be an affine G-variety. Then there is a G-module\nW and an equivariant embedding φ : Z →W.\nProof: Since C[Z] is a finitely generated C-algebra, C[Z] = C[f1, . . . , fk]\nwhere f1, . . . , fk are some specific elements of C[Z]. By expanding the list\nof generators, we may assume that the vector space C · {f1, . . . , fk} ⊆C[Z]\nis a finite-dimensional G-module.\nLet us construct W as an isomorphic\ncopy of this module with the dual basis W1, . . . , Wk. We construct the map\nφ∗: C[W] →C[Z] by defining φ(Wi) = fi. Since C[W] is a free algebra, we\nindeed have the surjection:\nφ∗: C[W] →C[Z]\nThis proves the proposition. □\nWe are now ready to prove:\nTheorem 6. Let G satisfy the Nagata hypothesis. Let Z be an affine G-\nvariety with coordinate ring C[Z]. Then C[Z]G is finitely generated as a\nC-algebra.\nProof: We construct the equivariant surjection φ∗: C[W] →C[Z]. Note\nthat C[W]G is already finitely generated over C, say C[W]G = C[h1, . . . , hk].\nWe claim that φ∗(h1), . . . , φ∗(hk) generate C[Z]G.\nNow, let f ∈C[Z]G. By the surjectivity of φ∗, there is an h ∈C[W] such\nthat φ∗(h) = f. Consider the space N(h). A typical generator of N(h) is\ns · h −t · h. Applying φ∗to this, we see that:\nφ∗(s · h −t · h)\n=\ns · φ∗(h) −t · φ∗(h)\n=\nf −f = 0\nBy an earlier lemma there is an invariant h∗such that h −h∗∈N(h). thus\napplying φ∗we see that φ∗(h∗) = φ(h) = f. Thus there is an invariant h∗\nsuch that φ∗(h∗) = f. Now h∗∈C[h1, . . . , hk] implies that f = φ∗(h∗) ∈\nC[φ∗(h1), . . . , φ∗(hk)]. □\n138\n\nChapter 24\nOrbit-closures\nReference: [Ke, N]\nIn this chapter we will analyse the validity of Theorem 3 for general\ngroups G with the Nagata hypothesis. The objective is to analyse the map\nπ : Z →Z/G.\nProposition 24.1. Let G satisfy the Nagata hypothesis and act on the affine\nvariety Z. Then the map π : Z →Z/G is surjective.\nProof: Let J ⊆C[Z]G be a maximal ideal. By lemma 23.2 J · C[Z] is a\nproper ideal of C[Z]. This implies that π is surjective. □\nTheorem 7. Let G satisfy the Nagata hypothesis and act on the affine\nvariety Z. Let W1 and W2 be G-invariant (Zariski-)closed subsets of Z such\nthat W1 ∩W2 is empty. Then there is an invariant f ∈C[Z]G such that\nf(W1) = 0 and f(W2) = 1.\nProof: Let I1 and I2 be the ideals of W1 and W2 in C[Z]. Since their inter-\nsection is empty, by Hilbert Nullstellensatz, we have I1 + I2 = 1. Whence\nthere are functions f1 ∈I1 and f2 ∈I2 such that f1 + f2 = 1. For arbitrary\ns, t ∈G we have:\n(s · f1 −t · f1) + (s · f2 −t · f2) = 1 −1 = 0\nNote that M(fi) and N(fi) are submodules of the G-invariant ideal Ii. Now\napplying lemma 23.1 to f1, we see that there are elements f ′\ni ∈N(fi) such\nthat f1 + f ′\n1 ∈C[Z]G. Whence we see that:\n(f1 + f ′\n1) + (f2 + f ′\n2) = 1\n139\n\nwhere f = f1 + f ′\n1 is an invariant. Since f1 + f ′\n1 ∈I1 we see that f(W1) = 0.\nOn the other hand, f = 1 −(f2 + f ′\n2) ∈1 + I2 and thus f(W2) = 1. □\nRecall that ∆(z) denotes the closure of the orbit O(z) in Z.\nTheorem 8. Let G satisfy the Nagata hypothesis and act on an affine va-\nriety Z. We define the relation ≈on Z as follows: z1 ≈z2 if and only if\n∆(z1) ∩∆(z2) is non-empty. Then\n(i) ≈is an equivalence relation.\n(ii) z1 ≈z2 ifff(z1) = f(z2) for all f ∈C[Z]G.\n(iii) Within each ∆(z) there is a unique closed orbit, and this is of minimum\ndimension among all orbits in ∆(z).\nProof: It is clear that (ii) proves (i). Towards (ii), if ∆(z1)∩∆(z2) is empty,\nthen by Theorem 7, there is an invariant separating the two points. Thus it\nremains to show that if ∆(z10 ∩∆(z2) is non-empty, and f is any invariant,\nthen f(z1) = f(z2). So let z be an element of the intersection. Since f is\nan invariant, we have f(O(z1)) is a constant, say α. Since O(z1) ⊆∆(z1) is\ndense, and f is continuous, we have f(z) = α. Thus f(z1) = f(z2) = f(z) =\nα.\nNow (iii) is easy. Clearly ∆(z) cannot have two closed distinct orbits,\nfor otherwise they woulbe separated by an invariant. But this must take the\nsame value on all points of ∆(z). That it is of minimum dimension follows\nalgebraic arguments. □\nDefinition 24.1. Let Z be a G-variety and z ∈Z. We say that z is stable\nif the orbit O(z) ⊆Z is closed in Z.\nBy the above theorem, every point x of Spec(C[Z]G) corresponds to\nexactly one stable point: the point whose orbit is of minimum dimension in\nπ−1(x).\nExercise 24.1. Consider the action of G = SLnon M, the space of n × n-\nmatrices by conjugation. Thus, given A ∈SLn and M ∈M, we have:\nA · M = AMA−1\nLet R = C[M] = C[X11, . . . , Xnn] be the ring of functions on M.\nThe\ninvariants RG is generated as a C-algebra by the forms ei(X) = Tr(Xi), for\ni = 1, . . . , n. The forms {ei|1 ≤i ≤n} are algebraically independent and\n140\n\nthus RG is the polynomial ring C[e1, . . . , en]. Clearly then Spec(RG) ∼= Cn\nand we have:\nπ : M ∼= Cn2 →Cn\nGiven a matrix M with eigenvalues {λ1, . . . , λn}, we have:\nei(M) = λi\n1 + . . . + λi\nn\nThus, by the fundamental theorem of algebra, the image π(M) determines\nthe set {λ1, . . . , λn} (with multiplicities).\nOn the other hand, given any\ntuple μ = (μ1, . . . , μn) there is a unique set λμ = {λ1, . . . , λn} such that\nP\nr λi\nr = μi. Clearly, for the diagonal matrix D(λμ) = diag(λ1, . . . , λn), we\nhave that π(D(λ)) = μ. This verifies that π is surjective.\nFor a given μ, the set π−1(μ) are all matrices M with Spec(M) = λ =\nλμ. By the Jordan canonical form (JCF), this set may be stratified by the\nvarious Jordan canonical blocks of spectrum λ. If λ has no multiplicities\nthen π−1(μ) consists of just one orbit: matrices M such that JCF(M) =\nD(λ). For a general λ, the orbit of M with JCF(M) = D(λ) is the unique\nclosed orbit of minimum dimension. All other orbits contain this orbit in its\nclosure. Thus stable points M ∈M are the diagonalizable matrices.\nAs an example, consider the case when n = 2 and the matrix:\nN =\n λ\n1\n0\nλ\n \nConsider the family A(t) = diag(t, t−1) ∈SL2. We see that:\nN(t) = A(t)NA(t)−1 =\n λ\nt2\n0\nλ\n \nThus limt→0 N(t) = diag(λ, λ), the diagonal matrix.\nThus, we see that the invariant ring C[Z]G puts a different equivalence\nrelation ≈on points in Z which is coarser than ∼=, the orbit equivalence\nrelation. The relation ≈is more ‘topological’ than group-theoretic and cor-\nrectly classifies orbits by their separability by invariants. The special case\nof Theorem 8 when Z is a representation was analysed by Hilbert in 1893.\nThe point 0 ∈Z is then the smallest closed orbit, and the equivalence class\n[0]≈is termed as the null-cone of Z. We see that the null-cone consists\nof all points z ∈Z such that 0 lies in the orbit-closure ∆(z) of z. It was\nHilbert who discovered that if 0 ∈∆(z) then 0 lies in the orbit-closure of a 1-\nparameter diagonal subgroup of SLn. To understand the intricay of Hilbert’s\nconstructions, it is essential that we understand diagonal subgroups of SLn.\n141\n\nChapter 25\nTori in SLn\nReference: [Ke, N]\nLet C∗denote the multiplicative group of non-zero complex numbers. A\ntorus is the abstract group (C∗)m for some m. Note that C∗is an abelian\nalgebraic group with C[G] = C[T, T −1]. Furthermore, C∗has a compact\nsubroup S1 = {z ∈C, |z| = 1}, the unit circle.\nNext, let us look at representations of tori. For C∗, the simplest repre-\nsentations are indexed by integers k ∈Z. So let k ∈Z. The representation\nC[k] corresponds to the 1-dimensional vector space C with the action:\nt · z = tkz\nThus a non-zero t ∈C∗acts on z by multiplication of the k-th power. Next,\nfor (C∗)m, let χ = (χ[1], . . . , χ[m]) be a sequence of integers. For such a χ,\nwe define the representation Cχ as follows: Let t = (t1, . . . , tm) ∈(C∗)m be\na general element and z ∈C. The action is given by:\nt · z = tχ[1]\n1\n. . . tχ[m]\nm\nz\nSuch a χ is called a character of (C∗)m.\nThese 1-dimensional representations of tori are crucial in the analysis of\nalgebraic group actions.\nLet us begin by understanding the structure of algebraic homomorphisms\nfrom C∗to SLn(C). So let\nλ : C∗→SLn(C)\nbe such a map such that λ(t) = [aij(t)] where aij(T) ∈C[T, T −1].\nAn\nimportant substitution is for t = eiθ, and we obtain a 2π-periodic map\nλ : C →SLn\n142\n\nWe see that λ(0) = I. Let the derivative at 0 for λ be X.\nWe have the following general lemma:\nLemma 25.1. Let f : C →SLn be a smooth map such that f(0) = I and\nf ′(0) = X, where X is an n × n-matrix. Then for θ ∈C,\nlim\nk→∞\n \nf\n θ\nk\n k\n= eθX\nThe proof follows from the local diffeomorphism of the exponential map\nin the neighborhood of the identity matrix.\nApplying this lemma to λ we see that eθX is in the image of λ for all θ.\nNow, since λ is 2π-periodic, we must have e(n2π+θ)X = eθX. This forces (i)\nX to be diagonalizable, and (ii) with integer eigenvalues. This proves:\nProposition 25.1. Let λ : C∗→SL(V ) be an algebraic homomorphism.\nThen the image of λ is closed and V ∼= C[m1] ⊕. . .⊕C[mn], for some integers\nm1, . . . , mn, where n = dimC(V ).\nBased on this, we have the generalization:\nProposition 25.2. Let λ : (C∗)r →SL(V ) be an algebraic homomorphism.\nThen the image of λ is closed and V ∼= Cχ1 ⊕. . . ⊕Cχn, for some integers\nχ1, . . . , χn, where n = dimC(V ).\nThus, in effect, for every homomorphism λ : (C∗)r →SLn, there is a\nfixed invertible matrix A such that for all t ∈(C∗)r, the conjugate Aλ(t)A−1\nis diagonal.\nA torus in SLn is defined as an abstract subgroup H of SLn which is\nisomorphic to (C∗)r for some r. The standard maximal torus D of SLn\nis the diagonal matrices diag(t1, . . . , tn) where ti ∈C∗and t1t2 . . . tn = 1.\nThis clears the way for the important theorem:\nTheorem 9.\n(i) Every torus is contained in a maximal torus. All maxi-\nmal tori in SLn are isomorphic to (C∗)n−1.\n(ii) If T and T ′ are two maximal tori then there is an A ∈SLn such that\nT ′ = ATA−1. Thus all maximal tori are conjugate to D above.\n(iii) Let N(D) be the normalizer of D and N(D)o be the connected compo-\nnent of N(D). Then N(D)o = D and N(D)/D is the Weyl group\nW, isomorphic to the symmetric group Sn.\n143\n\nDefinition 25.1. Let G be an algebraic group. Γ(G) will denote the col-\nlection of all 1-parameter subgroups of G, i.e., morphisms λ : C∗→G.\nX(G) will denote the collection of all characters of G, i.e., homomorphisms\nχ : G →C∗.\nWe consider the case when G = (C∗)r. Clearly, for a given λ : C∗→G,\nthere are integers m1, . . . , mr ∈Z such that:\nλ(t) = (tm1, . . . , tmr)\nIn the same vein, for the character χ : G →C∗, we have integers a1, . . . , ar\nsuch that\nχ(t1, . . . , tr) =\nY\ni\ntai\ni\nWe also have the composition λ ◦χ : C∗→C∗, where by:\nλ ◦χ(t) = tm1a1+...+mrar\nConsolidating all this, we have:\nTheorem 10. Let G = (C∗)r. Then Γ(G) ∼= Zr and X(G) ∼= Zr. Further-\nmore, there is the pairing\n< , >: Γ(G) × X(G) →Z\nwhich is a unimodular pairing on lattices.\nExercise 25.1. Let G = (C∗)3 and λ and χ be as follows:\nλ(t)\n=\n(t3, t−1, t2)\nχ(t1, t2, t3)\n=\nt−1\n1 t2t2\n3\nThen, λ ∼= [3, −1, 2] and χ ∼= [−1, 1, 2]. We evaluate the pairing:\n< λ, χ >= 3 · −1 + (−1) · 1 + 2 · 2 = 0\nWe now turn to the special case of D ⊆SLn, the maximal torus which\nis isomorphic to (C∗)n−1.\nBy the above theorem, Γ(D), X(D) ∼= Zn−1.\nHowever, it will more convenient to identify this space as a subset of Zn. So\nlet:\nYn = {[m1, . . . , mn] ∈Zn|m1 + . . . + mn = 0}\nIt is easy to see that Yn ∼= Zn−1. In fact, we will set up a special bijection\nθ : Yn →Zn−1 defined as:\nθ([m1, m2, . . . , mn]) = [m1, m1 + m2, . . . , m1 + . . . + mn−1]\n144\n\nThe inverse θ−1 is also easily computed:\nθ−1[a1, . . . , an−1] = [a1, a2 −a1, a3 −a2, . . . , an−1 −an−2, −an−1]\nThis θ corresponds to the Z-basis of Yn consisting of the vectors e1 −\ne2, . . . , en−1 −en where ei is the standard basis of Zn. This is also equivalent\nto the embedding θ∗: (C∗)n−1 →D as follows:\n(t1, . . . , tn−1) →\n \n \nt1\n0\n. . .\n0\n0\nt−1\n1 t2\n0\n. . .\n0\n...\n0\n. . .\n0\nt−1\nn−2tn−1\n0\n0\n. . .\n0\nt−1\nn−1\n \n \nA useful computation is to consider the inclusion D ⊆D∗, where D∗⊆GLn\nis subgroup of all diagonal matrices. Clearly Γ(D) ⊆Γ(D∗), however there\nis a surjection X(D∗) →X(D). It will be useful to work out this surjection\nexplicitly via θ and θ∗. If [m1, . . . .mn] ∈Zn ∼= X(D∗), then it maps to\n[m1 −m2, . . . , mn−1 −mn] ∈Zn−1 ∼= X((C∗)n−1) via θ∗. If we push this\nback into Yn via θ−1, we get:\n[m1, . . . , mn] →[m1−m2, 2m2−m1−m3, , . . . , 2mn−1−mn−2−mn, mn−mn−1]\nWe are now ready to define the weight spaces of an SLn-module W.\nSo let W be such a module.\nBy restricting this module to D ⊆G, via\nProposition 25.2, we see that W is a direct sum W = Cχ1 ⊕. . . ⊕CχN,\nwhere N = dimC(W). Collecting identical characters, we see that:\nW = ⊕χ∈X(D)Cmχ\nχ\nThus W is a sum of mχ copies of the module Cχ. Clearly mχ = 0 for all but\na finite number, and is called the multiplicity of χ. For a given module\nW, computing mχ is an intricate combinatorial exercise and is given by the\ncelebrated Weyl Character Formula.\nExercise 25.2. Let us look at SL3 and the weight-spaces for some modules\nof SL3.\nAll modules that we discuss will also be GL3-modules and thus\nD∗modules. The formula for converting D∗-modules to D-modules will be\nuseful. This map is Z3 →Y3 and is given by:\n[m1, m2, m3] →[m1 −m2, 2m2 −m1 −m3, m3 −m2]\n145\n\nThe simplest SLn module is C3 with the basis {X1, X2, X3} with D∗\nweights [1, 0, 0], [0, 1, 0] and [0, 0, 1].\nThis converted to D-weights give us\n{[1, −1, 0], [−1, 2, −1], [0, −1, 1]}, with C[1,−1,0] ∼= C · X1 and so on.\nThe next module is Sym2(C3) with the basis X2\ni and XiXj. The six D∗\nand D-weights with the weight-spaces are given below:\nD∗-wieghts\nD-weights\nweight-space\n[2, 0, 0]\n[2, −2, 0]\nX2\n1\n[0, 2, 0]\n[−2, 4, −2]\nX2\n2\n[0, 0, 2]\n[0, −2, 2]\nX2\n3\n[0, 1, 1]\n[−1, 1, 0]\nX2X3\n[1, 0, 1]\n[1, −2, 1]\nX1X3\n[1, 1, 0]\n[0, 1, −1]\nX1X2\nThe final example is the space of 3× 3-matrices M acted upon by conju-\ngation. We see at once that M = M0 ⊕C·I where M0 is the 8-dimensional\nspace of trace-zero matrices, and C · I is 1-dimensional space of multiples\nof the idenity matrix. Weight vectors are Eij, with 1 ≤i, j ≤3. The D∗\nweights are [1, −1, 0], [1, 0, −1], [0, 1, −1], [−1, 0, 1], [0, −1, 1], [−1, 1, 0] and [0, 0, 0].\nThe multiplicity of [0, 0, 0] in M is 3 and in M0 is 2. Note that Eii ̸∈M0.\nThe D-weights are [2, −3, 1], [1, 0, −1], [−1, 3, −2] and its negatives, and ob-\nviously [0, 0, 0].\nThe normalizer N(D) gives us an action of N(D) on the weight spaces.\nIf w is a weight-vector of weight χ, t ∈D and g ∈N(D), then g · w is also\na weight vector. Afterall t · (g · w) = g · t′ · w where t′ = g−1tg. Thus\nt · (g · w) = χ(t′)(g · w)\nwhence g · w must also be a weight-vector with some weight χ′. This χ′ is\neasily computed via the action of D∗. Here the action of N(D∗) is clear: if\nχ = [m1, . . . , mn], then χ′ = [mσ(1), . . . , mσ(n)] for some permutation σ ∈Sn\ndetermined by the component of N(D∗) containing g. Thus the map χ to\nχ′ for D-weights in the case of SL3 is as follows:\n[m1−m2, 2m2−m1−m3, m3−m2] →[mσ(1)−mσ(2), 2mσ(2)−mσ(1)−mσ(3), mσ(3)−mσ(2)]\nCaution: Note that though Y3 ⊆Z3 is an S3-invariant subset, the action\nof S3 on χ ∈Y3 is different. Note that, e.g., in the last example above,\n[2, −3, 1] is a weight but not the ‘permuted’ vector [−3, 2, 1]. This is because\nof our peculiar embedding of Zn−1 →Yn.\n146\n\nChapter 26\nThe Null-cone and the\nDestabilizing flag\nReference: [Ke, N]\nThe fundamental result of Hilbert states:\nTheorem 11. Let W be an SLn-module, and let w ∈W be an element of\nthe null-cone. Then there is a 1-parameter subgroup λ : C∗→SLn such that\nlim\nt→0 λ(t) · w = 0W\nIn other words, if the zero-vector 0W lies in the orbit-closure of w, then\nthere is a 1-parameter subgroup taking it there, in the limit. We will not\nprove this statement here. Our objective for this chapter is to interpret the\ngeometric content of the theorem. We will show that there is a standard\nform for an element of the null-cone. For well-known representations, this\nstandard form is easily identified by geometric concepts.\n26.1\nCharacters and the half-space criterion\nTo begin, let D be the fixed maximal torus. For any w ∈W, we may express:\nw = w1 + w2 + . . . + wr\nwhere wi ∈Wχi, the weight-space for character χi. Note the the above ex-\npression is unique if we insist that each wi be non-zero. The set of characters\n{χ1, . . . , χr} will be called the support of w and denoted as supp(w). Let\n147\n\nλ : C∗→SLn be such that Im(λ) ⊆D. In this case, the action of t ∈C∗\nvia λ is easily described:\nt · w = t(λ,χ1)w1 + . . . + t(λ,χr)wr\nThus, if limt→0 t · w exists (and is 0W ), then for all χ ∈supp(w), we have\n(λ, χ) ≥0 (and further (λ, χ) > 0).\nNote that (λ, χ) is implemented as a linear functional on Yn. Thus, if\nlimt→0 t · w exists (and is )W ) then there is a hyperplane in Yn such that\nthe support of w is on one side of the hyperplane (strictly on one side of\nthe hyperplane). The normal to this hyperplane is given by the conversion\nof λ into Yn notation.\nOn the other hand if the support supp(w) enjoys the geometric/combinatorial\nproperty, then by the approximability of reals by rationals, we see that there\nis a λ such that limt→0 t · w exists (and is zero).\nThus for 1-parameter subgroups of D, Hilbert’s theorem translates into\na combinatorial statement on the lattice subset supp(w) ⊂Yn.\nWe call\nthis the (strict) half-space property. In the general case, we know that\ngiven any λ : C∗→SLn, there is a maximal torus T containing Im(λ). By\nthe conjugacy result on maximal tori, we know that T = ADA−1 for some\nA ∈SLn. Thus, we may say that w is in the null-cone iffthere is a translate\nA · w such that supp(A · w) satisfies the strict half-space property.\nExercise 26.1. Let us consider SL3 acting of the space of forms of degree\n2. For the standard torus D, the weight-spaces are C · X2\ni and C · XiXj.\nConsider the form f = (X1 + X2 + X3)2. We see that supp(f) is set of all\ncharacters of Sym2(C3) and does not satisfy the combinatorial property.\nHowever, under a basis change A:\nX1\n→\nX1 + X2 + X3\nX2\n→\nX2\nX3\n→\nX3\nwe see that A·f = X2\n1. Thus A·f does satisfy the strict half-space property.\nIndeed consider the λ\nλ(t) =\n \n \nt\n0\n0\n0\nt−1\n0\n0\n0\n1\n \n \nWe see that\nlim\nt→0 t · (A · f) = t2X2\n1 = 0\n148\n\nThus we see that every form in the null-cone has a standard form with a\nvery limited sets of possible supports.\nLet us look at the module M of 3 × 3-matrices under conjugation. Let\nus fix a λ:\nλ(t) =\n \n \ntn1\n0\n0\n0\ntn2\n0\n0\n0\ntn3\n \n \nsuch that n1 + n2 + n3 = 0. We may assume that n1 ≥n2 ≥n3. Looking at\nthe action of λ(t) on a general matrix X, we see that:\nt · X = (tni−njxij)\nThus if limt→0 t · X is to be 0 then xij = 0 for all i > j. In other words,\nX is strictly upper-triangular. Considering the general 1-parameter group\ntells us that X is in the null-cone iffthere is an A such that AXA−1 is\nstrictly upper-triangular. In other words, X is nilpotent. The 1-parameter\nsubgroup identifies this transformation and thus the flag of nilpotency.\n26.2\nThe destabilizing flag\nIn this section we do a more refined analysis of elements of the null-cone. The\nbasic motivation is to identify a unique set of 1-parameter subgroups which\ndrive a null-point to zero. The simplest example is given by X2\n1 ∈Sym2(C3).\nLet λ, λ′ and λ′′ be as below:\nλ(t) =\n \n \nt\n0\n0\n0\nt−1\n0\n0\n0\n1\n \n \nλ′(t) =\n \n \nt\n0\n0\n0\n1\n0\n0\n0\nt−1\n \n \nλ′′(t) =\n \n \nt\n0\n0\n0\n0\n−1\n0\nt−1\n0\n \n \nWe see that all the three λ, λ′ and λ′′ drive X2\n1 to zero.\nThe question\nis whether these are related, and to classify such 1-parameter subgroups.\nAlternately, one may view this to a more refined classification of points in\nthe null-cone, such as the stratification of the nilpotent matrices by their\nJordan canonical form.\nThere are two aspects to this analysis. Firstly, to identify a metric by\nwhich to choose the ’best’ 1-parameter subgroup driving a null-point to\nzero. Next, to show that there is a unique equivalence class of such ’best’\nsubgroups.\n149\n\nTowards the first objective, let λ : C∗→SLn be a 1-parameter subgroup.\nWithout loss of generality, we may assume that Im(λ) ⊆D. If w is a null-\npoint then we have:\nt · w = tn1w1 + . . . + tnkwk\nwhere ni > 0 for all i. Clearly, a measure of how fast λ drives w to zero\nis m(λ) = min{n1, . . . , nk}. Verify that this really does not depend on the\nchoice of the maximal torus at all, and thus is well-defined.\nNext, we see that for a λ as above, we consider λ2 : C∗→SLn such that\nλ2(t) = λ(t2). It is easy to see that m(λ2) = 2 · m(λ). Clearly, λ and λ2 are\nintrinsically identical and we would like to have a measure invariant under\nsuch scaling. This comes about by associating a length to each λ. Let λ be\nas above and let Im(λ) ⊆D. Then, there are integers a1, . . . , an such that\nλ(t) =\n \n \nta1\n0\n0\n0\n0\nta2\n0\n0\n...\n0\n0\n0\ntan\n \n \nWe define ∥λ∥as\n∥λ∥=\nq\na2\n1 + . . . + a2n\nWe must show that this does not depend on the choice of the maximal\ntorus D. Let T (SLn) denote the collection of all maximal tori of SLn as\nabstract subgroups. For every A ∈SLn, we may define the map φA : T →T\ndefined by T →ATA−1. The stabilizer of a torus T for this action of SLn\nis clearly N(T), the normalizer of T.\nAlso recall that N(T)/T = W is\nthe (discrete) weyl group.\nLet Im(λ) ⊆D ∩D′ for some two maximal\ntori D and D′.\nSince there is an A such that AD′A−1 = D, it is clear\nthat ∥λ∥= ∥AλA−1∥. Thus, we are left to check if ∥λ′∥= ∥λ∥when (i)\nIm(λ), Im(λ′) ⊆D, and (ii) λ′ = AλA−1 for some A ∈SLn. This throws\nthe question to invariance of ∥λ∥under N(D), or in other words, symmetry\nunder the weyl group.\nSince W ∼= Sn, the symmetric group, and since\np\na2\n1 + . . . + a2n is a symmetric function on a1, . . . , an, we have that ∥λ∥is\nwell defined.\nWe now define the efficiency of λ on a null-point w to be\ne(λ) = m(λ)\n∥λ∥\nWe immediately see that e(λ) = e(λ2).\n150\n\nLemma 26.1. Let W be a representation of SLn and let w ∈W be a null-\npoint. Let N(w, D) be the collection of all λ : C∗→D such that limt→0 t ·\nw = 0W . If N(w, D) is non-empty then there is a unique λ′ ∈N(w, D)\nwhich maximizes the efficiency, i.e., e(λ′) > e(λ) for all λ ∈N(w, D) and\nλ ̸= (λ′)k for any k ∈Z. This 1-parameter subgroup will be denoted by\nλ(w, D).\nProof: Suppose that N(w, D) is non-empty.\nThen in the weight-space\nexpansion of w for the maximal torus D, we see that supp(w) staisfies the\nhalf-space property for some λ ∈Yn.\nNote that the λ ∈N(w, D) are\nparametrized by lattice points λ ∈Yn such that (λ, χ) > 0 for all χ ∈\nsupp(w).\nLet Cone(w) be the conical combination (over R) of all χ ∈\nsupp(w) and Cone(w)◦its polar. Thus, in other words, N(w, D) is precisely\nthe collection of lattice points in the cone Cone(w)◦. Next, we see that e(λ)\nis a convex function of Cone(w)◦which is constant over rays R+ · λ for all\nλ ∈Cone(w)◦. By a routine analysis, the maximum of such a function must\nbe a unique ray with rational entries. This proves the lamma. □\nThis covers one important part in our task of identifying the ’best’ 1-\nparameter subgroup driving a null-point to zero. The next part is to relate\nD to other maximal tori.\nLet λ : C∗→SLn and let P(λ) be defined as follows:\nP(λ) = {A ∈SLn| lim\nt→0 λ(t)Aλ(t−1) = I ∈SLn}\nHaving fixed a maximal torus D containing IM(λ), we easily identify\nP(λ) as a parabolic subgroup, i.e., block upper-triangular. Indeed, let\nλ(t) =\n \n \nta1\n0\n0\n0\n0\nta2\n0\n0\n...\n0\n0\n0\ntan\n \n \nwith a1 ≥a2 ≥. . . ≥an (obviously with a1 + . . . + an = 0). Then\nP(λ) = {(xij|xij = 0 for all i, j such that ai < aj}\nThe unipotent radical U(λ) is a normal subgroup of P(λ) defined as:\nU(λ) = (xij) where =\n \nxij = 0\nif ai < aj\nxij = δij\nif ai = aj\n151\n\nLemma 26.2. Let λ ∈N(w, D) and let g ∈P(λ), then (i) gλg−1 ⊆\nP(λ) and P(gλg−1) = P(λ), (ii) gλg−1 ∈N(w, gDg−1), and (iii) e(λ) =\ne(gλg−1).\nThis actually follows from the construction of the explicit SLn-modules\nand is left to the reader. We now come to the unique object that we will\ndefine for each w ∈W in the null-cone. This is the parabolic subgroup P(λ)\nfor any ’best’ λ. We have already seen above that if λ′ is a P(λ)-conjugate\nof a best λ then λ′ is ’equally best’ and P(λ) = P(λ′).\nWe now relate two general equally best λ and λ′. For this we need a\npreliminary definition and a lemma:\nDefinition 26.1. Let V be a vector space over C.\nA flag F of V is a\nsequence (V0, . . . , Vr) of nested subspaces 0 = V0 ⊂V1 ⊂. . . ⊂Vr = V .\nLemma 26.3. Let dimC(V ) = r and let F = (V0, . . . , Vr) and F′ =\n(V ′\n0, . . . , V ′\nr) be two (complete) flags for V . Then there is a basis b1, . . . , br\nof V and a permutation σ ∈Sr such that Vi = {b1, . . . , bi} and V ′\ni =\n{bσ(1), . . . , bσ(i)} for all i.\nThis is proved by induction on r.\nCorollary 26.1. Let λ and λ′ be two 1-parameter subgroups and P(λ) and\nP(λ′) be their corresponding parabolic subgroups. Then there is a maximal\ntorus T of SLn such that T ⊆P(λ) ∩P(λ′).\nProof: It is clear that there is a correspondence between parabolic sub-\ngroups of SLn and flags. We refine the flags associated to the parabolic\nsubgroups P(λ) and P(λ′) to complete flags and apply the above lemma. □\nWe are now prepared to prove Kempf’d theorem:\nTheorem 12. Let W be a representation of SLn and w ∈W a null-point.\nThen there is a 1-parameter subgroup λ ∈Γ(SLn) such that (i) for all λ′ ∈\nΓ(SLn), we have e(λ) ≥e(λ′), and (ii) for all λ′ such that e(λ) = e(λ′) we\nhave P(λ) = P(λ′) and that there is a g ∈P(λ) such that λ′ = gλg−1.\nProof: Let N(w) be all elements of Γ(SLn) which drive w to zero. Let Ξ(W)\nbe the (finite) collection of D-characters appearing in the representation W.\nFor every λ(w, T) such that Im(λ) ⊆D, we may consider an A ∈SLn such\nthat AλA−1 ∈N(A · w, D) and e(λ) = e(AλA−1). Since the ’best’ element\nof N(A · w, D) is determined by supp(A · w) ⊆Ξ, we see that there are only\nfinitely many possibilities for e(A · w, AλA−1) and therefore for e(λ) for the\n’best’ λ driving w to zero.\n152\n\nThus the length k of any sequence λ(w, T1), . . . , λ(w, Tk) such that e(λ(w, T1)) <\n. . . < e(λ(w, Tk)) must be bounded by the number 2Ξ. This proves (i).\nNext, let λ1 = λ(w, T1) and λ2 = λ(w, T2) be two ’best’ elements of\nN(w, T1) and N(w, T2) respectively. By corollary 26.1, we have a torus ,\nsay D, and P(λ(w, Ti))-conjugates λi such that (i) e(λi) = e(λ(w, Ti)) and\n(ii) Im(λi) ⊆D. By lemma 26.1, we have λ1 = λ2 and thus P(λ1) = P(λ2).\nOn the other hand, P(λ(w, Ti)) = P(λi) and this proves (ii). □\nThus 12 associates a unique parabolic subgroup P(w) to every point in\nthe null-cone. This subgroup is called the destabilizing flag of w. Clearly,\nif w is in the null-cone then so is A · w, where A ∈SLn. Furthermore, it is\nclear that P(A · w) = AP(w)A−1.\nCorollary 26.2. Let w ∈W be in the null-cone and let Gw ⊆SLn stabilize\nw. Then Gw ⊆P(w).\nProof: Let g ∈Gw. Since g · w = w, we see that gP(w)g−1 = P(w), and\nthat g normalizes P(w). Since the normalizer of any parabolic subgroup is\nitself, we see that g ∈P(w). □\n153\n\nChapter 27\nStability\nReference: [Ke, GCT1]\nRecall that z ∈W is stable iffits orbit O(z) is closed in W. In the last\nchapter, we tackled the points in the null-cone, i.e., points in the set [0W ]≈,\nor in other words, points which close onto the stable point 0W. A similar\nanalysis may be done for arbitrary stable points.\nFollowing kempf, let S ⊆W be a closed SLn-invariant subset.\nLet\nz ∈W be arbitrary. If the orbit-closure ∆(z) intersects S, then we associate\na unique parabolic subgroup Pz,S ⊆SLn as a witness to this fact.\nThe\nconstruction of this parabolic subgroup is in several steps.\nAs the first step, we construct a representation X of SLn and a closed\nSLn-invariant embedding φ : W →X such that φ−1(0X) = S, scheme-\ntheoretically. This may be done as follows: since S is a closed sub-variety\nof W, there is an ideal Is = (f1, . . . , fk) of definition for S. We may further\nassume that the vector space {f1, . . . , fk} is itself an SLn-module, say X.\nWe assume that X is k-dimensional.\nWe now construct the map φ : W →X as follows:\nφ(w) = (f1(x), . . . , fk(x))\nNote that φ(S) = 0X and that IS = (f1, . . . , fk) ensure that the requirements\non our φ do hold.\nNext, there is an adaptation of (Hilbert’s) Theorem 11 which we do not\nprove:\nTheorem 13. Let W be an SLn-module and let y ∈W be a stable point.\nLet z ∈[y]≈be an element which closes onto y. Then there is a 1-parameter\nsubgroup λ : C∗→SLn such that\nlim\nt→0 λ(t) · w ∈O(y)\n154\n\nThus the limit exists and lies in the closed orbit of y.\nNow suppose that ∆(z) ∩S is non-empty. Then there must be stable\ny ∈∆(z). We apply the theorem to O(y) and obtain the λ as above. This\nshows that there is indeed a 1-parameter subgroup driving z into S. Next,\nit is easy to see that\nlim\nt→0[λ(t) · φ(z)] = 0X\nThus φ(z) actually lies in the null-cone of X. We may now be tempted to\napply the techniques of the previous chapter to come up with the ’best’ λ\nand its parabolic, now called P(z, S). This is almost the technique to be\nadopted , except that this ’best’ λ drives φ(z) into 0X but limt→0[λ(t) · z]\n(which is supposed to be in S) may not exist! This is because we are using the\nunproved (and untrue) converse of the assertion that 1-parameter subgroups\nwhich drive z into S drive φ(z) into 0X.\nThis above argument is rectified by limiting the domain of allowed 1-\nparameter subgroups to (i) Cone(supp(φ(z))◦as before, and (ii) those λ\nsuch that limt→0[λ(t) · z] exists. This second condition is also a ’convex’\ncondition and then the ’best’ λ does exist. This completes the construction\nof P(z, S).\nAs before, if Gz ⊆SLn stabilizes z then it normalizes P(z, S) thus must\nbe contained in it:\nProposition 27.1. If Gz stabilizes z then Gz ⊆P(z, S).\nLet us now consider the permanent and the determinant. Let M\nbe the n2-dimensional space of all n × n-matrices.\nSince det and perm\nare homogeneous n-forms on M, we consider the SL(M)-module W =\nSymn(M∗). We recall now certain stabilizing groups of the det and the\nperm. We will need the definition of a certain group L′. This is defined as\nthe group generated by the permutation and diagonal matrices in GLn. In\nother words, L′ is the normalizer of the complete standard torus D∗⊆GLn.\nL is defined as that subgroup of L′ which is contained in SLn.\nProposition 27.2.\n(A) Consider the group K = SLn × SLn. We define\nthe action μK of typical element (A, B) ∈K on X ∈M as given by:\nX →AXB−1\nThen (i) M is an irreducible representation of K and Im(K) ⊆SL(M),\nand (ii) K stabilizes the determinant.\n155\n\n(B) Consider the group H = L × L. 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Lam,\nC.S. and Patera, J. (eds.) Symmetry in Physics: In Memory of Robert\nT. Sharp. Providence, USA, AMS OUP, 99-112, CRM Proceedings and\nLecture Notes 34, 2004.\n[Ki] A. Kirillov, An invitation to the generalized saturation conjecture,\nmath. CO/0404353, 20 Apr. 2004.\n[KS] A. Klimyck, and K. Schm ̈udgen, Quantum groups and their represen-\ntations, Springer, 1997.\n[KT] A. Knutson, T. Tao, The honeycomb model of GLn(C) tensor products\nI: proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999)\n1055-1090.\n[KT2] A. Knutson, T. Tao: Honeycombs and sums of Hermitian matrices,\nNotices Amer. Math. Soc. 48 (2001) No. 2, 175-186.\n[LV] D. Luna and T. Vust, Plongements d’espaces homogenes, Comment.\nMath. Helv. 58, 186(1983).\n[Lu2] G. Lusztig, Introduction to quantum groups, Birkh ̈auser, 1993.\n[Ml] K. Mulmuley, Lower bounds in a parallel model without bit operations.\nSIAM J. Comput. 28, 1460–1509, 1999.\n[Mm] D. Mumford, Algebraic Geometry I, Springer-Verlang, 1995.\n[N] M. Nagata, Polynomial Rings and Affine Spaces. CBMS Regional Con-\nference no. 37, American Mathematical Society, 1978.\n[S]\nR. Stanley,\nEnumerative combinatorics,\nvol. 1,\nWadsworth and\nBrooks/Cole, Advanced Books and Software, 1986.\n[Z]\nA.\nZelevinsky,\nLittlewood-Richardson\nsemigroups,\narXiv:math.CO/9704228 v1 30 Apr 1997.\n160\n\nGv\nClosure\nDv\n\nd\nZ\nn","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0709.0746v1 [cs.CC] 5 Sep 2007\nGeometric Complexity Theory: Introduction\nDedicated to Sri Ramakrishna\nKetan D. Mulmuley 1\nThe University of Chicago\nMilind Sohoni\nI.I.T., Mumbai\nTechnical Report TR-2007-16\nComputer Science Department\nThe University of Chicago\nSeptember 2007\nAugust 2, 2014\n1Part of the work on GCT was done while the first author was visiting I.I.T.\nMumbai to which he is grateful for its hospitality"},{"paragraph_id":"p2","order":2,"text":"Foreword\nThese are lectures notes for the introductory graduate courses on geo-\nmetric complexity theory (GCT) in the computer science department, the\nuniversity of Chicago. Part I consists of the lecture notes for the course\ngiven by the first author in the spring quarter, 2007. It gives introduction\nto the basic structure of GCT. Part II consists of the lecture notes for the\ncourse given by the second author in the spring quarter, 2003. It gives in-\ntroduction to invariant theory with a view towards GCT. No background\nin algebraic geometry or representation theory is assumed. These lecture\nnotes in conjunction with the article [GCTflip1], which describes in detail\nthe basic plan of GCT based on the principle called the flip, should provide\na high level picture of GCT assuming familiarity with only basic notions\nof algebra, such as groups, rings, fields etc. Many of the theorems in these\nlecture notes are stated without proofs, but after giving enough motivation\nso that they can be taken on faith. For the readers interested in further\nstudy, Figure 1 shows logical dependence among the various papers of GCT\nand a suggested reading sequence.\nThe first author is grateful to Paolo Codenotti, Joshua Grochow, Sourav\nChakraborty and Hari Narayanan for taking notes for his lectures.\n1"},{"paragraph_id":"p3","order":3,"text":"GCTabs\n|\n↓\nGCTflip1\n|\n↓\nThese lecture notes\n−−→\nGCT3\n|\n|\n↓\n|\nGCT1\n|\n|\n|\n↓\n|\nGCT2\n|\n|\n|\n↓\n↓\nGCT6\n←−−\nGCT5\n|\n↓\nGCT4\n−−→\nGCT9\n|\n↓\nGCT7\n|\n↓\nGCT8\n|\n↓\nGCT10\n|\n↓\nGCT11\n|\n↓\nGCTflip2\nFigure 1: Logical dependence among the GCT papers\n2"},{"paragraph_id":"p4","order":4,"text":"Contents\nI\nThe basic structure of GCT\nBy Ketan D. Mulmuley\n8\n1\nOverview\n9\n1.1\nOutline\n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n9\n1.2\nThe G ̈odelian Flip\n. . . . . . . . . . . . . . . . . . . . . . . .\n12\n1.3\nMore details of the GCT approach . . . . . . . . . . . . . . .\n13\n2\nRepresentation theory of reductive groups\n16\n2.1\nBasics of Representation Theory\n. . . . . . . . . . . . . . . .\n16\n2.1.1\nDefinitions\n. . . . . . . . . . . . . . . . . . . . . . . .\n16\n2.1.2\nNew representations from old . . . . . . . . . . . . . .\n17\n2.2\nReductivity of finite groups . . . . . . . . . . . . . . . . . . .\n19\n2.3\nCompact Groups and GLn(C) are reductive . . . . . . . . . .\n20\n2.3.1\nCompact groups\n. . . . . . . . . . . . . . . . . . . . .\n20\n2.3.2\nWeyl’s unitary trick and GLn(C) . . . . . . . . . . . .\n21\n3\nRepresentation theory of reductive groups (cont)\n22\n3.1\nProjection Formula . . . . . . . . . . . . . . . . . . . . . . . .\n23\n3.2\nThe characters of irreducible representations form a basis\n. .\n25\n3.3\nExtending to Infinite Compact Groups . . . . . . . . . . . . .\n27\n4\nRepresentations of the symmetric group\n29\n4.1\nRepresentations and characters of Sn . . . . . . . . . . . . . .\n30\n4.1.1\nFirst Construction . . . . . . . . . . . . . . . . . . . .\n30\n4.1.2\nSecond Construction . . . . . . . . . . . . . . . . . . .\n31\n4.1.3\nThird Construction . . . . . . . . . . . . . . . . . . . .\n32\n4.1.4\nCharacter of Sλ [Frobenius character formula] . . . . .\n32\n4.2\nThe first decision problem in GCT . . . . . . . . . . . . . . .\n33\n3"},{"paragraph_id":"p5","order":5,"text":"5\nRepresentations of GLn(C)\n35\n5.1\nFirst Approach [Deruyts]\n. . . . . . . . . . . . . . . . . . . .\n35\n5.1.1\nHighest weight vectors . . . . . . . . . . . . . . . . . .\n38\n5.2\nSecond Approach [Weyl] . . . . . . . . . . . . . . . . . . . . .\n39\n6\nDeciding nonvanishing of Littlewood-Richardson coefficients 41\n6.1\nLittlewood-Richardson coefficients\n. . . . . . . . . . . . . . .\n41\n7\nLittlewood-Richardson coefficients (cont)\n46\n7.1\nThe stretching function\n. . . . . . . . . . . . . . . . . . . . .\n47\n7.2\nOn(C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n48\n8\nDeciding nonvanishing of Littlewood-Richardson coefficients\nfor On(C)\n52\n9\nThe plethysm problem\n56\n9.1\nLittlewood-Richardson Problem [GCT 3,5] . . . . . . . . . . .\n57\n9.2\nKronecker Problem [GCT 4,6] . . . . . . . . . . . . . . . . . .\n57\n9.3\nPlethysm Problem [GCT 6,7] . . . . . . . . . . . . . . . . . .\n58\n10 Saturated and positive integer programming\n61\n10.1 Saturated, positive integer programming . . . . . . . . . . . .\n61\n10.2 Application to the plethysm problem . . . . . . . . . . . . . .\n63\n11 Basic algebraic geometry\n64\n11.1 Algebraic geometry definitions\n. . . . . . . . . . . . . . . . .\n64\n11.2 Orbit closures . . . . . . . . . . . . . . . . . . . . . . . . . . .\n66\n11.3 Grassmanians . . . . . . . . . . . . . . . . . . . . . . . . . . .\n67\n12 The class varieties\n70\n12.1 Class Varieties in GCT . . . . . . . . . . . . . . . . . . . . . .\n70\n13 Obstructions\n73\n13.1 Obstructions\n. . . . . . . . . . . . . . . . . . . . . . . . . . .\n74\n13.1.1 Why are the class varieties exceptional? . . . . . . . .\n75\n14 Group theoretic varieties\n78\n14.1 Representation theoretic data . . . . . . . . . . . . . . . . . .\n79\n14.2 The second fundamental theorem . . . . . . . . . . . . . . . .\n80\n14.3 Why should obstructions exist? . . . . . . . . . . . . . . . . .\n81\n4"},{"paragraph_id":"p6","order":6,"text":"15 The flip\n82\n15.1 The flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n83\n16 The Grassmanian\n86\n16.1 The second fundamental theorem . . . . . . . . . . . . . . . .\n87\n16.2 The Borel-Weil theorem . . . . . . . . . . . . . . . . . . . . .\n88\n17 Quantum group: basic definitions\n90\n17.1 Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . .\n90\n18 Standard quantum group\n97\n19 Quantum unitary group\n103\n19.1 A q-analogue of the unitary group\n. . . . . . . . . . . . . . . 103\n19.2 Properties of Uq . . . . . . . . . . . . . . . . . . . . . . . . . . 105\n19.3 Irreducible Representations of Gq . . . . . . . . . . . . . . . . 106\n19.4 Gelfand-Tsetlin basis . . . . . . . . . . . . . . . . . . . . . . . 106\n20 Towards positivity hypotheses via quantum groups\n108\n20.1 Littlewood-Richardson rule via standard quantum groups\n. . 108\n20.1.1 An embedding of the Weyl module . . . . . . . . . . . 109\n20.1.2 Crystal operators and crystal bases . . . . . . . . . . . 110\n20.2 Explicit decomposition of the tensor product\n. . . . . . . . . 112\n20.3 Towards nonstandard quantum groups for the Kronecker and\nplethysm problems . . . . . . . . . . . . . . . . . . . . . . . . 113\nII\nInvariant theory with a view towards GCT\nBy Milind Sohoni\n116\n21 Finite Groups\n117\n21.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117\n21.2 The finite group action . . . . . . . . . . . . . . . . . . . . . . 118\n21.3 The Symmetric Group . . . . . . . . . . . . . . . . . . . . . . 122\n22 The Group SLn\n124\n22.1 The Canonical Representation . . . . . . . . . . . . . . . . . . 124\n22.2 The Diagonal Representation . . . . . . . . . . . . . . . . . . 125\n22.3 Other Representations . . . . . . . . . . . . . . . . . . . . . . 128\n22.4 Full Reducibility\n. . . . . . . . . . . . . . . . . . . . . . . . . 130\n5"},{"paragraph_id":"p7","order":7,"text":"23 Invariant Theory\n131\n23.1 Algebraic Groups and affine actions\n. . . . . . . . . . . . . . 131\n23.2 Orbits and Invariants . . . . . . . . . . . . . . . . . . . . . . . 132\n23.3 The Nagata Hypothesis\n. . . . . . . . . . . . . . . . . . . . . 136\n24 Orbit-closures\n139\n25 Tori in SLn\n142\n26 The Null-cone and the Destabilizing flag\n147\n26.1 Characters and the half-space criterion . . . . . . . . . . . . . 147\n26.2 The destabilizing flag . . . . . . . . . . . . . . . . . . . . . . . 149\n27 Stability\n154\n6"},{"paragraph_id":"p8","order":8,"text":"7"},{"paragraph_id":"p9","order":9,"text":"Part I\nThe basic structure of GCT\nBy Ketan D. Mulmuley\n8"},{"paragraph_id":"p10","order":10,"text":"Chapter 1\nOverview\nScribe: Joshua A. Grochow\nGoal: An overview of GCT.\nThe purpose of this course is to give an introduction to Geometric Com-\nplexity Theory (GCT), which is an approach to proving P ̸= NP via al-\ngebraic geometry and representation theory. A basic plan of this approach\nis described in [GCTflip1, GCTflip2]. It is partially implemented in a se-\nries of articles [GCT1]-[GCT11]. The paper [GCTconf] is a conference an-\nnouncement of GCT. The paper [Ml] gives an unconditional lower bound\nin a PRAM model without bit operations based on elementary algebraic\ngeometry, and was a starting point for the GCT investigation via algebraic\ngeometry.\nThe only mathematical prerequisites for this course are a basic knowl-\nedge of abstract algebra (groups, ring, fields, etc.)\nand a knowledge of\ncomputational complexity. In the first month we plan to cover the represen-\ntation theory of finite groups, the symmetric group Sn, and GLn(C), and\nenough algebraic geometry so that in the remaining lectures we can cover\nbasic GCT. Most of the background results will only be sketched or omitted.\nThis lecture uses slightly more algebraic geometry and representation\ntheory than the reader is assumed to know in order to give a more complete\npicture of GCT. As the course continues, we will cover this material.\n1.1\nOutline\nHere is an outline of the GCT approach. Consider the P vs. NP question\nin characteristic 0; i.e., over integers. So bit operations are not allowed, and\n9"},{"paragraph_id":"p11","order":11,"text":"basic operations on integers are considered to take constant time. For a sim-\nilar approach in nonzero characteristic (characteristic 2 being the classical\ncase from a computational complexity point of view), see GCT 11.\nThe basic principle of GCT is the called the flip [GCTflip1]. It “reduces”\n(in essence, not formally) the lower bound problems such as P vs. NP in\ncharacteristic 0 to upper bound problems: showing that certain decision\nproblems in algebraic geometry and representation theory belong to P. Each\nof these decision problems is of the form: is a given (nonnegative) structural\nconstant associated to some algebro-geometric or representation theoretic\nobject nonzero? This is akin to the decision problem: given a matrix, is\nits permanent nonzero? (We know how to solve this particular problem in\npolynomial time via reduction to the perfect matching problem.)\nNext, the preceding upper bound problems are reduced to purely math-\nematical positivity hypotheses [GCT6]. The goal is to show that these and\nother auxilliary structural constants have positive formulae. By a positive\nformula we mean a formula that does not involve any alternating signs like\nthe usual positive formula for the permanent; in contrast the usual formula\nfor the determinant involves alternating signs.\nFinally, these positivity hypotheses are “reduced” to conjectures in the\ntheory of quantum groups [GCT6, GCT7, GCT8, GCT10] intimately related\nto the Riemann hypothesis over finite fields proved in [Dl2], and the related\nworks [BBD, KL2, Lu2].\nA pictorial summary of the GCT approach is\nshown in Figure 1.1, where the arrows represent reductions, rather than\nimplications.\nTo recap: we move from a negative hypothesis in complexity theory\n(that there does not exist a polynomial time algorithm for an NP-complete\nproblem) to a positive hypotheses in complexity theory (that there exist\npolynomial-time algorithms for certain decision problems) to positive hy-\npotheses in mathematics (that certain structural constants have positive\nformulae) to conjectures on quantum groups related to the Riemann hy-\npothesis over finite fields, the related works and their possible extensions.\nThe first reduction here is the flip: we reduce a question about lower bounds,\nwhich are notoriously difficult, to the one about upper bounds, which we\nhave a much better handle on. This flip from negative to positive is already\npresent in G ̈odel’s work: to show something is impossible it suffices to show\nthat something else is possible. This was one of the motivations for the GCT\napproach. The G ̈odelian flip would not work for the P vs. NP problem be-\ncause it relativizes. We can think of GCT as a form of nonrelativizable (and\nnon-naturalizable, if reader knows what that means) diagonalization.\nIn summary, this approach very roughly “reduces” the lower bound prob-\n10"},{"paragraph_id":"p12","order":12,"text":"P vs. NP\nchar. 0\nFlip\n=⇒\nDecision problems\nin alg. geom.\n& rep. thy.\n=⇒\nShow certain\nconstants in alg.\ngeom. and repr.\ntheory have\npositive formulae\nLower bounds\n(Neg. hypothesis\nin complexity thy.)\nUpper bounds\n(Pos. hypotheses\nin complexity thy.)\nPos. hypotheses\nin mathematics\n=⇒\nConjectures on\nquantum groups\nrelated to RH over\nfinite fields\nFigure 1.1: The basic approach of GCT\n11"},{"paragraph_id":"p13","order":13,"text":"lems such as P vs. NP in characteristic zero to as-yet-unproved quantum-\ngroup-conjectures related to the Riemann Hypothesis over finite fields. As\nwith the classical RH, there is experimental evidence to suggest these con-\njectures hold – which indirectly suggests that certain generalizations of the\nRiemann hypothesis over finite fields also hold – and there are hints on how\nthe problem might be attacked. See [GCTflip1, GCT6, GCT7, GCT8] for a\nmore detailed exposition.\n1.2\nThe G ̈odelian Flip\nWe now re-visit G ̈odel’s original flip in modern language to get the flavor of\nthe GCT flip.\nG ̈odel set out to answer the question:\nQ: Is truth provable?\nBut what “truth” and “provable” means here is not so obvious a priori. We\nstart by setting the stage: in any mathematical theory, we have the syntax\n(i.e. the language used) and the semantics (the domain of discussion). In\nthis case, we have:\nSyntax (language)\nSemantics (domain)\nFirst order logic\n(∀, ∃, ¬, ∨, ∧, . . . )\nConstants\n0,1\nVariables\nx, y, z, . . .\nBasic Predicates\n>, <, =\nFunctions\n+,−,×,exponentiation\nAxioms\nAxioms of the natural numbers N\nUniverse: N\nA sentence is a valid formula with all variables quantified, and by a\ntruth we mean a sentence that is true in the domain. By a proof we mean a\nvalid deduction based on standard rules of inference and the axioms of the\ndomain, whose final result is the desired statement.\nHilbert’s program asked for an algorithm that, given a sentence in num-\nber theory, decides whether it is true or false.\nA special case of this is\nHilbert’s 10th problem, which asked for an algorithm to decide whether a\nDiophantine equation (equation with only integer coefficients) has a nonzero\ninteger solution.\nG ̈odel showed that Hilbert’s general program was not\n12"},{"paragraph_id":"p14","order":14,"text":"achievable. The tenth problem remained unresolved until 1970, at which\npoint Matiyasevich showed its impossibility as well.\nHere is the main idea of G ̈odel’s proof, re-cast in modern language.\nFor a Turing Machine M, whether the empty string ε is in the language\nL(M) recognized by M is undecidable. The idea is to reduce a question\nof the form ε ∈L(M) to a question in number theory. If there were an\nalgorithm for deciding the truth of number-theoretic statements, it would\ngive an algorithm for the above Turing machine problem, which we know\ndoes not exist.\nThe basic idea of the reduction is similar to the one in Cook’s proof that\nSAT is NP-complete. Namely, ε ∈L(M) iffthere is a valid computation\nof M which accepts ε. Using Cook’s idea, we can use this to get a Boolean\nformula:\n∃m∃a valid computation of M with configurations of size ≤m\ns.t. the computation accepts ε.\nThen we use G ̈odel numbering – which assigns a unique number to each\nsentence in number theory – to translate this formula to a sentence in number\ntheory. The details of this should be familiar.\nThe key point here is: to show that truth is undecidable in number theory\n(a negative statement), we show that there exists a computable reduction\nfrom ε\n?∈L(M) to number theory (a positive statement). This is the essence\nof the G ̈odelian flip, which is analogous to – and in fact was the original\nmotivation for – the GCT flip.\n1.3\nMore details of the GCT approach\nTo begin with, GCT associates to each complexity class such as P and NP\na projective algebraic variety χP, χNP , etc. [GCT1]. In fact, it associates\na family of varieties χNP (n, m): one for each input length n and circuit\nsize m, but for simplicity we suppress this here. The languages L in the\nassociated complexity class will be points on these varieties, and the set\nof such points is dense in the variety. These varieties are thus called class\nvarieties. To show that NP ⊈P in characteristic zero, it suffices to show\nthat χNP cannot be imbedded in χP .\nThese class varieties are in fact G-varieties. That is, they have an action\nof the group G = GLn(C) on them. This action induces an action on the\nhomogeneous coordinate ring of the variety, given by (σf)(x) = f(σ−1x)\nfor all σ ∈G. Thus the coordinate rings RP and RNP of χP and χNP are\n13"},{"paragraph_id":"p15","order":15,"text":"G-algebras, i.e., algebras with G-action. Their degree d-components RP (d)\nand RNP (d) are thus finite dimensional G-representations.\nFor the sake of contradiction, suppose NP ⊆P in characteristic 0. Then\nthere must be an embedding of χNP into χP as a G-subvariety, which\nin turn gives rise (by standard algebraic geometry arguments) to a sur-\njection RP ։ RNP of the coordinate rings.\nThis implies (by standard\nrepresentation-theoretic arguments) that RNP (d) can be embedded as a G-\nsub-representation of RP (d). The following diagram summarizes the impli-\ncations.\ncomplexity\nclasses\nclass\nvarieties\ncoordinate\nrings\nrepresentations\nof GLn(C)\nNP _"},{"paragraph_id":"p16","order":16,"text":"//o/o/o/o/o/o/o/o/o/o\nχNP _"},{"paragraph_id":"p17","order":17,"text":"//o/o/o/o/o/o/o/o/o/o\nRNP\n//o/o/o/o/o/o/o/o/o\nRNP (d)\n _"},{"paragraph_id":"p18","order":18,"text":"P\n//o/o/o/o/o/o/o/o/o/o/o\nχP\n//o/o/o/o/o/o/o/o/o/o/o\nRP\nOO\n//o/o/o/o/o/o/o/o/o/o\nRP(d)\nWeyl’s theorem–that all finite-dimensional representations of G = GLn(C)\nare completely reducible, i.e. can be written as a direct sum of irreducible\nrepresentations–implies that both RNP (d) and RP (d) can be written as di-\nrect sums of irreducible G-representations. An obstruction [GCT2] of degree\nd is defined to be an irreducible G-representation occuring (as a subrepre-\nsentation) in RNP (d) but not in RP (d). Its existence implies that RNP (d)\ncannot be embedded as a subrepresentation of RP (d), and hence, χNP can-\nnot be embedded in χP as a G-subvariety; a contradiction.\nWe actually have a family of varieties χNP (n, m): one for each input\nlength n and circuit size m. Thus if an obstruction of some degree exists for\nall n →∞, assuming m = nlog n (say), then NP ̸= P in characteristic zero.\nConjecture 1.1. [GCTflip1] There is a polynomial-time algorithm for con-\nstructing such obstructions.\nThis is the GCT flip: to show that no polynomial-time algorithm exists\nfor an NP-complete problem, we hope to show that there is a polynomial\ntime algorithm for finding obstructions. This task then is further reduced to\nfinding polynomial time algorithms for other decision problems in algebraic\ngeometry and representation theory.\nMere existence of an obstruction for all n would actually suffice here. For\nthis, it suffices to show that there is an algorithm which, given n, outputs\n14"},{"paragraph_id":"p19","order":19,"text":"an obstruction showing that χNP (n, m) cannot be imbedded in χP(n, m),\nwhen m = nlog n. But the conjecture is not just that there is an algorithm,\nbut that there is a polynomial-time algorithm.\nThe basic principle here is that the complexity of the proof of existence\nof an object (in this case, an obstruction) is very closed tied to the computa-\ntional complexity of finding that object, and hence, techniques underneath\nan easy (i.e. polynomial time) time algorithm for deciding existence may\nyield an easy (i.e. feasible) proof of existence. This is supported by much\nanecdotal evidence:\n• An obstruction to planar embedding (a forbidden Kurotowski minor)\ncan be found in polynomial, in fact, linear time by variants of the\nusual planarity testing algorithms, and the underlying techniques, in\nretrospect, yield an algorithmic proof of Kurotowski’s theorem that\nevery nonplanar graph contains a forbidden minor.\n• Hall’s marriage theorem, which characterizes the existence of per-\nfect matchings, in retrospect, follows from the techniques underlying\npolynomial-time algorithms for finding perfect matchings.\n• The proof that a graph is Eulerian iffall vertices have even degree is,\nessentially, a polynomial-time algorithm for finding an Eulerian circuit.\n• In contrast, we know of no Hall-type theorem for Hamiltonians paths,\nessentially, because finding such a path is computationally difficult\n(NP-complete).\nAnalogously the goal is to find a polynomial time algorithm for deciding\nif there exists an obstruction for given n and m, and then use the underlying\ntechniques to show that an obstruction always exists for every large enough\nn if m = nlog n. The main mathematical work in GCT takes steps towards\nthis goal.\n15"},{"paragraph_id":"p20","order":20,"text":"Chapter 2\nRepresentation theory of\nreductive groups\nScribe: Paolo Codenotti\nGoal: Basic notions in representation theory.\nReferences: [FH, F]\nIn this lecture we review the basic representation theory of reductive\ngroups as needed in this course. Most of the proofs will be omitted, or just\nsketched.\nFor complete proofs, see the books by Fulton and Harris, and\nFulton [FH, F]. The underlying field throughout this course is C.\n2.1\nBasics of Representation Theory\n2.1.1\nDefinitions\nDefinition 2.1. A representation of a group G, also called a G-module, is\na vector space V with an associated homomorphism ρ : G →GL(V ). We\nwill refer to a representation by V .\nThe map ρ induces a natural action of G on V , defined by g·v = (ρ(g))(v).\nDefinition 2.2. A map φ : V →W is G-equivariant if the following dia-\ngram commutes:\nV\nφ\n−−−−→W\n yg\n yg\nV\nφ\n−−−−→W\n16"},{"paragraph_id":"p21","order":21,"text":"That is, if φ(g·v) = g·φ(v). A G-equivariant map is also called G-invariant\nor a G-homomorphism.\nDefinition 2.3. A subspace W ⊆V is said to be a subrepresentation, or a\nG-submodule of a representation V over a group G if W is G-equivariant,\nthat is if g · w ∈W for all w ∈W.\nDefinition 2.4. A representation V of a group G is said to be irreducible\nif it has no proper non-zero G-subrepresentations.\nDefinition 2.5. A group G is called reductive if every finite dimensional\nrepresentation V of G is a direct sum of irreducible representation.\nHere are some examples of reductive groups:\n• finite groups;\n• the n-dimensional torus (C∗)n;\n• linear groups:\n– the general linear group GLn(C),\n– the special linear group SLn(C),\n– the orthogonal group On(C) (linear transformations that preserve\na symmetric form),\n– and the symplectic group Spn(C) (linear transformations that\npreserve a skew symmetric form);\n• Exceptional Lie Groups\nTheir reductivity is a nontrivial fact.\nIt will be proved later in this\nlecture for finite groups, and the general and special linear groups. In some\nsense, the list above is complete: all reductive groups can be constructed\nby basic operations from the components which are either in this list or are\nrelated to them in a simple way.\n2.1.2\nNew representations from old\nGiven representations V and W of a group G, we can construct new repre-\nsentations in several ways, some of which are described below.\n• Tensor product: V ⊗W. g · (v ⊗w) = (g · v) ⊗(g · w).\n• Direct sum: V ⊕W.\n17"},{"paragraph_id":"p22","order":22,"text":"• Symmetric tensor representation: The subspace Symn(V ) ⊂V ⊗· · ·⊗\nV spanned by elements of the form\nX\nσ\n(v1 ⊗· · · ⊗vn) · σ =\nX\nσ\nvσ(1) ⊗· · · vσ(n),\nwhere σ ranges over all permutations in the symmetric group Sn.\n• Exterior tensor representation: The subspace Λn(V ) ⊂V ⊗· · · ⊗V\nspanned by elements of the form\nX\nσ\nsgn(σ)(v1 ⊗· · · ⊗vn) · σ =\nX\nσ\nsgn(σ)vσ(1) ⊗· · · vσ(n).\n• Let V and W be representations, then Hom(V, W) is also a represen-\ntation, where g · φ is defined so that the following diagram commutes:\nV\nφ\n−−−−→W\n yg\n yg\nV\ng·φ\n−−−−→W\nMore precisely,\n(g · φ)(v) = g · (φ(g−1 · v)).\n• In particular, V ∗: V →C is a representation, and is called the dual\nrepresentation.\n• Let G be a finite group. Let S be a finite G-set (that is, a finite set\nwith an associated action of G on its elements). We construct a vector\nspace over any field K (we will be mostly concerned with the case\nK = C), with a basis vector associated to each element in S. More\nspecifically, consider the set K[S] of formal sums P\ns∈S αses, where\nαs ∈K, and es is a vector associated with S ∈s. Note that this set\nhas a vector space structure over K, and there is a natural induced\naction of G on K[S], defined by:\ng ·\nX\ns∈S\nαses =\nX\ns∈S\nαseg·s.\nThis action gives rise to a representation of G.\n• In particular, G is a G-set under the action of left multiplication. The\nrepresentation we obtain in the manner described above from this G-\nset is called the regular representation.\n18"},{"paragraph_id":"p23","order":23,"text":"2.2\nReductivity of finite groups\nProposition 2.1. Let G be a finite group. If W is a subrepresentation of a\nrepresentation V , then there exists a representation W ⊥s.t. V = W ⊕W ⊥.\nProof. Choose any Hermitian form H0 of V , and construct a new Hermitian\nform H defined as:\nH(v, w) =\nX\ng∈G\nHo(g · v, g · w).\nAveraging is a useful trick that is used very often in representation theory,\nbecause it ensures G-invariance. In fact, H is G-invariant, that is,\nH(v, w) =\nX\ng∈G\nHo(g · v, g · w) = H(h · v, h · w)\nLet W ⊥be the perpendicular complement to W with respect to the Her-\nmitian form H.\nThen W ⊥is also G-invariant, and therefore it is a G-\nsubmodule.\nCorollary 2.1. Every representation of a finite group is a direct sum of\nirreducible representations.\nLemma 2.1. (Schur) If V and W are irreducible representations over C,\nand φ : V →W is a homomorphism (i.e. a G-invariant map), then:\n1. Either φ is an isomorphism or φ = 0.\n2. If V = W, φ = λI for some λ ∈C.\nProof.\n1. Since Ker(φ), and Imφ are G-submodules, either Im(φ) = V\nor Im(φ) = 0.\n2. Let φ : V →V . Since C algebraically closed, there exists an eigenvalue\nλ of φ. Look at the map φ−λI : V →V . By (1), φ−λI = 0 (it can’t\nbe an isomorphism because something maps to 0). So φ = λI.\nCorollary 2.2. Every representation is a unique direct sum of irreducible\nrepresentations. More precisely, given two decompositions into irreducible\nrepresentations,\nV =\nM\nV ai\ni\n=\nM\nW bj\nj ,\nthere is a one to one correspondence between the Vi’s and Wj’s, and the\nmultiplicities correspond.\nProof. exercise (follows from Schur’s lemma).\n19"},{"paragraph_id":"p24","order":24,"text":"gu\nu\nR\ngR\nFigure 2.1: Example of a left Haar measure for the circle (U1(C)). Left action by\na group element g on a small region R around u does not change the area.\n2.3\nCompact Groups and GLn(C) are reductive\nNow we prove reductivity of compact groups.\n2.3.1\nCompact groups\nExamples of compact groups:\n• Un(C) ⊆GLn(C), the unitary groups (all rows are normal and orthog-\nonal).\n• SUn(C) ⊆SLn(C), the special unitary group.\nGiven a compact group, a left-invariant Haar measure is a measure that\nis invariant under the left action of the group. In other words, multiplication\nby a group element does not change the area of a small region (i.e., the group\naction is an isometry, see figure 2.1).\nTheorem 2.1. Compact groups are reductive\nProof. We use the averaging trick again. In fact the proof is the same as\nin the case of finite groups, using integration instead of summation for the\naveraging trick. Let H0 be any Hermitian form on V. Then define H as:\nH(v, w) =\nZ\nG\nH(gv, gw)dG\n20"},{"paragraph_id":"p25","order":25,"text":"where dG is a left-invariant Haar measure.\nNote that H is G-invariant.\nLet W ⊥be the perpendicular complement to W. Then W ⊥is G-invariant.\nHence it is a G-submodule.\nThe same proof as before then gives us Schur’s lemma for compact\ngroups, from which follows:\nTheorem 2.2. If G is compact, then every finite dimensional representation\nof G is a unique direct sum of irreducible representations.\n2.3.2\nWeyl’s unitary trick and GLn(C)\nTheorem 2.3. (Weyl) GLn(C) is reductive\nProof. (general idea)\nLet V be a representation of GLn(C). Then GLn(C) acts on V :\nGLn(C) ֒→V.\nBut Un(C) is a subgroup of GLn(C). Therefore we have an induced action\nof Un(C) on V , and we can look at V as a representation of Un(C). As a\nrepresentation of Un(C), V breaks into irreducible representations of Un(C)\nby the theorem above. To summarize, we have:\nUn(C) ⊆GLn(C) ֒→V = ⊕iVi,\nwhere the Vi’s are irreducible representations of Un(C). Weyl’s unitary trick\nuses Lie algebra to show that every finite dimensional representation of\nUn(C) is also a representation of GLn(C), and irreducible representations of\nUn(C) correspond to irreducible representations of GLn(C). Hence each Vi\nabove is an irreducible representation of GLn(C).\nOnce we know these groups are reductive, the goal is to construct and\nclassify their irreducible finite dimensional representations.\nThis will be\ndone in the next lectures: Specht modules for Sn, and Weyl modules for\nGLn(C).\n21"},{"paragraph_id":"p26","order":26,"text":"Chapter 3\nRepresentation theory of\nreductive groups (cont)\nScribe: Paolo Codenotti\nGoal: Basic representation theory, continued from the last lecture.\nIn this lecture we continue our introduction to representation theory.\nAgain we refer the reader to the book by Fulton and Harris for full details\n[FH]. Let G be a finite group, and V a finite-dimensional G-representation\ngiven by a homomorphism ρ : G →GL(V ). We define the character of the\nrepresentation V (denoted χV ) by χV (g) = Tr(ρ(g)).\nSince Tr(A−1BA) = Tr(B), χV (hgh−1) = χV (g). This means charac-\nters are constant on conjugacy classes (sets of the form {hgh−1|h ∈G}, for\nany g ∈G). We call such functions class functions.\nOur goal for this lecture is to prove the following two facts:\nGoal 1 A finite dimensional representation is completely determined by its\ncharacter.\nGoal 2 The space of class functions is spanned by the characters of irreducible\nrepresentations. In fact, these characters form an orthonormal basis\nof this space.\nFirst, we prove some useful lemmas about characters.\nLemma 3.1. χV ⊕W = χV + χW\nProof. Let g ∈G, and let ρ, σ be homomorphisms from G into V and W,\nrespectively. Let λ1, . . . , λr be the eigenvalues of ρ(g), and μ1, . . . , μs the\n22"},{"paragraph_id":"p27","order":27,"text":"eigenvalues of σ(g). Then (ρ ⊕σ)(g) = (ρ(g), σ(g)), so the eigenvalues of\n(ρ ⊕σ)(g) are just the eigenvalues of ρ(g) together with the eigenvalues of\nσ(g).\nThen χV (g) = P\ni λi, χW (g) = P\ni μi, and χV ⊕W = P\ni λi + P\ni μi.\nLemma 3.2. χV ⊗W = χV χW\nProof. Let g ∈G, and let ρ, σ be homomorphisms into V and W, respec-\ntively. Let λ1, . . . , λr be the eigenvalues of ρ(g), and μ1, . . . , μs the eigenval-\nues of σ(g). Then (ρ ⊗σ)(g) is the Kronecker product of the matrices ρ(g)\nand σ(g). So its eigenvalues are all λiμj where 1 ≤i ≤r, 1 ≤j ≤s.\nThen, Tr((ρ ⊗σ)(g)) = P\ni,j λiμj = (P\ni λi)\n P\nj μj"},{"paragraph_id":"p28","order":28,"text":", which is equal to\nTr(ρ(g))Tr(σ(g)).\n3.1\nProjection Formula\nIn this section, we derive a projection formula needed for Goal 1 that allows\nus to determine the multiplicity of an irreducible representation in another\nrepresentation. Given a G-module V , let V G = {v|∀g ∈G, g · v = v}. We\nwill call these elements G-invariant. Let\nφ =\n1\n|G|\nX\ng∈G\ng ∈End(V ),\n(3.1)\nwhere each g, via ρ is considered an element of End(V ).\nLemma 3.3. The map φ : V\n→V is a G-homomorphism; i.e., φ ∈\nHomG(V, V ) = (Hom(V, V ))G.\nProof. The set End(V ) is a G-module, as we saw in last class, via the fol-\nlowing commutative diagram: for any π ∈End(V ), and h ∈G:\nV\nπ\n−−−−→V\n yh\n yh\nV\nh·π\n−−−−→V.\nTherefore π ∈HomG(V, V ) (i.e., π is a G-equivariant morphism) iff\nh · π = π for all h ∈G.\nWhen φ is defined as in equation (3.1) above,\nh · φ =\n1\n|G|\nX\ng\nhgh−1 =\n1\n|G|\nX\ng\ng = φ.\n23"},{"paragraph_id":"p29","order":29,"text":"Thus\nh · φ = φ, ∀h ∈G,\nand φ : V →V is a G-equivariant morphism, i.e. φ ∈HomG(V, V ).\nLemma 3.4. The map φ is a G-equivariant projection of V onto V G\nProof. For every w ∈W, let\nv = φ(w) =\n1\n|G|\nX\ng∈G\ng · w.\nThen\nh · v = h · φ(w) =\n1\n|G|\nX\ng∈G\nhg · w = v, for any h ∈G.\nSo v ∈V G. That is, Im(φ) ⊆V G. But if v ∈V G, then\nφ(v) =\n1\n|G|\nX\ng∈G\ng · v =\n1\n|G||G|v = v.\nSo V G ⊆Im(φ), and φ is the identity on V G. This means that φ is the\nprojection onto V G.\nLemma 3.5.\ndim(V G) =\n1\n|G|\nX\ng∈G\nχV (g).\nProof. We have: dim(V G) = Tr(φ), because φ is a projection (φ = φ|V G ⊕\nφ|Ker(φ)). Also,\nTr(φ) =\n1\n|G|\nX\ng∈G\nTrV (g) =\n1\n|G|\nX\ng∈G\nχV (g).\nThis gives us a formula for the multiplicity of the trivial representation\n(i.e., dim(V G)) inside V .\nLemma 3.6. Let V, W be G-representations. If V is irreducible, dim(HomG(V, W))\nis the multiplicity of V inside W. If W is irreducible, dim(HomG(V, W)) is\nthe multiplicity of W inside V .\nProof. By Schur’s Lemma.\n24"},{"paragraph_id":"p30","order":30,"text":"Let Cclass(G) be the space of class functions on (G), and let (α, β) =\n1\n|G|\nP\ng α(g)β(g) be the Hermitian form on Cclass\nLemma 3.7. If V and W are irreducible G representations, then\n(χV , χW ) =\n1\n|G|\nX\ng∈G\nχV (g)χW (g) =\n(\n1\nif V ∼= W\n0\nif V ≇W.\n(3.2)\nProof. Since Hom(V, W) ∼= V ∗⊗W, χHom(V,W ) = χV ∗χW = χV χW . Now\nthe result follows from Lemmas 3.5 and 3.6.\nLemma 3.8. The characters of the irreducible representations form an or-\nthonormal set.\nProof. Follows from Lemma 3.7.\nIf V ,W are irreducible, then ⟨χV , χW ⟩is 0 if V ̸= W and 1 otherwise.\nThis implies that:\nTheorem 3.1 (Goal 1). A representation is determined completely by its\ncharacter.\nProof. Let V = L\ni V ⊕ai\ni\n. So χV = P\ni aiχVi, and ai = (χV , χVi). This\ngives us a formula for the multiplicity of an irreducible representation in\nanother representation, solely in terms of their characters.\nTherefore, a\nrepresentation is completely determined by its character.\n3.2\nThe characters of irreducible representations\nform a basis\nIn this section, we address Goal 2.\nLet R be the regular representation of G, V an irreducible representation\nof G.\nLemma 3.9.\nR =\nM\nV\nEnd(V, V ),\nwhere V ranges over all irreducible representations of G.\n25"},{"paragraph_id":"p31","order":31,"text":"Proof. χR(g) is 0 if g is not the identity and |G| otherwise.\n(χR, χV ) =\n1\n|G|\nX\ng∈G\nχR(g)χV (g) =\n1\n|G||G|χV (e) = χV (e) = dim(V )\nLet α : G →C. For any G-module V , let φα,V = P\ng α(g)g : V →V\nExercise 3.1. φα,V is G equivariant (i.e. a G-homomorphism) iffα is a\nclass function.\nProposition 3.1. Suppose α : G →C is a class function, and (α, χV ) = 0\nfor all irreducible representations V . Then α is identically 0.\nProof. If V is irreducible, then, by Schur’s lemma, since φα,V is a G-homomorphism,\nand V is irreducible, φα,V = λId, where λ = 1\nnTr(φα,V ), n = dim(V ). We\nhave:\nλ = 1\nn\nX\ng\nα(g)χV (g) = 1\nn|G|(α, χV ∗).\nNow V is irreducible iffV ∗is irreducible. So λ = 1\nn|G|0 = 0. Therefore,\nφα,V = 0 for any irreducible representation, and hence for any representa-\ntion.\nNow let V be the regular representation. Since g as endomorphisms of\nV are linearly independent, φα,V = 0 implies that α(g) = 0.\nTheorem 3.2. Characters form an orthonormal basis for the space of class\nfunctions.\nProof. Follows from Proposition 3.1, and Lemma 3.8\nIf V = L\ni V ⊕ai\ni\n, and πi : V →V ⊕ai\ni\nis the projection operator. We have\na formula π =\n1\n|G|\nP\ng g for the trivial representation. Analogously:\nExercise 3.2. πi = dimVi\n|G|\nP\ng χVi(g)g.\n26"},{"paragraph_id":"p32","order":32,"text":"3.3\nExtending to Infinite Compact Groups\nIn this section, we extend the preceding results to infinite compact groups.\nWe must take some facts as given, since these theorems are much more\ncomplicated than those for finite groups.\nConsider compact G, specifically Un(C), the unitary subgroup of G(C).\nU1(C) is the circle group. Since U1(C) is abelian, all its representations are\none-dimensional.\nSince the group G is infinite, we can no longer sum over it. The idea\nis to replace the sum\n1\n|G|\nP\ng f(g) in the previous setting with\nR\nG f(g)dμ,\nwhere μ is a left-invariant Haar measure on G.\nIn this fashion, we can\nderive analogues of the preceding results for compact groups. We need to\nnormalize, so we set\nR\nG dμ = 1.\nLet ρ : G →GL(V ), where V is a finite dimensional G-representation.\nLet χV (g) = Tr(ρ(g)). Let V = L\ni V ai\ni\nbe the complete decomposition of\nV into irreducible representations.\nWe can again create a projection operator π : V →V G, by letting\nπ =\nR\nG ρ(g)dμ.\nLemma 3.10. We have:\ndim(V G) =\nZ\nG\nχV (g)dμ.\nProof. This result is analogous to Lemma 3.5 for finite groups.\nFor class functions α, β, define an inner product\n(α, β) =\nZ\nG\nα(g)β(g)dμ.\nLemma 3.10 applied to HomG(V, W) gives\n(χV , χW ) =\nZ\nG\nχV χW dμ = dim(HomG(V, W)).\nLemma 3.11. If V, W are irreducible, (χV , χW) = 1 if V and W are iso-\nmorphic, and (χV , χW ) = 0 otherwise.\nProof. This result is analogous to Lemma 3.7 for finite groups.\nLemma 3.12. The irreducible representations are orthonormal, just as in\nLemma 3.8 in the case of finite groups.\n27"},{"paragraph_id":"p33","order":33,"text":"If V is reducible, V = L\ni V ⊕ai\ni\n, then\nai = (χV , χVi) =\nZ\nG\nχV χVidμ.\nHence\nTheorem 3.3. A finite dimensional representation is completely determined\nby its character.\nThis achieves Goal 1 for compact groups. Goal 2 is much harder:\nTheorem 3.4 (Peter-Weyl Theorem). (1) The characters of the irreducible\nrepresentations of G span a dense subset of the space of continuous class\nfunctions.\n(2) The coordinate functions of all irreducible matrix representations of\nG span a dense subset of all continuous functions on G.\nBy a coordinate function of a representation ρ : G →GL(V ), we mean\nthe function on G corresponding to a fixed entry of the matrix form of\nρ(g). For G = U1(C), (2) gives the Fourier series expansion on the circle.\nHence, the Peter-Weyl theorem constitutes a far reaching generalization of\nthe harmonic analyis from the circle to general Un(C).\n28"},{"paragraph_id":"p34","order":34,"text":"Chapter 4\nRepresentations of the\nsymmetric group\nScribe: Sourav Chakraborty\nGoal:\nTo determine the irreducible representations of the Symmetric\ngroup Sn and their characters.\nReference:\n[FH, F]\nRecall\nLet G be a reductive group. Then\n1. Every finite dimensional representation of G is completely reducible,\nthat is, can be written as a direct sum of irreducible representations.\n2. Every irreducible representation is determined by its character.\nExamples of reductive groups:\n• Continuous: algebraic torus (C∗)m, general linear group GLn(C), spe-\ncial linear group Sln(C), symplectic group Spn(C), orthogonal group\nOn(C).\n• Finite: alternating group An, symmetric group Sn, Gln(Fp), simple lie\ngroups of finite type.\n29"},{"paragraph_id":"p35","order":35,"text":"4.1\nRepresentations and characters of Sn\nThe number of irreducible representations of Sn is the same as the the\nnumber of conjugacy classes in Sn since the irreducible characters form a\nbasis of the space of class functions.\nEach permutation can be written\nuniquely as a product of disjoint cycles. The collection of lengths of the\ncycles in a permutation is called the cycle type of the permutation. So a\ncycle type of a permutation on n elements is a partition of n. And in Sn\neach conjugacy class is determined by the cycle type, which, in turn, is\ndetermined by the partition of n. So the number of conjugacy class is same\nas the number of partitions of n. Hence:\nNumber of irreducible representations of Sn = Number of partitions of n\n(4.1)\nLet λ = {λ1 ≥λ2 ≥. . . } be a partition of n; i.e., the size |λ| = P λi is\nn. The Young diagram corresponding to λ is a table shown in Figure 1. It\nis like an inverted staircase. The top row has λ1 boxes, the second row has\nλ2 boxes and so on. There are exactly n boxes.\nrow 1\nrow 5\nrow 6\nrow 2\nrow 3\nrow 4\nFigure 4.1: Row i has λi number of boxes\nFor a given partition λ, we want to construct an irreducible representa-\ntion Sλ, called the Specht-module of Sn for the partition λ, and calculate\nthe character of Sλ. We shall give three constructions of Sλ.\n4.1.1\nFirst Construction\nA numbering T of a Young diagram is a filling of the boxes in its table\nwith distinct numbers from 1, . . . , n. A numbering of a Young diagram is\n30"},{"paragraph_id":"p36","order":36,"text":"also called a tableau. It is called a standard tableaux if the numbers are\nstrictly increasing in each row and column. By Tij we mean the value in the\ntableaux at i-th row and j-th column. We associate with each tableaux T a\npolynomial in C[X1, X2, . . . , Xn]:\nfT = ΠjΠi<i′(XTij −XTi′j).\nLet Sλ be the subspace of C[X1, X2, . . . , Xn] spanned by fT ’s, where T\nranges over all tableaux of shape λ. It is a representation of Sn. Here Sn\nacts on C[X1, X2, . . . , Xn] as:\n(σ.f)(X1, X2, . . . , Xn) = f(Xσ(1), Xσ(2), . . . , Xσ(n))\nTheorem 4.1.\n1. Sλ is irreducible.\n2. Sλ ̸≈Sλ′ if λ ̸= λ′\n3. The set {fT }, where T ranges over standard tableau of shape λ, is a\nbasis of Sλ.\n4.1.2\nSecond Construction\nLet T be a numbering of a Young diagram with distinct numbers from\n{1, . . . , n}. An element σ in Sn acts on T in the usual way by permuting\nthe numbers. Let R(T), C(T) ⊂Sn be the sets of permutations that fix\nthe rows and columns of T, respectively. We have R(σT) = σR(T)σ−1 and\nC(σT) = σR(T)σ−1. We say T ≡T ′ if the rows of T and T ′ are the same\nup to ordering. The equivalence class of T, called the tabloid, is denoted by\n{T}. Its orbit is isomorphic to Sn/R(T).\nLet C[Sn] be the group algebra of Sn. Representations of Sn are the same\nas the representations of C[Sn]. The element aT = P\np∈R(T) p in C[Sn] is\ncalled the row symmetrizer, bT = P\nq∈C(T) sign(q)q the column symmetrizer,\nand cT = aT bT the Young symmetrizer.\nLet\nVT = bT .{T} =\nX\nq∈C(T)\nsign(q){qT}.\nThen σ.VT = VσT . Let Sλ be the span of all VT ’s, where T ranges over\nall numberings of shape λ.\nTheorem 4.2.\n1. Sλ is irreducible\n31"},{"paragraph_id":"p37","order":37,"text":"2. Sλ ̸≈Sλ′ if λ ̸= λ′.\n3. The set {VT |T standard} forms a basis for Sλ.\n4.1.3\nThird Construction\nLet T be a canonical numbering of shape λ = (λ1, . . . , λk). By this, we mean\nthe first row is numbered by 1, . . . , λ1, the second row by λ1 + 1, . . . λ1 + λ2,\nand so on, and the rows are increasing. Let aλ = aT , bλ = bT , and cλ =\ncT = aT .bT .\nThen Sλ = C[Sn].cλ is a representation of Sn from the left.\nTheorem 4.3.\n1. Sλ is irreducible\n2. Sλ ̸∼= Sλ′ if λ ̸= λ′.\n3. The basis: an exercise.\n4.1.4\nCharacter of Sλ [Frobenius character formula]\nLet i = (i1, i2, . . . , ik) be such that P\nj jij = n. Let Ci be the conjugacy\nclass consisting of permutations with ij cycles of length j. Let χλ be the\ncharacter of Sλ. The goal is to find χλ(Ci).\nLet λ : λ1 ≥· · · ≥λk be a partition of length k. Given k variables\nX1, X2, . . . , Xk, let\nPj(X) =\nX\nXj\ni ,\nbe the power sum, and\n∆(X) = Πi<j(Xj −Xi)\nthe discriminant. Let f(X) be a formal power series on Xi’s. Let [f(X)]l1,l2,...,lk\ndenote the coefficient of Xl1\n1 Xl2\n2 . . . Xlk\nk in f(X). Let li = λi + k −i.\nTheorem 4.4 (Frobenius Character Formula).\nχλ(Ci) ="},{"paragraph_id":"p38","order":38,"text":"∆(X) · ΠjPj(X)ij \nl1,l2,...,lk\n32"},{"paragraph_id":"p39","order":39,"text":"4.2\nThe first decision problem in GCT\nNow we can state the first hard decision problem in representation theory\nthat arises in the context of the flip. Let Sα and Sβ be two Specht modules\nof Sn. Since Sn is reductive, Sα ⊗Sβ decomposes as\nSα ⊗Sβ =\nM\nkλ\nαβSλ\nHere kλ\nαβ is called the Kronecker coefficient.\nProblem 4.1. (Kronecker problem) Given λ, α and β decide if kλ\nαβ > 0.\nConjecture 4.1 (GCT6). This can be done in polynomial time; i.e. in time\npolynomial in the bit lengths of the inputs λ, α and β.\n33"},{"paragraph_id":"p40","order":40,"text":"55\n34"},{"paragraph_id":"p41","order":41,"text":"Chapter 5\nRepresentations of GLn(C)\nScribe: Joshua A. Grochow\nGoal: To determine the irreducible representations of GLn(C) and their\ncharacters.\nReferences: [FH, F]\nThe goal of today’s lecture is to classify all irreducible representations\nof GLn(C) and compute their characters. We will go over two approaches,\nthe first due to Deruyts and the second due to Weyl.\nA polynomial representation of GLn(C) is a representation ρ : GLn(C) →\nGL(V ) such that each entry in the matrix ρ(g) is a polynomial in the entries\nof the matrix g ∈GLn(C).\nThe main result is that the polynomial irreducible representations of\nGLn(C) are in bijective correspondence with Young diagrams λ of height\nat most n, i.e. λ1 ≥λ2 ≥· · · ≥λn ≥0. Because of the importance of\nWeyl’s construction (similar constructions can be used on many other Lie\ngroups besides GLn(C)), the irreducible representation corresponding to λ\nis known as the Weyl module Vλ.\n5.1\nFirst Approach [Deruyts]\nLet X = (xij) be a generic n × n matrix with variable entries xij. Consider\nthe polynomial ring C[X] = C[x11, x12, . . . , xnn].\nThen GLn(C) acts on\nC[X] by (A ◦f)(X) = f(ATX) (it is easily checked that this is in fact a left\naction).\nLet T be a tableau of shape λ. To each column C of T of length r, we\nassociate an r × r minor of X as follows: if C has the entries i1, . . . , ir, then\n35"},{"paragraph_id":"p42","order":42,"text":"take from the first r columns of X the rows i1, . . . , ir. Visually:\nC ="},{"paragraph_id":"p43","order":43,"text":"i1\n...\nir"},{"paragraph_id":"p44","order":44,"text":"−→eC =\n1\n· · ·\nr\n↓\n↓\ni1 →\ni2 →\n...\nir →"},{"paragraph_id":"p45","order":45,"text":"xi1,1\n· · ·\nxi1,r\n· · ·\nxi1,n\nxi2,1\n· · ·\nxi2,r\n· · ·\nxi2,n\n...\n...\n...\nxir,1\n· · ·\nxir,r\n· · ·\nxir,n"},{"paragraph_id":"p46","order":46,"text":"(Thus if there is a repeated number in the column C, eC = 0, since\nthe same row will get chosen twice.) Using these monomials eC for each\ncolumn C of the tableau T, we associate a monomial to the entire tableau,\neT = Q\nC eC. (Thus, if in any column of T there is a repeated number,\neT = 0. Furthermore, the numbers must all come from {1, . . . , n} if they\nare to specify rows of an n × n matrix.\nSo we restrict our attention to\nnumberings of T from {1, . . . , n} in which the numbers in any given column\nare all distinct.)\nLet Vλ be the vector space generated by the set {eT }, where T ranges over\nall such numberings of shape λ. Then GLn(C) acts on Vλ: for g ∈GLn(C),\neach row of gX is a linear combination of the rows of X, and since eC is a\nminor of X, g · eC is a linear combination of minors of X of the same size,\ni.e. g(eC) = P\nD ag\nC,DeD (this follows from standard linear algebra). Then\ng(eT )\n=\ng(eC1eC2 · · · eCk)\n=\n X\nD\nag\nC1,DeD\n!\n· · ·\n X\nD\nag\nCk,DeD\n!\nIf we expand this product out, we find that each term is in fact eT ′ for some\nT ′ of the appropriate shape. We then have the following theorem:\nTheorem 5.1.\n1. Vλ is an irreducible representation of GLn(C).\n2. The set {eT |T is a semistandard tableau of shape λ} is a basis for Vλ.\n(Recall that a semistandard tableau is one whose numbering is weakly\nincreasing across each row and strictly increasing down each column.)\n3. Every polynomial irreducible representation of GLn(C) of degree d is\nisomorphic to Vλ for some partition λ of d of height at most n.\n36"},{"paragraph_id":"p47","order":47,"text":"4. Every rational irreducible representation of GLn(C) (each entry of ρ(g)\nis a rational function in the entries of g ∈GLn(C)) is isomorphic to\nVλ⊗detk for some partition λ of height at most n and for some integer\nk (where det is the determinant representation).\n5. (Weyl’s character formula) Define the character χλ of Vλ by χλ(g) =\nTr(ρ(g)), where ρ : GLn(C) →GL(Vλ) is the representation map.\nThen, for g ∈GLn(C) with eigenvalues x1, . . . , xn,\nχλ(g) = Sλ(x1, . . . , xn) :=\n xλi+n−i\nj"},{"paragraph_id":"p48","order":48,"text":"xn−i\nj"},{"paragraph_id":"p49","order":49,"text":"(where |yi\nj| is the determinant of the n × n matrix whose entries are\nyij = yi\nj, so, e.g., the determinant in the denominator is the usual van\nder Monde determinant, which is equal to Q\ni<j(xi −xj)). Here Sλ is\na polynomial, called the Schur polynomial.\nNB: It turns out that all holomorphic representations of GLn(C) are\nrational, and, by part (4) of the theorem, the Weyl modules classify all such\nrepresentations up to scalar multiplication by powers of the determinant.\nWe’ll give here a very brief introduction to the Schur polynomial in-\ntroduced in the above theorem, and explain why the Schur polynomial Sλ\nassociated to λ gives the character of Vλ.\nLet λ be a partition, and T a semistandard tableau of shape λ. Define\nx(T) = Q\ni xμi(T)\ni\n∈C[x1, . . . , xn], where μi(T) is the number of times i\nappears in T. Then it can be shown [F] that\nSλ(x1, . . . , xn) =\nX\nT\nx(T),\nwhere the sum is taken over all semistandard tableau of shape λ.\nProposition 5.1. Sλ(x1, . . . , xn) is the character of Vλ, where x1, . . . , xn\ndenote the eigenvalues of an element of GLn(C).\nProof. It suffices to show this diagonalizable g ∈GLn(C), since the diago-\nnalizable matrices are dense in GLn(C).\nSo let g ∈GLn(C) be diagonalizable with eigenvalues x1, . . . , xn. We\ncan assume that g is diagonal. If not, let A be a matrix that diagonalizes\ng.\nSo AgA−1 is diagonal with x1, . . . , xn as its diagonal entries.\nIf ρ :\nGLn(C) →GL(Vλ) is the representation corresponding to the module Vλ,\n37"},{"paragraph_id":"p50","order":50,"text":"then conjugate ρ by A to get ρ′ : GLn(C) →GL(Vλ) defined by ρ′(h) =\nAρ(h)A−1. In particular, since trace is invariant under conjugation, ρ and\nρ′ have the same character. The module corresponding to ρ′ is simply A·Vλ,\nwhich is clearly isomorphic to Vλ since A is invertible. Thus to compute the\ncharacter χλ(g), it suffices to compute the character of g under ρ′, i.e., when\ng is diagonal, as we shall assume now.\nWe will show that eT is an eigenvector of g with eigenvalue x(T), i.e.\ng(eT ) = x(T)eT . Then since {eT |T is a semistandard tableau of shape λ} is\na basis for Vλ, the trace of g on Vλ will just be P\nT x(T), where the sum is\nover semistandard T of shape λ; this is exactly Sλ(x1, . . . , xn).\nWe reduce to the case where T is a single column. Suppose the claim\nis true for all columns C. Then since eT is a product of eC where C is\na column, the corresponding eigenvalue of eT will be Q\nC x(C) (where the\nproduct is taken over the columns C of T), which is exactly x(T).\nSo assume T is a single column, say with entries i1, . . . , ir. Then eT is\nsimply the above-mentioned r×r minor of the generic n×n matrix X = (xij)\n(do not confuse the double-indexed entries of the matrix X with the single-\nindexed eigenvalues of g). Since g is diagonal, gt = g. So (g ◦eT )(X) =\neT (gtX) = eT (gX). Thus g multiplies the ij-th column by xij. Thus its\neffect on eT is simply to multiply it by Qr\nj=1 xij, which is exactly x(T).\n5.1.1\nHighest weight vectors\nThe subgroup B ⊂GLn(C) of lower triangular invertible matrices, called\nthe Borel subgroup, is solvable.\nSo every irreducible representation of B\nis one-dimensional. A weight vector for GLn(C) is a vector v which is an\neigenvector for every matrix b ∈B.\nIn other words, there is a function\nλ : B →C such that b · v = λ(b)v for all b ∈B. The restriction of λ to the\nsubgroup of diagonal matrices in B is known as the weight of v.\nAs we showed in the proof of the above proposition,"},{"paragraph_id":"p51","order":51,"text":"x1\n...\nxn"},{"paragraph_id":"p52","order":52,"text":"eT = x(T)eT = xλ1\n1 · · · xλn\nn eT .\nSo eT is a weight vector with weight x(T). Thus Theorem 5.1 (2) gives a\nbasis consisting entirely of weight vectors. We abbreviate the weight x(T)\nby the sequence of exponents (λ1, . . . , λn). We say eT is a highest weight\nvector if its weight is the highest in the lexicographic ordering (using the\nabove sequence notation for the weight).\n38"},{"paragraph_id":"p53","order":53,"text":"Each Vλ has a unique (up to scalars) B-invariant vector, which turns\nout to be the highest weight vector: namely eT , where T is canonical. For\nexample, for λ = (5, 3, 2, 2, 1), the canonical T is:\nT =\n1 1 1 1 1\n2 2 2\n3 3\n4 4\n5\nNote that the weight of such eT is (λ1, . . . , λn), so that the highest weight\nvector uniquely determines λ, and thus the entire representation Vλ. (This\nis a general feature of highest weight vectors in the representation theory of\nLie algebras and Lie groups.) Thus the irreducible representations GLn(C)\nare in bijective correspondence wih the highest weights of GLn(C), i.e. the\nsequences of exponents of the eigenvalues of the B-invariant eigenvectors.\n5.2\nSecond Approach [Weyl]\nLet V = Cn and consider the d-th tensor power V ⊗d. The group GLn(C)\nacts on V ⊗d on the left by the diagonal action\ng(v1 ⊗· · · ⊗vd) = gv1 ⊗· · · ⊗gvd (g ∈GL(V ))\nwhile the symmetric group Sd acts on the right by\n(v1 ⊗· · · ⊗vd)τ = v1τ ⊗· · · ⊗vdτ (τ ∈Sd) .\nThese two actions commute, so V ⊗d is a representation of H = GLn(C) ×\nSd. Every irreducible representation of H is of the form U ⊗W for some\nirreducible representation U of GLn(C) and some irreducible representation\nW of Sd. Since both GLn(C) and Sd are reductive (every finite-dimensional\nrepresentation is a direct sum of irreducible representations), their product\nH is reductive as well. So there are some partitions α and β and integers\nmαβ such that\nV ⊗d =\nM\n(Vα ⊗Sβ)mαβ,\nwhere Vα are Weyl modules and Sβ are Specht modules (irreducible repre-\nsentations of the symmetric group Sd).\nTheorem 5.2. V ⊗d = L\nλ Vλ⊗Sλ, where the sum is taken over partitions λ\nof d of height at most n. (Note that each summand appears with multiplicity\none, so this is a “multiplicity-free” decomposition.)\n39"},{"paragraph_id":"p54","order":54,"text":"Now, let T be any standard tableau of shape λ, and recall the Young\nsymmetrizer cT from our discussion of the irreducible representations of Sd.\nThen V ⊗dcT is a representation of GLn(C) from the left (since cT ∈C[Sd]\nacts on the right, and the left action of GLn(C) and the right action of Sd\ncommute.)\nTheorem 5.3. V ⊗dcT ∼= Vλ\nThus\nV ⊗d =\nM\nλ:|λ|=d\nM\nstd. tableau\nT of shape λ\nV ⊗dcT ,\nwhere |λ| = P λi denotes the size of λ. In particular, Vλ occurs in V ⊗d with\nmultiplicity dim(Sλ).\nFinally, we construct a basis for V ⊗dcT . A bitableau of shape λ is a pair\n(U, T) where U is a semistandard tableau of shape λ and T is a standard\ntableau of shape λ. (Recall that the the semistandard tableau of shape λ\nare in natural bijective correspondence with a basis for the Weyl module Vλ,\nwhile the standard tableau of shape λ are in natural bijective correspondence\nwith a basis for the Specht module Sλ.)\nTo each bitableau we associate a vector e(U,T) = ei1 ⊗· · · ⊗eid where\nij is defined as follows. Each number 1, . . . , n appears in T exactly once.\nThe number ij is the entry of U in the same location as the number j in T;\npictorially:\nU\nT\nij\nj\nThen:\nTheorem 5.4. The set {e(U,T)cT } is a basis for V ⊗dcT .\n40"},{"paragraph_id":"p55","order":55,"text":"Chapter 6\nDeciding nonvanishing of\nLittlewood-Richardson\ncoefficients\nScribe: Hariharan Narayanan\nGoal:\nTo show that nonvanishing of Littlewood-Richardson coefficients\ncan be decided in polynomial time.\nReferences: [DM2, GCT3, KT2]\n6.1\nLittlewood-Richardson coefficients\nFirst we define Littlewood-Richardson coefficients, which are basic quanti-\nties encountered in representation theory. Recall that the irreducible rep-\nresentations Vλ of GLn(C), the Weyl modules, are indexed by partitions λ,\nand:\nTheorem 6.1 (Weyl). Every finite dimensional representation of GLn(C)\nis completely reducible.\nLet G = GLn(C). Consider the diagonal embedding of G ֒→G × G.\nThis is a group homomorphism. Any G × G module, in particular, Vα ⊗Vβ\ncan also be viewed as a G module via this homomorphism. It then splits\ninto irreducible G-submodules:\n41"},{"paragraph_id":"p56","order":56,"text":"Vα ⊗Vβ = ⊕γcγ\nαβVλ.\n(6.1)\nHere cγ\nαβ is the multiplicity of Vγ in Vα ⊗Vβ and is known as the Littlewood-\nRichardson coefficient.\nThe character of Vλ is the Schur polynomial Sλ. Hence, it follows from\n(6.1) that the Schur polynomials satisfy the following relation:\nSαSβ = ⊕γcγ\nαβSλ.\n(6.2)\nTheorem 6.2. cγ\nαβ is in PSPACE.\nProof: This easily follows from eq.(6.2) and the definition of Schur poly-\nnomials.\nAs a matter of fact, a stronger result holds:\nTheorem 6.3. cγ\nαβ is in #P.\nRecall that #P ⫅PSPACE.\nProof:\nThis is an immediate consequence of the following Littlewood-\nRichardson rule (formula) for cγ\nαβ. To state it, we need a few definitions.\nGiven partitions γ and α, a skew Young diagram of shape γ/α is the dif-\nference between the Young diagrams for γ and α, with their top-left corners\naligned; cf. Figure 6.2. A skew tableau of shape γ/α is a numbering of the\nboxes in this diagram. It is called semi-standard (SST) if the entries in each\ncolumn are strictly increasing top to bottom and the entries in each row\nare weakly increasing left to right; see Figures 6.1 and 6.2. The row word\nrow(T) of a skew-tableau T is the sequence of numbers obtained by reading\nT left to right, bottom to top; e.g. row(T) for Figure 6.2 is 13312211. It is\ncalled a reverse lattice word, if when read right to left, for each i, the number\nof i’s encountered at any point is at least the number of i + 1’s encountered\ntill that point; thus the row word for Figure 6.2 is a reverse lattice word.\nWe say that T is an LR tableau for given α, β, γ of shape γ/α and content\nβ if\n1. T is an SST,\n2. row(T) is a reverse lattice word,\n3. T has shape γ/α, and\n4. the content of T is β, i. e. the number of i’s in T is βi.\n42"},{"paragraph_id":"p57","order":57,"text":"1 1 2 5\n2 2 3\n4\nFigure 6.1: semi-standard Young tableau\n:\n:\n:\n:\n:\n: 1 1\n:\n:\n: 1 2 2\n:\n: 3\n1 3\nFigure 6.2: Littlewood–Richardson skew tableau\nFor example, Figure 6.2 shows an LR tableau with α = (6, 3, 2), β =\n(4, 2, 2) and γ = (8, 6, 3, 2).\nThe Littlewood-Richardson rule [F, FH]: cγ\nαβ is equal to the number\nof LR skew tableaux of shape γ/α and content β.\nRemark: It may be noticed that the Littlewood-Richardson rule depends\nonly on the partitions α, β and γ and not on n, the rank of GLn(C) (as long\nas it is greater than or equal to the maximum height of α, β or γ). For this\nreason, we can assume without loss of generalitity that n is the maximum\nof the heights of α, β and γ, as we shall henceforth.\nNow we express cγ\nαβ as the number of integer points in some polytope\nP γ\nαβ using the Littlewood-Richardson rule:\nLemma 6.1. There exists a polytope P = P γ\nαβ of dimension polynomial in\nn such that the number of integer points in it is cγ\nαβ.\nProof: Let ri\nj(T), i ≤n, j ≤n, denote the number of j’s in the i-th row of\nT. If T is an LR-tableau of shape γ/α with content β then these integers\nsatisfy the following constraints:\n1. Nonnegativity: ri\nj ≥0.\n2. Shape constraints: For i ≤n,\nαi +\nX\nj\nri\nj = γi.\n43"},{"paragraph_id":"p58","order":58,"text":"3. Content constraints: For j ≤n:\nX\ni\nri\nj = βj.\n4. Tableau constraints:\nαi+1 +\nX\nk≤j\nri+1\nk\n≤αi +\nX\nk′<j\nri\nk′.\n5. Reverse lattice word constraints: ri\nj = 0 for i < j, and for i ≤n,\n1 < j ≤n:\nX\ni′≤i\nri′\nj ≤\nX\ni′<i\nri′\nj−1.\nLet P γ\nαβ be the polytope defined by these constraints. Then cγ\nαβ is the\nnumber of integer points in this polytope. This proves the lemma.\n□\nThe membership function of the polytope P γ\nαβ is clearly computable in\ntime that is polynomial in the bitlengths of α, β and γ. Hence cγ\nαβ belongs\nto #P. This proves the theorem.\n□\nThe complexity-theoretic content of the Littlewood-Richardson rule is\nthat it puts a quantity, which is a priori only in PSPACE, in #P. We also\nhave:\nTheorem 6.4 ([H]). cγ\nαβ is #P-complete.\nFinally, the main complexity-theoretic result that we are interested in:\nTheorem 6.5 (GCT3, Knutson-Tao, De Loera-McAllister). The problem\nof deciding nonvanishing of cγ\nαβ is in P, i. e. , it can be solved in time that is\npolynomial in the bitlengths of α, β and γ. In fact, it can solved in strongly\npolynomial time [GCT3].\nHere, by a strongly polynomial time algorithm, we mean that the num-\nber of arithmetic steps +, −, ∗, ≤, . . . in the algorithm is polynomial in the\nnumber of parts of α, β and γ regardless of their bitlengths, and the bit-\nlength of each intermediate operand is polynomial in the bitlengths of α, β\nand γ.\nProof: Let P = P γ\nαβ be the polytope as in Lemma 6.1.\nAll vertices of\nP have rational coefficients. Hence, for some positive integer q, the scaled\npolytope qP has an integer point. It follows that, for this q, cqγ\nqα,qβ is positive.\nThe saturation Theorem [KT] says that, in this case, cγ\nα,β is positive. Hence,\nP contains an integer point. This implies:\n44"},{"paragraph_id":"p59","order":59,"text":"Lemma 6.2. If P ̸= ∅then cγ\nαβ > 0.\nBy this lemma, to decide if cγ\nαβ > 0, it suffices to test if P is nonempty.\nThe polytope P is given by Ax ≤b where the entries of A are 0 or 1–\nsuch linear programs are called combinatorial. Hence, this can be done in\nstrongly polynomial time using Tardos’ algorithm [GLS] for combinatorial\nlinear programming. This proves the theorem.\n□\nThe integer programming problem is NP-complete, in general. However,\nlinear programming works for the specific integer programming problem here\nbecause of the saturation property [KT].\nProblem: Find a genuinely combinatorial poly-time algorithm for deciding\nnon-vanishing of cγ\nαβ.\n45"},{"paragraph_id":"p60","order":60,"text":"Chapter 7\nLittlewood-Richardson\ncoefficients (cont)\nScribe: Paolo Codenotti\nGoal: We continue our study of Littlewood-Richardson coefficients and\ndefine Littlewood-Richardson coefficients for the orthogonal group On(C).\nReferences: [FH, F]\nRecall\nLet us first recall some definitions and results from the last class. Let cγ\nα,β\ndenote the Littlewood-Richardson coefficient for GLn(C).\nTheorem 7.1 (last class). Non-vanishing of cγ\nα,β can be decided in poly(⟨α⟩, ⟨β⟩, ⟨γ⟩)\ntime, where ⟨⟩denotes the bit length.\nThe positivity hypotheses which hold here are:\n• cγ\nα,β ∈#P, and more strongly,\n• Positivity Hypothesis 1 (PH1): There exists a polytope P γ\nα,β of\ndimension polynomial in the heights of α, β and γ such that cγ\nα,β =\nφ(P γ\nα,β), where φ indicates the number of integer points.\n• Saturation Hypothesis (SH): If ckγ\nkα,kβ ̸= 0 for some k ≥1, then\ncγ\nα,β ̸= 0 [Saturation Theorem].\nProof. (of theorem)\nPH1 + SH + Linear programming.\n46"},{"paragraph_id":"p61","order":61,"text":"This is the general form of algorithms in GCT. The main principle is\nthat linear programming works for integer programming when PH1 and SH\nhold.\n7.1\nThe stretching function\nWe define ecγ\nα,β(k) = ckγ\nkα,kβ.\nTheorem 7.2 (Kirillov, Derkesen Weyman [Der, Ki]). ecγ\nα,β(k) is a polyno-\nmial in k.\nHere we prove a weaker result. For its statement, we will quickly review\nthe theory of Ehrhart quasipolynomials (cf. Stanley [S]).\nDefinition 7.1. (Quasipolynomial) A function f(k) is called a quasipoly-\nnomial if there exist polynomials fi, 1 ≤i ≤l, for some lsuch that\nf(k) = fi(k) if k ≡i mod l.\nWe denote such a quasipolynomial f by f = (fi). Here lis called the period\nof f(k) (we can assume it is the smallest such period).\nThe degree of a\nquasipolynomial f is the max of the degrees of the fi’s.\nNow let P ⊆Rm be a polytope given by Ax ≤b. Let φ(P) be the number\nof integer points inside P. We define the stretching function fP(k) = φ(kP),\nwhere kP is the dilated polytope defined by Ax ≤kb.\nTheorem 7.3. (Ehrhart) The stretching function fP(k) is a quasipolyno-\nmial. Furthermore, fP (k) is a polynomial if P is an integral polytope (i.e.\nall vertices of P are integral).\nIn view of this result, fP(k) is called the Ehrhart quasi-polynomial of\nP. Now ecγ\nα,β(k) is just the Ehrhart quasipolynomial of P γ\nα,β, and cγ\nα,β =\nφ(P γ\nα,β), the number of integer points in P γ\nα,β. Moreover P γ\nα,β is defined by\nthe inequality Ax ≤b, where A is constant, and b is a homogeneous linear\nform in the coefficients of α, β, and γ.\nHowever, P γ\nα,β need not be integral. Therefore Theorem (7.2) does not\nfollow from Ehrhart’s result. Its proof needs representation theory.\nDefinition 7.2. A quasipolynomial f(k) is said to be positive if all the\ncoefficients of fi(k) are nonnegative. In particular, if f(k) is a polynomial,\nthen it’s positive if all its coefficients are nonnegative.\n47"},{"paragraph_id":"p62","order":62,"text":"The Ehrhart quasipolynomial of a polytope is positive only in exceptional\ncases. In this context:\nPH2 (positivity hypothesis 2) [KTT]: The polynomial ecγ\nα,β(k) is positive.\nThere is considerable computer evidence for this.\nProposition 7.1. PH2 implies SH.\nProof. Look at:\nc(k) = ecγ\nα,β(k) =\nX\naiki.\nIf all the coefficients ai are nonnegative (by PH2), and c(k) ̸= 0, then c(1) ̸=\n0.\nSH has a proof involving algebraic geometry [B]. Therefore we suspect\nthat the stronger PH2 is a deep phenomenon related to algebraic geometry.\n7.2\nOn(C)\nSo far we have talked about GLn(C).\nNow we move on to the orthogo-\nnal group On(C). Fix Q, a symmetric bilinear form on Cn; for example,\nQ(V, W) = V T W.\nDefinition 7.3. The orthogonal group On(C) ⊆GLn(C) is the group con-\nsisting of all A ∈GLn(C) s.t.\nQ(AV, AW) = Q(V, W) for all V and\nW ∈Cn. The subgroup SOn(C) ⊆SLn(C), where SLn(C) is the set of\nmatrices with determinant 1, is defined similarly.\nTheorem 7.4 (Weyl). The group On(C) is reductive\nProof. The proof is similar to the reductivity of GLn(C), based on Weyl’s\nunitary trick.\nThe next step is to classify all irreducible polynomial representations of\nOn(C).\nFix a partition λ = (λ1 ≥λ2 ≥. . . ) of length at most n.\nLet\n|λ| = d = P λi be its size. Let V = Cn, V ⊗d = V ⊗· · · ⊗V d times, and\nembed the Weyl module Vλ of GLn(C) in V ⊗d as per Theorem 5.3. Define\na contraction map\nφp,q : V ⊗d →V ⊗(d−2)\nfor 1 ≤p ≤q ≤d by:\nφp,q(vi1 ⊗· · · ⊗vid) = Q(vip, viq)(vi1 ⊗· · · ⊗c\nvip ⊗· · · ⊗c\nviq ⊗· · · ⊗vid),\n48"},{"paragraph_id":"p63","order":63,"text":"λ\nFigure 7.1: The first two columns of the partition λ are highlighted.\nwhere c\nvip means omit vip.\nIt is On(C)-equivariant, i.e. the following diagram commutes:\nV ⊗d\nφp,q\n−−−−→V ⊗d−2\n yσ∈On(C)\n yσ∈On(C)\nV ⊗d\nφp,q\n−−−−→V ⊗d−2\nLet\nV [d] =\n\\\npq\nker(φp,q).\nBecause the maps are equivariant, each kernel is an On(C)-module, and V [d]\nis an On(C)-module. Let V[λ] = V [d] T Vλ, where Vλ ⊆V ⊗d is the embedded\nWeyl module as above. Then V[λ] is an On(C)-module.\nTheorem 7.5 (Weyl). V[λ] is an irreducible representation of On(C). More-\nover, the following two conditions hold:\n1. If n is odd, then V[λ] is non-zero if and only if the sum of the lengths\nof the first two columns of λ is ≤n (see figure 7.1).\n2. If n is odd, then each polynomial irreducible representation is isomor-\nphic to V[λ] for some λ.\nLet\nV[λ] ⊗V[μ] = ⊗γdγ\nλ,μV[γ]\n49"},{"paragraph_id":"p64","order":64,"text":"be the decomposition of V[λ]⊗V[μ] into irreducibles. Here dγ\nλ,μ is called the\nLittlewood-Richardson coefficient of type B. The types of various connected\nreductive groups are defined as follows:\n• GLn(C): type A\n• On(C), n odd: type B\n• Spn(C): type C\n• On(C), n even: type D\nThe Littlewood-Richardson coefficient can be defined for any type in a sim-\nilar fashion.\nTheorem 7.6 (Generalized Littlewood-Richardson rule). The Littlewood-\nRichardson coefficient dγ\nλ,μ ∈#P. This also holds for any type.\nProof. The most transparent proof of this theorem comes through the theory\nof quantum groups [K]; cf. Chapter 20.\nAs in type A this leads to:\nHypothesis 7.1 (PH1). There exists a polytope P γ\nλ,μ of dimension polyno-\nmial in the heights of λ, μ and γ such that:\n1. dγ\nλ,μ = φ(P γ\nλ,μ), the number of integer points in P γ\nλ,μ, and\n2. edγ\nλ,μ(k) = dkγ\nkλ,kμ is the Ehrhart quasipolynomial of P γ\nλ,μ.\nThere are several choices for such polytopes; e.g. the BZ-polytope [BZ].\nTheorem 7.7 (De Loera, McAllister [DM2]). The stretching function edγ\nλ,μ(k)\nis a quasipolynomial of degree at most 2; so also for types C and D.\nA verbatim translation of the saturation property fails here [Z]): there\nexist λ, μ and γ such that d2γ\n2λ,2μ ̸= 0 but dγ\nλ,μ = 0. Therefore we change the\ndefinition of saturation:\nDefinition 7.4. Given a quasipolynomial f(k) = (fi), index(f) is the\nsmallest i such that fi(k) is not an identically zero polynomial.\nIf f(k)\nis identically zero, index(f) = 0.\nDefinition 7.5. A quasipolynomial f(k) is saturated if f(index(f)) ̸= 0.\nIn particular, if index(f) = 1, then f(k) is saturated if f(1) ̸= 0.\n50"},{"paragraph_id":"p65","order":65,"text":"A positive quasi-polynomial is clearly saturated.\nPositivity Hypothesis 2 (PH2) [DM2]: The stretching quasipolyomial\nedγ\nλ,μ(k) is positive.\nThere is considerable evidence for this.\nSaturation Hypothesis (SH): The stretching quasipolynomial edγ\nλ,μ(k) is\nsaturated.\nPH2 implies SH.\nTheorem 7.8. [GCT5] Assuming SH (or PH2), positivity of the Littlewood-\nRichardson coefficient dγ\nλ,μ of type B can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨γ⟩)\ntime.\nThis is also true for all types.\nProof. next class.\n51"},{"paragraph_id":"p66","order":66,"text":"Chapter 8\nDeciding nonvanishing of\nLittlewood-Richardson\ncoefficients for On(C)\nScribe: Hariharan Narayanan\nGoal: A polynomial time algorithm for deciding nonvanishing of Littlewood-\nRichardson coefficients for the orthogonal group assuming SH.\nReference: [GCT5]\nLet dν\nλ,μ denote the Littlewood-Richardson coefficient of type B (i.e. for\nthe orthogonal group On(C), n odd) as defined in the earlier lecture. In this\nlecture we describe a polynomial time algorithm for deciding nonvanishing\nof dν\nλ,μ assuming the following positivity hypothesis PH2.\nSimilar result\nalso holds for all types, though we shall only concentrate on type B in this\nlecture.\nLet ̃dν\nλ,μ(k) = dkν\nkλ,kμ denote the associated stretching function.\nIt is\nknown to be a quasi-polynomial of period at most two [DM2]. This means\nthere are polynomials f1(k) and f2(k) such that\ndkν\nkλ,kμ =\n f1(k),\nif k is odd;\nf2(k),\nif k is even.\nPositivity Hypothesis (PH2) [DM2]: The stretching quasi-polynomial\n ̃dν\nλ,μ(k) is positive.\nThis means the coefficients of f1 and f2 are all non-\nnegative.\nThe main result in this lecture is:\n52"},{"paragraph_id":"p67","order":67,"text":"Theorem 8.1. [GCT5] If PH2 holds, then the problem of deciding the posi-\ntivity (nonvanishing) of dν\nλμ belongs to P. That is, this problem can be solved\nin time polynomial in the bitlengths of λ, μ and ν.\nWe need a few lemmas for the proof.\nLemma 8.1. If PH2 holds, the following are equivalent:\n(1) dν\nλμ ≥1.\n(2) There exists an odd integer k such that dkν\nkλ kμ ≥1.\nProof: Clearly (1) implies (2). By PH2, there exists a polynomial f1\nwith non-negative coefficients such that\n∀odd k, f1(k) = dkν\nkλ kμ.\nSuppose that for some odd k, dkν\nkλ kμ ≥1. Then f1(k) ≥1. Therefore f1 has\nat least one non-zero coefficient. Since all coefficients of f1 are nonnegative,\ndν\nλμ = f1(1) > 0. Since dν\nλμ is an integer, (1) follows.\n□\nDefinition 8.1. Let Z<2> be the subring of Q obtained by localizing Z at\n2:\nZ<2> :=\n p\nq | p, q −1\n2\n∈Z"},{"paragraph_id":"p68","order":68,"text":".\nThis ring consists of all fractions whose denominators are odd.\nLemma 8.2. Let P ∈Rd be a convex polytope specified by Ax ≤B, xi ≥0\nfor all i, where A and B are integral. Let Aff(P) denote its affine span. The\nfollowing are equivalent:\n(1) P contains a point in Zd\n<2>.\n(2) Aff(P) contains a point in Zd\n<2>.\nProof: Since P ⊆Aff(P), (1) implies (2). Now suppose (2) holds. We\nhave to show (1). Let z ∈Zd\n<2> ∩Aff(P).\nFirst, consider the case when Aff(P) is one dimensional. In this case, P\nis the line segment joining two points x and y in Qd. The point z can be\nexpressed as an affine linear combination, z = ax + (1−a)y for some a ∈Q.\nThere exists q ∈Z such that qx ∈Zd\n<2> and qy ∈Zd\n<2>. Note that\n{z + λ(qx −qy) | λ ∈Z<2>} ⊆Aff(P) ∩Zd\n<2>.\n53"},{"paragraph_id":"p69","order":69,"text":"Since Z<2> is a dense subset of Q, the l.h.s. and hence the r.h.s. is a dense\nsubset of Aff(P). Consequently, P ∩Zd\n<2> ̸= ∅.\nNow consider the general case. Let u be any point in the interior of P\nwith rational coordinates, and L the line through u and z. By restricting to\nL, the lemma reduces to the preceding one dimensional case.\n□\nLemma 8.3. Let\nP = {x | Ax ≤B, (∀i)xi ≥0} ⊆Rd\nbe a convex polytope where A and B are integral. Then, it is possible to\ndetermine in polynomial time whether or not Aff(P) ∩Zd\n<2> = ∅.\nProof: Using Linear Programming [Kha79, Kar84], a presentation of\nthe form Cx = D can be obtained for Aff(P) in polynomial time, where C\nis an integer matrix and D is a vector with integer coordinates. We may\nassume that C is square since this can be achieved by padding it with 0’s\nif necessary, and extending D. The Smith Normal Form over Z of C is a\nmatrix S such that C = USV where U and V are unimodular and S has\nthe form"},{"paragraph_id":"p70","order":70,"text":"s11\n0\n. . .\n0\n0\ns22\n. . .\n0\n...\n...\n...\n0\n0\n0\n. . .\nsdd"},{"paragraph_id":"p71","order":71,"text":"where for 1 ≤i ≤d−1, sii divides si+1 i+1. It can be computed in polynomial\ntime [KB79].\nThe question now reduces to whether USV x = D has a\nsolution x ∈Zd\n<2>. Since V is unimodular, its inverse has integer entries\ntoo, and y := V x ∈Zd\n<2> ⇔x ∈Zd\n<2>.\nThis is equivalent to whether\nSy = U−1D has a solution y ∈Zd\n<2>.\nSince S is diagonal, this can be\nanswered in polynomial time simply by checking each coordinate.\n□\nProof of Theorem 8.1: By [BZ], there exists a polytope P = P ν\nλ,μ\nsuch that the Littlewood-Richardson coefficient dν\nλμ is equal to the number\nof integer points in P. This polytope is such that the number of integer\npoints in the dilated polytope kP is dkν\nkλ kμ. Assuming PH2, we know from\nLemma 8.1 that\nP ∩Zd ̸= ∅⇔(∃odd k), kP ∩Zd ̸= ∅.\nThe latter is equivalent to\nP ∩Zd\n<2> ̸= ∅.\n54"},{"paragraph_id":"p72","order":72,"text":"The theorem now follows from Lemma 8.2 and Lemma 8.3.\n□\nIn combinatorial optimization, LP works if the polytope is integral. In\nour setting, this is not necessarily the case [DM1]: the denominators of the\ncoordinates of the vertices of P can be Ω(l), where l is the total height of\nλ, μ and ν. LP works here nevertheless because of PH2; it can be checked\nthat SH is also sufficient.\n55"},{"paragraph_id":"p73","order":73,"text":"Chapter 9\nThe plethysm problem\nScribe: Joshua A. Grochow\nGoal: In this lecture we describe the general plethysm problem, state anal-\nogous positivity and saturation hypotheses for it, and state the results from\nGCT 6 which imply a polynomial time algorithm for deciding positivity of\na plethysm constant assuming these hypotheses.\nReference: [GCT6]\nRecall\nRecall that a function f(k) is quasipolynomial if there are functions fi(k)\nfor i = 1, . . . , lsuch that f(k) = fi(k) whenever k ≡i mod l. The number\nlis then the period of f. The index of f is the least i such that fi(k) is\nnot identically zero.\nIf f is identically zero, then the index of f is zero\nby convention. We say f is positive if all the coefficients of each fi(k) are\nnonnegative. We say f is saturated if f(index(f)) ̸= 0. If f is positive, then\nit is saturated.\nGiven any function f(k), we associate to it the rational series F(t) =\nP\nk≥0 f(k)tk.\nProposition 9.1. [S] The following are equivalent:\n1. f(k) is a quasipolynomial of period l.\n2. F(t) is a rational function of the form A(t)\nB(t) where deg A < deg B and\nevery root of B(t) is an l-th root of unity.\n56"},{"paragraph_id":"p74","order":74,"text":"9.1\nLittlewood-Richardson Problem [GCT 3,5]\nLet G = GLn(C) and cγ\nα,β the Littlewood-Richardson coefficient – i.e. the\nmultiplicity of the Weyl module Vγ in Vα ⊗Vβ. We saw that the positivity\nof cγ\nα,β can be decided in poly(⟨α⟩, ⟨β⟩, ⟨γ⟩) time, where ⟨·⟩denotes the bit-\nlength. Furthermore, we saw that the stretching function ecγ\nα,β(k) = ckγ\nkα,kβ\nis a polynomial, and the analogous stretching function for type B is a\nquasipolynomial of period at most 2.\n9.2\nKronecker Problem [GCT 4,6]\nNow we study the analogous problem for the representations of the symmet-\nric group (the Specht modules), called the Kronecker problem.\nLet Sα be the Specht module of the symmetric group Sm associated to\nthe partition α. Define the Kronecker coefficient κπ\nλ,μ to be the multiplicity\nof Sπ in Sλ ⊗Sμ (considered as an Sm-module via the diagonal action). In\nother words, write Sλ ⊗Sμ = L\nπ κπ\nλ,μSπ. We have κπ\nλ,μ = (χλχμ, χπ), where\nχλ denotes the character of Sλ. By the Frobenius character formula, this\ncan be computed in PSPACE. More strongly, analogous to the Littlewood-\nRichardson problem:\nConjecture 9.1. [GCT4, GCT6] The Kronecker coefficient κπ\nλ,μ ∈# P.\nIn other words, there is a positive #P-formula for κπ\nλ,μ.\nThis is a fundamental problem in representation theory. More concretely,\nit can be phrased as asking for a set of combinatorial objects I and a char-\nacteristic function χ : {I} →{0, 1} such that χ ∈FP and κπ\nλ,μ = P\nI χ(I).\nContinuing our analogy:\nConjecture 9.2. [GCT6] The problem of deciding positivity of κπ\nλ,μ belongs\nto P.\nTheorem 9.1. [GCT6] The stretching function eκπ\nλ,μ(k) = κkπ\nkλ,kμ is a quasipoly-\nnomial.\nNote that κkπ\nkλ,kμ is a Kronecker coefficient for Skm.\nThere is also a dual definition of the Kronecker coefficients. Namely,\nconsider the embedding\nH = GLn(C) × GLn(C) ֒→G = GL(Cn ⊗Cn),\nwhere (g, h)(v ⊗w) = (gv ⊗hw). Then\n57"},{"paragraph_id":"p75","order":75,"text":"Proposition 9.2. [FH] The Kronecker coefficient κπ\nλ,μ is the multiplicity of\nthe tensor product of Weyl modules Vλ(GLn(C)) ⊗Vμ(GLn(C)) (this is an\nirreducible H-module) in the Weyl module Vπ(G) considered as an H-module\nvia the embedding above.\n9.3\nPlethysm Problem [GCT 6,7]\nNext we consider the more general plethysm problem.\nLet H = GLn(C), V = Vμ(H) the Weyl module of H corresponding to\na partition μ, and ρ : H →G = GL(V ) the corresponding representation\nmap. Then the Weyl module Vλ(G) of G for a given partition λ can be\nconsidered an H-module via the map ρ. By complete reducibility, we may\ndecompose this H-representation as\nVλ(G) =\nM\nπ\naπ\nλ,μVπ(H).\nThe coefficients aπ\nλ,μ are known as plethsym constants (this definition can\neasily be generalized to any reductive group H). The Kronecker coefficient\nis a special case of the plethsym constant [Ki].\nTheorem 9.2 (GCT 6). The plethysm constant aπ\nλ,μ ∈PSPACE.\nThis is based on a parallel algorithm to compute the plethysm constant\nusing Weyl’s character formula. Continuing in our previous trend:\nConjecture 9.3. [GCT6] aπ\nλ,μ ∈# P and the problem of deciding positivity\nof aπ\nλ,μ belongs to P.\nFor the stretching function, we need to be a bit careful. Define eaπ\nλ,μ =\nakπ\nkλ,μ. Here the subscript μ is not stretched, since that would change G,\nwhile stretching λ and π only alters the representations of G.\nAs in the beginning of the lecture, we can associate a function Aπ\nλ,μ(t) =\nP\nk≥0 eaπ\nλ,μ(k)tk to the plethysm constant. Kirillov conjectured that Aπ\nλ,μ(t) is\nrational. In view of Proposition 9.1, this follows from the following stronger\nresult:\nTheorem 9.3 (GCT 6). The stretching function eaπ\nλ,μ(k) is a quasipolyno-\nmial.\nThis is the main result of GCT 6, which in some sense allows GCT to\ngo forward.\nWithout it, there would be little hope for proving that the\n58"},{"paragraph_id":"p76","order":76,"text":"positivity of plethysm constants can be decided in polynomial time.\nIts\nproof is essentially algebro-geometric. The basic idea is to show that the\nstretching function is the Hilbert function of some algebraic variety with nice\n(i.e. “rational”) singularities. Similar results are shown for the stretching\nfunctions in the algebro-geometric problems arising in GCT.\nThe main complexity-theoretic result in [GCT6] shows that, under the\nfollowing positivity and saturation hypotheses (for which there is much ex-\nperimental evidence), the positivity of the plethysm constants can indeed\nbe decided in polynomial time (cf. Conjecture 9.3).\nThe first positivity hypothesis is suggested by Theorem 9.3: since the\nstretching function is a quasipolynomial, we may suspect that it is captured\nby some polytope:\nPositivity Hypothesis 1 (PH1). There exists a polytope P = P π\nλ,μ\nsuch that:\n1. aπ\nλ,μ = φ(P), where φ denotes the number of integer points inside the\npolytope,\n2. The stretching quasipolynomial (cf. Thm. 9.3) eaπ\nλ,μ(k) is equal to the\nEhrhart quasipolynomial fP(k) of P,\n3. The dimension of P is polynomial in ⟨λ⟩, ⟨μ⟩, and ⟨π⟩,\n4. the membership in P π\nλ,μ can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) time, and\nthere is a polynomial time separation oracle [GLS] for P.\nHere (4) does not imply that the polytope P has only polynomially\nmany constraints. In fact, in the plethysm problem there may be a super-\npolynomial number of constraints.\nPositivity Hypothesis 2 (PH2).\nThe stretching quasipolynomial\neaπ\nλ,μ(k) is positive.\nThis implies:\nSaturation Hypothesis (SH). The stretching quasipolynomial is sat-\nurated.\nTheorem 9.3 is essential to state these hypotheses, since positivity and\nsaturation are properties that only apply to quasipolynomials. Evidence for\nPH1, PH2, and SH can be found in GCT 6.\nTheorem 9.4. [GCT6] Assuming PH1 and SH (or PH2), positivity of the\nplethysm constant aπ\nλ,μ can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) time.\n59"},{"paragraph_id":"p77","order":77,"text":"This follows from the polynomial time algorithm for saturated integer\nprogramming described in the next class. As with Theorem 9.3, this also\nholds for more general problems in algebraic geometry.\n60"},{"paragraph_id":"p78","order":78,"text":"Chapter 10\nSaturated and positive\ninteger programming\nScribe: Sourav Chakraborty\nGoal :\nA polynomial time algorithm for saturated integer programming\nand its application to the plethysm problem.\nReference: [GCT6]\nNotation :\nIn this class we denote by ⟨a⟩the bit-length of the a.\n10.1\nSaturated, positive integer programming\nLet Ax ≤b be a set of inequalities.\nThe number of constraints can be\nexponential. Let P ⊂Rn be the polytope defined by these inequalities. The\nbit length of P is defined to be ⟨P⟩= n + ψ, where ψ is the maximum\nbit-length of a constraint in the set of inequalities. We assume that P is\ngiven by a separating oracle. This means membership in P can be decided\nin poly(⟨P⟩) time, and if x ̸∈P then a separating hyperplane is given as a\nproof as in [GLS].\nLet fP(k) be the Ehrhart quasi-polynomial of P. Quasi-polynomiality\nmeans there exist polynomials fi(k), 1 ≤i ≤l, l the period, so that fP(k) =\nfi(k) if k = i modulo l. Then\nIndex(fP) = min{i|fi(k)not identically 0 as a polynomial}\nThe integer programming problem is called positive if fP(k) is positive\nwhenever P is non-empty, and saturated if fP(k) is saturated whenever P is\nnon-empty.\n61"},{"paragraph_id":"p79","order":79,"text":"Theorem 10.1 (GCT6).\n1. Index(fP ) can be computed in time polyno-\nmial in the bit length ⟨P⟩of P assuming that the separation oracle\nworks in poly-⟨P⟩-time.\n2. Saturated and hence positive integer programming problem can be solved\nin poly-⟨P⟩-time.\nThe second statement follow from the first.\nProof. Let Aff(P) denote the affine span of P. By [GLS] we can compute\nthe specifications Cx = d, C and d integral, of Aff(P) in poly(⟨P⟩) time.\nWithout loss of generality, by padding, we can assume that C is square. By\n[KB79] we find the Smith-normal form of C in polynomial time. Let it be\n ̄C. So,\n ̄C = ACB\nwhere A and B are unimodular, and ̄C is a diagonal matrix, where the\ndiagonal entries c1, c2, . . . are such that with ci|ci+1.\nClearly Cx = d iff ̄Cz = ̄d where z = B−1x and ̄d = Ad.\nSo all equations here are of form\n ̄cizi = di\n(10.1)\nWithout loss of generality we can assume that ci and di are relatively\nprime. Let ̃c = lcm(ci).\nClaim 10.1. Index(fP ) = ̃c.\nFrom this claim the theorem clearly follows.\nProof of the claim. Let fP(t) = P\nk≥0 fP(k)tk be the Ehrhart Series of P.\nNow kP will not have an integer point unless ̃c divides k because of\n(10.1).\nHence fP(t) = f ̄P(t ̃c) where ̄P is the stretched polytope ̃cP and f ̄P(s) is\nthe Ehrhart series of ̄P. From this it follows that\nIndex(fP ) = ̃cIndex(f ̄P )\nNow we show that Index(f ̄P) = 1.\nThe equations of ̄P are of the form\nzi = ̃c\nci\ndi\n62"},{"paragraph_id":"p80","order":80,"text":"where each\n ̃c\nci is an integer.\nTherefore without loss of generality we can\nignore these equations and assume the ̄P is full dimensional.\nThen it suffices to show that ̄P contains a rational point whose denomi-\nnators are all 1 modulo l( ̄P), the period of the quasi-polynomial f ̄P (s).\nThis follows from a simple density argument that we saw earlier (cf. the\nproof of Lemma 8.2).\nFrom this the claim follows.\n10.2\nApplication to the plethysm problem\nNow we can prove the result stated in the last class:\nTheorem 10.2. Assuming PH1 and SH, positivity of the plethysm constant\naπ\nλ,μ can be decided in time polynomial in ⟨λ⟩, ⟨μ⟩and ⟨π⟩.\nProof. Let P = P π\nλ,μ be the polytope as in PH1 such that aπ\nλ,μ is the number\nof integer points in P. The goal is to decide if P contains an integer point.\nThis integer programming problem is saturated because of SH. Hence the\nresult follows from Theorem 10.1.\n63"},{"paragraph_id":"p81","order":81,"text":"Chapter 11\nBasic algebraic geometry\nScribe: Paolo Codenotti\nGoal: So far we have focussed on purely representation-theoretic aspects of\nGCT. Now we have to bring in algebraic geometry. In this lecture we review\nthe basic definitions and results in algebraic geometry that will be needed\nfor this purpose. The proofs will be omitted or only sketched. For details,\nsee the books by Mumford [Mm] and Fulton [F].\n11.1\nAlgebraic geometry definitions\nLet V = Cn, and v1, . . . , vn the coordinates of V .\nDefinition 11.1.\n• Y is an affine algebraic set in V if Y is the set of\nsimultaneous zeros of a set of polynomials in vi’s.\n• An algebraic set that cannot be written as the union of two proper\nalgebraic sets Y1 and Y2 is called irreducible.\n• An irreducible affine algebraic set is called an affine variety.\n• The ideal of an affine algebraic set Y is I(Y ), the set of all polynomials\nthat vanish on Y .\nFor example, Y = (v1 −v2\n2 + v3, v2\n3 −v2 + 4v1) is an irreducible affine\nalgebraic set (and therefore an affine variety).\nTheorem 11.1 (Hilbert). I(Y ) is finitely generated, i.e. there exist poly-\nnomials g1, . . . , gk that generate I(Y ). This means every f ∈I(Y ) can be\nwritten as f = P figi for some polynomials fi.\n64"},{"paragraph_id":"p82","order":82,"text":"Let C[V ], the coordinate ring of V , be the ring of polynomials over\nthe variables v1, . . . , vn. The coordinate ring of Y is defined to be C[Y ] =\nC[V ]/I(Y ). It is the set of polynomial functions over Y .\nDefinition 11.2.\n• P(V ) is the projective space associated with V , i.e.\nthe set of lines through the origin in V .\n• V is called the cone of P(V ).\n• C[V ] is called the homogeneous coordinate ring of P(V ).\n• Y ⊆P(V ) is a projective algebraic set if it is the set of simultaneous\nzeros of a set of homogeneous forms (polynomials) in the variables\nv1, . . . , vn. It is necessary that the polynomials be homogeneous because\na point in P(V ) is a line in V .\n• A projective algebraic set Y is irreducible if it can not be expressed as\nthe union of two proper algebraic sets in P(V ).\n• An irreducible projective algebraic set is called a projective variety.\nLet Y ⊆P(V ) be a projective variety, and define I(Y ), the ideal of Y\nto be the set of all homogeneous forms that vanish on Y . Hilbert’s result\nimplies that I(Y ) is finitely generated.\nDefinition 11.3. The cone C(Y ) ⊆V of a projective variety Y ⊆P(V ) is\ndefined to be the set of all points on the lines in Y .\nDefinition 11.4. We define the homogeneous coordinate ring of Y as\nR(Y ) = C[V ]/I(Y ), the set of homogeneous polynomial forms on the cone\nof Y .\nDefinition 11.5. A Zariski open subset of Y is the complement of a pro-\njective algebraic subset of Y . It is called a quasi-projective variety.\nLet G = GLn(C), and V a finite dimensional representation of G. Then\nC[V ] is a G-module, with the action of σ ∈G defined by:\n(σ · f)(v) = f(σ−1v), v ∈V.\nDefinition 11.6. Let Y ⊆P(V ) be a projective variety with ideal I(Y ). We\nsay that Y is a G-variety if I(Y ) is a G-module, i.e., I(Y ) is a G-submodule\nof C[V ].\n65"},{"paragraph_id":"p83","order":83,"text":"If Y is a projective variety, then R(Y ) = C[V ]/I(Y ) is also a G-module.\nTherefore Y is G-invariant, i.e.\ny ∈Y ⇒σy ∈Y, ∀σ ∈G.\nThe algebraic geometry of G-varieties is called geometric invariant theory\n(GIT).\n11.2\nOrbit closures\nWe now define special classes of G-varieties called orbit closures. Let v ∈\nP(V ) be a point, and Gv the orbit of v:\nGv = {gv|g ∈G}.\nLet the stabilizer of v be\nH = Gv = {g ∈G|gv = v}.\nThe orbit Gv is isomorphic to the space G/H of cosets, called the ho-\nmogeneous space. This is a very special kind of algebraic variety.\nDefinition 11.7. The orbit closure of v is defined by:\n∆V [v] = Gv ⊆P(V ).\nHere Gv is the closure of the orbit Gv in the complex topology on P(V ) (see\nfigure 11.1).\nA basic fact of algebraic geometry:\nTheorem 11.2. The orbit closure ∆V [v] is a projective G-variety\nIt is also called an almost homogeneous space.\nLet IV [v] be the ideal of ∆V [v], and RV [v] the homogeneous coordinate\nring of ∆V [v]. The algebraic geometry of general orbit closures is hopeless,\nsince the closures can be horrendous (see figure 11.1). Fortunately we shall\nonly be interested in very special kinds of orbit closures with good algebraic\ngeometry.\nWe now define the simplest kind of orbit closure, which is obtained when\nthe orbit itself is closed. Let Vλ be an irreducible Weyl module of GLn(C),\nwhere λ = (λ1 ≥λ2 ≥· · · ≥λn ≥0) is a partition. Let vλ be the highest\nweight point in P(Vλ), i.e., the point corresponding to the highest weight\n66"},{"paragraph_id":"p84","order":84,"text":"v\nGv\n∆V [v]\nLimit points of Gv\nFigure 11.1: The limit points of Gv in ∆V [v] can be horrendous.\nvector in Vλ. This means bvλ = vλ for all b ∈B, where B ⊆GLn(C) is the\nBorel subgroup of lower triangular matrices. Recall that the highest weight\nvector is unique.\nConsider the orbit Gvλ of vλ. Basic fact:\nProposition 11.1. The orbit Gvλ is already closed in P(V ).\nIt can be shown that the stabilizer Pλ = Gvλ is a group of block lower\ntriangular matrices, where the block lengths only depend on λ (see figure\n11.2). Such subgroups of GLn(C) are called parabolic subgroups, and will be\ndenoted by P. Clearly Gvλ ∼= G/Pλ = G/P.\n11.3\nGrassmanians\nThe simplest examples of G/P are Grassmanians.\nDefinition 11.8. Let G = Gln(C), and V = Cn. The Grassmanian Grn\nd is\nthe space of d-dimensional subspaces (containing the origin) of V .\nExamples:\n1. Gr2\n1 is the set of lines in C2 (see figure 11.3).\n2. More generally, P(V ) = Grn\n1 .\nProposition 11.2. The Grassmanian Grn\nd is a projective variety (just like\nP(V ) = Grn\n1 ).\n67"},{"paragraph_id":"p85","order":85,"text":"*\nA1\nA2\nA3\nA4\nA5\nm1\nm2\nm3\nm4\nm5\nFigure 11.2: The parabolic subgroup of block lower triangular matrices. The sizes\nmi only depend on λ.\nFigure 11.3: Gr2\n1 is the set of lines in C2.\n68"},{"paragraph_id":"p86","order":86,"text":"It is easy to see that Grn\nd is closed (since the limit of a sequence of d-\ndimensional subspaces of V is a d-dimensional subspace). Hence this follows\nfrom:\nProposition 11.3. Let λ = (1, . . . , 1) be the partition of d, whose all parts\nare 1. Then Grn\nd ∼= Gvλ ⊆P(Vλ).\nProof. For the given λ, Vλ can be identified with the dth wedge product\nΛd(V ) = span{(vi1 ∧· · ·∧vid)|i1, . . . , id are distinct} ⊆V ⊗· · ·⊗V (d times),\nwhere\n(vi1 ∧· · · ∧vid) = 1\nd!\nX\nσ∈Sd\nsgn(σ)(vσ(i1) ⊗· · · ⊗vσ(id)).\nLet Z be a variable d × n matrix.\nThen C[Z] is a G-module: given\nf ∈C[Z] and σ ∈GLn(C), we define the action of σ by\n(σ · f)(Z) = f(Zσ).\nNow Λd(V ), as a G-module, is isomorphic to the span in C[Z] of all d × d\nminors of Z.\nLet A ∈Grn\nd be a d-dimensional subspace of V . Take any basis {v1, . . . , vd}\nof A. The point v1 ∧· · · ∧vd ∈Λd(V ) depends only on the subspace A, and\nnot on the basis, since the change of basis does not change the wedge prod-\nuct. Let ZA be the d × n complex matrix whose rows are the basis vectors\nv1, . . . , vd of A. The Plucker map associates with A the tuple of all d × d\nminors Aj1,...,jd of ZA, where Aj1,...,jd denotes the minor of ZA formed by\nthe columns j1, . . . , jd. This depends only on A, and not on the choice of\nbasis for A.\nThe proposition follows from:\nClaim 11.1. The Plucker map is a G-equivariant map from Grn\nd to Gvλ ⊆\nP(Vλ) and Grn\nd ≈Gvλ ⊆P(Vλ).\nProof. Exercise. Hint: take the usual basis, and note that the highest weight\npoint vλ corresponds to v1 ∧· · · ∧vd.\n69"},{"paragraph_id":"p87","order":87,"text":"Chapter 12\nThe class varieties\nScribe: Hariharan Narayanan\nGoal: Associate class varieties with the complexity classes #P and NC\nand reduce the NC ̸= P #P conjecture over C to a conjecture that the class\nvariety for #P cannot be embedded in the class variety for NC.\nreference: [GCT1]\nThe NC ̸= P #P conjecture over C says that the permanent of an n × n\ncomplex matrix X cannot be expressed as a determinant of an m × m com-\nplex matrix Y , m = poly(n), whose entries are (possibly nonhomogeneous)\nlinear forms in the entries of X. This obviously implies the NC ̸= P #P\nconjecture over Z, since multivariate polynomials over Cn are determined\nby the values that they take over the subset Zn. The conjecture over Z is\nimplied by the usual NC ̸= P #P conjecture over a finite field Fp, p ̸= 2,\nand hence, has to be proved first anyway.\nFor this reason, we concentrate on the NC ̸= P #P conjecture over C in\nthis lecture. The goal is to reduce this conjecture to a statement in geometric\ninvariant theory.\n12.1\nClass Varieties in GCT\nTowards that end, we associate with the complexity classes #P and NC\ncertain projective algebraic varieties, which we call class varieties. For this,\nwe need a few definitions.\nLet G = GLl(C), V a finite dimensional representation of G. Let P(V )\nbe the associated projective space, which inherits the group action. Given a\npoint v ∈P(V ), let ∆V [v] = Gv ⊆P(V ) be its orbit closure. Here Gv is the\n70"},{"paragraph_id":"p88","order":88,"text":"closure of the orbit Gv in the complex topology on P(v). It is a projective\nG-variety; i.e., a projective variety with the action of G.\nAll class varieties in GCT are orbit closures (or their slight generaliza-\ntions), where v ∈P(V ) corresponds to a complete function for the class in\nquestion. The choice of the complete function is crucial, since it determines\nthe algebraic geometry of ∆V [v].\nWe now associate a class variety with NC. Let g = det(Y ), Y an m × m\nvariable matrix. This is a complete function for NC. Let V = symm(Y )\nbe the space of homogeneous forms in the entries of Y of degree m. It is a\nG-module, G = GLm2(C), with the action of σ ∈G given by:\nσ : f(Y ) 7−→f(σ−1Y ).\nHere σ−1Y is defined thinking of Y as an m2-vector.\nLet ∆V [g] = ∆V [g, m] = Gg, where we think of g as an element of\nP(V ).\nThis is the class variety associated with NC.\nIf g is a different\nfunction instead of det(Y ), the algebraic geometry of ∆V [g] would have been\nunmanageable. The main point is that the algebraic geometry of ∆V [g] is\nnice, because of the very special nature of the determinant function.\nWe next associate a class variety with #P. Let h = perm(X), X an\nn × n variable matrix. Let W = symn(X). It is similarly an H-module,\nH = GLk(C), k = n2. Think of h as an element of P(W), and let ∆W[h] =\nHh be its orbit closure. It is called the class variety associated with #P.\nNow assume that m > n, and think of X as a submatrix of Y , say the\nlower principal submatrix. Fix a variable entry y of Y outside of X. Define\nthe map φ : W →V which takes w(x) ∈W to ym−nw(x) ∈V . This induces\na map from P(V ) to P(W) which we call φ as well. Let φ(h) = f ∈P(V )\nand ∆V [f, m, n] = Gf its orbit closure. It is called the extended class variety\nassociated with #P.\nProposition 12.1 (GCT 1).\n1. If h(X) ∈W can be computed by a cir-\ncuit (over C) of depth ≤logc(n), c a constant, then f = φ(h) ∈\n∆V [g, m], for m = O(2logc n).\n2. Conversely if f ∈∆V [g, m] for m = 2logc n, then h(X) can be approx-\nimated infinitesimally closely by a circuit of depth log2c m. That is,\n∀ǫ > 0, there exists a function ̃h(X) that can be computed by a circuit\nof depth ≤log2c m such that ∥ ̃h −h∥< ǫ in the usual norm on P(V ).\nIf the permanent h(X) can be approximated infinitesimally closely by\nsmall depth circuits, then every function in #P can be approximated in-\nfinitesimally closely by small depth circuits. This is not expected. Hence:\n71"},{"paragraph_id":"p89","order":89,"text":"Conjecture 12.1 (GCT 1). Let h(X) = perm(X), X an n × n variable\nmatrix. Then f = φ(h) ̸∈∆V [g; m] if m = 2polylog(n) and n is sufficiently\nlarge.\nThis is equivalent to:\nConjecture 12.2 (GCT 1). The G-variety ∆V [f; m, n] cannot be embedded\nas a G-subvariety of ∆V [g, m], symbolically\n∆V [f; m, n] ̸֒→∆V [g, m],\nif m = 2polylog(n) and n →∞.\nThis is the statement in geometric invariant theory (GIT) that we sought.\n72"},{"paragraph_id":"p90","order":90,"text":"Chapter 13\nObstructions\nScribe: Paolo Codenotti\nGoal: Define an obstruction to the embedding of the #P-class variety in\nthe NC-class-variety and describe why it should exist.\nReferences: [GCT1, GCT2]\nRecall\nLet us first recall some definitions and results from the last class. Let Y be\na generic m × m variable matrix, and X an n × n minor of Y (see figure\n13.1).\nLet g = det(Y ), h = perm(X), f = φ(h) = ym−nperm(X), and\nV = Symm[Y ] the set of homogeneous forms of degree m in the entries of Y .\nThen V is a G-module for G = GL(Y ) = GLl(C), l = m2, with the action\nY\nX\nn\nn\nm\nm\nFigure 13.1: Here Y is a generic m by m matrix, and X is an n by n minor.\n73"},{"paragraph_id":"p91","order":91,"text":"of σ ∈G given by\nσ : f(Y ) →f(σ−1Y ),\nwhere Y is thought of as an l-vector, and P(V ) a G-variety. Let\n∆V [f; m, n] = Gf ⊆P(V ),\nand\n∆V [g; m] = Gg ⊆P(V )\nbe the class varieties associated with #P and NC.\n13.1\nObstructions\nConjecture 13.1. [GCT1] There does not exist an embedding ∆V [f; m, n] ֒→\n∆V [g; m] with m = 2polylog(n), n →∞.\nThis implies Valiant’s conjecture that the permanent cannot be com-\nputed by circuits of polylog depth. Now we discuss how to go about proving\nthe conjecture.\nSuppose to the contrary,\n∆[f; m, n] ֒→∆V [g; m].\n(13.1)\nWe denote ∆V [f; m, n] by ∆V [f], and ∆V [g; m] by ∆V [g]. Let RV [g] be\nthe homogeneous coordinate ring of ∆V [g]. The embedding (13.1) implies\nexistence of a surjection:\nRV [f] եւ RV [g]\n(13.2)\nThis is a basic fact from algebraic geometry. The reason is that RV [g] is the\nset of homogeneous polynomial functions on the cone C of ∆V [g], and any\nsuch function τ can be restricted to ∆V [f] (see figure 13.2). Conversely, any\npolynomial function on ∆V [f] can be extended to a homogeneous polynomial\nfunction on the cone C.\nLet RV [f]d and RV [g]d be the degree d components of RV [f] and RV [g].\nThese are G-modules since ∆V [f] and ∆V [g] are G-varieties. The surjection\n(13.2) is degree preserving. So there is a surjection\nRV [f]d եւ RV [g]d\n(13.3)\nfor every d. Since G is reductive, both RV [f]d and RV [g]d are direct sums\nof irreducible G-modules. Hence the surjection (13.3) implies that RV [f]d\ncan be embedded as a G submodule of RV [g]d.\n74"},{"paragraph_id":"p92","order":92,"text":"C\nτ\n∆V [f]\nFigure 13.2: C denotes the cone of ∆V [g].\nDefinition 13.1. We say that a Weyl-module S = Vλ(G) is an obstruction\nfor the embedding (13.1) (or, equivalently, for the pair (f, g)) if Vλ(G) occurs\nin RV [f; m, n]d, but not in RV [g; m]d, for some d. Here occurs means the\nmultiplicity of Vλ(G) in the decomposition of RV [f; m, n]d is nonzero.\nIf an obstruction exists for given m, n, then the embedding (13.1) does\nnot exist.\nConjecture 13.2 (GCT2). An obstruction exists for the pair (f, g) for all\nlarge enough n if m = 2polylog(n).\nThis implies Conjecture 13.1. In essence, this turns a nonexistence prob-\nlem (of polylog depth circuit for the permanent) into an existence problem\n(of an obstruction).\nIf we replace the determinant here by any other complete function in\nNC, an obstruction need not exist. Because, as we shall see in the next\nlecture, the existence of an obstruction crucially depends on the exceptional\nnature of the class variety constructed from the determinant.\nThe main\ngoals of GCT in this context are:\n1. understand the exceptional nature of the class varieties for NC and\n#P, and\n2. use it to prove the existence of obstructions.\n13.1.1\nWhy are the class varieties exceptional?\nWe now elaborate on the exceptional nature of the class varieties. Its signif-\nicance for the existence of obstructions will be discussed in the next lecture.\nLet V be a G-module, G = GLn(C). Let P(V ) be a projective variety\nover V . Let v ∈P(V ), and recall ∆V [v] = Gv. Let H = Gv be the stabilizer\nof v, that is, Gv = {σ ∈G|σv = v}.\n75"},{"paragraph_id":"p93","order":93,"text":"Definition 13.2. We say that v is characterized by its stabilizer H = Gv\nif v is the only point in P(V ) such that hv = v, ∀h ∈H.\nIf v is characterized by its stabilizer, then ∆V [v] is completely determined\nby the group triple H ֒→G ֒→K = GL(V ).\nDefinition 13.3. The orbit closure ∆V [v], when v is characterized by its\nstabilizer, is called a group-theoretic variety.\nProposition 13.1. [GCT1]\n1. The determinant g = det(Y ) ∈P(V ) is characterized by its stabilizer.\nTherefore ∆V [g] is group theoretic.\n2. The permanent h = perm(X) ∈P(W), where W = Symn(X), is also\ncharacterized by its stabilizer. Therefore ∆W[h] is also group theoretic.\n3. Finally, f = φ(h) ∈P(V ) is also characterized by its stabilizer. Hence\n∆V [f] is also group theoretic.\nProof. (1) It is a fact in classical representation theory that the stabilizer\nof det(Y ) in G = GL(Y ) = GLm2(C) is the subgroup Gdet that consists of\nlinear transformations of the form Y →AY ∗B, where Y ∗= Y or Y t, for any\nA, B ∈GLm(C). It is clear that linear transformation of this form stabilize\nthe determinant since:\n1. det(AY B) = det(A)det(B)det(Y ) = c det(Y ), where c = det(A) det(B).\nNote that the constant c doesn’t matter because we get the same point\nin the projective space.\n2. det(Y ∗) = det(Y ).\nIt is a basic fact in classical invariant theory that det(Y ) is the only point in\nP(V ) stabilized by Gdet. Furthermore, the stabilizer Gdet is reductive, since\nits connected part is (Gdet)◦≈GLm × GLm with the natural embedding\n(Gdet)◦= GLm × GLm ֒→GL(Cm ⊗Cm) = GLm2(C) = G.\n(2) The stabilizer of perm(x) is the subgroup Gperm ⊆GL(X) = GLn2(C)\ngenerated by linear transformations of the form X →λX∗μ, where X∗=\nXorXt, and λ and μ are diagonal (which change the permanent by a con-\nstant factor) or permutation matrices (which do not change the permanent).\nFinally, the discrete component of Gperm is isomorphic to S2 ⋊Sn ×Sn,\nwhere ⋊denotes semidirect product. The continuous part is (C∗)n × (C∗)n.\nSo Gperm is reductive.\n76"},{"paragraph_id":"p94","order":94,"text":"(3) Similar.\nThe main significance of this proposition is the following.\nBecause\n∆V [g], ∆V [f], and ∆W[h] are group theoretic, the algebraic geometric prob-\nlems concerning these varieties can be “reduced” to problems in the theory\nof quantum groups. So the plan is:\n1. Use the theory of quantum groups to understand the structure of the\ngroup triple associated with the algebraic variety.\n2. Translate this understanding to the structure of the algebraic variety.\n3. Use this to show the existence of obstructions.\n77"},{"paragraph_id":"p95","order":95,"text":"Chapter 14\nGroup theoretic varieties\nScribe: Joshua A. Grochow\nGoal: In this lecture we continue our discussion of group-theoretic varieties.\nWe describe why obstructions should exist, and why the exceptional group-\ntheoretic nature of the class varieties is crucial for this existence.\nRecall\nLet G = GLn(C), V a G-module, and P(V ) the associated projective space.\nLet v ∈P(V ) be a point characterized by its stabilizer H = Gv ⊂G. In\nother words, v is the only point in P(V ) stabilized by H. Then ∆V [v] = Gv\nis called a group-theoretic variety because it is completely determined by the\ngroup triple\nH ֒→G ֒→GL(V ).\nThe simplest example of a group-theoretic variety is a variety of the form\nG/P that we described in the earlier lecture. Let V = Vλ(G) be a Weyl\nmodule of G and vλ ∈P(V ) the highest weight point (recall: the unique\npoint stabilized by the Borel subgroup B ⊂G of lower triangular matrices).\nThen the stabilizer of vλ consists of block-upper triangular matrices, where\n78"},{"paragraph_id":"p96","order":96,"text":"the block sizes are determined by λ:\nPλ := Gvλ ="},{"paragraph_id":"p97","order":97,"text":"∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n0\n0\n0\n∗\n∗\n∗\n∗\n∗\n0\n0\n0\n∗\n∗\n∗\n∗\n∗\n0\n0\n0\n0\n0\n∗\n∗\n∗\n0\n0\n0\n0\n0\n∗\n∗\n∗\n0\n0\n0\n0\n0\n∗\n∗\n∗"},{"paragraph_id":"p98","order":98,"text":"The orbit ∆V [vλ] = Gvλ ∼= G/Pλ is a group-theoretic variety determined\nentirely by the triple\nPλ = Gvλ ֒→G ֒→K = GL(V ).\nThe group-theoretic varieties of main interest in GCT are the class va-\nrieties associated with the various complexity classes.\n14.1\nRepresentation theoretic data\nThe main principle guiding GCT is that the algebraic geometry of a group-\ntheoretic variety ought to be completely determined by the representation\ntheory of the corresponding group triple.\nThis is a natural extension of\nwork already pursued in mathematics by Deligne and Milne on Tannakien\ncategories [DeM], showing that an algebraic group is completely determined\nby its representation theory. So the goal is to associate to a group-theoretic\nvariety some representation-theoretic data that will analogously capture the\ninformation in the variety completely. We shall now illustrate this for the\nclass variety for NC. First a few definitions.\nLet v ∈P(V ) be the point as above characterized by its stabilizer Gv.\nThis means the line Cv ⊆V corresponding to v is a one-dimensional repre-\nsentation of Gv. Thus (Cv)⊗d is a one-dimensional degree d representation,\ni.e. the representation ρ : G →GL(Cv) ∼= C∗is polynomial of degree d\nin the entries of the matrix of an element in G. Recall that C[V ] is the\ncoordinate ring of V , and C[V ]d is its degree d homogeneous component, so\n(Cv)⊗d ⊆C[V ]d.\nTo each v ∈P(V ) that is characterized by its stabilizer, we associate a\nrepresentation-theoretic data, which is the set of G-modules\nΠv =\n[\nd\nΠv(d),\n79"},{"paragraph_id":"p99","order":99,"text":"where Πv(d) is the set of all irreducible G-submodules S of C[V ]d whose\nduals S∗do not contain a Gv-submodule isomorphic to (Cv)⊗d∗(the dual of\n(Cv)⊗d). The following proposition elucidates the importance of this data:\nProposition 14.1. [GCT2] Πv ⊆IV [v] (where IV [v] is the ideal of the\nprojective variety ∆V [v] ⊆P(V )).\nProof. Fix S ∈Πv(d). Suppose, for the sake of contradiction, that S ⊈\nIV [v]. Since S ⊆C[V ], S consists of “functions” on the variety P(V ) (ac-\ntually homogeneous polynomials on V ). The coordinate ring of ∆V [v] is\nC[V ]/IV [v], and since S ⊈IV [v], S must not vanish identically on ∆V [v].\nSince the orbit Gv is dense in ∆V [v], S must not vanish identically on this\nsingle orbit Gv. Since S is a G-module, if S were to vanish identically on the\nline Cv, then it would vanish on the entire orbit Gv, so S does not vanish\nidentically on Cv.\nNow S consists of functions of degree d. Restrict them to the line Cv.\nThe dual of this restriction gives an injection of (Cv)⊗d∗as a Gv-submodule\nof S∗, contradicting the definition of Πv(d).\n14.2\nThe second fundamental theorem\nWe now ask essentially the reverse question: when does the representation\ntheoretic data Πv generate the ideal IV [v]? For if Πv generates IV [v], then Πv\ncompletely captures the coordinate ring C[V ]/IV [v], and hence the variety\n∆V [v].\nTheorem 14.1 (Second fundamental theorem of invariant theory for G/P).\nThe G-modules in Πvλ(2) generate the ideal IV [vλ] of the orbit Gvλ ∼= G/Pλ,\nwhen V = Vλ(G).\nThis theorem justifies the main principle for G/P, so we can hope that\nsimilar results hold for the class varieties in GCT (though not always exactly\nin the same form).\nNow, let ∆V [g] be the class variety for NC (in other words, take g =\ndet(Y ) for a matrix Y of indeterminates).\nBased on the main principle,\nwe have the following conjecture, which essentially generalizes the second\nfundamental theorem of invariant theory for G/P to the class variety for\nNC:\nConjecture 14.1 (GCT 2). ∆V [g] = X(Πg) where X(Πg) is the zero-set\nof all forms in the G-modules contained in Πg.\n80"},{"paragraph_id":"p100","order":100,"text":"Theorem 14.2 (GCT 2). A weaker version of the above conjecture holds.\nSpecifically, assuming that the Kronecker coefficients satisfy a certain sep-\naration property, there exists a G-invariant (Zariski) open neighbourhood\nU ⊆P(V ) of the orbit Gg such that X(Πg) ∩U = ∆V [g] ∩U.\nThere is a notion of algebro-geometric complexity called Luna-Vust com-\nplexity which quantifies the gap between G/P and class varieties. The Luna-\nVust complexity of G/P is 0. The Luna-Vust complexity of the NC class\nvariety is Ω(dim(Y )). This is analogous to the difference between circuits\nof constant depth and circuits of superpolynomial depth. This is why the\nprevious conjecture and theorem turn out to be far harder than the corre-\nsponding facts for G/P.\n14.3\nWhy should obstructions exist?\nThe following proposition explains why obstructions should exist to separate\nNC from P #P.\nProposition 14.2 (GCT 2). Let g = det(Y ), h = perm(X), f = φ(h),\nn = dim(X), m = dim(Y ). If Conjecture 14.1 holds and the permanent can-\nnot be approximated arbitrarily closely by circuits of poly-logarithmic depth\n(hardness assumption), then an obstruction for the pair (f, g) exists for all\nlarge enough n, when m = 2logc n for some constant c. Hence, under these\nconditions, NC ̸= P #P over C.\nThis proposition may seem a bit circular at first, since it relies on a hard-\nness assumption. But we do not plan to prove the existence of obstructions\nby proving the assumptions of this proposition. Rather, this proposition\nshould be taken as evidence that obstructions exist (since we expect the\nhardness assumption therein to hold, given that the permanent is # P-\ncomplete), and we will develop other methods to prove their existence.\nProof. The hardness assumption implies that f /∈∆V [g] if m = 2logc n [GCT\n1].\nConjecture 14.1 says that X(Πg) = ∆V [g]. So there exists an irreducible\nG-module S ∈Πg such that S does not vanish on f. So S occurs in RV [f]\nas a G-submodule.\nOn the other hand, since S ∈Πg, S ⊆IV [g] by Proposition 14.1. So\nS does not occur in RV [g] = C[V ]/IV [g]. Thus S is not a G-submodule of\nRV [g], but it is a G-submodule of RV [f], i.e., S is an obstruction.\n81"},{"paragraph_id":"p101","order":101,"text":"Chapter 15\nThe flip\nScribe: Hariharan Narayanan\nGoal: Describe the basic principle of GCT, called the flip, in the context of\nthe NC vs. P #P problem over C.\nreferences: [GCTflip1, GCT1, GCT2, GCT6]\nRecall\nAs in the previous lectures, let g = det(Y ) ∈P(V ), Y an m × m vari-\nable matrix, G = GLm2(C), and ∆V (g) = ∆V [g; m] = Gg ⊆P(V ) the\nclass variety for NC. Let h = perm(X), X an n × n variable matrix,\nf = φ(h) = ym−nh ∈P(V ), and ∆V (f) = ∆V [f; m, n] = Gf ⊆P(V ) the\nclass variety for P #P. Let RV [f; m, n] denote the homogeneous coordinate\nring of ∆V [f; m, n], RV [g; m] the homogeneous coordinate ring of ∆V [g; m],\nand RV [f; m, n]d and RV [g; m]d their degree d-components. A Weyl module\nS = Vλ(G) of G is an obstruction of degree d for the pair (f, g) if Vλ occurs\nin RV [f; m, n]d but not RV [g; m]d.\nConjecture 15.1. [GCT2] An obstruction (of degree polynomial in m) ex-\nists if m = 2polylog(n) as n →∞.\nThis implies NC ̸= P #P over C.\n82"},{"paragraph_id":"p102","order":102,"text":"15.1\nThe flip\nIn this lecture we describe an approach to prove the existence of such ob-\nstructions. It is based on the following complexity theoretic positivity hy-\npothesis:\nPHflip [GCTflip1]:\n1. Given n, m and d, whether an obstruction of degree d for m and n\nexists can be decided in poly(n, m, ⟨d⟩) time, and if it exists, the label\nλ of such an obstruction can be constructed in poly(n, m, ⟨d⟩) time.\nHere ⟨d⟩denotes the bitlength of d.\n2.\n(a) Whether Vλ occurs in RV [f; m, n]d can be decided in poly(n, m, ⟨d⟩, ⟨λ⟩)\ntime.\n(b) Whether Vλ occurs in RV [g; m]d can be decided in poly(n, m, ⟨d⟩, ⟨λ⟩)\ntime.\nThis suggests the following approach for proving Conjecture 15.1:\n1. Find polynomial time algorithms sought in PHflip-2 for the basic de-\ncision problems (a) and (b) therein.\n2. Using these find a polynomial time algorithm sought in PHflip-1 for\ndeciding if an obstruction exists.\n3. Transform (the techniques underlying) this “easy” (polynomial time)\nalgorithm for deciding if an obstruction exists for given n and m into\nan “easy” (i.e., feasible) proof of existence of an obstruction for every\nn →∞when d is large enough and m = 2polylog(n).\nThe first step here is the crux of the matter. The main results of [GCT6]\nsay that the polynomial time algorithms for the basic decision problems as\nsought in PHflip-2 indeed exist assuming natural analogues of PH1 and SH\n(PH2) that we have seen earlier in the context of the plethysm problem. To\nstate them, we need some definitions.\nLet Sλ\nd [f] = Sλ\nd [f; m, n] be the multiplicity of Vλ = Vλ(G) in RV [f; m, n].\nThe stretching function ̃Sd[f] = ̃Sλ\nd [f; m, n] is defined by\n ̃Sλ\nd [f](k) := Skλ\nkd[f].\nThe stretching function for g, ̃Sλ\nd [g] = ̃Sλ\nd [g; m], is defined analogously.\nThe main mathematical result of [GCT6] is:\n83"},{"paragraph_id":"p103","order":103,"text":"Theorem 15.1. [GCT6] The stretching functions ̃Sλ\nd [g] and ̃Sλ\nd [f] are quasipoly-\nnomials assuming that the singularities of ∆V [f; m, n] and ∆V [g; m] are ra-\ntional.\nHere rational means “nice”; we shall not worry about the exact defini-\ntion.\nThe main complexity-theoretic result is:\nTheorem 15.2. [GCT6] Assuming the following mathematical positivity\nhypothesis PH1 and the saturation hypothesis SH (or the stronger positivity\nhypothesis PH2), PHflip-2 holds.\nPH1: There exists a polytope P = P λ\nd [f] such that\n1. The Ehrhart quasi-polynomial of P, fP(k), is ̃Sλ\nd [f](k).\n2. dim(P) = poly(n, m, ⟨d⟩).\n3. Membership in P can be answered in polynomial time.\n4. There is a polynomial time separation oracle [GLS] for P.\nSimilarly, there exists a polytope Q = Qλ\nd[g] such that\n1. The Ehrhart quasi-polynomial of Q, fQ(k), is ̃Sλ\nd [g](k).\n2. dim(Q) = poly(m, ⟨d⟩).\n3. Membership in Q can be answered in polynomial time.\n4. There is a polynomial time separation oracle for Q.\nPH2: The quasi-polynomials ̃Sλ\nd [g] and ̃Sλ\nd [f] are positive.\nThis implies:\nSH: The quasi-polynomials ̃Sλ\nd [g] and ̃Sλ\nd [f] are saturated.\nPH1 and SH imply that the decision problems in PHflip-2 can be trans-\nformed into saturated positive integer programming problems. Hence The-\norem 15.2 follows from the polynomial time algorithm for saturated linear\nprogramming that we described in an earlier class.\nThe decision problems in PHflip-2 are “hyped” up versions of the plethysm\nproblem discussed earlier. The article [GCT6] provides evidence for PH1\n84"},{"paragraph_id":"p104","order":104,"text":"and PH2 for the plethysm problem. This constitutes the main evidence for\nPH1 and PH2 for the class varieties in view of their group-theoretic nature;\ncf. [GCTflip1].\nThe following problem is important in the context of PHflip-2:\nProblem 15.1. Understand the G-module structure of the homogeneous\ncoordinate rings RV [f]d and RV [g]d.\nThis is an instance of the following abstract:\nProblem 15.2. Let X be a projective group-theoretic G-variety. Let R =\nL∞\nd=0 Rd be its homogeneous coordinate ring.\nUnderstand the G-module\nstructure of Rd.\nThe simplest group-theoretic variety is G/P. For it, a solution to this\nabstract problem is given by the following results:\n1. The Borel-Weil theorem.\n2. The Second Fundamental theorem of invariant theory [SFT].\nThese will be covered in the next class for the simplest case of G/P, the\nGrassmanian.\n85"},{"paragraph_id":"p105","order":105,"text":"Chapter 16\nThe Grassmanian\nScribe: Hariharan Narayanan\nGoal: The Borel-Weil and the second fundamental theorem of invariant\ntheory for the Grassmanian.\nReference: [F]\nRecall\nLet V = Vλ(G) be a Weyl module of G = GLn(C) and vλ ∈P(V ) the point\ncorresponding to its highest weight vector. The orbit ∆V [vλ] := Gvλ, which\nis already closed, is of the form G/P, where P is the parabolic stabilizer of\nvλ. When λ is a single column, it is called the Grassmannian.\nAn alternative description of the Grassmanian is as follows. Assume that\nλ is a single column of length d. Let Z be a d × n matrix of variables zij.\nThen V = Vλ(G) can be identified with the span of d × d minors of Z with\nthe action of σ ∈G given by:\nσ : f(z) 7→f(zσ).\nLet Grn\nd be the space of all d-dimensional subspaces of Cn. Let W be a\nd-dimensional subspace of Cn. Let B = B(W) be a basis of W. Construct\nthe d × n matrix zB, whose rows are vectors in B. Consider the Pl ̈ucker\nmap from Grn\nd to P(V ) which maps any W ∈Grn\nd to the tuple of d × d\nminors of ZB. Here the choice of B = B(W) does not matter, since any\nchoice gives the same point in P(V ). Then the image of Grn\nd is precisely the\nGrassmanian Gvλ ⊆P(V ).\n86"},{"paragraph_id":"p106","order":106,"text":"16.1\nThe second fundamental theorem\nNow we ask:\nQuestion 16.1. What is the ideal of Grn\nd ≈Gvλ ⊆P(V )?\nThe homogeneous coordinate ring of P(V ) is C[V ].\nWe want an ex-\nplicit set of generators of this ideal in C[V ]. This is given by the second\nfundamental theorem of invariant theory, which we describe next.\nThe coordinates of P(V ) are in one-to-one correspondence with the d×d\nminors of the matrix Z. Let each minor of Z be indexed by its columns. Thus\nfor 1 ≤i1 < · · · < id ≤n, Zi1,...,id is a coordinate of P(V ) corresponding\nto the minor of Z formed by the columns i1, i2, . . ..\nLet Λ(n, d) be the\nset of ordered d-tuples of {1, . . . , n}. The tuple [i1, . . . , id] in this set will\nbe identified with the coordinate Zi1,...,id of P(V ).\nThere is a bijection\nbetween the elements of Λ(n, d) and of Λ(n, n −d) obtained by associating\ncomplementary sets:\nΛ(n, d) ∋λ ↭λ∗∈Λ(n, n −d).\nWe define sgn(λ, λ∗) to be the sign of the permutation that takes [1, . . . , n]\nto [λ1, . . . , λd, λ∗\n1, . . . , λ∗\nn−d].\nGiven s ∈{1. . . . , d}, α ∈Λ(n, s−1), β ∈Λ(n, d+1), and γ ∈Λ(n, d−s),\nwe now define the Van der Waerden Syzygy [[α, β, γ]], which is an element\nof the degree two component C[V ]2 of C[V ], as follows:\n[[α, β, γ]] =\nP\nτ∈Λ(d+1,s) sgn(τ, τ ∗)[α1, . . . , αs−1, βτ ∗\n1 , . . . , βτ ∗\nd+1−s][βτ1, . . . , βτs, γ1, . . . , γd−s].\nIt is easy to show that this syzygy vanishes on the Grassmanian Grn\nd :\nbecause it is an alternating (d+1)-multilinear-form, and hence has to vanish\non any d-dimensional space W ∈Grn\nd . Thus it belongs to the ideal of the\nGrassmanian. Moreover:\nTheorem 16.1 (Second fundamental theorem). The ideal of the Grassma-\nnian Grn\nd is generated by the Van-der-Waerden syzygies.\nAn alternative formulation of this result is as follows. Let Pλ ⊆G be the\nstabilizer of vλ. Let Πvλ(2) be the set of irreducible G-submodules of C[V ]2\nwhose duals do not contain a Pλ-submodule isomorphic to Cv⊗2∗\nλ\n(the dual\nof Cv⊗2\nλ ). Here Cvλ denotes the line in P(V ) corresponding to vλ, which is\na one-dimensional representation of Pλ since it stabilizes vλ ∈P(V ). It can\n87"},{"paragraph_id":"p107","order":107,"text":"be shown that the span of the G-modules in Πvλ(2) is equal to the span of\nthe Van-der-Waerden syzygies. Hence, Theorem 16.1 is equivalent to:\nTheorem 16.2 (Second Fundamental Theorem(SFT)). The G-modules in\nΠvλ(2) generate the ideal of Grn\nd .\nThis formulation of SFT for the Grassmanian looks very similar to the\ngeneralized conjectural SFT for the NC-class variety described in the ear-\nlier class. This indicates that the class varieties in GCT are “qualitatively\nsimilar” to G/P.\n16.2\nThe Borel-Weil theorem\nWe now describe the G-module structure of the homogeneous coordinate\nring R of the Grassmannian Gvλ ⊆P(V ), where λ is a single column of\nheight d. The goal is to give an explicit basis for R. Let Rs be the degree\ns component of R. Corresponding to any numbering T of the shape sλ,\nwhich is a d × s rectangle, whose columns have strictly increasing elements\ntop to bottom, we have a monomial mT = Q\nc Zc ∈C[V ]s, were Zc is the\ncoordinate of P(V ) indexed by the d-tuple c, and c ranges over the s columns\nof T. We say that mT is (semi)-standard if the rows of T are nondecreasing,\nwhen read left to right. It is called nonstandard otherwise.\nLemma 16.1 (Straightening Lemma). Each non-standard mT can be straight-\nened to a normal form, as a linear combination of standard monomials, by\nusing Van der Waerden Syzygies as straightening relations (rewriting rules).\nFor any numbering T as above, express mT in a normal form as per the\nlemma:\nmT =\nX\n(Semi)-Standard Tableau S\nα(S, T), mS\nwhere α(S, T) ∈C.\nTheorem 16.3 (Borel-Weil Theorem for Grassmannians). Standard mono-\nmials {mT } form a basis of Rs, where T ranges over all semi-standard\ntableaux of rectangular shape sλ. Hence, Rs ∼= V ∗\nsλ, the dual of the Weyl\nmodule Vsλ.\nThis gives the G-module structure of R completely. It follows that the\nproblem of deciding if Vβ(G) occurs in Rs can be solved in polynomial time:\nthis is so if and only if (sλ)∗= β, where (sλ)∗denotes the dual partition,\nwhose description is left as an exercise.\n88"},{"paragraph_id":"p108","order":108,"text":"The second fundamental theorem as well as the Borel-Weil theorem easily\nfollow from the straightening lemma and linear independence of the standard\nmonomials (as functions on the Grassmanian).\n89"},{"paragraph_id":"p109","order":109,"text":"Chapter 17\nQuantum group: basic\ndefinitions\nScribe: Paolo Codenotti\nGoal: The basic plan to implement the flip in [GCT6] is to prove PH1 and\nSH via the theory of quantum groups. We introduce the basic concepts in\nthis theory in this and the next two lectures, and briefly show their relevance\nin the context of PH1 in the final lecture.\nReference: [KS]\n17.1\nHopf Algebras\nLet G be a group, and K[G] the ring of functions on G with values in the\nfield K, which will be C in our applications. The group G is defined by the\nfollowing operations:\n• multiplication: G × G →G,\n• identity e: e →G,\n• inverse: G →G.\nIn order for G to be a group, the following properties have to hold:\n• eg = ge = g,\n• g1(g2g3) = (g1g2)g3,\n• g−1g = gg−1 = e.\n90"},{"paragraph_id":"p110","order":110,"text":"We now want to translate these properties to properties of K[G]. This\nshould be possible since K[G] contains all the information that G has. In\nother words, we want to translate the notion of a group in terms of K[G].\nThis translate is called a Hopf algebra. Thus if G is a group, K[G] is a Hopf\nalgebra. Let us first define the dual operations.\n• Multiplication is a map:\n· : G × G →G.\nSo co-multiplication ∆will be a map as follows:\nK[G × G] = K[G] ⊗K[G] ←K[G].\nWe want ∆to be the pullback of multiplication. So for a given f ∈\nK[G] we define ∆(f) ∈K[G] ⊗K[G] by:\n∆(f)(g1, g2) = f(g1g2).\nPictorially:\nG × G\n·\n−−−−→G\n∆(f)\n y\n yf\nk\nk\n• The unit is a map:\ne →G.\nTherefore we want the co-unit ǫ to be a map:\nK ǫ←−K[G],\ndefined by: for f ∈K[G], ǫ(f) = f(e).\n• Inverse is a map:\n( )−1 : G →G.\nWe want the dual antipode S to be the map:\nK[G] ←K[G]\ndefined by: for f ∈K[G], S(f)(g) = f(g−1).\nThe following are the abstract axioms satisfied by ∆, ǫ and S.\n91"},{"paragraph_id":"p111","order":111,"text":"1. ∆and ǫ are algebra homomorphisms.\n∆: K[G] →K[G] ⊗K[G]\nǫ : K[G] →K.\n2. co-associativity: Associativity is defined so that the following diagram\ncommutes:\nG × G×G\nG×G × G\n·\n y\nid\n y\n yid\n y·\nG\n×G\nG×\nG\n·\n y\n y·\nG\nG\nSimilarly, we define co-associativity so that the following dual diagram\ncommutes:\nK[G] ⊗K[G] ⊗K[G]\nK[G] ⊗K[G] ⊗K[G]\n∆\nx \nid\nx \nx id\nx ∆\nK[G]\n⊗K[G]\nK[G] ⊗\nK[G]\n∆\nx \nx ∆\nK[G]\nK[G]\nTherefore co-associativity says:\n(∆⊗id) ◦∆= (id ⊗∆) ◦∆.\n3. The property ge = g is defined so that the following diagram com-\nmutes:\ne ×G\nG\ne\n y\n yid\n y\nG×G\nid\n y·\n y\nG\nG\n92"},{"paragraph_id":"p112","order":112,"text":"We define the co of this property so that the following diagram com-\nmutes:\nK\n× K[G]\nK[G]\nǫ\nx \nx id\nx \nK[G] × K[G]\nid\nx ∆\nx \nK[G]\nK[G]\nThat is, id = (ǫ⊗id)◦∆. Similarly, ge = g translates to: id = (id⊗ǫ)◦∆.\nTherefore we get\nid = (ǫ ⊗id) ◦∆= (id ⊗ǫ) ◦∆.\n4. The last property is gg−1 = e = g−1g. The first equality is equivalent\nto requiring that the following diagram commute:\nG\nG\ndiag\n y\n y\nG×G\n y\n()−1\n y\n yid\ne\nG×G\n y\n y·\n y\nG\nG\nWhere diag : G →G× G is the diagonal embedding. The co of diag is\nm : K[G] ←K[G] ⊗K[G] defined by m(f1, f2)(g) = f1(g) · f2(g). So\nthe co of this property will hold when the following diagram commutes:\n93"},{"paragraph_id":"p113","order":113,"text":"K[G]\nK[G]\nm\nx \nx \nK[G] ⊗\nk[G]\nν\nx \nS\nx \nx id\nK\nK[G] ⊗K[G]\nx \nx ∆\nǫ\nx \nK[G]\nK[G]\nWhere ν is the embedding of K into K[G]. Therefore the last property\nwe want to be satisfied is:\nm ◦(S ⊗id) ◦∆= ν ◦ǫ.\nFor e = g−1g, we similarly get:\nm ◦(id ⊗S) ◦∆= ν ◦ǫ.\nDefinition 17.1 (Hopf algebra). A K-algebra A is called a Hopf algebra if\nthere exist homomorphisms ∆: A ⊗A →A, S : A →A, ǫ : A →K, and\nν : A →K that satisfy (1) −(4) above, with A in place of K[G].\nWe have shown that if G is a group, the ring K[G] of functions on\nG is a (commutative) Hopf algebra, which is non-co-commutative if G is\nnon-commutative. Thus for every usual group, we get a commutative Hopf\nalgebra. However, in general, Hopf algebras may be non-commutative.\nDefinition 17.2. A quantum group is a (non-commutative and non-co-\ncommutative) Hopf algebra.\nA nontrivial example of a quantum group will be constructed in the next\nlecture.\nNext we want to look at what happens to group theoretic notions such\nas representations, actions, and homomorphisms, in the context of Hopf\nalgebras. These will correspond to co-representations, co-actions, and co-\nhomomorphisms.\nLet us look closely at the notion of co-representation. A representation\nis a map · : G × V →V , such that\n• (h1h2) · v = h1 · (h2 · v), and\n94"},{"paragraph_id":"p114","order":114,"text":"• e · v = v.\nTherefore a (right) co-representation of A will be a linear mapping φ : V →\nV ⊗A, where V is a K-vector space, and φ satisfies the following:\n• The following diagram commutes:\nV ⊗A ⊗A\nid⊗∆\n←−−−−V ⊗A\nφ⊗id\nx \nx φ\nV ⊗A\n←−−−−\nφ\nV\nThat is, the following equality holds:\n(φ ⊗id) ◦φ = (id ⊗∆) ◦φ.\n• The following diagram commutes:\nV ⊗K\nid\n←−−−−V ⊗K\nid⊗ǫ\nx"},{"paragraph_id":"p115","order":115,"text":"V ⊗A ←−−−−\nφ\nV\nThat is, the following equality holds:\n(id ⊗ǫ) ◦φ = id\nIn fact all usual group theoretic notions can be “Hopfified” in this sense\n[exercise].\nLet us look now at an example. Let\nG = GLn(C) = GL(Cn) = GL(V ),\nwhere V = Cn. Let Mn be the matrix space of n×n C-matrices, and O(Mn)\nthe coordinate ring of Mn,\nO(Mn) = C[U] = C[{ui\nj}],\nwhere U is an n × n variable matrix with entries ui\nj. Let C[G] = O(G) be\nthe coordinate ring of G obtained by adjoining det(U)−1 to O(Mn). That\nis, C[G] = O(G) = C[U][det(U)−1], which is the C algebra generated by ui\nj’s\nand det(U)−1.\n95"},{"paragraph_id":"p116","order":116,"text":"Proposition 17.1. C[G] is a Hopf algebra, with ∆, ǫ, and S as follows.\n• Recall that the axioms of a Hopf algebra require that\n∆: C[G] →C[G] ⊗C[G],\n∆(f)(g1, g2) = f(g1g2).\nTherefore we define\n∆(ui\nj) =\nX\nk\nui\nk ⊗uk\nj,\nwhere U denotes the generic matrix in Mn as above.\n• Again, it is required that\nǫ(f) = f(e).\nTherefore we define\nǫ(ui\nj) = δij,\nwhere δij is the Kronecker delta function.\n• Finally, the antipode is required to satisfy S(f)(g) = f(g−1). Let eU\nbe the cofactor matrix of U, U−1 =\n1\ndet(U) eU, and eui\nj the entries of eU.\nThen we define S by:\nS(ui\nj) =\n1\ndet(U) eui\nj = (U−1)i\nj.\n96"},{"paragraph_id":"p117","order":117,"text":"Chapter 18\nStandard quantum group\nScribe: Paolo Codenotti\nGoal: In this lecture we construct the standard (Drinfeld-Jimbo) quantum\ngroup, which is a q-deformation of the general linear group GLn(C) with\nremarkable properties.\nReference: [KS]\nLet G = GL(V ) = GL(Cn), and V = Cn. In the earlier lecture, we\nconstructed the commutative and non co-commutative Hopf algebra C[G].\nIn this lecture we quantize C[G] to get a non-commutative and non-co-\ncommutative Hopf algebra Cq[G], and then define the standard quantum\ngroup Gq = GLq(V ) = GLq(n) as the virtual object whose coordinate ring\nis Cq[G].\nWe start by defining GLq(2) and SLq(2), for n = 2.\nThen we will\ngeneralize this construction to arbitrary n. Let O(M2) be the coordinate\nring of M2, the set of 2 × 2 complex matrices, C[V ] the coordinate ring of\nV generated by the coordinates x1 and x2 of V which satisfy x1x2 = x2x1.\nLet\nU =\n a\nb\nc\nd"},{"paragraph_id":"p118","order":118,"text":"be the generic (variable) matrix in M2. It acts on V = C2 from the left and\nfrom the right. Let\nx =\n x1\nx2"},{"paragraph_id":"p119","order":119,"text":".\nThe left action is defined by\nx →x′ := Ux.\n97"},{"paragraph_id":"p120","order":120,"text":"Let\nx′ =\n x′\n1\nx′\n2"},{"paragraph_id":"p121","order":121,"text":".\nSimilarly, the right action is defined by\nxT →(x′′)T := xT U.\nLet\nx′′ =\n x′′\n1\nx′′\n2"},{"paragraph_id":"p122","order":122,"text":".\nThe action of M2 on V satisfies\nx′\n1x′\n2 = x′\n2x′\n1, and\nx′′\n1x′′\n2 = x′′\n2x′′\n1.\nNow instead of V , we take its q-deformation Vq, a quantum space, whose\ncoordinates x1 and x2 satisfy\nx1x2 = qx2x1,\n(18.1)\nwhere q ∈C is a parameter. Intuitively, in quantum physics if x1 and x2\nare position and momentum, then q = eiħwhen ħis Planck’s constant. Let\nCq[V ] be the ring generated by x1 and x2 with the relation (18.1). That is,\nCq[V ] = C[x1, x2]/ < x1x2 −qx2x1 > .\nIt is the coordinate ring of the quantum space Vq. Now we want to quantize\nM(2) to get Mq(2), the space of quantum 2 × 2 matrices, and GL(2) to\nGLq(2), the space of quantum 2×2 nonsingular matrices. Intuitively, Mq(2)\nis the space of linear transformations of the quantum space Vq which preserve\nthe equation (18.1) under the left and right actions, and similarly, GLq(2) is\nthe space of non-singular linear transformation that preserve the equation\n(18.1) under the left and right actions. We now formalize this intuition.\nLet U =\n a\nb\nc\nd"},{"paragraph_id":"p123","order":123,"text":"be a quantum matrix whose coordinates do not\ncommute. The left and right actions of U must preserve 18.1.\n[Left action:] Let the left action be φL : x →Ux, and Ux = x′. Then we\nmust have:\n a\nb\nc\nd\n x1\nx2"},{"paragraph_id":"p124","order":124,"text":"=\n ax1 + bx2\ncx1 + dx2"},{"paragraph_id":"p125","order":125,"text":"=\n x′\n1\nx′\n2"},{"paragraph_id":"p126","order":126,"text":".\n98"},{"paragraph_id":"p127","order":127,"text":"[Right action:] Let the right action be φR : xT →xT U, and let x′′ =\n(xT U)T = UT x. Then we must have:\n x1\nx2\n a\nb\nc\nd"},{"paragraph_id":"p128","order":128,"text":"=\n ax1 + cx2\nbx1 + dx2"},{"paragraph_id":"p129","order":129,"text":"=\n x′′\n1\nx′′\n2"},{"paragraph_id":"p130","order":130,"text":".\nThe preservation of x1x2 = qx2x1 under left multiplication means\nx′\n1x′\n2 = qx′\n2x′\n1.\nThat is,\n(ax1 + bx2)(cx1 + dx2) = q(cx1 + dx2)(ax1 + bx2).\n(18.2)\nThe left hand side of (18.2) is\nacx2\n1 + bcx2x1 + adx1x2 + bdx2\n2 = acx2\n1 + (bc + adq)x2x1 + bdx2\n2.\nSimilarly, the right hand side of (18.2) is\nq(cax2\n1 + (da + cbq)x2x1 + bdx2\n2).\nTherefore equation (18.2) implies:\nac = qca\nbd = qdb\nbc + adq = da + qcb.\nThat is,\nac = qca\nbd = qdb\nad −da −qcb + q−1bc = 0.\nSimilarly, since x′′\n1x′′\n2 = qx′′\n2x′′\n1, we get:\nab = qba\ncd = qdc\nad −da −qbc + q−1cb = 0.\nThe last equations from each of these sets imply bc = cb.\nSo we define O(Mq(2)), the coordinate ring of the space of 2×2 quantum\nmatrices Mq(2), to be the C-algebra with generators a, b, c, and d, satisfying\nthe relations:\nab = qba,\nac = qca,\nbd = qdb,\ncd = qdc,\nbc = cb,\nad −da = (q −q−1)bc.\n99"},{"paragraph_id":"p131","order":131,"text":"Let\nU =\n a\nb\nc\nd"},{"paragraph_id":"p132","order":132,"text":"=\n u1\n1\nu1\n2\nu2\n1\nu2\n2"},{"paragraph_id":"p133","order":133,"text":".\nDefine the quantum determinant of U to be\nDq = det(U) = ad −qbc = da −q−1bc.\nDefine Cq[G] = O(GLq(2)), the coordinate ring of the virtual quantum group\nGLq(2) of invertible 2 × 2 quantum matrices, to be\nO(GLq(2)) = O(Mq(2))[D−1\nq ],\nwhere the square brackets indicate adjoining.\nProposition 18.1. The coordinate ring O(GLq(2)) is a Hopf algebra, with\n∆(ui\nj) =\nX\nk\nui\nk ⊗uk\nj ,\nS(ui\nj) = 1\nDq\neui\nj = (U−1)i\nj,\nǫ(ui\nj) = δij,\nwhere eU = [ ̃ui\nj] is the cofactor matrix\neU =\n d\n−q−1b\n−qc\na"},{"paragraph_id":"p134","order":134,"text":".\n(defined so that U eU = DqI) and U−1 = ̃U/Dq is the inverse of U.\nThis is a non-commutative and non-co-commutative Hopf algebra.\nNow we go to the general n. Let Vq be the n-dimensional quantum space,\nthe q-deformation of V , with coordinates xi’s which satisfy\nxixj = qxjxi\n∀i < j.\n(18.3)\nLet Cq[V ] be the coordinate ring of Vq defined by\nCq[V ] = C[x1, . . . , xn]/ < xixj −qxjxi > .\n100"},{"paragraph_id":"p135","order":135,"text":"Let Mq(n) be the space of quantum n × n matrices, that is the set of linear\ntransformations on Vq which preserve (18.3) under the left as well as the\nright action. The left action is given by:"},{"paragraph_id":"p136","order":136,"text":"x1\n...\nxn"},{"paragraph_id":"p137","order":137,"text":"= x →Ux = x′,\nwhere U is the n × n generic quantum matrix. Similarly, the right action is\ngiven by:\nxT →xT U = (x′′)T .\nPreservation of (18.3) under the left and right actions means:\nx′\niy′\nj = qx′\njx′\ni,\nfor\ni < j\nx′′\ni y′′\nj = qx′′\njx′′\ni ,\nfor\ni < j.\nAfter straightforward calculations, these yield the following relations on\nthe entries uij = ui\nj of U:\nujkuik = q−1uikujk\n(i < j)\nukjuki = q−1ukiukj\n(i < j)\nujkuil= uilujk\n(i < j, k < l)\nujluik = uikujl−(q −q−1)ujkuil\n(i < j, k < l).\n(18.4)\nThe quantum determinant is defined as\nDq =\nX\nσ∈Sn\n(−q)l(σ)u\niσ(1)\nj1\n. . . u\niσ(n)\njn\n,\nwhere l(σ) denotes the length of the permutation σ, that is, the number of\ninversions in σ. This determinant formula is the same as the usual formula\nsubstituting (−q) for (−1).\nWe define the coordinate ring of the space Mq(n) of quantum n × n\nmatrices by\nO(Mq(n)) = C[U]/ < (18.4) >, and\nand the coordinate ring of the virtual quantum group GLq(n) by\nCq[G] = O(GLq(n)) = O(Mq(n))[D−1\nq ].\nWe define the quantum minors and, using these, the quantum co-factor\nmatrix eU and the quantum inverse matrix U−1 = eU/Dq in a straightforward\nfashion (these constructions are left as exercises).\n101"},{"paragraph_id":"p138","order":138,"text":"Theorem 18.1. The algebra O(GLq(n)) is a Hopf algebra, with\n∆(ui\nj) =\nX\nk\nui\nk ⊗uk\nj\nǫ(ui\nj) = δij\nS(ui\nj) = 1\nDq\neui\nj = (U−1)i\nj\nS(D−1\nq ) = Dq.\nWe also denote the quantum group GLq(n) by Gq, GLq(Cn) or GLq(V ).\nIt has to be emphasized that this is only a virtual object. Only its coordinate\nring Cq[G] is real. Henceforth, whenever we say representation or action of\nGq, we actually mean corepresentation or coaction of Cq[G], and so forth.\n102"},{"paragraph_id":"p139","order":139,"text":"Chapter 19\nQuantum unitary group\nScribe: Joshua A. Grochow\nGoal:\nDefine the quantum unitary subgroup of the standard quantum\ngroup.\nReference: [KS]\nRecall\nLet V = Cn, G = GLn(C) = GL(V ) = GL(Cn), and O(G) the coordinate\nring of G. The quantum group Gq = GLq(V ) is the virtual object whose\ncoordinate ring is\nO(Gq) = C[U]/⟨relations ⟩,\nwhere U is the generic n × n matrix of indeterminates, and the relations\nare the quadratic relations on the coordinates uj\ni defined in the last class so\nas to preserve the non-commuting relations among the coordinates of the\nquantum vector space Vq on which Gq acts. This coordinate ring is a Hopf\nalgebra.\n19.1\nA q-analogue of the unitary group\nIn this lecture we define a q-analogue of the unitary subgroup U = Un(C) =\nU(V ) ⊆GLn(C) = GL(V ) = G. This is a q-deformation Uq = Uq(V ) ⊆Gq\nof U(V ). Since Gq is only a virtual object, Uq will also be virtual. To define\nUq, we must determine how to capture the notion of unitarity in the setting\nof Hopf algebras. As we shall see, it is captured by the notion of a Hopf\n∗-algebra.\n103"},{"paragraph_id":"p140","order":140,"text":"Definition 19.1. A ∗-vector space is a vector space V with an involution\n∗: V →V satisfying\n(αv + βw)∗= αv∗+ βw∗\n(v∗)∗= v\nfor all v, w ∈V , and α, β ∈C.\nWe think of ∗as a generalization of complex conjugation; and in fact\nevery complex vector space is a ∗-vector space, where ∗is exactly complex\nconjugation.\nDefinition 19.2. A Hopf ∗-algebra is a Hopf algebra (A, ∆, ǫ, S) with an\ninvolution ∗: A →A such that (A, ∗) is a ∗-vector space, and:\n1. (ab)∗= b∗a∗, 1∗= 1\n2. ∆(a∗) = ∆(a)∗(where ∗acts diagonally on the tensor product A ⊗A:\n(v ⊗w)∗= (v∗⊗w∗))\n3. ǫ(a∗) = ǫ(a)\nThere is no explicit condition here on how ∗interacts with the antipode\nS.\nLet O(G) = C[G] be the coordinate ring of G as defined earlier.\nProposition 19.1. Then O(G) is a Hopf ∗-algebra.\nProof. We think of the elements in O(G) as C-valued functions on G and\ndefine ∗: O(G) →O(G) so that it satisfies the three conditions for a Hopf\n∗-algebra, and\n(4) For all f ∈O(G) and g ∈U ⊆G, f ∗(g) = f(g)\nLet uj\ni be the coordinate functions which, together with D−1, D = det(U),\ngenerate O(G). Because of the first condition on a Hopf ∗-algebra (relating\nthe involution ∗to multiplication), specifying (uj\ni)∗and D∗suffices to define\n∗completely. We define\n(uj\ni)∗= S(ui\nj) = (U−1)i\nj\nand D∗= D−1. We can check that this satifies (1)-(4). Here we will only\ncheck (4), and leave the remaining verification as an exercise. Let g be an\nelement of the unitary group U. Then (uj\ni)∗(g) = S(ui\nj)(g) = (g−1)i\nj = (g)j\ni,\nwhere the last equality follows from the fact that g is unitary (i.e. g−1 = g†,\nwhere † denotes conjugate transpose).\n104"},{"paragraph_id":"p141","order":141,"text":"Thus, we have defined a map f 7→f ∗purely algebraically in such a way\nthat the restriction of f ∗to the unitary group U is the same as taking the\ncomplex conjugate f on U.\nProposition 19.2. The coordinate ring Cq[G] = O(Gq) of the quantum\ngroup Gq = GLq(V ) is also a Hopf ∗-algebra.\nProof. The proof is syntactically identical to the proof for O(G), except\nthat the coordinate function uj\ni now lives in O(Gq) and the determinant D\nbecomes the q-determinant Dq. The definition of ∗is: (uj\ni)∗= S(ui\nj) and\nD∗\nq = D−1\nq , essentially the same as in the classical case.\nIntuitively, the “quantum subgroup” Uq of Gq is the virtual object such\nthat the restriction to Uq of the involution ∗just defined coincides with the\ncomplex conjugate.\n19.2\nProperties of Uq\nWe would like the nice properties of the classical unitary group to transfer\nover to the quantum unitary group, and this is indeed the case. Some of the\nnice properties of U are:\n1. It is compact, so we can integrate over U.\n2. we can do harmonic analysis on U (viz.\nthe Peter-Weyl Theorem,\nwhich is an analogue for U of the Fourier analysis on the circle U1).\n3. Every finite dimensional representation of U has a G-invariant Hermi-\ntian form, and thus a unitary basis – we say that every finite dimen-\nsional representation of U is unitarizable.\n4. Every finite dimensional representation X of U is completely reducible;\nthis follows from (3), since any subrepresentation W ⊆X has a per-\npendicular subrepresentation W ⊥under the G-invariant Hermitian\nform.\nCompactness is in some sense the key here. The question is how to define\nit in the quantum setting. Following Woronowicz, we define compactness to\nmean that every finite dimensional representation of Uq is unitarizable. Let\nus see what this means formally.\nLet A be a Hopf ∗-algebra, and W a corepresentation of A. Let ρ : W →\nW ⊗A be the corepresentation map. Let {bi} be a basis of W. Then, under\n105"},{"paragraph_id":"p142","order":142,"text":"ρ, bi 7→P\nj bj ⊗mj\ni for some mj\ni ∈A. We can thus define the matrix of\nthe (co)representation M = (mj\ni) in the basis {bi}. We define M∗such that\n(M∗)j\ni = (Mi\nj)∗. Thus, in the classical case (i.e. when q = 1), M∗= M†.\nWe say that the corepresentation W is unitarizable if it has a basis B =\n{bi} such that the corresponding matrix MB of corepresentation satisfies the\nunitarity condition: MBM∗\nB = I. In this case, we say B is a unitary basis\nof the corepresentation W.\nDefinition 19.3. A Hopf ∗-algebra A is compact if every finite dimensional\ncorepresentation of A is unitarizable.\nTheorem 19.1 (Woronowicz). The coordinte ring Cq[G] = O(Gq) is a\ncompact Hopf ∗-algebra. This implies that every finite dimensional repre-\nsentation of Gq, by which we mean a finite dimensional coorepresentation\nof Cq[G], is completely reducible.\nWoronowicz goes further to show that we can q-integrate on Uq, and that\nwe can do quantized harmonic analysis on Uq; i.e., a quantum analogue of\nthe Peter-Weyl theorem holds.\nNow that we know the finite dimensional representations of Gq are com-\npletely reducible, we can ask what the irreducible representations are.\n19.3\nIrreducible Representations of Gq\nWe proceed by analogy with the Weyl modules Vλ(G) for G. Recall that\nevery polynomial irreducible representation of G = GLn(C) is of this form.\nTheorem 19.2.\n1. For all partitions λ of length at most n, there exists\na q-Weyl module Vq,λ(Gq) which is an irreducible representation of Gq\nsuch that\nlim\nq→1 Vq,λ(Gq) = Vλ(G).\n2. The q-Weyl modules give all polynomial irreducible representations of\nGq.\n19.4\nGelfand-Tsetlin basis\nTo understand the q-Weyl modules better, we wish to get an explicit basis for\neach module Vq,λ. We begin by defining a very useful basis – the Gel’fand-\nTestlin basis – in the classical case for Vλ(G), and then describe the q-\nanalogue of this basis.\n106"},{"paragraph_id":"p143","order":143,"text":"By Pieri’s rule [FH]\nVλ(GLn(C)) =\nM\nλ′\nVλ′(GLn−1(C))\nwhere the sum is taken over all λ′ obtained from λ by removing any number\nof boxes (in a legal way) such that no two removed boxes come from the\nsame column. This is an orthogonal decomposition (relative to the GLn(C)-\ninvariant Hermitian form on Vλ) and it is also multiplicity-free, i.e., each Vλ′\nappears only once.\nFix a G-invariant Hermitian form on Vλ. Then the Gel’fand-Tsetlin basis\nfor Vλ(GLn(C)), denoted GT n\nλ , is the unique orthonormal basis for Vλ such\nthat\nGT n\nλ =\n[\nλ′\nGT n−1\nλ′\n,\nwhere the disjoint union is over the λ′ as in Pieri’s rule, and GT n−1\nλ′\nis defined\nrecursively, the case n = 1 being trivial.\nThe dimension of Vλ is the number of semistandard tableau of shape λ.\nWith any tableau T of this shape, one can also explicitly associate a basis\nelement GT(T) ∈GT n\nλ ; we shall not worry about how.\nWe can define the Gel’fand-Tsetlin basis GT n\nq,λ for Vq,λ(Gq(Cn)) analo-\ngously. We have the q-analogue of Pieri’s rule:\nVq,λ(Gq(Cn)) =\nM\nλ′\nVq,λ′(Gq(Cn−1))\nwhere the decomposition is orthogonal and multiplicity-free, and the sum\nranges over the same λ′ as above. So we can define GT n\nq,λ to be the unique\nunitary basis of Vq,λ such that\nGT n\nq,λ =\n[\nλ′\nGT n−1\nq,λ′ .\nWith any semistandard tableau T, one can also explicitly associate a basis\nelement GTq(T) ∈GT n\nq,λ′; details omitted.\n107"},{"paragraph_id":"p144","order":144,"text":"Chapter 20\nTowards positivity\nhypotheses via quantum\ngroups\nScribe: Joshua A. Grochow\nGoal: In this final brisk lecture, we indicate the role of quantum groups in\nthe context of the positivity hypothesis PH1. Specifically, we sketch how the\nLittlewood-Richardson rule – the gist of PH1 in the Littlewood-Richardson\nproblem – follows from the theory of standard quantum groups. We then\nbriefly mention analogous (nonstandard) quantum groups for the Kronecker\nand plethysm problems defined in [GCT4, GCT7], and the theorems and\nconjectures for them that would imply PH1 for these problems.\nReferences: [KS, K, Lu2, GCT4, GCT6, GCT7, GCT8]\nLet V = Cn, G = GLn(C) = GL(V ), Vλ = Vλ(G) a Weyl module of\nG, Gq = GLq(V ) the standard quantum group, Vq the q-deformation of V\non which GLq(V ) acts, Vq,λ = Vq,λ(Gq) the q-deformation of Vλ(G), and\nGTq,λ = GT n\nq,λ the Gel’fand-Tsetlin basis for Vq,λ.\n20.1\nLittlewood-Richardson rule via standard quan-\ntum groups\nWe now sketch how the Littlewood-Richardson rule falls out of the standard\nquantum group machinery, specifically the properties of the Gelfand-Tsetlin\nbasis.\n108"},{"paragraph_id":"p145","order":145,"text":"20.1.1\nAn embedding of the Weyl module\nFor this, we have to embed the q-Weyl module Vq,λ in V ⊗d\nq\n, where d = |λ| =\nP λi is the size of λ. We first describe how to embed the Weyl module Vλ\nof G in V ⊗d in a standard way that can be quantized.\nIf d = 1, then Vλ(G) = V = V ⊗1. Otherwise, obtain a Young diagram\nμ from λ by removing its top-rightmost box that can be removed to get a\nvalid Young diagram, e.g.:\nx\n⇝\nλ\nμ\nIn the following, the box must be removed from the second row, since\nremoving from the first row would result in an illegal Young diagram:\nx\n⇝\nλ\nμ\nBy induction on d, we have a standard embedding Vμ(G) ֒→V ⊗d−1. This\ngives us an embedding Vμ(G) ⊗V ֒→V ⊗d. By Pieri’s rule [FH]\nVμ(G) ⊗V =\nM\nβ\nVβ(G),\nwhere the sum is over all β obtained from μ by adding one box in a legal way.\nIn particular, Vλ(G) ⊂Vμ(G) ⊗V . By restricting the above embedding, we\nget a standard embedding Vλ(G) ֒→V ⊗d.\nNow Pieri’s rule also holds in a quantized setting:\nVq,μ ⊗Vq =\nM\nβ\nVq,β(G),\nwhere β is as above.\nHence, the standard embedding Vλ ֒→V ⊗d above\ncan be quantized in a straightforward fashion to get a standard embedding\nVq,λ ֒→V ⊗d\nq\n. We shall denote it by ρ. Here the tensor product is meant to\nbe over Q(q). Actually, Q(q) doesn’t quite work. We have to allow square\nroots of elements of Q(q), but we won’t worry about this. For a semistandard\ntableau b of shape λ, we denote the image of a Gelfand-Tsetlin basis element\nGTq,λ(b) ∈GTq,λ under ρ by GT ρ\nq,λ(b) = ρ(GTq,λ(b)) ∈V ⊗d\nq\n.\n109"},{"paragraph_id":"p146","order":146,"text":"20.1.2\nCrystal operators and crystal bases\nTheorem 20.1 (Crystallization). [DJM] The Gelfand-Tsetlin basis ele-\nments crystallize at q = 0. This means:\nlim\nq→0 GT ρ\nq,λ(b) = vi1(b) ⊗· · · ⊗vid(b),\n(20.1)\nfor some integer functions i1(b), . . . , id(b), and\nlim\nq→∞GT ρ\nq,λ(b) = vj1(b) ⊗· · · ⊗vjd(b),\n(20.2)\nfor some integer functions j1(b), . . . , jd(b).\nThe phenomenon that these limits consists of monomials, i.e., simple\ntensors is known as crystallization. It is related to the physical phenomenon\nof crystallization, hence the name. The maps b 7→i(b) = (i1(b), . . . , id(b))\nand b 7→j(b) = (j1(b), . . . , jd(b)) are computable in poly(⟨b⟩) time (where\n⟨b⟩is the bit-length of b).\nNow we want to define a special crystal basis of Vq,λ based on this phe-\nnomenon of crystallization. Towards that end, consider the following family\nof n × n matrices:\nEi ="},{"paragraph_id":"p147","order":147,"text":"0\n0\n· · ·\n0\n...\n...\n...\n0\n1\n· · ·\n0\n0\n· · ·\n0\n...\n...\n0"},{"paragraph_id":"p148","order":148,"text":",\nwhere the only nonzero entry is a 1 in the i-th row and (i + 1)-st column.\nLet Fi = ET\ni . Corresponding to Ei and Fi, Kashiwara associates certain\noperators ˆEi and ˆFi on Vq,λ(Gq). We shall not worry about their actual\nconstruction here (for the readers familiar with Lie algebras: these are closely\nrelated to the usual operators in the Lie algebra of G associated with Ei and\nFi).\nIf we let ˆEi act on GT ρ\nq,λ(b), we get some linear combination\nˆEi(GT ρ\nq,λ(b)) =\nX\nb′\nab\nb′(q)GT ρ\nq,λ(b′),\nwhere ab\nb′(q) ∈Q(q) (actually an algebraic extension of Q(q) as mentioned\nabove). Essentially because of crystallization (Theorem 20.1), it turns out\n110"},{"paragraph_id":"p149","order":149,"text":"that limq→0 ab\nb′(q) is always either 0 or 1, and for a given b, this limit is 1 for\nat most one b′, if any. A similar result holds for ˆFi(GT ρ\nq,λ(b)). This allows\nus to define the crystal operators (due to Kashiwara):\neei · b =\n b′\nif limq→0 ab\nb′(q) = 1,\n0\nif no such b′ exists,\nand similarly for efi. Although these operators are defined according to a\nparticular embedding Vq,λ ֒→V ⊗d\nq\nand a basis, they can be defined intrinsi-\ncally, i.e., without reference to the embedding or the Gel’fand-Tsetlin basis.\nNow, let W be a finite-dimensional representation of Gq, and R the\nsubring of functions in Q(q) regular at q = 0 (i.e. without a pole at q = 0).\nA lattice within W is an R-submodule of W such that Q(q) ⊗R L = W.\n(Intuition behind this definition: R ⊂Q(q) is analogous to Z ⊂Q. A lattice\nin Rn is a Z-submodule L of Rn such that R ⊗Z L = Rn.)\nDefinition 20.1. An (upper) crystal basis of a representation W of Gq is\na pair (L, B) such that\n• L is a lattice in W preserved by the Kashiwara operators ˆEi and ˆFi,\ni.e. ˆEi(L) ⊆L and ˆFi(L) ⊆L.\n• B is a basis of L/qL preserved by the crystal operators eei and efi, i.e.,\neei(B) ⊆B ∪{0} and efi(B) ⊆B ∪{0}.\n• The crystal operators eei and efi are inverse to each other wherever\npossible, i.e., for all b, b′ ∈B, if eei(b) = b′ ̸= 0 then efi(b′) = b, and\nsimilarly, if efi(b) = b′ ̸= 0 then eei(b′) = b.\nIt can be shown that if W = Vq,λ(Gq), then there exists a unique b ∈B\nsuch that eei(b) = 0 for all i; this corresponds to the highest weight vector\nof Vq,λ (the weight vectors in Vq,λ are analogous to the weight vectors in Vλ;\nwe do not give their exact definition here). By the work of Kashiwara and\nDate et al [K, DJM] above, the Gel’fand-Tsetlin basis (after appropriate\nrescaling) is in fact a crystal basis: just let\nL = LGT\n=\nthe R-module generated by GTq,λ, and\nBGT\n=\nGTq,λ(b),\nwhere GTq,λ(b) is the image under the projection L 7→L/qL of the set of\nbasis vectors in GTq,λ(b).\n111"},{"paragraph_id":"p150","order":150,"text":"Theorem 20.2 (Kashiwara).\n1. Every finite-dimensional Gq-module has\na unique crystal basis (up to isomorphism).\n2. Let (Lλ, Bλ) be the unique crystal basis corresponding to Vq,λ. Then\n(Lα, Bα) ⊗(Lβ, Bβ) = (Lα ⊗Lβ, Bα ⊗Bβ) is the unique crystal basis\nof Vq,α ⊗Vq,β, where Bα ⊗Bβ denotes {ba ⊗bb|ba ∈Bα, bb ∈Bβ}.\nIt can be shown that every b ∈Bλ has a weight; i.e., it is the image of a\nweight vector in Lλ under the projection Lλ →Lλ/qLλ.\nNow let us see how the Littlewood-Richardson rule falls out of the prop-\nerties of the crystal bases. Recall that the specialization of Vq,α at q = 1 is\nthe Weyl module Vα of G = GLn(C), and\nVα ⊗Vβ =\nM\nγ\ncγ\nα,βVγ\n(20.3)\nwhere cγ\nα,β are the Littlewood-Richardson coefficients.\nThe Littlewood-\nRichardson rule now follows from the following fact:\ncγ\nα,β = #{b ⊗b′ ∈Bα ⊗Bβ|∀i, eei(b ⊗b′) = 0 and b ⊗b′ has weight γ}.\nIntuitively, b ⊗b′ here correspond to the highest weight vectors of the G-\nsubmodules of Vα ⊗Vβ isomorphic to Vγ.\n20.2\nExplicit decomposition of the tensor product\nThe decomposition (20.3) is only an abstract decomposition of Vα ⊗Vβ as a\nG-module. Next we consider the explicit decomposition problem. The goal\nis to find an explicit basis B = Bα⊗β of Vα ⊗Vβ that is compatible with this\nabstract decomposition. Specifically, we want to construct an explicit basis\nB of Vα ⊗Vβ in terms of suitable explicit bases of Vα and Vβ such that B\nhas a filtration\nB = B0 ⊇B1 ⊇· · · ⊇∅\nwhere each ⟨Bi⟩/⟨Bi+1⟩is an irreducible representation of G and ⟨Bi⟩denotes\nthe linear span of Bi.\nFurthermore, each element b ∈B should have a\nsufficiently explicit representation in terms of the basis Bα ⊗Bβ of Vα ⊗Vβ.\nThe explicit decomposition problem for the q-analogue Vq,α ⊗Vq,β is similar.\nFor example, we have already constructed explicit Gelfand-Tsetlin bases\nof Weyl modules. But it is not known how to construct an explicit basis B\n112"},{"paragraph_id":"p151","order":151,"text":"with filtration as above in terms of the Gelfand-Tsetlin bases of Vα and Vβ\n(except when the Young diagram of either α or β is a single row).\nKashiwara and Lusztig [K, Lu2] construct certain canonical bases Bq,α\nand Bq,β of Vq,α and Vq,β, and Lusztig furthermore constructs a canonical\nbasis Bq = Bq,α⊗β of Vq,α ⊗Vq,β such that:\n1. Bq has a filtration as above,\n2. Each b ∈Bq has an expansion of the form\nb =\nX\nbα∈Bq,α,bβ∈Bq,β\nabα,bβ\nb\nbα ⊗bβ,\nwhere each abα,bβ\nb\nis a polynomial in q and q−1 with nonnegative inte-\ngral coefficients,\n3. Crystallization: For each b, as q →0, exactly one coefficient abα,bβ\nb\n→1,\nand the remaining all vanish.\nThe proof of nonnegativity of the coefficients of abα,bβ\nb\nis based on the Rie-\nmann hypothesis (theorem) over finite fields [Dl2], and explicit formulae for\nthese coefficients are known in terms of perverse sheaves [BBD] (which are\ncertain types of algebro-geometric objects).\nThis then provides a satisfactory solution to the explicit decomposition\nproblem, which is far harder and deeper than the abstract decomposition\nprovided by the Littlewood-Richardson rule. By specializing at q = 1, we\nalso get a solution to the explicit decomposition problem for Vα ⊗Vβ. This\n(i.e. via quantum groups) is the only known solution to the explicit decom-\nposition problem even at q = 1. This may give some idea of the power of\nthe quantum group machinery.\n20.3\nTowards nonstandard quantum groups for the\nKronecker and plethysm problems\nNow the goal is to construct quantum groups which can be used to de-\nrive PH1 and explicit decomposition for the Kronecker and plethysm prob-\nlems just as the standard quantum group can be used for the same in the\nLittlewood-Richardson problem.\nIn the Kronecker problem, we let H = GL(Cn) and G = GL(Cn ⊗Cn).\nThe Kronecker coefficient κγ\nα,β is the multiplicity of Vα(H)⊗Vβ(H) in Vγ(G):\nVγ(G) =\nM\nα,β\nκγ\nα,βVα(H) ⊗Vβ(H).\n113"},{"paragraph_id":"p152","order":152,"text":"The goal is to get a positive # P-formula for κγ\nα,β; this is the gist of PH1\nfor the Kronecker problem.\nIn the plethysm problem, we let H = GL(Cn) and G = GL(Vμ(H)).\nThe plethysm constant aπ\nλ,μ is the multiplicity of Vπ(H) in Vλ(G):\nVλ(G) =\nM\nπ\naπ\nλ,μVπ(H).\nAgain, the goal is to get a positive # P-formula for the plethysm constant;\nthis is the gist of PH1 for the plethysm problem.\nTo apply the quantum group approach, we need a q-analogue of the\nembedding H ֒→G. Unfortunately, there is no such q-analogue in the theory\nof standard quantum groups. Because there is no nontrivial quantum group\nhomomorphism from the standard quantum group Hq = GLq(Cn) and to\nthe standard quantum group Gq.\nTheorem 20.3. (1) [GCT4]: Let H and G be as in the Kronecker problem.\nThen there exists a quantum group ˆGq such that the homomorphism H →G\ncan be quantized in the form Hq ֒→ˆGq. Furthermore, ˆGq has a unitary\nquantum subgroup ˆUq which corresponds to the maximal unitary subgroup\nU ⊆G, and a q-analogue of the Peter-Weyl theorem holds for ˆGq. The\nlatter implies that every finite dimensional representation of ˆGq is completely\ndecomposible into irreducibles.\n(2) [GCT7] There is an analogous (possibly singular) quantum group ˆGq\nwhen H and G are as in the plethysm problem. This also holds for general\nconnected reductive (classical) H.\nSince the Kronecker problem is a special case of the (generalized) plethysm\nproblem, the quantum group in GCT 4 is a special case of the quantum group\nin GCT 7. The quantum group in the plethysm problem can be singular, i.e.,\nits determinant can vanish and hence the antipode need not exist. We still\ncall it a quantum group because its properties are very similar to those of the\nstandard quantum group; e.g. q-analogue of the Peter-Weyl theorem, which\nallows q-harmonic analysis on these groups. We call the quantum group\nˆGq nonstandard, because though it is qualitatively similar to the standard\n(Drinfeld-Jimbo) quantum group Gq, it is also, as expected, fundamentally\ndifferent.\nThe article [GCT8] gives a conjecturally correct algorithm to construct a\ncanonical basis of an irreducible polynomial representation of ˆGq which gen-\neralizes the canonical basis for a polynomial representation of the standard\nquantum group as per Kashiwara and Lusztig. It also gives a conjecturally\n114"},{"paragraph_id":"p153","order":153,"text":"correct algorithm to construct a canonical basis of a certain q-deformation of\nthe symmetric group algebra C[Sr] which generalizes the Kazhdan-Lusztig\nbasis [KL] of the Hecke algebra (a standard q-deformation of C[Sr]). It is\nshown in [GCT7, GCT8] that PH1 for the Kronecker and plethysm problems\nfollows assuming that these canonical bases in the nonstandard setting have\nproperties akin to the ones in the standard setting. For a discussion on SH,\nsee [GCT6].\n115"},{"paragraph_id":"p154","order":154,"text":"Part II\nInvariant theory with a view\ntowards GCT\nBy Milind Sohoni\n116"},{"paragraph_id":"p155","order":155,"text":"Chapter 21\nFinite Groups\nReferences: [FH, N]\n21.1\nGeneralities\nLet V be a vector space over C, and let GL(V ) denote the group of all\nisomorphisms on V . For a fixed basis of V , GL(V ) is isomorphic to the\ngroup GLn(C), the group of all n × n invertible matrices.\nLet G be a group and ρ : G →GL(V ) be a representation. We also\ndenote this by the tuple (ρ, V ) or say that V is a G-module. Let Z ⊆V be\na subspace such that ρ(g)(Z) ⊆Z for all g ∈G. Then, we say that Z is\nan invariant subspace. We say that (ρ, V ) is irreducible if there is no\nproper subspace W ⊂V such that ρ(g)(W) ⊆W for all g ∈G. We say that\n(ρ, V ) is indecomposable is there is no expression V = W1 ⊕W2 such that\nρ(g)(Wi) ⊆Wi, for all g ∈G.\nFor a point v ∈V , the orbit O(v), and the stabilizer Stab(v) are\ndefined as:\nO(v)\n=\n{v′ ∈V |∃g ∈G with ρ(g)(v) = v′}\nStab(v)\n=\n{g ∈G|ρ(g)(v) = v}\nOne may also define v ∼v′ if there is a g ∈G such that ρ(g)(v) = v′. It is\nthen easy to show that [v]∼= O(v).\nLet V ∗be the dual-space of V . The representation (ρ, V ) induces the\ndual representation (ρ∗, V ∗) defined as ρ∗(v∗)(v) = v∗(ρ(g−1)(v)). It will\nbe convenient for ρ∗to act on the right, i.e., ((v∗)(ρ∗))(v) = v∗(ρ(g−1)(v)).\nWhen ρ is fixed, we abbrieviate ρ(g)(v) as just g · v. Along with this,\nthere are the standard constructions of the tensor T d(V ), the symmetric\npower Symd(V ) and the alternating power ∧d(V ) representations.\n117"},{"paragraph_id":"p156","order":156,"text":"Of special significance is Symd(V ∗), the space of homogeneous polyno-\nmial functions on V of degree d. Let dim(V ) = n and let X1, . . . , Xn be a\nbasis of V ∗. We define as follows:\nR = C[X1, . . . , Xn] = ⊕∞\nd=0Rd = ⊕∞\nd=0Symd(V ∗)\nThus R is the ring of all polynomial functions on V and is isomorphic\nto the algebra (over C) of n indeterminates. Since G acts on the domain V ,\nG also acts on all functions f : V →C as follows:\n(f · g)(v) = f(g−1 · v)\nThis action of G on all functions extends the action of G on polynomial\nfunctions above.\nIndeed, for any g ∈G, the map tg : R →R given by\nf →f · g is an algebra isomorphism. This is called the translation map.\nFor an f ∈R, we say that f is an invariant if f · g = f for all g ∈G.\nThe following are equivalent:\n• f ∈R is an invariant.\n• Stab(f) = G.\n• f(g · v) = f(v) for all g ∈G and v ∈V .\n• For all v, v′ such that v′ ∈Orbit(v), we have f(v) = f(v′).\nIf W1 and W2 are two modules of G and φ : W1 →W2 is a linear map\nsuch that g · φ(w1) = φ(g · w1) for all g ∈G and w1 ∈W1 then we say that\nφ is G-equivariant or that φ is a morphism of G-modules.\n21.2\nThe finite group action\nLet G be a finite group and (μ, W) be a representation.\nRecall that a complex inner product on W is a map h : W × W →C\nsuch that:\n• h(αw + βw′, w′′) = αh(w, w′′) + βh(w′, w′′) for all α, β ∈C and all\nw, w′, w′′ ∈W.\n• h(w′′, αw + βw′) = αh(w′′, w) + βh(w′′, w′) for all α, β ∈C and all\nw, w′, w′′ ∈W.\n• h(w, w) > 0 for all w ̸= 0.\n118"},{"paragraph_id":"p157","order":157,"text":"Also recall that if Z ⊆W is a subspace, then Z⊥is defined as:\nZ⊥= {w ∈W|h(w, z) = 0 ∀z ∈Z}\nAlso recall that W = Z ⊕Z⊥.\nWe say that an inner product h is G-invariant if h(g·w, g·w′) = h(w, w′)\nfor all w, w′ ∈W and g ∈G.\nProposition 21.1. Let W be as above, and Z be an invariant subspace of\nW. Then Z⊥is also an invariant subspace. Thus every reducible represen-\ntation of G is also decomposable.\nProof: Let x ∈Z⊥, z ∈Z and let us examine (g · x, z). Applying g−1 to\nboth sides, we see that:\nh(g · x, z) = h(g−1 · g · x, g−1 · z) = h(x, g−1 · z) = 0\nThus, G preserves Z⊥as claimed. □\nLet h be a complex inner product on W. We define the inner product\nhG as follows:\nhG(w, w′) =\n1\n|G|\nX\ng′∈G\nh(g′ · w, g′ · w′)\nLemma 21.1. hG is a G-invariant inner product.\nProof: First we see that\nhG(w, w) =\n1\n|G|\nX\ng′∈G\nh(w′, w′)\nwhere w′ = g′ · w. Thus hG(w, w) > 0 unless w = 0. Secondly, by the\nlinearity of the action of G, we see that hG is indeed an inner product.\nFinally, we see that:\nhG(g · w, g · w′) =\n1\n|G|\nX\ng′∈G\nh(g′ · g · w, g′ · g · w′)\nSince as g′ ranges over G, so will g′ · g for any fixed g, we have that hG is\nG-invariant. □\nTheorem 1.\n• Let G be a finite group and (ρ, V ) be an indecomposable\nrepresentation, then it is also irreducible.\n119"},{"paragraph_id":"p158","order":158,"text":"• Every representation (ρ, V ) may be decomposed into irreducible repre-\nsentations Vi. Thus V = ⊕iVi, where (ρi, Vi) is an irreducible repre-\nsentation.\nProof: Suppose that Z ⊆V is an invariant subspace, then V = Z ⊕Z⊥is\na non-trivial decomposition of V contradicting the hypothesis. The second\npart is proved by applying the first, recursively. □\nWe have seen the operation of averaging over the group in going\nfrom the inner product h to the G-invariant inner product hG. A similar\napproach may be used for constructing invariant polynomials functions. So\nlet p(X) ∈R = C[X1, . . . , Xn] be a polynomial function. We define the\nfunction pG : V →C as:\npG(v) =\n1\n|G|\nX\ng∈G\np(g · v)\nThe transition from p to pG is called the Reynold’s operator.\nProposition 21.2. Let p ∈R be of degree atmost d, then pG is also a\npolynomial function of degree atmost d. Next, pG is an invariant.\nLet RG denote the set of all invariant polynomial functions on the space\nV . It is easy to see that RG ⊆R is actually a subring of R.\nLet Z ⊆V be an arbitrary subset of V . We say that Z is G-closed if\ng ·z ∈Z for all g ∈G and z ∈Z. Thus Z is a union of orbits of points in V .\nLemma 21.2. Let p ∈RG be an invariant and let Z = V (p) be the variety\nof p. Then Z is G-closed.\nWe have already seen that O(v), the orbit of v arises from the equivalence\nclass ∼on V . Since the group is finite, |O(v)| ≤|G| for any v. Let O1 and\nO2 be disjoint orbits.\nIt is essential to determine if elements of RG can\nseparate O1 and O2.\nLemma 21.3. Let O1 and O2 be as above, and I1 and I2 be their ideals in\nR. Then there are p1 ∈I1 and p2 ∈I2 so that p1 + p2 = 1.\nProof: This follows from the Hilbert Nullstellensatz. Since the point sets\nare finite, there is an explicit construction based on Lagrange interpolation.\n□\nLet G be a finite group and (ρ, V ) be a representation as above. We\nhave see that this induces an action on C[X1, . . . , Xn]. Also note that this\naction is homogeneous: for a g ∈G and p ∈Rd, we have that p · g ∈Rd\n120"},{"paragraph_id":"p159","order":159,"text":"as well. Thus RG, the ring of invariants, is a homogeneous subring of R. In\nother words:\nRG = ⊕∞\nd=0RG\nd\nwhere RG\nd are invariants which are homogeneous of degree d. The existence\nof the above decomposition implies that every invariant is a sum of homo-\ngeneous invariants. Now, since RG\nd ⊆Rd as a vector space over C. Thus\ndimC(RG\nd ) ≤dimC(Rd) ≤\n n + d −1\nn −1"},{"paragraph_id":"p160","order":160,"text":"We define the hilbert function h(RG) of RG (or for that matter, of\nany homogeneous ring) as:\nh(RG) =\n∞\nX\nd=0\ndimC(RG\nd )zd\nWe will see now that h(RG) is actually a rational function which is easily\ncomputed. We need a lemma.\nLet (ρ, W) be a representation of the finite group G. Let\nW G = {w ∈W|g · w = w}\nbe the set of all vector invariants in W, and this is a subspace of W. .\nLemma 21.4. Let (ρ, W) be as above. We have:\ndimC(W G) =\n1\n|G|\nX\ng∈G\ntrace(ρ(g))\nProof: Define P =\n1\n|G|\nP\ng∈G ρ(g), as the average of the representation\nmatrices. We see that ρ(g) · P = P · ρ(g) and that P 2 = P. Thus P is\ndiagonalizable and the eigenvalues of P are in the set {1, 0}. Let W1 and\nW0 be the corresponding eigen-spaces. It is clear that W G ⊆W1 and that\nW1 is fixed by each g ∈G. We now argue that every w ∈W1 is actually an\ninvariant. For that, let wg = g · w. We then have that Pw = w implies that\nw =\n1\n|G|\nX\ng∈G\nwg\nNote that a change-of-basis does not affect the hypothesis nor the assertion.\nWe may thus assume that each ρ(g) is unitary, we have that wg = w for all\ng ∈G. Now, the claim follows by computing trace(P). □\nWe are now ready to state Molien’s Theorem:\n121"},{"paragraph_id":"p161","order":161,"text":"Theorem 2. Let (ρ, W) be as above. We have:\nh(RG) =\n1\n|G|\nX\ng∈G\n1\ndet(I −zρ(g))\nProof: Let dimC(W) = n and let {X1, . . . , Xn} be a basis of W ∗. Since\nRG = P\nd RG\nd and each RG\nd ⊆C[X1, . . . , Xn]d. Note that each C[X1, . . . , Xn]d\nis also a representation ρd of G.\nFurthermore, it is easy to see that if\n{λ1, . . . , λn} are the eigenvalues of ρ(g), then the eigen-values of the matrix\nρd(g) are precisely (including multiplicity)\n{\nY\ni\nλdi\ni |\nX\ni\ndi = d}\nThus\ntrace(ρd(g)) =\nX\nd:|d|=d\nY\ni\nλdi\ni\nWe then have:\nh(RG)\n=\nP\nd zddimC(RG\nd )\n=\nP\nd zd[ 1\n|G|\nP\ng trace(ρd(g))]\n=\n1\n|G|\nP\ng\n1\n(1−λ1(g)z)...(1−λn(g)z)\n=\n1\n|G|\nP\ng\n1\ndet(I−zρ(g))\nThis proves the theorem. □\n21.3\nThe Symmetric Group\nSn will denote the symmetric group of all bijections on the set [n]. The\nstandard representation of Sn is obviously on V = Cn with\nσ · (v1, . . . , vn) = (vσ(1), . . . , vσ(n))\nThus, regarding V as column vectors, and Sn as the group of n × n-\npermutation matrices, we see that the action of permutation P on vector v\nis given by the matrix multiplication P · v.\nLet X1, . . . , Xn be a basis of V ∗.\nSn acts on R = C[X1, . . . , Xn] by\nXi · σ = Xσ(i). The orbit of any point v = (v1, . . . , vn) is the collection of\nall permutation of the entries of the vector v and thus the size of the orbit\nis bounded by n!.\n122"},{"paragraph_id":"p162","order":162,"text":"The invariants for this action are given by the elementary symmetric\npolynomials ek(X), for k = 1, . . . , n, where\nek(X) =\nX\ni1<i2<...<ik\nXi1Xi2 . . . Xik\nGiven two vector v and w, if w ̸∈O(v), then there is a k such that\nek(v) ̸= ek(w). This follows from the theory of equations in one variable.\nThe ring RG equals C[e1, . . . , en] and has no algebraic dependencies. The\nhilbert function of RG may then be expressed as:\nh(RG) =\n1\n(1 −z)(1 −z2) . . . (1 −zn)\nIt is an exercise to verify that Molien’s expression agrees with the above.\nA related action of Sn is the diagonal action: Let X = {X1, . . . , Xn},\nY = {Y1, . . . , Yn}, and so on upto W = {W1, . . . , Wn} be a family of r\n(disjoint) variables. Let B = C[X, Y, . . . , W] be the ring of polynomials in\nthe variables of the disjoint union.\nWe define the action of Sn on X ∪Y ∪. . . ∪W as Xi · σ = Xσ(i),\nYi · σ = Yσ(i), and so on.\nThe matrix equivalence of this action is the action of the permutation\nmatrices on n × r matrices A, where the action of P on A is given by P · A.\nThe invariants BG is obtained from the r = 1 case by a curious operation:\nLet DXY denote the operator:\nDXY = Y1\n∂\n∂X1\n+ . . . + Yn\n∂\n∂Xn\nThe ring BG is obtained from RG by applying the operators DXY , DXW , DW X\nand so on, to elements of RG. As an example, we have\ne2(X) = X1X2 + X1X3 + . . . + Xn−1Xn\nWe have DXY (e2) as:\nDXY (e2(X)) =\nX\ni̸=j\nXiYj\nThis is clearly an element of BG.\n123"},{"paragraph_id":"p163","order":163,"text":"Chapter 22\nThe Group SLn\nReferences: [FH, N]\n22.1\nThe Canonical Representation\nLet V be a vector space of dimension n, and let x1, . . . , xn be a basis for V .\nLet X1, . . . , Xn be the dual basis of V ∗.\nSL(V ) will denote the group of all unimodular linear transformations on\nV . In the above basis, this group is isomorphic to that of all n × n matrices\nof determinant 1, or in other words SLn(C).\nThe standard representation of SL(V ) is obviously V itself: Given φ ∈\nSL(V ) and v ∈V , we have φ · v = φ(v) is the action of φ on v.\nIn terms of the basis x above we may write v = [x1, . . . , xn][α1, . . . , αn]T\nand thus φ·v as [φ·x1, . . . , φ·xn][α1, . . . , αn]T . If φ·xi = [x1, . . . , xn][a1i, . . . , ani]T ,\nthen we have\nφ · v = [x1, . . . , xn]"},{"paragraph_id":"p164","order":164,"text":"a11\n. . .\na1n\n...\n...\nan1\n. . .\nann"},{"paragraph_id":"p165","order":165,"text":"α1\n...\nαn"},{"paragraph_id":"p166","order":166,"text":"We denote this matrix as Aφ.\nWe may now work with SLn(C) or simply SLn. Given a vector a =\n[α1, . . . , αn]T , we see that the matrix multiplication A · a is the action of A\non the column vector a.\nLet us now understand the orbits of typical elements in the column space\nCn. The typical v ∈Cn is a non-zero column vector. For any non-zero vector\nw, we see that there is an element A ∈SLn such that w = Av. Furthermore,\n124"},{"paragraph_id":"p167","order":167,"text":"for any B ∈SLn, clearly Bv ̸= 0. Thus we see that Cn has exactly two\norbits:\nO0 = {0} O1{v ∈Cn|v ̸= 0}\nNote that O1 is dense in V = Cn and its closure includes the orbit O0 and\ntherefore, the whole of V .\nLet R = C[X1, . . . Xn] be the ring of polynomial functions on Cn. We\nexamine the action of SLn on C[X]. Recall that the action of A on X should\nbe such that the evaluations Xi(xj) = δij must be conserved. Thus if the\ncolumn vectors xi = [0, . . . , 0, 1, 0 . . . , 0]T are the basis vectors of V and the\nrow vectors Xi = [0, . . . , 0, 1, 0 . . . , 0] that of V ∗, then a matrix A ∈SLn\ntransforms xi to Axi and Xj to XjA−1. Thus Xj(xi) = Xj/cdotxi goes to\nXjA−1Axi = Xj · xi.\nNext, we examine R = C[X] for invariants. First note that the action of\nSLn is homogeneous and thus we may assume that an invariant p is actually\nhomogeneous of degree d. Next we see that p must be constant on O1 and\nO0. In this case, if p(O1) = α then by the density of O1 in V , we see that p\nis actually constant on V . Thus RG = R0 = C.\n22.2\nThe Diagonal Representation\nLet us now consider the diagonal representation of the above representation.\nIn other words, let V r be the space of all complex n × r matrices x. The\naction of A on x is obvious A · x. Let X be the r × n-matrix dual to x. It\ntransforms according to X →XA−1.\nLet us examine the case when r < n and compute the orbits in V r. Let\nx ∈V r be a matrix and y be a column vector such that x · y = 0. We see\nthat (A · x) · y = 0 as well. Thus if ann(x) = {y ∈Cr|x · y = 0} is the\nannihilator space of x, then we see that ann(x) = ann(A · x).\nWe show that the converse is also true:\nProposition 22.1. Let r < n and x, x′ ∈V r be such that ann(x) = ann(x′).\nThen there is an A ∈SLn such that x = A · x′.\nProof: Use the row-echelon form construction. Make the pivots as 1 using\nappropriate diagonal matrices. Since r < n these can be chosen from SLn.\n□\nThis decides the orbit structure of V r for r < n. The largest orbit Or is\nof course when ann(x) = 0. Thus, when x ∈Vr is a mtrix of rank r, we see\nthat ann(x) = 0 is trivial. The generic element of V r is of this form. Thus\nOr is dense in V r. For this reason, there are no non-constant invariants.\n125"},{"paragraph_id":"p168","order":168,"text":"Another calculation is the computation of the closure of orbits. Let O, O′\nbe two arbitrary orbits of V r. We have:\nProposition 22.2. O′ lies in the closure of O if and only if ann(O′) ⊇\nann(O).\nProof: One direction is clear. We prove the other direction when ann(O) ⊆\nann(O′) and dim(ann(O)) = dim(ann(O′)) −1.\nThus, we may assume\nthat O = [x] and O′ = [x′] with both x and x′ such that rowspace(x) ⊇\nrowspace(x′) with rank(x′) = rank(x)−1. Then, upto SLn, we may assume\nthat the rows x′[1], . . . , x′[k] match the first k rows of x, and that x′[k+1] =\nx′[k + 2] = . . . = x′[n] = 0. Note that x[k + 1] is non-zero and x[k + 2] exists\nand is zero. We then construct the matrix A(t) as follows:\nA(t) ="},{"paragraph_id":"p169","order":169,"text":"Ik\n0\n0\n0\n0\nt\n0\n0\n0\n0\nt−1\n0\n0\n0\n0\nIn−k−2"},{"paragraph_id":"p170","order":170,"text":"We see that A(t) ∈SLn for all t ̸= 0. Next, if we let x(t) = A(t) · x, then\nwe see that:\nlim\nt→0 x(t) = x′\nThis shows that x′ lies in the closure of O. □\nWe now look at the case when r ≥n. We have:\nProposition 22.3. Let r ≥n and x, x′ ∈V r be such that ann(x) = ann(x′).\nIf (i) rank(x) < n then there is an A ∈SLn such that x′ = Ax, (ii) if\nrank(x) = n there is a unique A ∈SLn and a λ ∈C∗such that if z = A · x,\nthen the first n −1 rows of z equal those of x′ and z[n] = λx′[n].\nThe proof is easy. Let x be matrix in V r of rank n, and C be a subset\nof [r] = {1, 2, . . . , r} such that det(x[C]) ̸= 0.\nProposition 22.4. Let x be as above. Then O(x), the orbit of x, equals\nall points x′ ∈V r such that (i) ann(x) = ann(x′) and (ii) det(X′[C]) =\ndet(x[C]). The set of all rank n points in V r is dense in V r.\nThe proof is easy.\nProposition 22.5. The orbit O(x) as above is closed.\n126"},{"paragraph_id":"p171","order":171,"text":"Proof: Notice that if A ∈SLn and z = Ax then det(x[C]) = det(z[C]).\nWe may rewrite condition (i) above as ann(x′) ⊇ann(x). Condition (ii)\nensures that rank(x′) = n and that condition (i) holds with equality. Thus\nis y1, . . . , yr−n are column vectors generating ann(x), then the equations\nx′ys = 0 and det(x′[C]) = det(x[C]) determines the orbit O(x). Thus O(x)\nis the zero-set of some algebraic equations and thus is closed. □\nProposition 22.6. Let x ∈V r be such that rank(x) < n. Then (i) O(x)\nequals those x′ such that ann(x) = ann(x′), (ii) O(x) is not closed and its\nclosure O(x) equals points x′ such that ann(x′) ⊇ann(x).\nWe now move to the computation of invariants. The space C[V r] equals\nthe space of all polynomials in the variable matrix X = (Xij) where i =\n1, . . . , n and j = 1, . . . , r. It is clear that for any set C of n columns of X,\nwe see that det(X[C]) is an invariant. We will denote C as C = c1 < c2 <\n. . . < cn and det(X[C]) as pC, the C-th Plucker coordinate. We aim to\nshow now that these are the only invariants.\nLet C0 = {1 < 2 < . . . < n} and W ⊆V r be the space of all matrices\nx ∈V r such that x[C0] = diag(1, . . . , 1, λ). Let W ′ be those elements of W\nfor which λ ̸= 0.\nLemma 22.1. Let W ′ be as above. (i) If x ∈W ′, then O(x) ∩W ′ = x, (ii)\nfor any x ∈V r such that det(x[C0]) ̸= 0, there is a unique A ∈SLn such\nthat Ax ∈W ′.\nLet us call Z′ ⊆V r as those x such that det(x[C0]) ̸= 0. We then have\nthe projection map\nπ : Z′ →W ′\ngiven by the above lemma. Note that Z′ is SLn-invariant: if x ∈Z′ and\nA ∈SLn then A · x ∈Z′ as well.\nThe ring C[Z′] of regular functions on Z′ is precisely C[X]det([X[C0]), the\nlocalization of C[X] at det(X[C0]).\nWe may parametrize W ′ as:\nW ="},{"paragraph_id":"p172","order":172,"text":"1\n. . .\n0\nw1,n+1\n. . .\nw1,r\n...\n...\n...\n...\n0\n. . .\nwn,n\nwn,n+1\n. . .\nwn,r"},{"paragraph_id":"p173","order":173,"text":"Let\nW = {Wi,j|i = 1, . . . , n, j = n + 1, . . . , r} ∪{Wn,n}\nbe a set of (n −r) · r + 1 variables. The ring C[W ′] of regular function on\nW ′ is precisely C[W]Wn,n.\n127"},{"paragraph_id":"p174","order":174,"text":"Let x ∈Z′ be an arbitrary point and A = x[C0]. Let Ci,j be the set\nC0 −i + j. The map π is given by:\nπ(x)n,n\n=\ndet(A)\nπ(x)i,j\n=\n det(x[Ci,j])/det(A)\nfor i ̸= n\ndet(x[Ci,j])\notherwise\nThe map π causes the map:\nπ∗: C[W ′] →C[Z′]\ngiven by:\nπ∗(Wn,n)\n=\ndet(X[C0])\nπ∗(Wi,j)\n=\n det(X[Ci,j])/det(X[C0])\nfor i ̸= n\ndet(X[Ci,j])\notherwise\nNow we note that Z′ is dense in V r. Let p ∈C[X]SLn be an invariant.\nClearly p restricted to W ′ defines an element pW ′ of C[W ′].\nThis then\nextends to Z′ via π∗. Clearly π∗(pW ′) must match p on Z′. Thus we have\nthat p is a polynomial in det(X[Ci,j]) possibly localized at det(X[C0]).\nNote that for a general C, det(X[C]) is already expressible in det(X[C0])\nand det(X[Ci,j]).\n22.3\nOther Representations\nWe discuss two other representations:\nThe Conjugate Action: Let M be the space of all n × n matrices with\ncomplex entries. we define the action of A ∈SLn on M as follows. For an\nM ∈M, A acts on it by conjugation.\nA · M = AMA−1\nNote that dimC(M) = n2. Let X = {Xij|1 ≤i, j ≤n} be the dual space\nto M. The invariants for this action are Tr(X), . . . , Tr(Xi), . . . , Tr(Xn).\nThat these are invariants ic clear, for Tr(AMiA−1) = Tr(M) for any A, M.\nAlso note that Tr(X) = X11 + . . . + Xnn is linear in the X’s.\nThis\nindicates an invariant hyperplane in M.\nThis is precisely the trace zero\nmatrices. Thus M = M0 ⊕M1 where M0 are all matrices M such that\nTr(M) = 0. The one-dimensional complementary space M1 is composed of\nmultiples of the identity matrix.\n128"},{"paragraph_id":"p175","order":175,"text":"The orbits are parametrized by the Jordan canonical form JCF(M).\nIn other words, if JCF(M) = JCF(M′) them M′ ∈O(M). Furthermore,\nif JCF(M) is diagonal then the orbit of M is closed in M.\nThe Space of Forms: Let V be a complex vector space of dimension n\nandV ∗its dual. Let X1, . . . , Xn be the dual basis. The space Symd(V ∗)\nconsists of degree d forms are formal linear combinations of the monomials\nXi1\n1 . . . Xin\nn with i1 + . . . + in = d. The typical form may be written as:\nf(X) =\nX\ni1+...+in=d\nBi1...inXi1\n1 . . . Xin\nn\nLet A ∈SLn be such that\nA−1 ="},{"paragraph_id":"p176","order":176,"text":"a11\n. . .\nan1\n...\n...\na1n\n. . .\nann"},{"paragraph_id":"p177","order":177,"text":"The space Symd(V ∗) is the formal linear combination of the generaic coef-\nficients {Bi||i| = d}. The action of A is obtained by substituting\nXi →\nX\nj\naijXi\nin f(X) and recoomputing the coefficients. we illustrate this for n = 2 and\nd = 2. Then, the generic form is given by:\nf(X1, X2) = B20X2\n1 + B11X1\n1X1\n2 + B02X2\n2\nUpon substitution, we get f(a11X1 + a12X2, a21X1 + a22X2): Thus the new\ncoefficients are:\nB20\n→\nB20a2\n11+\nB11a11a21+\nB02a2\n21\nB11\n→\nB202a11a12+\nB11(a11a22 + a12a21)+\nB022a21a22\nB02\n→\nB20a2\n12+\nB11a12a22+\nB02a2\n22\nWe thus see that the variables B move linearly, with coefficients as homo-\ngeneous polynomials of degree 2 in the coefficients of the group elements. In\nthe above case, we know that the discriminant B2\n11−4B20B02 is an invariant,\nThese spaces have been the subject of intense analysis and their study by\nGordan, Hilbert and other heralded the beginning of commutative algebra\nand invariant theory.\n129"},{"paragraph_id":"p178","order":178,"text":"22.4\nFull Reducibility\nLet W be a representation of SLn and let Z ⊆W be an invariant subspace.\nThe reducibility question is whether there exists a complement Z′ such that\nW = Z ⊕Z′ and Z′ is SLn-invariant as well.\nThe above result is indeed true although we will not prove it here. There\nare many proofs known, each with a specific objective in a specific situation,\nand each extremely instructive.\nThe simplest is possibly through the Weyl Unitary trick.\nIn this, a\nsuitable compact subgroup U ⊆SLn is chosen. The theory of compact\ngroups is much like that of finite groups and a complement Z′ may easily\nbe found. It is then shown that Z′ is SLn-invariant.\nThe second attack is through showing the fill reducibility of the module\n⊗dV , the d-th tensor product representation of V . This goes through the\nconstruction of the commutator of the same module regarded as a right\nSd-module, with the symmetric group permutating the contents of the d\npositions. The full reducibility then follows from Maschke’s theorem and\nthat of the finite group Sd.\nThe oldest approach was through the construction of a symbolic reynold’s\noperator, which is the Cayley Ω-process. Let C[W] be the ring of polyno-\nmial functions on the space W. The operator Ωis a map:\nΩ: C[W] →C[W]SLn\nsuch that if p is an invariant then Ω(p · p′) = p · Ω(p′). The definition of Ω\nis nothing but curious.\n130"},{"paragraph_id":"p179","order":179,"text":"Chapter 23\nInvariant Theory\nReferences: [FH, N]\n23.1\nAlgebraic Groups and affine actions\nAn algebraic group (over C) is an affine variety G equipped with (i) the\ngroup product, i.e., a morphism of algebraic varieties · : G×G →G which is\nassociative, i.e. (g1·g2)·g3 = g1·(g2·g3) for all g1, g2, g3 ∈G, (ii) and algebraic\ninverse i : G →G, i.e, g · i(g) = i(g) · g = 1G, where 1G is (iii) a special\nelement 1G ∈G, which functions as the identity, i.e., 1G · g = g · 1G = g for\nall g ∈G. Let C[G] be the ring of regular functions on G. The requirements\n(i)-(iii) above are via morphisms of algebraic varities, and thus the group\nproduct and the inverse must be defined algebraically. Thus the product\n· : G × G →G results in a morphism of C-algebras ·∗: C[G] →C[G] ⊗C[G]\nand the inverse into another morphism i∗: C[G] →C[G].\nThe essential example is obviously SLn. Clearly for G = SLn, we have\nC[G] = C[X]/(det(X)−1), where X is the n×n indeterminate matrix. The\nmorphism ·∗and i∗are clearly:\n·∗(Xij)\n=\nP\nk Xik ⊗Xkj\ni∗(Xij)\n=\ndet(Mji)\nwhere Mji is the corresponding minor of X.\nNext, let Z be another affine variety with C[Z] as its ring of regular\nfunctions. We say that Z is a G-variety if there is a morphism μ : G×Z →Z\nwhich is a group action. Thus, not only must G act on Z, it must do so\nalgebraically. This μ induces the map μ∗:\nμ∗: C[Z] →C[G] ⊗C[Z]\n131"},{"paragraph_id":"p180","order":180,"text":"Thus every function f ∈C[Z] goes to a finite sum:\nμ∗(f) =\nk\nX\ni=1\nhi ⊗fi\nwhere fi ∈C[Z] and hi ∈C[G] for all i.\nTo continue with our example, consider SL2 and Z = Symd(V ∗). C[Z] =\nC[B20, B11, B02] and C[G] = C[A11, A12, A21, A22]/(A11A22 −A21A12 −1).\nWe have for example:\nμ∗(B20) = A2\n11 ⊗B20 + A11A21 ⊗B11 + A2\n21 ⊗B02\nLet μ : G × Z →Z and μ′ : G × Z′ →Z′ be two G-varities and let\nφ : Z →Z′ be a morphism. We say that φ is G-equivariant if φ(μ(g, z)) =\nμ′(g, φ(z)) for all g ∈G and z ∈Z. Thus φ commutes with the action of G.\n23.2\nOrbits and Invariants\nEvery g ∈G induces an algebraic bijection on Z by restricting the map\nμ : G × Z →Z to g. We denote this map by μ(g) and call it translation\nby g. The map:\nμ(g) : Z →Z\ninduces the isomorphism of C-algebras:\nμ(g)∗: C[Z] →C[Z]\nGiven any function f ∈C[Z], μ(g)∗(f) ∈C[Z] is the translated function\nand denotes the action of G on f. This makes C[Z] into a G-module. If\nφ : Z →Z′ is a G-equivariant map, then the map φ∗: C[Z′] →C[Z] is also\nG-equivariant, for the G action on C[Z] and C[Z′] as above.\nWe next examine the equation:\nμ∗(f) =\nk\nX\ni=1\nhi ⊗fi\nwhere fi ∈C[Z] and hi ∈C[G] for all i. For a fixed g, we see that\nμ(g)∗(f) =\nk\nX\ni=1\nhi(g) ⊗fi\n132"},{"paragraph_id":"p181","order":181,"text":"Thus we see that every translate of f lies in the k-dimensional vector space\nC · {f1, . . . , fk}. Let\nM(f) = C · {μ(g)∗(f)|g ∈G} ⊆C · {f1, . . . , fk}\nClearly, M(f) is a G-invariant subspace of C[Z]. This may be generalized:\nProposition 23.1. Let S = {s1, . . . , sm} be a finite subset of C[Z]. Then\nthere is a finite-dimensional G-invariant subspace M(S) of C[Z] containing\ns1, . . . , sm.\nNext, let us consider μ : G × Z →Z and fix a z ∈Z. We get the map\nμz : G →X. Since G is an affine variety, we see that the image μz(G) is a\nconstructible set, whose closure is an affine variety. The image is precisely\nthe orbit O(z) ⊆Z. The closure of O(z) will be denoted by ∆(z).\nIf O(z) = ∆(z), then we have the G-equivariant embedding iz : O(z) →\nZ. This gives us the map i∗\nz : C[Z] →C[O(z)] which is a surjection. Thus\nthere is an ideal I(z) = ker(i∗\nz) such that C[O(z)] ∼= C[Z]/I(z). Since the\nmap iz is G-equivariant, I(z) is a G-submodule of C[Z] and is the ideal of\ndefinition of the orbit O(z).\nIn general, if I ⊆C[Z] is an ideal which is G-invariant, then the variety\nof I is also G-invariant and is the union of orbits.\nThe second construction that we make is that of the quotient Z/G.\nSince C[Z] is a G-module, we examine C[Z]G, the subring of G-invariant\nfunctions in C[Z]. We define Z/G as the spectrum Spec(C[Z]G). The inclu-\nsion C[Z]G →C[Z] gives us the quotient map:\nπ : Z →Z/G\nExercise 23.1. Let us consider Z = Sym2(V ∗) ∼= C3 where V is the stan-\ndard representation of G = SL2. As we have seen, C[Z] = C[B20, B11, B02].\nThere is only one invariant δ = B2\n11 −4B02B20. Thus C[Z]G = C[δ]. Thus\nZ/G is precisely Spec(C[δ]) = C, the complex plane. The map π is executed\nas follows: given a form aX2\n1 + bX1X2 + cX2\n2 ≡(a, b, c) ∈Z, we eveluate\nthe invariant δ at the point (a, b, c). Thus π : C3 →C is given by:\nπ(a, b, c) = b2 −4ac\nClearly, if f, f ′ ∈Z such that f = g · f ′ for g ∈SL2 then δ(f) = δ(f ′). We\nlook at the converse: if f, f ′ are such that δ(f) = δ(f ′) then is it that f ∈\nO(f ′)? We begin with f = aX2\n1 +bX1X2 +cX2\n2. We assume for the moment\nthat a ̸= 0. If that is the case, we make the substitution X1 →X1 −αX2\n133"},{"paragraph_id":"p182","order":182,"text":"and X2 →X2. Note that this transformation is unimodular for all α. This\ntransforms f to:\na(X1 −αX2)2 + b(X1 −αX2)X2 + cX2\n2\nThe coefficient of X1X2 is −2aα+b. Thus by choosing α as b/2a we see that\nf is transformed into a′′X2\n1 −c′′X2\n2 for some a′′, c′′. By a similar token, even\nif a = 0 one may do a similar transform. Thus in general, if f is not the\nzero form, there is a point in O(f) which is of the form aX2\n1 −cX2\n2. Thus,\nwe may assume that both f and f ′ are in this form. We can simplify the\nform further by a diagonal element of SL2 to put both f and f ′ as X2\n1 −cX2\n2\nand X2\n1 −c′X2\n2. It is now clear that δ(f) = δ(f ′) implies that c = c′. Thus\nthe general answer is that if f, f ′ ̸= 0 and δ(f) = δ(f ′) then f ∈O(f ′).\nNext, let us examine the form 0. we see that δ(0) = 0. Thus we see that\n(i) for any point d ∈C, if d ̸= 0, then π−1(d) consists of a single orbit, (ii)\nπ−1(0) consists of two orbits, O(0) and O(X2\n1), the perfect square, (iii) the\norbits O(f) are closed when δ(f) ̸= 0, (iv) O(X2\n1) is not closed. Its closure\nincludes the closed orbit 0.\nThe above example illustrates the utility of contructing the quotient Z/G\nas a variety parametrizing closed orbits albeit with some deviant points. In\nthe example above, the discrepancy was at the point 0, wherein the pre-\nimage is closed but decomposes into two orbits.\nWe state the all-important theorem linking a space and its quotient in\nthe restricted case when G is finite and Z is a finite-dimensional G-module.\nThus C[Z] is a polynomial ring and the action of G is homogeneous.\nTheorem 3. Let G be a finite group and act on the space Z. Let R = C[Z]\nand RG = C[Z]G be the ring of invariants. Let π : Z →Z/G be the quotient\nmap. Then\n(i) For any ideal J ⊆RG, we have (J · R) ∩RG = J.\n(ii) The map π is surjective. Further, for any x ∈Z/G, π−1(x) is a single\norbit in Z.\nProof: Let f1, . . . , fk be elements of RG and f ∈RG be such that:\nf = r1f1 + . . . + rkfk\nwhere ri are elements of R. Since G is finite, we apply the Reynolds operator\n134"},{"paragraph_id":"p183","order":183,"text":"p →pG. In other words, we have:\nf\n=\n1\n|G|\nX\ng∈G\nf\n=\n1\n|G|\nX\ng∈G\nX\ni\nrifi\n=\n1\n|G|\nX\ni\nfi\nX\ng∈G\nri\n=\nX\ni\nfi\n1\n|G|\nX\ng∈G\nri\n=\nX\ni\nfirG\ni\nNote that we have used the fact that if h ∈RG and p ∈R then (h · p)G =\nh · pG. Thus we see that any element f of RG which is expressible as an\nR-linear combination of elements fi’s of RG is already expressible as an\nRG-linear combination of the same elements. This proves (i) above.\nNow we prove (ii). Firstly, let J be a maximal ideal of RG. By part (i)\nabove, J · R is a proper ideal of R and thus π is surjective. Let x ∈Z/G\nand Jx ⊆RG be the maximal ideal for the point x. Let z be such that\nπ(z) = x and let IO(x) ⊆R be the ideal of all functions in R vanishing at\nall points of the orbit O(z) of z. We show that rad(Jx · R) = IO(z) which\nproves (ii). Towards that, it is clear that (a) JxR ⊆IO(z) and (b) the variety\nof JxR is G-invariant. Let O(z′) is another orbit in the variety of Jx · R.\nIf O(z′) ̸= O(z) then we already have that there is a p ∈RG such that\np(O(z)) = 0 and p(O(z′)) = 1. Since this p ∈Jx · R, the variety of Jx · R\nexcludes the orbit O(z′) proving our claim. □\nTheorem 4. Let Z be a finite-dimesional G-module for a finite group G.\nThen C[Z]G, the ring of G-invariants, is finitely generated as a C-algebra.\nProof: Since the action of G is homogeneous, every invariant f ∈C[Z]G is\nthe sum of homogeneous invariants. Let IG ⊆C[Z] be the ideal generated\nby the positive degree invariants.\nBy the Hilbert Basis theorem, IG =\n(f1, . . . , fk)·C[Z], where each fi is itself an invariant. we claim that C[Z]G =\nC[f1, . . . , fk] ⊆C[Z], or in other words, every invariant is a polynomial over\nC in the terms f1, . . . , fk. To this end, let f be a homogeneous invariant.\nWe prove this by induction over the degree d of f. Since f ∈IG, we have\nf =\nX\ni\nrifi where for all i, ri ∈C[Z]\n135"},{"paragraph_id":"p184","order":184,"text":"By applying the reynolds operator, we have:\nf =\nX\ni\nhifi where for all i, hi ∈C[Z]G\nNow since each hi has degree less than d, by our induction hypothesis, each\nis an element of C[f1, . . . , fk]. This then shows that f too is an element of\nC[f1, . . . , fk]. □\n23.3\nThe Nagata Hypothesis\nWe now generalize the above two theorems to the case of more general\ngroups. This generlization is possible if the group G satisfies what we call\nthe Nagata Hypothesis.\nDefinition 23.1. Let G be an algebraic group. We say that G satisfies the\nNagata hypothesis if for every finite-dimensional module M over C and a\nG-invariant hyperplane N ⊆M, there is a decomposition M = H ⊕P as\nG-modules, where H is a hyperplane and P is an invariant (i.e., P is the\ntrivial representation.\nThe group SLn satisfies the hypothesis, and so do the so-called reductive\ngroups over C.\nTheorem 5. Let G satisfy the Nagata hypothesis and let Z be an affine\nG-variety. The ring C[Z]G is finitely generated as a C-algebra.\nThe proof will go through several steps.\nLet f ∈C[Z] be an arbitrary element. We define two modules M(f) and\nN(f).\nM(f)\n=\nC · {s · f|s ∈G}\nN(f)\n=\nC · {s · f −t · f|s, t ∈G}\nThus M(f) ⊇N(f) are finite-dimensional submodules of C[Z].\nLemma 23.1. Let f ∈C[Z] be an arbitrary element. Then there exists an\nf ∗∈C[Z]G ∩M(f) such that f −f ∗∈N(f).\nProof: We prove this by induction over dim(M(f)).\nNote that s · f =\nf+(s·f−1·f) and thus M(f) = N(f)⊕C·f as vector spaces. Thus N(f) is a\nG-invariant hyperplane of M(f). By the Nagata hypothesis, M(f) = f ′⊕H,\nwhere f ′ is an invariant. If f ′ ̸∈N(f) then M(f) = f ′⊕N(f) and the lemma\nfollows. However, if f ′ ∈N(f), then f = f ′ + h where h ∈H. Since H is\n136"},{"paragraph_id":"p185","order":185,"text":"a G-module, we see that M(h) ⊆H which is of dimension less than that of\nM(f). Thus there is an invariant h∗such that h −h∗∈N(h). Note that\nf −f ′ = h with f ′ invariant, implies that s · h −t · h = s · f −t · f. Thus\nN(h) ⊆N(f). We take f ∗= h∗. Examining f −f ∗, we see that\nf −h∗= f ′ + h −h∗∈N(f)\nThis proves the lemma. □\nLemma 23.2. Let f1, . . . , fr ∈C[Z]G.\nThen C[Z]G ∩(P\ni C[Z] · fi) =\nP\ni C[Z]G · fi. Thus if an invariant f is a C[Z]linear combination of invari-\nants, then it is already a C[Z]G linear combination.\nProof: This is proved by induction on r. Say f = Pr\ni=1 hifi with hi ∈C[Z].\nApplying the above lemma to hr, there is an h′′ ∈C[Z]G and an h′ ∈N(hk)\nsuch that hr = h′ + h′′. We tackle h′ as follows. Since f is an invariant, we\nhave for all s, t ∈G:\nr\nX\ni=1\n(s · hi −t · hi)fi = s · f −t · f = 0\nHence:\n(s · hr −t · hr)fr =\nr−1\nX\ni=1\n(s · hi −t · hi)fi\nIt follows from this that h′fr = Pr−1\ni=1 h′\nifi for some h′\ni ∈C[Z]. Substituting\nthis in the expression\nf =\nr−1\nX\ni=1\nhifi + (h′ + h′′)fr\nwe get:\nf −h′′fr =\nr−1\nX\ni=1\n(hi + h′\ni)fi\nThus the invariant f −h′′fr is C[Z]-linear combination of r −1 invariants,\nand thus the induction hypothesis applies. This then results in an expression\nfor f as a C[Z]G-linear combination for f. □\nThe above lemma proves part (i) of Theorem 3 for groups G for which\nthe Nagata hypothesis holds. The route to Theorem 4 is now a straight-\nforward adaptation of its proof for finite groups. In the case when Z is a\nG-module, C[Z]G would then be homogeneous and the proof of Theorem 4\nholds. For general Z here is a trick which converts it to the homogeneous\ncase:\n137"},{"paragraph_id":"p186","order":186,"text":"Proposition 23.2. Let Z be an affine G-variety. Then there is a G-module\nW and an equivariant embedding φ : Z →W.\nProof: Since C[Z] is a finitely generated C-algebra, C[Z] = C[f1, . . . , fk]\nwhere f1, . . . , fk are some specific elements of C[Z]. By expanding the list\nof generators, we may assume that the vector space C · {f1, . . . , fk} ⊆C[Z]\nis a finite-dimensional G-module.\nLet us construct W as an isomorphic\ncopy of this module with the dual basis W1, . . . , Wk. We construct the map\nφ∗: C[W] →C[Z] by defining φ(Wi) = fi. Since C[W] is a free algebra, we\nindeed have the surjection:\nφ∗: C[W] →C[Z]\nThis proves the proposition. □\nWe are now ready to prove:\nTheorem 6. Let G satisfy the Nagata hypothesis. Let Z be an affine G-\nvariety with coordinate ring C[Z]. Then C[Z]G is finitely generated as a\nC-algebra.\nProof: We construct the equivariant surjection φ∗: C[W] →C[Z]. Note\nthat C[W]G is already finitely generated over C, say C[W]G = C[h1, . . . , hk].\nWe claim that φ∗(h1), . . . , φ∗(hk) generate C[Z]G.\nNow, let f ∈C[Z]G. By the surjectivity of φ∗, there is an h ∈C[W] such\nthat φ∗(h) = f. Consider the space N(h). A typical generator of N(h) is\ns · h −t · h. Applying φ∗to this, we see that:\nφ∗(s · h −t · h)\n=\ns · φ∗(h) −t · φ∗(h)\n=\nf −f = 0\nBy an earlier lemma there is an invariant h∗such that h −h∗∈N(h). thus\napplying φ∗we see that φ∗(h∗) = φ(h) = f. Thus there is an invariant h∗\nsuch that φ∗(h∗) = f. Now h∗∈C[h1, . . . , hk] implies that f = φ∗(h∗) ∈\nC[φ∗(h1), . . . , φ∗(hk)]. □\n138"},{"paragraph_id":"p187","order":187,"text":"Chapter 24\nOrbit-closures\nReference: [Ke, N]\nIn this chapter we will analyse the validity of Theorem 3 for general\ngroups G with the Nagata hypothesis. The objective is to analyse the map\nπ : Z →Z/G.\nProposition 24.1. Let G satisfy the Nagata hypothesis and act on the affine\nvariety Z. Then the map π : Z →Z/G is surjective.\nProof: Let J ⊆C[Z]G be a maximal ideal. By lemma 23.2 J · C[Z] is a\nproper ideal of C[Z]. This implies that π is surjective. □\nTheorem 7. Let G satisfy the Nagata hypothesis and act on the affine\nvariety Z. Let W1 and W2 be G-invariant (Zariski-)closed subsets of Z such\nthat W1 ∩W2 is empty. Then there is an invariant f ∈C[Z]G such that\nf(W1) = 0 and f(W2) = 1.\nProof: Let I1 and I2 be the ideals of W1 and W2 in C[Z]. Since their inter-\nsection is empty, by Hilbert Nullstellensatz, we have I1 + I2 = 1. Whence\nthere are functions f1 ∈I1 and f2 ∈I2 such that f1 + f2 = 1. For arbitrary\ns, t ∈G we have:\n(s · f1 −t · f1) + (s · f2 −t · f2) = 1 −1 = 0\nNote that M(fi) and N(fi) are submodules of the G-invariant ideal Ii. Now\napplying lemma 23.1 to f1, we see that there are elements f ′\ni ∈N(fi) such\nthat f1 + f ′\n1 ∈C[Z]G. Whence we see that:\n(f1 + f ′\n1) + (f2 + f ′\n2) = 1\n139"},{"paragraph_id":"p188","order":188,"text":"where f = f1 + f ′\n1 is an invariant. Since f1 + f ′\n1 ∈I1 we see that f(W1) = 0.\nOn the other hand, f = 1 −(f2 + f ′\n2) ∈1 + I2 and thus f(W2) = 1. □\nRecall that ∆(z) denotes the closure of the orbit O(z) in Z.\nTheorem 8. Let G satisfy the Nagata hypothesis and act on an affine va-\nriety Z. We define the relation ≈on Z as follows: z1 ≈z2 if and only if\n∆(z1) ∩∆(z2) is non-empty. Then\n(i) ≈is an equivalence relation.\n(ii) z1 ≈z2 ifff(z1) = f(z2) for all f ∈C[Z]G.\n(iii) Within each ∆(z) there is a unique closed orbit, and this is of minimum\ndimension among all orbits in ∆(z).\nProof: It is clear that (ii) proves (i). Towards (ii), if ∆(z1)∩∆(z2) is empty,\nthen by Theorem 7, there is an invariant separating the two points. Thus it\nremains to show that if ∆(z10 ∩∆(z2) is non-empty, and f is any invariant,\nthen f(z1) = f(z2). So let z be an element of the intersection. Since f is\nan invariant, we have f(O(z1)) is a constant, say α. Since O(z1) ⊆∆(z1) is\ndense, and f is continuous, we have f(z) = α. Thus f(z1) = f(z2) = f(z) =\nα.\nNow (iii) is easy. Clearly ∆(z) cannot have two closed distinct orbits,\nfor otherwise they woulbe separated by an invariant. But this must take the\nsame value on all points of ∆(z). That it is of minimum dimension follows\nalgebraic arguments. □\nDefinition 24.1. Let Z be a G-variety and z ∈Z. We say that z is stable\nif the orbit O(z) ⊆Z is closed in Z.\nBy the above theorem, every point x of Spec(C[Z]G) corresponds to\nexactly one stable point: the point whose orbit is of minimum dimension in\nπ−1(x).\nExercise 24.1. Consider the action of G = SLnon M, the space of n × n-\nmatrices by conjugation. Thus, given A ∈SLn and M ∈M, we have:\nA · M = AMA−1\nLet R = C[M] = C[X11, . . . , Xnn] be the ring of functions on M.\nThe\ninvariants RG is generated as a C-algebra by the forms ei(X) = Tr(Xi), for\ni = 1, . . . , n. The forms {ei|1 ≤i ≤n} are algebraically independent and\n140"},{"paragraph_id":"p189","order":189,"text":"thus RG is the polynomial ring C[e1, . . . , en]. Clearly then Spec(RG) ∼= Cn\nand we have:\nπ : M ∼= Cn2 →Cn\nGiven a matrix M with eigenvalues {λ1, . . . , λn}, we have:\nei(M) = λi\n1 + . . . + λi\nn\nThus, by the fundamental theorem of algebra, the image π(M) determines\nthe set {λ1, . . . , λn} (with multiplicities).\nOn the other hand, given any\ntuple μ = (μ1, . . . , μn) there is a unique set λμ = {λ1, . . . , λn} such that\nP\nr λi\nr = μi. Clearly, for the diagonal matrix D(λμ) = diag(λ1, . . . , λn), we\nhave that π(D(λ)) = μ. This verifies that π is surjective.\nFor a given μ, the set π−1(μ) are all matrices M with Spec(M) = λ =\nλμ. By the Jordan canonical form (JCF), this set may be stratified by the\nvarious Jordan canonical blocks of spectrum λ. If λ has no multiplicities\nthen π−1(μ) consists of just one orbit: matrices M such that JCF(M) =\nD(λ). For a general λ, the orbit of M with JCF(M) = D(λ) is the unique\nclosed orbit of minimum dimension. All other orbits contain this orbit in its\nclosure. Thus stable points M ∈M are the diagonalizable matrices.\nAs an example, consider the case when n = 2 and the matrix:\nN =\n λ\n1\n0\nλ"},{"paragraph_id":"p190","order":190,"text":"Consider the family A(t) = diag(t, t−1) ∈SL2. We see that:\nN(t) = A(t)NA(t)−1 =\n λ\nt2\n0\nλ"},{"paragraph_id":"p191","order":191,"text":"Thus limt→0 N(t) = diag(λ, λ), the diagonal matrix.\nThus, we see that the invariant ring C[Z]G puts a different equivalence\nrelation ≈on points in Z which is coarser than ∼=, the orbit equivalence\nrelation. The relation ≈is more ‘topological’ than group-theoretic and cor-\nrectly classifies orbits by their separability by invariants. The special case\nof Theorem 8 when Z is a representation was analysed by Hilbert in 1893.\nThe point 0 ∈Z is then the smallest closed orbit, and the equivalence class\n[0]≈is termed as the null-cone of Z. We see that the null-cone consists\nof all points z ∈Z such that 0 lies in the orbit-closure ∆(z) of z. It was\nHilbert who discovered that if 0 ∈∆(z) then 0 lies in the orbit-closure of a 1-\nparameter diagonal subgroup of SLn. To understand the intricay of Hilbert’s\nconstructions, it is essential that we understand diagonal subgroups of SLn.\n141"},{"paragraph_id":"p192","order":192,"text":"Chapter 25\nTori in SLn\nReference: [Ke, N]\nLet C∗denote the multiplicative group of non-zero complex numbers. A\ntorus is the abstract group (C∗)m for some m. Note that C∗is an abelian\nalgebraic group with C[G] = C[T, T −1]. Furthermore, C∗has a compact\nsubroup S1 = {z ∈C, |z| = 1}, the unit circle.\nNext, let us look at representations of tori. For C∗, the simplest repre-\nsentations are indexed by integers k ∈Z. So let k ∈Z. The representation\nC[k] corresponds to the 1-dimensional vector space C with the action:\nt · z = tkz\nThus a non-zero t ∈C∗acts on z by multiplication of the k-th power. Next,\nfor (C∗)m, let χ = (χ[1], . . . , χ[m]) be a sequence of integers. For such a χ,\nwe define the representation Cχ as follows: Let t = (t1, . . . , tm) ∈(C∗)m be\na general element and z ∈C. The action is given by:\nt · z = tχ[1]\n1\n. . . tχ[m]\nm\nz\nSuch a χ is called a character of (C∗)m.\nThese 1-dimensional representations of tori are crucial in the analysis of\nalgebraic group actions.\nLet us begin by understanding the structure of algebraic homomorphisms\nfrom C∗to SLn(C). So let\nλ : C∗→SLn(C)\nbe such a map such that λ(t) = [aij(t)] where aij(T) ∈C[T, T −1].\nAn\nimportant substitution is for t = eiθ, and we obtain a 2π-periodic map\nλ : C →SLn\n142"},{"paragraph_id":"p193","order":193,"text":"We see that λ(0) = I. Let the derivative at 0 for λ be X.\nWe have the following general lemma:\nLemma 25.1. Let f : C →SLn be a smooth map such that f(0) = I and\nf ′(0) = X, where X is an n × n-matrix. Then for θ ∈C,\nlim\nk→∞"},{"paragraph_id":"p194","order":194,"text":"f\n θ\nk\n k\n= eθX\nThe proof follows from the local diffeomorphism of the exponential map\nin the neighborhood of the identity matrix.\nApplying this lemma to λ we see that eθX is in the image of λ for all θ.\nNow, since λ is 2π-periodic, we must have e(n2π+θ)X = eθX. This forces (i)\nX to be diagonalizable, and (ii) with integer eigenvalues. This proves:\nProposition 25.1. Let λ : C∗→SL(V ) be an algebraic homomorphism.\nThen the image of λ is closed and V ∼= C[m1] ⊕. . .⊕C[mn], for some integers\nm1, . . . , mn, where n = dimC(V ).\nBased on this, we have the generalization:\nProposition 25.2. Let λ : (C∗)r →SL(V ) be an algebraic homomorphism.\nThen the image of λ is closed and V ∼= Cχ1 ⊕. . . ⊕Cχn, for some integers\nχ1, . . . , χn, where n = dimC(V ).\nThus, in effect, for every homomorphism λ : (C∗)r →SLn, there is a\nfixed invertible matrix A such that for all t ∈(C∗)r, the conjugate Aλ(t)A−1\nis diagonal.\nA torus in SLn is defined as an abstract subgroup H of SLn which is\nisomorphic to (C∗)r for some r. The standard maximal torus D of SLn\nis the diagonal matrices diag(t1, . . . , tn) where ti ∈C∗and t1t2 . . . tn = 1.\nThis clears the way for the important theorem:\nTheorem 9.\n(i) Every torus is contained in a maximal torus. All maxi-\nmal tori in SLn are isomorphic to (C∗)n−1.\n(ii) If T and T ′ are two maximal tori then there is an A ∈SLn such that\nT ′ = ATA−1. Thus all maximal tori are conjugate to D above.\n(iii) Let N(D) be the normalizer of D and N(D)o be the connected compo-\nnent of N(D). Then N(D)o = D and N(D)/D is the Weyl group\nW, isomorphic to the symmetric group Sn.\n143"},{"paragraph_id":"p195","order":195,"text":"Definition 25.1. Let G be an algebraic group. Γ(G) will denote the col-\nlection of all 1-parameter subgroups of G, i.e., morphisms λ : C∗→G.\nX(G) will denote the collection of all characters of G, i.e., homomorphisms\nχ : G →C∗.\nWe consider the case when G = (C∗)r. Clearly, for a given λ : C∗→G,\nthere are integers m1, . . . , mr ∈Z such that:\nλ(t) = (tm1, . . . , tmr)\nIn the same vein, for the character χ : G →C∗, we have integers a1, . . . , ar\nsuch that\nχ(t1, . . . , tr) =\nY\ni\ntai\ni\nWe also have the composition λ ◦χ : C∗→C∗, where by:\nλ ◦χ(t) = tm1a1+...+mrar\nConsolidating all this, we have:\nTheorem 10. Let G = (C∗)r. Then Γ(G) ∼= Zr and X(G) ∼= Zr. Further-\nmore, there is the pairing\n< , >: Γ(G) × X(G) →Z\nwhich is a unimodular pairing on lattices.\nExercise 25.1. Let G = (C∗)3 and λ and χ be as follows:\nλ(t)\n=\n(t3, t−1, t2)\nχ(t1, t2, t3)\n=\nt−1\n1 t2t2\n3\nThen, λ ∼= [3, −1, 2] and χ ∼= [−1, 1, 2]. We evaluate the pairing:\n< λ, χ >= 3 · −1 + (−1) · 1 + 2 · 2 = 0\nWe now turn to the special case of D ⊆SLn, the maximal torus which\nis isomorphic to (C∗)n−1.\nBy the above theorem, Γ(D), X(D) ∼= Zn−1.\nHowever, it will more convenient to identify this space as a subset of Zn. So\nlet:\nYn = {[m1, . . . , mn] ∈Zn|m1 + . . . + mn = 0}\nIt is easy to see that Yn ∼= Zn−1. In fact, we will set up a special bijection\nθ : Yn →Zn−1 defined as:\nθ([m1, m2, . . . , mn]) = [m1, m1 + m2, . . . , m1 + . . . + mn−1]\n144"},{"paragraph_id":"p196","order":196,"text":"The inverse θ−1 is also easily computed:\nθ−1[a1, . . . , an−1] = [a1, a2 −a1, a3 −a2, . . . , an−1 −an−2, −an−1]\nThis θ corresponds to the Z-basis of Yn consisting of the vectors e1 −\ne2, . . . , en−1 −en where ei is the standard basis of Zn. This is also equivalent\nto the embedding θ∗: (C∗)n−1 →D as follows:\n(t1, . . . , tn−1) →"},{"paragraph_id":"p197","order":197,"text":"t1\n0\n. . .\n0\n0\nt−1\n1 t2\n0\n. . .\n0\n...\n0\n. . .\n0\nt−1\nn−2tn−1\n0\n0\n. . .\n0\nt−1\nn−1"},{"paragraph_id":"p198","order":198,"text":"A useful computation is to consider the inclusion D ⊆D∗, where D∗⊆GLn\nis subgroup of all diagonal matrices. Clearly Γ(D) ⊆Γ(D∗), however there\nis a surjection X(D∗) →X(D). It will be useful to work out this surjection\nexplicitly via θ and θ∗. If [m1, . . . .mn] ∈Zn ∼= X(D∗), then it maps to\n[m1 −m2, . . . , mn−1 −mn] ∈Zn−1 ∼= X((C∗)n−1) via θ∗. If we push this\nback into Yn via θ−1, we get:\n[m1, . . . , mn] →[m1−m2, 2m2−m1−m3, , . . . , 2mn−1−mn−2−mn, mn−mn−1]\nWe are now ready to define the weight spaces of an SLn-module W.\nSo let W be such a module.\nBy restricting this module to D ⊆G, via\nProposition 25.2, we see that W is a direct sum W = Cχ1 ⊕. . . ⊕CχN,\nwhere N = dimC(W). Collecting identical characters, we see that:\nW = ⊕χ∈X(D)Cmχ\nχ\nThus W is a sum of mχ copies of the module Cχ. Clearly mχ = 0 for all but\na finite number, and is called the multiplicity of χ. For a given module\nW, computing mχ is an intricate combinatorial exercise and is given by the\ncelebrated Weyl Character Formula.\nExercise 25.2. Let us look at SL3 and the weight-spaces for some modules\nof SL3.\nAll modules that we discuss will also be GL3-modules and thus\nD∗modules. The formula for converting D∗-modules to D-modules will be\nuseful. This map is Z3 →Y3 and is given by:\n[m1, m2, m3] →[m1 −m2, 2m2 −m1 −m3, m3 −m2]\n145"},{"paragraph_id":"p199","order":199,"text":"The simplest SLn module is C3 with the basis {X1, X2, X3} with D∗\nweights [1, 0, 0], [0, 1, 0] and [0, 0, 1].\nThis converted to D-weights give us\n{[1, −1, 0], [−1, 2, −1], [0, −1, 1]}, with C[1,−1,0] ∼= C · X1 and so on.\nThe next module is Sym2(C3) with the basis X2\ni and XiXj. The six D∗\nand D-weights with the weight-spaces are given below:\nD∗-wieghts\nD-weights\nweight-space\n[2, 0, 0]\n[2, −2, 0]\nX2\n1\n[0, 2, 0]\n[−2, 4, −2]\nX2\n2\n[0, 0, 2]\n[0, −2, 2]\nX2\n3\n[0, 1, 1]\n[−1, 1, 0]\nX2X3\n[1, 0, 1]\n[1, −2, 1]\nX1X3\n[1, 1, 0]\n[0, 1, −1]\nX1X2\nThe final example is the space of 3× 3-matrices M acted upon by conju-\ngation. We see at once that M = M0 ⊕C·I where M0 is the 8-dimensional\nspace of trace-zero matrices, and C · I is 1-dimensional space of multiples\nof the idenity matrix. Weight vectors are Eij, with 1 ≤i, j ≤3. The D∗\nweights are [1, −1, 0], [1, 0, −1], [0, 1, −1], [−1, 0, 1], [0, −1, 1], [−1, 1, 0] and [0, 0, 0].\nThe multiplicity of [0, 0, 0] in M is 3 and in M0 is 2. Note that Eii ̸∈M0.\nThe D-weights are [2, −3, 1], [1, 0, −1], [−1, 3, −2] and its negatives, and ob-\nviously [0, 0, 0].\nThe normalizer N(D) gives us an action of N(D) on the weight spaces.\nIf w is a weight-vector of weight χ, t ∈D and g ∈N(D), then g · w is also\na weight vector. Afterall t · (g · w) = g · t′ · w where t′ = g−1tg. Thus\nt · (g · w) = χ(t′)(g · w)\nwhence g · w must also be a weight-vector with some weight χ′. This χ′ is\neasily computed via the action of D∗. Here the action of N(D∗) is clear: if\nχ = [m1, . . . , mn], then χ′ = [mσ(1), . . . , mσ(n)] for some permutation σ ∈Sn\ndetermined by the component of N(D∗) containing g. Thus the map χ to\nχ′ for D-weights in the case of SL3 is as follows:\n[m1−m2, 2m2−m1−m3, m3−m2] →[mσ(1)−mσ(2), 2mσ(2)−mσ(1)−mσ(3), mσ(3)−mσ(2)]\nCaution: Note that though Y3 ⊆Z3 is an S3-invariant subset, the action\nof S3 on χ ∈Y3 is different. Note that, e.g., in the last example above,\n[2, −3, 1] is a weight but not the ‘permuted’ vector [−3, 2, 1]. This is because\nof our peculiar embedding of Zn−1 →Yn.\n146"},{"paragraph_id":"p200","order":200,"text":"Chapter 26\nThe Null-cone and the\nDestabilizing flag\nReference: [Ke, N]\nThe fundamental result of Hilbert states:\nTheorem 11. Let W be an SLn-module, and let w ∈W be an element of\nthe null-cone. Then there is a 1-parameter subgroup λ : C∗→SLn such that\nlim\nt→0 λ(t) · w = 0W\nIn other words, if the zero-vector 0W lies in the orbit-closure of w, then\nthere is a 1-parameter subgroup taking it there, in the limit. We will not\nprove this statement here. Our objective for this chapter is to interpret the\ngeometric content of the theorem. We will show that there is a standard\nform for an element of the null-cone. For well-known representations, this\nstandard form is easily identified by geometric concepts.\n26.1\nCharacters and the half-space criterion\nTo begin, let D be the fixed maximal torus. For any w ∈W, we may express:\nw = w1 + w2 + . . . + wr\nwhere wi ∈Wχi, the weight-space for character χi. Note the the above ex-\npression is unique if we insist that each wi be non-zero. The set of characters\n{χ1, . . . , χr} will be called the support of w and denoted as supp(w). Let\n147"},{"paragraph_id":"p201","order":201,"text":"λ : C∗→SLn be such that Im(λ) ⊆D. In this case, the action of t ∈C∗\nvia λ is easily described:\nt · w = t(λ,χ1)w1 + . . . + t(λ,χr)wr\nThus, if limt→0 t · w exists (and is 0W ), then for all χ ∈supp(w), we have\n(λ, χ) ≥0 (and further (λ, χ) > 0).\nNote that (λ, χ) is implemented as a linear functional on Yn. Thus, if\nlimt→0 t · w exists (and is )W ) then there is a hyperplane in Yn such that\nthe support of w is on one side of the hyperplane (strictly on one side of\nthe hyperplane). The normal to this hyperplane is given by the conversion\nof λ into Yn notation.\nOn the other hand if the support supp(w) enjoys the geometric/combinatorial\nproperty, then by the approximability of reals by rationals, we see that there\nis a λ such that limt→0 t · w exists (and is zero).\nThus for 1-parameter subgroups of D, Hilbert’s theorem translates into\na combinatorial statement on the lattice subset supp(w) ⊂Yn.\nWe call\nthis the (strict) half-space property. In the general case, we know that\ngiven any λ : C∗→SLn, there is a maximal torus T containing Im(λ). By\nthe conjugacy result on maximal tori, we know that T = ADA−1 for some\nA ∈SLn. Thus, we may say that w is in the null-cone iffthere is a translate\nA · w such that supp(A · w) satisfies the strict half-space property.\nExercise 26.1. Let us consider SL3 acting of the space of forms of degree\n2. For the standard torus D, the weight-spaces are C · X2\ni and C · XiXj.\nConsider the form f = (X1 + X2 + X3)2. We see that supp(f) is set of all\ncharacters of Sym2(C3) and does not satisfy the combinatorial property.\nHowever, under a basis change A:\nX1\n→\nX1 + X2 + X3\nX2\n→\nX2\nX3\n→\nX3\nwe see that A·f = X2\n1. Thus A·f does satisfy the strict half-space property.\nIndeed consider the λ\nλ(t) ="},{"paragraph_id":"p202","order":202,"text":"t\n0\n0\n0\nt−1\n0\n0\n0\n1"},{"paragraph_id":"p203","order":203,"text":"We see that\nlim\nt→0 t · (A · f) = t2X2\n1 = 0\n148"},{"paragraph_id":"p204","order":204,"text":"Thus we see that every form in the null-cone has a standard form with a\nvery limited sets of possible supports.\nLet us look at the module M of 3 × 3-matrices under conjugation. Let\nus fix a λ:\nλ(t) ="},{"paragraph_id":"p205","order":205,"text":"tn1\n0\n0\n0\ntn2\n0\n0\n0\ntn3"},{"paragraph_id":"p206","order":206,"text":"such that n1 + n2 + n3 = 0. We may assume that n1 ≥n2 ≥n3. Looking at\nthe action of λ(t) on a general matrix X, we see that:\nt · X = (tni−njxij)\nThus if limt→0 t · X is to be 0 then xij = 0 for all i > j. In other words,\nX is strictly upper-triangular. Considering the general 1-parameter group\ntells us that X is in the null-cone iffthere is an A such that AXA−1 is\nstrictly upper-triangular. In other words, X is nilpotent. The 1-parameter\nsubgroup identifies this transformation and thus the flag of nilpotency.\n26.2\nThe destabilizing flag\nIn this section we do a more refined analysis of elements of the null-cone. The\nbasic motivation is to identify a unique set of 1-parameter subgroups which\ndrive a null-point to zero. The simplest example is given by X2\n1 ∈Sym2(C3).\nLet λ, λ′ and λ′′ be as below:\nλ(t) ="},{"paragraph_id":"p207","order":207,"text":"t\n0\n0\n0\nt−1\n0\n0\n0\n1"},{"paragraph_id":"p208","order":208,"text":"λ′(t) ="},{"paragraph_id":"p209","order":209,"text":"t\n0\n0\n0\n1\n0\n0\n0\nt−1"},{"paragraph_id":"p210","order":210,"text":"λ′′(t) ="},{"paragraph_id":"p211","order":211,"text":"t\n0\n0\n0\n0\n−1\n0\nt−1\n0"},{"paragraph_id":"p212","order":212,"text":"We see that all the three λ, λ′ and λ′′ drive X2\n1 to zero.\nThe question\nis whether these are related, and to classify such 1-parameter subgroups.\nAlternately, one may view this to a more refined classification of points in\nthe null-cone, such as the stratification of the nilpotent matrices by their\nJordan canonical form.\nThere are two aspects to this analysis. Firstly, to identify a metric by\nwhich to choose the ’best’ 1-parameter subgroup driving a null-point to\nzero. Next, to show that there is a unique equivalence class of such ’best’\nsubgroups.\n149"},{"paragraph_id":"p213","order":213,"text":"Towards the first objective, let λ : C∗→SLn be a 1-parameter subgroup.\nWithout loss of generality, we may assume that Im(λ) ⊆D. If w is a null-\npoint then we have:\nt · w = tn1w1 + . . . + tnkwk\nwhere ni > 0 for all i. Clearly, a measure of how fast λ drives w to zero\nis m(λ) = min{n1, . . . , nk}. Verify that this really does not depend on the\nchoice of the maximal torus at all, and thus is well-defined.\nNext, we see that for a λ as above, we consider λ2 : C∗→SLn such that\nλ2(t) = λ(t2). It is easy to see that m(λ2) = 2 · m(λ). Clearly, λ and λ2 are\nintrinsically identical and we would like to have a measure invariant under\nsuch scaling. This comes about by associating a length to each λ. Let λ be\nas above and let Im(λ) ⊆D. Then, there are integers a1, . . . , an such that\nλ(t) ="},{"paragraph_id":"p214","order":214,"text":"ta1\n0\n0\n0\n0\nta2\n0\n0\n...\n0\n0\n0\ntan"},{"paragraph_id":"p215","order":215,"text":"We define ∥λ∥as\n∥λ∥=\nq\na2\n1 + . . . + a2n\nWe must show that this does not depend on the choice of the maximal\ntorus D. Let T (SLn) denote the collection of all maximal tori of SLn as\nabstract subgroups. For every A ∈SLn, we may define the map φA : T →T\ndefined by T →ATA−1. The stabilizer of a torus T for this action of SLn\nis clearly N(T), the normalizer of T.\nAlso recall that N(T)/T = W is\nthe (discrete) weyl group.\nLet Im(λ) ⊆D ∩D′ for some two maximal\ntori D and D′.\nSince there is an A such that AD′A−1 = D, it is clear\nthat ∥λ∥= ∥AλA−1∥. Thus, we are left to check if ∥λ′∥= ∥λ∥when (i)\nIm(λ), Im(λ′) ⊆D, and (ii) λ′ = AλA−1 for some A ∈SLn. This throws\nthe question to invariance of ∥λ∥under N(D), or in other words, symmetry\nunder the weyl group.\nSince W ∼= Sn, the symmetric group, and since\np\na2\n1 + . . . + a2n is a symmetric function on a1, . . . , an, we have that ∥λ∥is\nwell defined.\nWe now define the efficiency of λ on a null-point w to be\ne(λ) = m(λ)\n∥λ∥\nWe immediately see that e(λ) = e(λ2).\n150"},{"paragraph_id":"p216","order":216,"text":"Lemma 26.1. Let W be a representation of SLn and let w ∈W be a null-\npoint. Let N(w, D) be the collection of all λ : C∗→D such that limt→0 t ·\nw = 0W . If N(w, D) is non-empty then there is a unique λ′ ∈N(w, D)\nwhich maximizes the efficiency, i.e., e(λ′) > e(λ) for all λ ∈N(w, D) and\nλ ̸= (λ′)k for any k ∈Z. This 1-parameter subgroup will be denoted by\nλ(w, D).\nProof: Suppose that N(w, D) is non-empty.\nThen in the weight-space\nexpansion of w for the maximal torus D, we see that supp(w) staisfies the\nhalf-space property for some λ ∈Yn.\nNote that the λ ∈N(w, D) are\nparametrized by lattice points λ ∈Yn such that (λ, χ) > 0 for all χ ∈\nsupp(w).\nLet Cone(w) be the conical combination (over R) of all χ ∈\nsupp(w) and Cone(w)◦its polar. Thus, in other words, N(w, D) is precisely\nthe collection of lattice points in the cone Cone(w)◦. Next, we see that e(λ)\nis a convex function of Cone(w)◦which is constant over rays R+ · λ for all\nλ ∈Cone(w)◦. By a routine analysis, the maximum of such a function must\nbe a unique ray with rational entries. This proves the lamma. □\nThis covers one important part in our task of identifying the ’best’ 1-\nparameter subgroup driving a null-point to zero. The next part is to relate\nD to other maximal tori.\nLet λ : C∗→SLn and let P(λ) be defined as follows:\nP(λ) = {A ∈SLn| lim\nt→0 λ(t)Aλ(t−1) = I ∈SLn}\nHaving fixed a maximal torus D containing IM(λ), we easily identify\nP(λ) as a parabolic subgroup, i.e., block upper-triangular. Indeed, let\nλ(t) ="},{"paragraph_id":"p217","order":217,"text":"ta1\n0\n0\n0\n0\nta2\n0\n0\n...\n0\n0\n0\ntan"},{"paragraph_id":"p218","order":218,"text":"with a1 ≥a2 ≥. . . ≥an (obviously with a1 + . . . + an = 0). Then\nP(λ) = {(xij|xij = 0 for all i, j such that ai < aj}\nThe unipotent radical U(λ) is a normal subgroup of P(λ) defined as:\nU(λ) = (xij) where ="},{"paragraph_id":"p219","order":219,"text":"xij = 0\nif ai < aj\nxij = δij\nif ai = aj\n151"},{"paragraph_id":"p220","order":220,"text":"Lemma 26.2. Let λ ∈N(w, D) and let g ∈P(λ), then (i) gλg−1 ⊆\nP(λ) and P(gλg−1) = P(λ), (ii) gλg−1 ∈N(w, gDg−1), and (iii) e(λ) =\ne(gλg−1).\nThis actually follows from the construction of the explicit SLn-modules\nand is left to the reader. We now come to the unique object that we will\ndefine for each w ∈W in the null-cone. This is the parabolic subgroup P(λ)\nfor any ’best’ λ. We have already seen above that if λ′ is a P(λ)-conjugate\nof a best λ then λ′ is ’equally best’ and P(λ) = P(λ′).\nWe now relate two general equally best λ and λ′. For this we need a\npreliminary definition and a lemma:\nDefinition 26.1. Let V be a vector space over C.\nA flag F of V is a\nsequence (V0, . . . , Vr) of nested subspaces 0 = V0 ⊂V1 ⊂. . . ⊂Vr = V .\nLemma 26.3. Let dimC(V ) = r and let F = (V0, . . . , Vr) and F′ =\n(V ′\n0, . . . , V ′\nr) be two (complete) flags for V . Then there is a basis b1, . . . , br\nof V and a permutation σ ∈Sr such that Vi = {b1, . . . , bi} and V ′\ni =\n{bσ(1), . . . , bσ(i)} for all i.\nThis is proved by induction on r.\nCorollary 26.1. Let λ and λ′ be two 1-parameter subgroups and P(λ) and\nP(λ′) be their corresponding parabolic subgroups. Then there is a maximal\ntorus T of SLn such that T ⊆P(λ) ∩P(λ′).\nProof: It is clear that there is a correspondence between parabolic sub-\ngroups of SLn and flags. We refine the flags associated to the parabolic\nsubgroups P(λ) and P(λ′) to complete flags and apply the above lemma. □\nWe are now prepared to prove Kempf’d theorem:\nTheorem 12. Let W be a representation of SLn and w ∈W a null-point.\nThen there is a 1-parameter subgroup λ ∈Γ(SLn) such that (i) for all λ′ ∈\nΓ(SLn), we have e(λ) ≥e(λ′), and (ii) for all λ′ such that e(λ) = e(λ′) we\nhave P(λ) = P(λ′) and that there is a g ∈P(λ) such that λ′ = gλg−1.\nProof: Let N(w) be all elements of Γ(SLn) which drive w to zero. Let Ξ(W)\nbe the (finite) collection of D-characters appearing in the representation W.\nFor every λ(w, T) such that Im(λ) ⊆D, we may consider an A ∈SLn such\nthat AλA−1 ∈N(A · w, D) and e(λ) = e(AλA−1). Since the ’best’ element\nof N(A · w, D) is determined by supp(A · w) ⊆Ξ, we see that there are only\nfinitely many possibilities for e(A · w, AλA−1) and therefore for e(λ) for the\n’best’ λ driving w to zero.\n152"},{"paragraph_id":"p221","order":221,"text":"Thus the length k of any sequence λ(w, T1), . . . , λ(w, Tk) such that e(λ(w, T1)) <\n. . . < e(λ(w, Tk)) must be bounded by the number 2Ξ. This proves (i).\nNext, let λ1 = λ(w, T1) and λ2 = λ(w, T2) be two ’best’ elements of\nN(w, T1) and N(w, T2) respectively. By corollary 26.1, we have a torus ,\nsay D, and P(λ(w, Ti))-conjugates λi such that (i) e(λi) = e(λ(w, Ti)) and\n(ii) Im(λi) ⊆D. By lemma 26.1, we have λ1 = λ2 and thus P(λ1) = P(λ2).\nOn the other hand, P(λ(w, Ti)) = P(λi) and this proves (ii). □\nThus 12 associates a unique parabolic subgroup P(w) to every point in\nthe null-cone. This subgroup is called the destabilizing flag of w. Clearly,\nif w is in the null-cone then so is A · w, where A ∈SLn. Furthermore, it is\nclear that P(A · w) = AP(w)A−1.\nCorollary 26.2. Let w ∈W be in the null-cone and let Gw ⊆SLn stabilize\nw. Then Gw ⊆P(w).\nProof: Let g ∈Gw. Since g · w = w, we see that gP(w)g−1 = P(w), and\nthat g normalizes P(w). Since the normalizer of any parabolic subgroup is\nitself, we see that g ∈P(w). □\n153"},{"paragraph_id":"p222","order":222,"text":"Chapter 27\nStability\nReference: [Ke, GCT1]\nRecall that z ∈W is stable iffits orbit O(z) is closed in W. In the last\nchapter, we tackled the points in the null-cone, i.e., points in the set [0W ]≈,\nor in other words, points which close onto the stable point 0W. A similar\nanalysis may be done for arbitrary stable points.\nFollowing kempf, let S ⊆W be a closed SLn-invariant subset.\nLet\nz ∈W be arbitrary. If the orbit-closure ∆(z) intersects S, then we associate\na unique parabolic subgroup Pz,S ⊆SLn as a witness to this fact.\nThe\nconstruction of this parabolic subgroup is in several steps.\nAs the first step, we construct a representation X of SLn and a closed\nSLn-invariant embedding φ : W →X such that φ−1(0X) = S, scheme-\ntheoretically. This may be done as follows: since S is a closed sub-variety\nof W, there is an ideal Is = (f1, . . . , fk) of definition for S. We may further\nassume that the vector space {f1, . . . , fk} is itself an SLn-module, say X.\nWe assume that X is k-dimensional.\nWe now construct the map φ : W →X as follows:\nφ(w) = (f1(x), . . . , fk(x))\nNote that φ(S) = 0X and that IS = (f1, . . . , fk) ensure that the requirements\non our φ do hold.\nNext, there is an adaptation of (Hilbert’s) Theorem 11 which we do not\nprove:\nTheorem 13. Let W be an SLn-module and let y ∈W be a stable point.\nLet z ∈[y]≈be an element which closes onto y. Then there is a 1-parameter\nsubgroup λ : C∗→SLn such that\nlim\nt→0 λ(t) · w ∈O(y)\n154"},{"paragraph_id":"p223","order":223,"text":"Thus the limit exists and lies in the closed orbit of y.\nNow suppose that ∆(z) ∩S is non-empty. Then there must be stable\ny ∈∆(z). We apply the theorem to O(y) and obtain the λ as above. This\nshows that there is indeed a 1-parameter subgroup driving z into S. Next,\nit is easy to see that\nlim\nt→0[λ(t) · φ(z)] = 0X\nThus φ(z) actually lies in the null-cone of X. We may now be tempted to\napply the techniques of the previous chapter to come up with the ’best’ λ\nand its parabolic, now called P(z, S). This is almost the technique to be\nadopted , except that this ’best’ λ drives φ(z) into 0X but limt→0[λ(t) · z]\n(which is supposed to be in S) may not exist! This is because we are using the\nunproved (and untrue) converse of the assertion that 1-parameter subgroups\nwhich drive z into S drive φ(z) into 0X.\nThis above argument is rectified by limiting the domain of allowed 1-\nparameter subgroups to (i) Cone(supp(φ(z))◦as before, and (ii) those λ\nsuch that limt→0[λ(t) · z] exists. This second condition is also a ’convex’\ncondition and then the ’best’ λ does exist. This completes the construction\nof P(z, S).\nAs before, if Gz ⊆SLn stabilizes z then it normalizes P(z, S) thus must\nbe contained in it:\nProposition 27.1. If Gz stabilizes z then Gz ⊆P(z, S).\nLet us now consider the permanent and the determinant. Let M\nbe the n2-dimensional space of all n × n-matrices.\nSince det and perm\nare homogeneous n-forms on M, we consider the SL(M)-module W =\nSymn(M∗). We recall now certain stabilizing groups of the det and the\nperm. We will need the definition of a certain group L′. This is defined as\nthe group generated by the permutation and diagonal matrices in GLn. In\nother words, L′ is the normalizer of the complete standard torus D∗⊆GLn.\nL is defined as that subgroup of L′ which is contained in SLn.\nProposition 27.2.\n(A) Consider the group K = SLn × SLn. We define\nthe action μK of typical element (A, B) ∈K on X ∈M as given by:\nX →AXB−1\nThen (i) M is an irreducible representation of K and Im(K) ⊆SL(M),\nand (ii) K stabilizes the determinant.\n155"},{"paragraph_id":"p224","order":224,"text":"(B) Consider the group H = L × L. We define the action μH of typical\nelement (A, B) ∈H on X ∈M as given by:\nX →AXB−1\nThen (i) M is an irreducible representation of H and Im(H) ⊆SL(M),\nand (ii) H stabilizes the permanent.\nWe are now ready to show:\nTheorem 14. The points det and perm in the SL(M-module W = Symn(M∗)\nare stable.\nProof: Lets look at det, the perm being similar. If det were not stable,\nthen there would be a closed SL(M)-invariant subset S ⊂W such that\ndet ̸∈S but closes onto S: just take S to be the unique closed orbit in\n[det]≈. Whence there is a parabolic P(det, S) which, by Proposition 27.1,\nwould contain K. This would mean that there is a K-invariant flag in M\ncorresponding to P(det, S). This contradicts the irreducibility of M as a\nK-module. □\n156"},{"paragraph_id":"p225","order":225,"text":"Bibliography\n[BBD] A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Ast ́erisque\n100, (1982), Soc. Math. France.\n[B]\nP. 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Sohoni, Geometric complexity theory, P vs.\nNP and explicit obstructions, in “Advances in Algebra and Geometry”,\nEdited by C. Musili, the proceedings of the International Conference\non Algebra and Geometry, Hyderabad, 2001.\n[GCT1] K. Mulmuley, M. Sohoni, Geometric complexity theory I: an ap-\nproach to the P vs. NP and related problems, SIAM J. Comput., vol\n31, no 2, pp 496-526, 2001.\n[GCT2] K. Mulmuley, M. Sohoni, Geometric complexity theory II: towards\nexplicit obstructions for embeddings among class varieties, to appear in\nSIAM J. Comput., cs. ArXiv preprint cs. CC/0612134, December 25,\n2006.\n[GCT3] K. Mulmuley, M. Sohoni, Geometric complexity theory III, on\ndeciding positivity of Littlewood-Richardson coefficients, cs. ArXiv\npreprint cs. CC/0501076 v1 26 Jan 2005.\n[GCT4] K. Mulmuley,\nM. Sohoni,\nGeometric complexity theory IV:\nquantum group for the Kronecker problem, cs. ArXiv preprint cs.\nCC/0703110, March, 2007.\n[GCT5] K. Mulmuley, H. Narayanan, Geometric complexity theory V: on\ndeciding nonvanishing of a generalized Littlewood-Richardson coeffi-\ncient, Technical report TR-2007-05, Comp. Sci. Dept. The university of\nchicago, May, 2007.\n[GCT6] K. Mulmuley,\nGeometric complexity theory VI: the flip via\nsaturated and positive integer programming in representation the-\nory and algebraic geometry, Technical report TR 2007-04, Comp.\n158"},{"paragraph_id":"p227","order":227,"text":"Sci. Dept., The University of Chicago, May, 2007. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu. Revised version to be available\nhere.\n[GCT7] K. Mulmuley, Geometric complexity theory VII: nonstandard quan-\ntum group for the plethysm problem (Extended Abstract), Technical\nreport TR-2007-14, Comp. Sci. Dept., The University of Chicago, Sept.\n2007. Available at: http://ramakrishnadas.cs.uchicago.edu.\n[GCT8] K. Mulmuley, Geometric complexity theory VIII: On canonical\nbases for the nonstandard quantum groups (Extended Abstract), Tech-\nnical report TR-2007-15, Comp. Sci. Dept., The University of Chicago,\nSept. 2007. Available at: http://ramakrishnadas.cs.uchicago.edu.\n[GCT9] B. Adsul, M. Sohoni, K. Subrahmanyam, Geometric complexity\ntheory IX: algbraic and combinatorial aspects of the Kronecker prob-\nlem, under preparation.\n[GCT10] K. Mulmuley, Geometric complexity theory X: On class varieties,\nand the natural proof barrier, under preparation.\n[GCT11] K. Mulmuley, Geometric complexity theory XI: on the flip over fi-\nnite or algebraically closed fields of positive characteristic, under prepa-\nration.\n[GLS] M. Gr ̈otschel, L. Lov ́asz, A. Schrijver, Geometric algorithms and com-\nbinatorial optimzation, Springer-Verlag, 1993.\n[H] H. Narayanan, On the complexity of computing Kostka numbers and\nLittlewood-Richardson coefficients Journal of Algebraic Combinatorics,\nVolume 24 , Issue 3 (November 2006) 347 - 354, 2006\n[KB79] R. Kannan and A. Bachem. Polynomial algorithms for computing\nthe Smith and Hermite normal forms of an integer matrix, SIAM J.\nComput., 8(4), 1979.\n[Kar84] N. Karmarkar. A new polynomial-time algorithm for linear pro-\ngramming. Combinatorica, 4(4):373–395, 1984.\n[KL] D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke\nalgebras, Invent. Math. 53 (1979), 165-184.\n[KL2] D. Kazhdan, G. Lusztig, Schubert varieties and Poincare duality,\nProc. Symp. Pure Math., AMS, 36 (1980), 185-203.\n159"},{"paragraph_id":"p228","order":228,"text":"[Kha79] L. G. Khachian. A polynomial algorithm for linear programming.\nDoklady Akedamii Nauk SSSR, 244:1093–1096, 1979. In Russian.\n[K] M. Kashiwara, On crystal bases of the q-analogue of universal envelop-\ning algebras, Duke Math. J. 63 (1991), 465-516.\n[Ke] G. Kempf: Instability in invariant theory, Annals of Mathematics, 108\n(1978), 299-316.\n[KTT] R. King, C. Tollu, F. Toumazet, Tretched Littlewood-Richardson\ncoefficients and Kostak coefficients. In, Winternitz, P. Harnard, J. Lam,\nC.S. and Patera, J. (eds.) Symmetry in Physics: In Memory of Robert\nT. Sharp. Providence, USA, AMS OUP, 99-112, CRM Proceedings and\nLecture Notes 34, 2004.\n[Ki] A. Kirillov, An invitation to the generalized saturation conjecture,\nmath. CO/0404353, 20 Apr. 2004.\n[KS] A. Klimyck, and K. Schm ̈udgen, Quantum groups and their represen-\ntations, Springer, 1997.\n[KT] A. Knutson, T. Tao, The honeycomb model of GLn(C) tensor products\nI: proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999)\n1055-1090.\n[KT2] A. Knutson, T. Tao: Honeycombs and sums of Hermitian matrices,\nNotices Amer. Math. Soc. 48 (2001) No. 2, 175-186.\n[LV] D. Luna and T. Vust, Plongements d’espaces homogenes, Comment.\nMath. Helv. 58, 186(1983).\n[Lu2] G. Lusztig, Introduction to quantum groups, Birkh ̈auser, 1993.\n[Ml] K. Mulmuley, Lower bounds in a parallel model without bit operations.\nSIAM J. Comput. 28, 1460–1509, 1999.\n[Mm] D. Mumford, Algebraic Geometry I, Springer-Verlang, 1995.\n[N] M. Nagata, Polynomial Rings and Affine Spaces. CBMS Regional Con-\nference no. 37, American Mathematical Society, 1978.\n[S]\nR. Stanley,\nEnumerative combinatorics,\nvol. 1,\nWadsworth and\nBrooks/Cole, Advanced Books and Software, 1986.\n[Z]\nA.\nZelevinsky,\nLittlewood-Richardson\nsemigroups,\narXiv:math.CO/9704228 v1 30 Apr 1997.\n160"},{"paragraph_id":"p229","order":229,"text":"Gv\nClosure\nDv"},{"paragraph_id":"p230","order":230,"text":"d\nZ\nn"}],"pages":[{"page":1,"text":"arXiv:0709.0746v1 [cs.CC] 5 Sep 2007\nGeometric Complexity Theory: Introduction\nDedicated to Sri Ramakrishna\nKetan D. Mulmuley 1\nThe University of Chicago\nMilind Sohoni\nI.I.T., Mumbai\nTechnical Report TR-2007-16\nComputer Science Department\nThe University of Chicago\nSeptember 2007\nAugust 2, 2014\n1Part of the work on GCT was done while the first author was visiting I.I.T.\nMumbai to which he is grateful for its hospitality"},{"page":2,"text":"Foreword\nThese are lectures notes for the introductory graduate courses on geo-\nmetric complexity theory (GCT) in the computer science department, the\nuniversity of Chicago. Part I consists of the lecture notes for the course\ngiven by the first author in the spring quarter, 2007. It gives introduction\nto the basic structure of GCT. Part II consists of the lecture notes for the\ncourse given by the second author in the spring quarter, 2003. It gives in-\ntroduction to invariant theory with a view towards GCT. No background\nin algebraic geometry or representation theory is assumed. These lecture\nnotes in conjunction with the article [GCTflip1], which describes in detail\nthe basic plan of GCT based on the principle called the flip, should provide\na high level picture of GCT assuming familiarity with only basic notions\nof algebra, such as groups, rings, fields etc. Many of the theorems in these\nlecture notes are stated without proofs, but after giving enough motivation\nso that they can be taken on faith. For the readers interested in further\nstudy, Figure 1 shows logical dependence among the various papers of GCT\nand a suggested reading sequence.\nThe first author is grateful to Paolo Codenotti, Joshua Grochow, Sourav\nChakraborty and Hari Narayanan for taking notes for his lectures.\n1"},{"page":3,"text":"GCTabs\n|\n↓\nGCTflip1\n|\n↓\nThese lecture notes\n−−→\nGCT3\n|\n|\n↓\n|\nGCT1\n|\n|\n|\n↓\n|\nGCT2\n|\n|\n|\n↓\n↓\nGCT6\n←−−\nGCT5\n|\n↓\nGCT4\n−−→\nGCT9\n|\n↓\nGCT7\n|\n↓\nGCT8\n|\n↓\nGCT10\n|\n↓\nGCT11\n|\n↓\nGCTflip2\nFigure 1: Logical dependence among the GCT papers\n2"},{"page":4,"text":"Contents\nI\nThe basic structure of GCT\nBy Ketan D. Mulmuley\n8\n1\nOverview\n9\n1.1\nOutline\n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n9\n1.2\nThe G ̈odelian Flip\n. . . . . . . . . . . . . . . . . . . . . . . .\n12\n1.3\nMore details of the GCT approach . . . . . . . . . . . . . . .\n13\n2\nRepresentation theory of reductive groups\n16\n2.1\nBasics of Representation Theory\n. . . . . . . . . . . . . . . .\n16\n2.1.1\nDefinitions\n. . . . . . . . . . . . . . . . . . . . . . . .\n16\n2.1.2\nNew representations from old . . . . . . . . . . . . . .\n17\n2.2\nReductivity of finite groups . . . . . . . . . . . . . . . . . . .\n19\n2.3\nCompact Groups and GLn(C) are reductive . . . . . . . . . .\n20\n2.3.1\nCompact groups\n. . . . . . . . . . . . . . . . . . . . .\n20\n2.3.2\nWeyl’s unitary trick and GLn(C) . . . . . . . . . . . .\n21\n3\nRepresentation theory of reductive groups (cont)\n22\n3.1\nProjection Formula . . . . . . . . . . . . . . . . . . . . . . . .\n23\n3.2\nThe characters of irreducible representations form a basis\n. .\n25\n3.3\nExtending to Infinite Compact Groups . . . . . . . . . . . . .\n27\n4\nRepresentations of the symmetric group\n29\n4.1\nRepresentations and characters of Sn . . . . . . . . . . . . . .\n30\n4.1.1\nFirst Construction . . . . . . . . . . . . . . . . . . . .\n30\n4.1.2\nSecond Construction . . . . . . . . . . . . . . . . . . .\n31\n4.1.3\nThird Construction . . . . . . . . . . . . . . . . . . . .\n32\n4.1.4\nCharacter of Sλ [Frobenius character formula] . . . . .\n32\n4.2\nThe first decision problem in GCT . . . . . . . . . . . . . . .\n33\n3"},{"page":5,"text":"5\nRepresentations of GLn(C)\n35\n5.1\nFirst Approach [Deruyts]\n. . . . . . . . . . . . . . . . . . . .\n35\n5.1.1\nHighest weight vectors . . . . . . . . . . . . . . . . . .\n38\n5.2\nSecond Approach [Weyl] . . . . . . . . . . . . . . . . . . . . .\n39\n6\nDeciding nonvanishing of Littlewood-Richardson coefficients 41\n6.1\nLittlewood-Richardson coefficients\n. . . . . . . . . . . . . . .\n41\n7\nLittlewood-Richardson coefficients (cont)\n46\n7.1\nThe stretching function\n. . . . . . . . . . . . . . . . . . . . .\n47\n7.2\nOn(C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n48\n8\nDeciding nonvanishing of Littlewood-Richardson coefficients\nfor On(C)\n52\n9\nThe plethysm problem\n56\n9.1\nLittlewood-Richardson Problem [GCT 3,5] . . . . . . . . . . .\n57\n9.2\nKronecker Problem [GCT 4,6] . . . . . . . . . . . . . . . . . .\n57\n9.3\nPlethysm Problem [GCT 6,7] . . . . . . . . . . . . . . . . . .\n58\n10 Saturated and positive integer programming\n61\n10.1 Saturated, positive integer programming . . . . . . . . . . . .\n61\n10.2 Application to the plethysm problem . . . . . . . . . . . . . .\n63\n11 Basic algebraic geometry\n64\n11.1 Algebraic geometry definitions\n. . . . . . . . . . . . . . . . .\n64\n11.2 Orbit closures . . . . . . . . . . . . . . . . . . . . . . . . . . .\n66\n11.3 Grassmanians . . . . . . . . . . . . . . . . . . . . . . . . . . .\n67\n12 The class varieties\n70\n12.1 Class Varieties in GCT . . . . . . . . . . . . . . . . . . . . . .\n70\n13 Obstructions\n73\n13.1 Obstructions\n. . . . . . . . . . . . . . . . . . . . . . . . . . .\n74\n13.1.1 Why are the class varieties exceptional? . . . . . . . .\n75\n14 Group theoretic varieties\n78\n14.1 Representation theoretic data . . . . . . . . . . . . . . . . . .\n79\n14.2 The second fundamental theorem . . . . . . . . . . . . . . . .\n80\n14.3 Why should obstructions exist? . . . . . . . . . . . . . . . . .\n81\n4"},{"page":6,"text":"15 The flip\n82\n15.1 The flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n83\n16 The Grassmanian\n86\n16.1 The second fundamental theorem . . . . . . . . . . . . . . . .\n87\n16.2 The Borel-Weil theorem . . . . . . . . . . . . . . . . . . . . .\n88\n17 Quantum group: basic definitions\n90\n17.1 Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . .\n90\n18 Standard quantum group\n97\n19 Quantum unitary group\n103\n19.1 A q-analogue of the unitary group\n. . . . . . . . . . . . . . . 103\n19.2 Properties of Uq . . . . . . . . . . . . . . . . . . . . . . . . . . 105\n19.3 Irreducible Representations of Gq . . . . . . . . . . . . . . . . 106\n19.4 Gelfand-Tsetlin basis . . . . . . . . . . . . . . . . . . . . . . . 106\n20 Towards positivity hypotheses via quantum groups\n108\n20.1 Littlewood-Richardson rule via standard quantum groups\n. . 108\n20.1.1 An embedding of the Weyl module . . . . . . . . . . . 109\n20.1.2 Crystal operators and crystal bases . . . . . . . . . . . 110\n20.2 Explicit decomposition of the tensor product\n. . . . . . . . . 112\n20.3 Towards nonstandard quantum groups for the Kronecker and\nplethysm problems . . . . . . . . . . . . . . . . . . . . . . . . 113\nII\nInvariant theory with a view towards GCT\nBy Milind Sohoni\n116\n21 Finite Groups\n117\n21.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117\n21.2 The finite group action . . . . . . . . . . . . . . . . . . . . . . 118\n21.3 The Symmetric Group . . . . . . . . . . . . . . . . . . . . . . 122\n22 The Group SLn\n124\n22.1 The Canonical Representation . . . . . . . . . . . . . . . . . . 124\n22.2 The Diagonal Representation . . . . . . . . . . . . . . . . . . 125\n22.3 Other Representations . . . . . . . . . . . . . . . . . . . . . . 128\n22.4 Full Reducibility\n. . . . . . . . . . . . . . . . . . . . . . . . . 130\n5"},{"page":7,"text":"23 Invariant Theory\n131\n23.1 Algebraic Groups and affine actions\n. . . . . . . . . . . . . . 131\n23.2 Orbits and Invariants . . . . . . . . . . . . . . . . . . . . . . . 132\n23.3 The Nagata Hypothesis\n. . . . . . . . . . . . . . . . . . . . . 136\n24 Orbit-closures\n139\n25 Tori in SLn\n142\n26 The Null-cone and the Destabilizing flag\n147\n26.1 Characters and the half-space criterion . . . . . . . . . . . . . 147\n26.2 The destabilizing flag . . . . . . . . . . . . . . . . . . . . . . . 149\n27 Stability\n154\n6"},{"page":8,"text":"7"},{"page":9,"text":"Part I\nThe basic structure of GCT\nBy Ketan D. Mulmuley\n8"},{"page":10,"text":"Chapter 1\nOverview\nScribe: Joshua A. Grochow\nGoal: An overview of GCT.\nThe purpose of this course is to give an introduction to Geometric Com-\nplexity Theory (GCT), which is an approach to proving P ̸= NP via al-\ngebraic geometry and representation theory. A basic plan of this approach\nis described in [GCTflip1, GCTflip2]. It is partially implemented in a se-\nries of articles [GCT1]-[GCT11]. The paper [GCTconf] is a conference an-\nnouncement of GCT. The paper [Ml] gives an unconditional lower bound\nin a PRAM model without bit operations based on elementary algebraic\ngeometry, and was a starting point for the GCT investigation via algebraic\ngeometry.\nThe only mathematical prerequisites for this course are a basic knowl-\nedge of abstract algebra (groups, ring, fields, etc.)\nand a knowledge of\ncomputational complexity. In the first month we plan to cover the represen-\ntation theory of finite groups, the symmetric group Sn, and GLn(C), and\nenough algebraic geometry so that in the remaining lectures we can cover\nbasic GCT. Most of the background results will only be sketched or omitted.\nThis lecture uses slightly more algebraic geometry and representation\ntheory than the reader is assumed to know in order to give a more complete\npicture of GCT. As the course continues, we will cover this material.\n1.1\nOutline\nHere is an outline of the GCT approach. Consider the P vs. NP question\nin characteristic 0; i.e., over integers. So bit operations are not allowed, and\n9"},{"page":11,"text":"basic operations on integers are considered to take constant time. For a sim-\nilar approach in nonzero characteristic (characteristic 2 being the classical\ncase from a computational complexity point of view), see GCT 11.\nThe basic principle of GCT is the called the flip [GCTflip1]. It “reduces”\n(in essence, not formally) the lower bound problems such as P vs. NP in\ncharacteristic 0 to upper bound problems: showing that certain decision\nproblems in algebraic geometry and representation theory belong to P. Each\nof these decision problems is of the form: is a given (nonnegative) structural\nconstant associated to some algebro-geometric or representation theoretic\nobject nonzero? This is akin to the decision problem: given a matrix, is\nits permanent nonzero? (We know how to solve this particular problem in\npolynomial time via reduction to the perfect matching problem.)\nNext, the preceding upper bound problems are reduced to purely math-\nematical positivity hypotheses [GCT6]. The goal is to show that these and\nother auxilliary structural constants have positive formulae. By a positive\nformula we mean a formula that does not involve any alternating signs like\nthe usual positive formula for the permanent; in contrast the usual formula\nfor the determinant involves alternating signs.\nFinally, these positivity hypotheses are “reduced” to conjectures in the\ntheory of quantum groups [GCT6, GCT7, GCT8, GCT10] intimately related\nto the Riemann hypothesis over finite fields proved in [Dl2], and the related\nworks [BBD, KL2, Lu2].\nA pictorial summary of the GCT approach is\nshown in Figure 1.1, where the arrows represent reductions, rather than\nimplications.\nTo recap: we move from a negative hypothesis in complexity theory\n(that there does not exist a polynomial time algorithm for an NP-complete\nproblem) to a positive hypotheses in complexity theory (that there exist\npolynomial-time algorithms for certain decision problems) to positive hy-\npotheses in mathematics (that certain structural constants have positive\nformulae) to conjectures on quantum groups related to the Riemann hy-\npothesis over finite fields, the related works and their possible extensions.\nThe first reduction here is the flip: we reduce a question about lower bounds,\nwhich are notoriously difficult, to the one about upper bounds, which we\nhave a much better handle on. This flip from negative to positive is already\npresent in G ̈odel’s work: to show something is impossible it suffices to show\nthat something else is possible. This was one of the motivations for the GCT\napproach. The G ̈odelian flip would not work for the P vs. NP problem be-\ncause it relativizes. We can think of GCT as a form of nonrelativizable (and\nnon-naturalizable, if reader knows what that means) diagonalization.\nIn summary, this approach very roughly “reduces” the lower bound prob-\n10"},{"page":12,"text":"P vs. NP\nchar. 0\nFlip\n=⇒\nDecision problems\nin alg. geom.\n& rep. thy.\n=⇒\nShow certain\nconstants in alg.\ngeom. and repr.\ntheory have\npositive formulae\nLower bounds\n(Neg. hypothesis\nin complexity thy.)\nUpper bounds\n(Pos. hypotheses\nin complexity thy.)\nPos. hypotheses\nin mathematics\n=⇒\nConjectures on\nquantum groups\nrelated to RH over\nfinite fields\nFigure 1.1: The basic approach of GCT\n11"},{"page":13,"text":"lems such as P vs. NP in characteristic zero to as-yet-unproved quantum-\ngroup-conjectures related to the Riemann Hypothesis over finite fields. As\nwith the classical RH, there is experimental evidence to suggest these con-\njectures hold – which indirectly suggests that certain generalizations of the\nRiemann hypothesis over finite fields also hold – and there are hints on how\nthe problem might be attacked. See [GCTflip1, GCT6, GCT7, GCT8] for a\nmore detailed exposition.\n1.2\nThe G ̈odelian Flip\nWe now re-visit G ̈odel’s original flip in modern language to get the flavor of\nthe GCT flip.\nG ̈odel set out to answer the question:\nQ: Is truth provable?\nBut what “truth” and “provable” means here is not so obvious a priori. We\nstart by setting the stage: in any mathematical theory, we have the syntax\n(i.e. the language used) and the semantics (the domain of discussion). In\nthis case, we have:\nSyntax (language)\nSemantics (domain)\nFirst order logic\n(∀, ∃, ¬, ∨, ∧, . . . )\nConstants\n0,1\nVariables\nx, y, z, . . .\nBasic Predicates\n>, <, =\nFunctions\n+,−,×,exponentiation\nAxioms\nAxioms of the natural numbers N\nUniverse: N\nA sentence is a valid formula with all variables quantified, and by a\ntruth we mean a sentence that is true in the domain. By a proof we mean a\nvalid deduction based on standard rules of inference and the axioms of the\ndomain, whose final result is the desired statement.\nHilbert’s program asked for an algorithm that, given a sentence in num-\nber theory, decides whether it is true or false.\nA special case of this is\nHilbert’s 10th problem, which asked for an algorithm to decide whether a\nDiophantine equation (equation with only integer coefficients) has a nonzero\ninteger solution.\nG ̈odel showed that Hilbert’s general program was not\n12"},{"page":14,"text":"achievable. The tenth problem remained unresolved until 1970, at which\npoint Matiyasevich showed its impossibility as well.\nHere is the main idea of G ̈odel’s proof, re-cast in modern language.\nFor a Turing Machine M, whether the empty string ε is in the language\nL(M) recognized by M is undecidable. The idea is to reduce a question\nof the form ε ∈L(M) to a question in number theory. If there were an\nalgorithm for deciding the truth of number-theoretic statements, it would\ngive an algorithm for the above Turing machine problem, which we know\ndoes not exist.\nThe basic idea of the reduction is similar to the one in Cook’s proof that\nSAT is NP-complete. Namely, ε ∈L(M) iffthere is a valid computation\nof M which accepts ε. Using Cook’s idea, we can use this to get a Boolean\nformula:\n∃m∃a valid computation of M with configurations of size ≤m\ns.t. the computation accepts ε.\nThen we use G ̈odel numbering – which assigns a unique number to each\nsentence in number theory – to translate this formula to a sentence in number\ntheory. The details of this should be familiar.\nThe key point here is: to show that truth is undecidable in number theory\n(a negative statement), we show that there exists a computable reduction\nfrom ε\n?∈L(M) to number theory (a positive statement). This is the essence\nof the G ̈odelian flip, which is analogous to – and in fact was the original\nmotivation for – the GCT flip.\n1.3\nMore details of the GCT approach\nTo begin with, GCT associates to each complexity class such as P and NP\na projective algebraic variety χP, χNP , etc. [GCT1]. In fact, it associates\na family of varieties χNP (n, m): one for each input length n and circuit\nsize m, but for simplicity we suppress this here. The languages L in the\nassociated complexity class will be points on these varieties, and the set\nof such points is dense in the variety. These varieties are thus called class\nvarieties. To show that NP ⊈P in characteristic zero, it suffices to show\nthat χNP cannot be imbedded in χP .\nThese class varieties are in fact G-varieties. That is, they have an action\nof the group G = GLn(C) on them. This action induces an action on the\nhomogeneous coordinate ring of the variety, given by (σf)(x) = f(σ−1x)\nfor all σ ∈G. Thus the coordinate rings RP and RNP of χP and χNP are\n13"},{"page":15,"text":"G-algebras, i.e., algebras with G-action. Their degree d-components RP (d)\nand RNP (d) are thus finite dimensional G-representations.\nFor the sake of contradiction, suppose NP ⊆P in characteristic 0. Then\nthere must be an embedding of χNP into χP as a G-subvariety, which\nin turn gives rise (by standard algebraic geometry arguments) to a sur-\njection RP ։ RNP of the coordinate rings.\nThis implies (by standard\nrepresentation-theoretic arguments) that RNP (d) can be embedded as a G-\nsub-representation of RP (d). The following diagram summarizes the impli-\ncations.\ncomplexity\nclasses\nclass\nvarieties\ncoordinate\nrings\nrepresentations\nof GLn(C)\nNP _\n \n//o/o/o/o/o/o/o/o/o/o\nχNP _\n \n//o/o/o/o/o/o/o/o/o/o\nRNP\n//o/o/o/o/o/o/o/o/o\nRNP (d)\n _\n \nP\n//o/o/o/o/o/o/o/o/o/o/o\nχP\n//o/o/o/o/o/o/o/o/o/o/o\nRP\nOO\n//o/o/o/o/o/o/o/o/o/o\nRP(d)\nWeyl’s theorem–that all finite-dimensional representations of G = GLn(C)\nare completely reducible, i.e. can be written as a direct sum of irreducible\nrepresentations–implies that both RNP (d) and RP (d) can be written as di-\nrect sums of irreducible G-representations. An obstruction [GCT2] of degree\nd is defined to be an irreducible G-representation occuring (as a subrepre-\nsentation) in RNP (d) but not in RP (d). Its existence implies that RNP (d)\ncannot be embedded as a subrepresentation of RP (d), and hence, χNP can-\nnot be embedded in χP as a G-subvariety; a contradiction.\nWe actually have a family of varieties χNP (n, m): one for each input\nlength n and circuit size m. Thus if an obstruction of some degree exists for\nall n →∞, assuming m = nlog n (say), then NP ̸= P in characteristic zero.\nConjecture 1.1. [GCTflip1] There is a polynomial-time algorithm for con-\nstructing such obstructions.\nThis is the GCT flip: to show that no polynomial-time algorithm exists\nfor an NP-complete problem, we hope to show that there is a polynomial\ntime algorithm for finding obstructions. This task then is further reduced to\nfinding polynomial time algorithms for other decision problems in algebraic\ngeometry and representation theory.\nMere existence of an obstruction for all n would actually suffice here. For\nthis, it suffices to show that there is an algorithm which, given n, outputs\n14"},{"page":16,"text":"an obstruction showing that χNP (n, m) cannot be imbedded in χP(n, m),\nwhen m = nlog n. But the conjecture is not just that there is an algorithm,\nbut that there is a polynomial-time algorithm.\nThe basic principle here is that the complexity of the proof of existence\nof an object (in this case, an obstruction) is very closed tied to the computa-\ntional complexity of finding that object, and hence, techniques underneath\nan easy (i.e. polynomial time) time algorithm for deciding existence may\nyield an easy (i.e. feasible) proof of existence. This is supported by much\nanecdotal evidence:\n• An obstruction to planar embedding (a forbidden Kurotowski minor)\ncan be found in polynomial, in fact, linear time by variants of the\nusual planarity testing algorithms, and the underlying techniques, in\nretrospect, yield an algorithmic proof of Kurotowski’s theorem that\nevery nonplanar graph contains a forbidden minor.\n• Hall’s marriage theorem, which characterizes the existence of per-\nfect matchings, in retrospect, follows from the techniques underlying\npolynomial-time algorithms for finding perfect matchings.\n• The proof that a graph is Eulerian iffall vertices have even degree is,\nessentially, a polynomial-time algorithm for finding an Eulerian circuit.\n• In contrast, we know of no Hall-type theorem for Hamiltonians paths,\nessentially, because finding such a path is computationally difficult\n(NP-complete).\nAnalogously the goal is to find a polynomial time algorithm for deciding\nif there exists an obstruction for given n and m, and then use the underlying\ntechniques to show that an obstruction always exists for every large enough\nn if m = nlog n. The main mathematical work in GCT takes steps towards\nthis goal.\n15"},{"page":17,"text":"Chapter 2\nRepresentation theory of\nreductive groups\nScribe: Paolo Codenotti\nGoal: Basic notions in representation theory.\nReferences: [FH, F]\nIn this lecture we review the basic representation theory of reductive\ngroups as needed in this course. Most of the proofs will be omitted, or just\nsketched.\nFor complete proofs, see the books by Fulton and Harris, and\nFulton [FH, F]. The underlying field throughout this course is C.\n2.1\nBasics of Representation Theory\n2.1.1\nDefinitions\nDefinition 2.1. A representation of a group G, also called a G-module, is\na vector space V with an associated homomorphism ρ : G →GL(V ). We\nwill refer to a representation by V .\nThe map ρ induces a natural action of G on V , defined by g·v = (ρ(g))(v).\nDefinition 2.2. A map φ : V →W is G-equivariant if the following dia-\ngram commutes:\nV\nφ\n−−−−→W\n yg\n yg\nV\nφ\n−−−−→W\n16"},{"page":18,"text":"That is, if φ(g·v) = g·φ(v). A G-equivariant map is also called G-invariant\nor a G-homomorphism.\nDefinition 2.3. A subspace W ⊆V is said to be a subrepresentation, or a\nG-submodule of a representation V over a group G if W is G-equivariant,\nthat is if g · w ∈W for all w ∈W.\nDefinition 2.4. A representation V of a group G is said to be irreducible\nif it has no proper non-zero G-subrepresentations.\nDefinition 2.5. A group G is called reductive if every finite dimensional\nrepresentation V of G is a direct sum of irreducible representation.\nHere are some examples of reductive groups:\n• finite groups;\n• the n-dimensional torus (C∗)n;\n• linear groups:\n– the general linear group GLn(C),\n– the special linear group SLn(C),\n– the orthogonal group On(C) (linear transformations that preserve\na symmetric form),\n– and the symplectic group Spn(C) (linear transformations that\npreserve a skew symmetric form);\n• Exceptional Lie Groups\nTheir reductivity is a nontrivial fact.\nIt will be proved later in this\nlecture for finite groups, and the general and special linear groups. In some\nsense, the list above is complete: all reductive groups can be constructed\nby basic operations from the components which are either in this list or are\nrelated to them in a simple way.\n2.1.2\nNew representations from old\nGiven representations V and W of a group G, we can construct new repre-\nsentations in several ways, some of which are described below.\n• Tensor product: V ⊗W. g · (v ⊗w) = (g · v) ⊗(g · w).\n• Direct sum: V ⊕W.\n17"},{"page":19,"text":"• Symmetric tensor representation: The subspace Symn(V ) ⊂V ⊗· · ·⊗\nV spanned by elements of the form\nX\nσ\n(v1 ⊗· · · ⊗vn) · σ =\nX\nσ\nvσ(1) ⊗· · · vσ(n),\nwhere σ ranges over all permutations in the symmetric group Sn.\n• Exterior tensor representation: The subspace Λn(V ) ⊂V ⊗· · · ⊗V\nspanned by elements of the form\nX\nσ\nsgn(σ)(v1 ⊗· · · ⊗vn) · σ =\nX\nσ\nsgn(σ)vσ(1) ⊗· · · vσ(n).\n• Let V and W be representations, then Hom(V, W) is also a represen-\ntation, where g · φ is defined so that the following diagram commutes:\nV\nφ\n−−−−→W\n yg\n yg\nV\ng·φ\n−−−−→W\nMore precisely,\n(g · φ)(v) = g · (φ(g−1 · v)).\n• In particular, V ∗: V →C is a representation, and is called the dual\nrepresentation.\n• Let G be a finite group. Let S be a finite G-set (that is, a finite set\nwith an associated action of G on its elements). We construct a vector\nspace over any field K (we will be mostly concerned with the case\nK = C), with a basis vector associated to each element in S. More\nspecifically, consider the set K[S] of formal sums P\ns∈S αses, where\nαs ∈K, and es is a vector associated with S ∈s. Note that this set\nhas a vector space structure over K, and there is a natural induced\naction of G on K[S], defined by:\ng ·\nX\ns∈S\nαses =\nX\ns∈S\nαseg·s.\nThis action gives rise to a representation of G.\n• In particular, G is a G-set under the action of left multiplication. The\nrepresentation we obtain in the manner described above from this G-\nset is called the regular representation.\n18"},{"page":20,"text":"2.2\nReductivity of finite groups\nProposition 2.1. Let G be a finite group. If W is a subrepresentation of a\nrepresentation V , then there exists a representation W ⊥s.t. V = W ⊕W ⊥.\nProof. Choose any Hermitian form H0 of V , and construct a new Hermitian\nform H defined as:\nH(v, w) =\nX\ng∈G\nHo(g · v, g · w).\nAveraging is a useful trick that is used very often in representation theory,\nbecause it ensures G-invariance. In fact, H is G-invariant, that is,\nH(v, w) =\nX\ng∈G\nHo(g · v, g · w) = H(h · v, h · w)\nLet W ⊥be the perpendicular complement to W with respect to the Her-\nmitian form H.\nThen W ⊥is also G-invariant, and therefore it is a G-\nsubmodule.\nCorollary 2.1. Every representation of a finite group is a direct sum of\nirreducible representations.\nLemma 2.1. (Schur) If V and W are irreducible representations over C,\nand φ : V →W is a homomorphism (i.e. a G-invariant map), then:\n1. Either φ is an isomorphism or φ = 0.\n2. If V = W, φ = λI for some λ ∈C.\nProof.\n1. Since Ker(φ), and Imφ are G-submodules, either Im(φ) = V\nor Im(φ) = 0.\n2. Let φ : V →V . Since C algebraically closed, there exists an eigenvalue\nλ of φ. Look at the map φ−λI : V →V . By (1), φ−λI = 0 (it can’t\nbe an isomorphism because something maps to 0). So φ = λI.\nCorollary 2.2. Every representation is a unique direct sum of irreducible\nrepresentations. More precisely, given two decompositions into irreducible\nrepresentations,\nV =\nM\nV ai\ni\n=\nM\nW bj\nj ,\nthere is a one to one correspondence between the Vi’s and Wj’s, and the\nmultiplicities correspond.\nProof. exercise (follows from Schur’s lemma).\n19"},{"page":21,"text":"gu\nu\nR\ngR\nFigure 2.1: Example of a left Haar measure for the circle (U1(C)). Left action by\na group element g on a small region R around u does not change the area.\n2.3\nCompact Groups and GLn(C) are reductive\nNow we prove reductivity of compact groups.\n2.3.1\nCompact groups\nExamples of compact groups:\n• Un(C) ⊆GLn(C), the unitary groups (all rows are normal and orthog-\nonal).\n• SUn(C) ⊆SLn(C), the special unitary group.\nGiven a compact group, a left-invariant Haar measure is a measure that\nis invariant under the left action of the group. In other words, multiplication\nby a group element does not change the area of a small region (i.e., the group\naction is an isometry, see figure 2.1).\nTheorem 2.1. Compact groups are reductive\nProof. We use the averaging trick again. In fact the proof is the same as\nin the case of finite groups, using integration instead of summation for the\naveraging trick. Let H0 be any Hermitian form on V. Then define H as:\nH(v, w) =\nZ\nG\nH(gv, gw)dG\n20"},{"page":22,"text":"where dG is a left-invariant Haar measure.\nNote that H is G-invariant.\nLet W ⊥be the perpendicular complement to W. Then W ⊥is G-invariant.\nHence it is a G-submodule.\nThe same proof as before then gives us Schur’s lemma for compact\ngroups, from which follows:\nTheorem 2.2. If G is compact, then every finite dimensional representation\nof G is a unique direct sum of irreducible representations.\n2.3.2\nWeyl’s unitary trick and GLn(C)\nTheorem 2.3. (Weyl) GLn(C) is reductive\nProof. (general idea)\nLet V be a representation of GLn(C). Then GLn(C) acts on V :\nGLn(C) ֒→V.\nBut Un(C) is a subgroup of GLn(C). Therefore we have an induced action\nof Un(C) on V , and we can look at V as a representation of Un(C). As a\nrepresentation of Un(C), V breaks into irreducible representations of Un(C)\nby the theorem above. To summarize, we have:\nUn(C) ⊆GLn(C) ֒→V = ⊕iVi,\nwhere the Vi’s are irreducible representations of Un(C). Weyl’s unitary trick\nuses Lie algebra to show that every finite dimensional representation of\nUn(C) is also a representation of GLn(C), and irreducible representations of\nUn(C) correspond to irreducible representations of GLn(C). Hence each Vi\nabove is an irreducible representation of GLn(C).\nOnce we know these groups are reductive, the goal is to construct and\nclassify their irreducible finite dimensional representations.\nThis will be\ndone in the next lectures: Specht modules for Sn, and Weyl modules for\nGLn(C).\n21"},{"page":23,"text":"Chapter 3\nRepresentation theory of\nreductive groups (cont)\nScribe: Paolo Codenotti\nGoal: Basic representation theory, continued from the last lecture.\nIn this lecture we continue our introduction to representation theory.\nAgain we refer the reader to the book by Fulton and Harris for full details\n[FH]. Let G be a finite group, and V a finite-dimensional G-representation\ngiven by a homomorphism ρ : G →GL(V ). We define the character of the\nrepresentation V (denoted χV ) by χV (g) = Tr(ρ(g)).\nSince Tr(A−1BA) = Tr(B), χV (hgh−1) = χV (g). This means charac-\nters are constant on conjugacy classes (sets of the form {hgh−1|h ∈G}, for\nany g ∈G). We call such functions class functions.\nOur goal for this lecture is to prove the following two facts:\nGoal 1 A finite dimensional representation is completely determined by its\ncharacter.\nGoal 2 The space of class functions is spanned by the characters of irreducible\nrepresentations. In fact, these characters form an orthonormal basis\nof this space.\nFirst, we prove some useful lemmas about characters.\nLemma 3.1. χV ⊕W = χV + χW\nProof. Let g ∈G, and let ρ, σ be homomorphisms from G into V and W,\nrespectively. Let λ1, . . . , λr be the eigenvalues of ρ(g), and μ1, . . . , μs the\n22"},{"page":24,"text":"eigenvalues of σ(g). Then (ρ ⊕σ)(g) = (ρ(g), σ(g)), so the eigenvalues of\n(ρ ⊕σ)(g) are just the eigenvalues of ρ(g) together with the eigenvalues of\nσ(g).\nThen χV (g) = P\ni λi, χW (g) = P\ni μi, and χV ⊕W = P\ni λi + P\ni μi.\nLemma 3.2. χV ⊗W = χV χW\nProof. Let g ∈G, and let ρ, σ be homomorphisms into V and W, respec-\ntively. Let λ1, . . . , λr be the eigenvalues of ρ(g), and μ1, . . . , μs the eigenval-\nues of σ(g). Then (ρ ⊗σ)(g) is the Kronecker product of the matrices ρ(g)\nand σ(g). So its eigenvalues are all λiμj where 1 ≤i ≤r, 1 ≤j ≤s.\nThen, Tr((ρ ⊗σ)(g)) = P\ni,j λiμj = (P\ni λi)\n P\nj μj\n \n, which is equal to\nTr(ρ(g))Tr(σ(g)).\n3.1\nProjection Formula\nIn this section, we derive a projection formula needed for Goal 1 that allows\nus to determine the multiplicity of an irreducible representation in another\nrepresentation. Given a G-module V , let V G = {v|∀g ∈G, g · v = v}. We\nwill call these elements G-invariant. Let\nφ =\n1\n|G|\nX\ng∈G\ng ∈End(V ),\n(3.1)\nwhere each g, via ρ is considered an element of End(V ).\nLemma 3.3. The map φ : V\n→V is a G-homomorphism; i.e., φ ∈\nHomG(V, V ) = (Hom(V, V ))G.\nProof. The set End(V ) is a G-module, as we saw in last class, via the fol-\nlowing commutative diagram: for any π ∈End(V ), and h ∈G:\nV\nπ\n−−−−→V\n yh\n yh\nV\nh·π\n−−−−→V.\nTherefore π ∈HomG(V, V ) (i.e., π is a G-equivariant morphism) iff\nh · π = π for all h ∈G.\nWhen φ is defined as in equation (3.1) above,\nh · φ =\n1\n|G|\nX\ng\nhgh−1 =\n1\n|G|\nX\ng\ng = φ.\n23"},{"page":25,"text":"Thus\nh · φ = φ, ∀h ∈G,\nand φ : V →V is a G-equivariant morphism, i.e. φ ∈HomG(V, V ).\nLemma 3.4. The map φ is a G-equivariant projection of V onto V G\nProof. For every w ∈W, let\nv = φ(w) =\n1\n|G|\nX\ng∈G\ng · w.\nThen\nh · v = h · φ(w) =\n1\n|G|\nX\ng∈G\nhg · w = v, for any h ∈G.\nSo v ∈V G. That is, Im(φ) ⊆V G. But if v ∈V G, then\nφ(v) =\n1\n|G|\nX\ng∈G\ng · v =\n1\n|G||G|v = v.\nSo V G ⊆Im(φ), and φ is the identity on V G. This means that φ is the\nprojection onto V G.\nLemma 3.5.\ndim(V G) =\n1\n|G|\nX\ng∈G\nχV (g).\nProof. We have: dim(V G) = Tr(φ), because φ is a projection (φ = φ|V G ⊕\nφ|Ker(φ)). Also,\nTr(φ) =\n1\n|G|\nX\ng∈G\nTrV (g) =\n1\n|G|\nX\ng∈G\nχV (g).\nThis gives us a formula for the multiplicity of the trivial representation\n(i.e., dim(V G)) inside V .\nLemma 3.6. Let V, W be G-representations. If V is irreducible, dim(HomG(V, W))\nis the multiplicity of V inside W. If W is irreducible, dim(HomG(V, W)) is\nthe multiplicity of W inside V .\nProof. By Schur’s Lemma.\n24"},{"page":26,"text":"Let Cclass(G) be the space of class functions on (G), and let (α, β) =\n1\n|G|\nP\ng α(g)β(g) be the Hermitian form on Cclass\nLemma 3.7. If V and W are irreducible G representations, then\n(χV , χW ) =\n1\n|G|\nX\ng∈G\nχV (g)χW (g) =\n(\n1\nif V ∼= W\n0\nif V ≇W.\n(3.2)\nProof. Since Hom(V, W) ∼= V ∗⊗W, χHom(V,W ) = χV ∗χW = χV χW . Now\nthe result follows from Lemmas 3.5 and 3.6.\nLemma 3.8. The characters of the irreducible representations form an or-\nthonormal set.\nProof. Follows from Lemma 3.7.\nIf V ,W are irreducible, then ⟨χV , χW ⟩is 0 if V ̸= W and 1 otherwise.\nThis implies that:\nTheorem 3.1 (Goal 1). A representation is determined completely by its\ncharacter.\nProof. Let V = L\ni V ⊕ai\ni\n. So χV = P\ni aiχVi, and ai = (χV , χVi). This\ngives us a formula for the multiplicity of an irreducible representation in\nanother representation, solely in terms of their characters.\nTherefore, a\nrepresentation is completely determined by its character.\n3.2\nThe characters of irreducible representations\nform a basis\nIn this section, we address Goal 2.\nLet R be the regular representation of G, V an irreducible representation\nof G.\nLemma 3.9.\nR =\nM\nV\nEnd(V, V ),\nwhere V ranges over all irreducible representations of G.\n25"},{"page":27,"text":"Proof. χR(g) is 0 if g is not the identity and |G| otherwise.\n(χR, χV ) =\n1\n|G|\nX\ng∈G\nχR(g)χV (g) =\n1\n|G||G|χV (e) = χV (e) = dim(V )\nLet α : G →C. For any G-module V , let φα,V = P\ng α(g)g : V →V\nExercise 3.1. φα,V is G equivariant (i.e. a G-homomorphism) iffα is a\nclass function.\nProposition 3.1. Suppose α : G →C is a class function, and (α, χV ) = 0\nfor all irreducible representations V . Then α is identically 0.\nProof. If V is irreducible, then, by Schur’s lemma, since φα,V is a G-homomorphism,\nand V is irreducible, φα,V = λId, where λ = 1\nnTr(φα,V ), n = dim(V ). We\nhave:\nλ = 1\nn\nX\ng\nα(g)χV (g) = 1\nn|G|(α, χV ∗).\nNow V is irreducible iffV ∗is irreducible. So λ = 1\nn|G|0 = 0. Therefore,\nφα,V = 0 for any irreducible representation, and hence for any representa-\ntion.\nNow let V be the regular representation. Since g as endomorphisms of\nV are linearly independent, φα,V = 0 implies that α(g) = 0.\nTheorem 3.2. Characters form an orthonormal basis for the space of class\nfunctions.\nProof. Follows from Proposition 3.1, and Lemma 3.8\nIf V = L\ni V ⊕ai\ni\n, and πi : V →V ⊕ai\ni\nis the projection operator. We have\na formula π =\n1\n|G|\nP\ng g for the trivial representation. Analogously:\nExercise 3.2. πi = dimVi\n|G|\nP\ng χVi(g)g.\n26"},{"page":28,"text":"3.3\nExtending to Infinite Compact Groups\nIn this section, we extend the preceding results to infinite compact groups.\nWe must take some facts as given, since these theorems are much more\ncomplicated than those for finite groups.\nConsider compact G, specifically Un(C), the unitary subgroup of G(C).\nU1(C) is the circle group. Since U1(C) is abelian, all its representations are\none-dimensional.\nSince the group G is infinite, we can no longer sum over it. The idea\nis to replace the sum\n1\n|G|\nP\ng f(g) in the previous setting with\nR\nG f(g)dμ,\nwhere μ is a left-invariant Haar measure on G.\nIn this fashion, we can\nderive analogues of the preceding results for compact groups. We need to\nnormalize, so we set\nR\nG dμ = 1.\nLet ρ : G →GL(V ), where V is a finite dimensional G-representation.\nLet χV (g) = Tr(ρ(g)). Let V = L\ni V ai\ni\nbe the complete decomposition of\nV into irreducible representations.\nWe can again create a projection operator π : V →V G, by letting\nπ =\nR\nG ρ(g)dμ.\nLemma 3.10. We have:\ndim(V G) =\nZ\nG\nχV (g)dμ.\nProof. This result is analogous to Lemma 3.5 for finite groups.\nFor class functions α, β, define an inner product\n(α, β) =\nZ\nG\nα(g)β(g)dμ.\nLemma 3.10 applied to HomG(V, W) gives\n(χV , χW ) =\nZ\nG\nχV χW dμ = dim(HomG(V, W)).\nLemma 3.11. If V, W are irreducible, (χV , χW) = 1 if V and W are iso-\nmorphic, and (χV , χW ) = 0 otherwise.\nProof. This result is analogous to Lemma 3.7 for finite groups.\nLemma 3.12. The irreducible representations are orthonormal, just as in\nLemma 3.8 in the case of finite groups.\n27"},{"page":29,"text":"If V is reducible, V = L\ni V ⊕ai\ni\n, then\nai = (χV , χVi) =\nZ\nG\nχV χVidμ.\nHence\nTheorem 3.3. A finite dimensional representation is completely determined\nby its character.\nThis achieves Goal 1 for compact groups. Goal 2 is much harder:\nTheorem 3.4 (Peter-Weyl Theorem). (1) The characters of the irreducible\nrepresentations of G span a dense subset of the space of continuous class\nfunctions.\n(2) The coordinate functions of all irreducible matrix representations of\nG span a dense subset of all continuous functions on G.\nBy a coordinate function of a representation ρ : G →GL(V ), we mean\nthe function on G corresponding to a fixed entry of the matrix form of\nρ(g). For G = U1(C), (2) gives the Fourier series expansion on the circle.\nHence, the Peter-Weyl theorem constitutes a far reaching generalization of\nthe harmonic analyis from the circle to general Un(C).\n28"},{"page":30,"text":"Chapter 4\nRepresentations of the\nsymmetric group\nScribe: Sourav Chakraborty\nGoal:\nTo determine the irreducible representations of the Symmetric\ngroup Sn and their characters.\nReference:\n[FH, F]\nRecall\nLet G be a reductive group. Then\n1. Every finite dimensional representation of G is completely reducible,\nthat is, can be written as a direct sum of irreducible representations.\n2. Every irreducible representation is determined by its character.\nExamples of reductive groups:\n• Continuous: algebraic torus (C∗)m, general linear group GLn(C), spe-\ncial linear group Sln(C), symplectic group Spn(C), orthogonal group\nOn(C).\n• Finite: alternating group An, symmetric group Sn, Gln(Fp), simple lie\ngroups of finite type.\n29"},{"page":31,"text":"4.1\nRepresentations and characters of Sn\nThe number of irreducible representations of Sn is the same as the the\nnumber of conjugacy classes in Sn since the irreducible characters form a\nbasis of the space of class functions.\nEach permutation can be written\nuniquely as a product of disjoint cycles. The collection of lengths of the\ncycles in a permutation is called the cycle type of the permutation. So a\ncycle type of a permutation on n elements is a partition of n. And in Sn\neach conjugacy class is determined by the cycle type, which, in turn, is\ndetermined by the partition of n. So the number of conjugacy class is same\nas the number of partitions of n. Hence:\nNumber of irreducible representations of Sn = Number of partitions of n\n(4.1)\nLet λ = {λ1 ≥λ2 ≥. . . } be a partition of n; i.e., the size |λ| = P λi is\nn. The Young diagram corresponding to λ is a table shown in Figure 1. It\nis like an inverted staircase. The top row has λ1 boxes, the second row has\nλ2 boxes and so on. There are exactly n boxes.\nrow 1\nrow 5\nrow 6\nrow 2\nrow 3\nrow 4\nFigure 4.1: Row i has λi number of boxes\nFor a given partition λ, we want to construct an irreducible representa-\ntion Sλ, called the Specht-module of Sn for the partition λ, and calculate\nthe character of Sλ. We shall give three constructions of Sλ.\n4.1.1\nFirst Construction\nA numbering T of a Young diagram is a filling of the boxes in its table\nwith distinct numbers from 1, . . . , n. A numbering of a Young diagram is\n30"},{"page":32,"text":"also called a tableau. It is called a standard tableaux if the numbers are\nstrictly increasing in each row and column. By Tij we mean the value in the\ntableaux at i-th row and j-th column. We associate with each tableaux T a\npolynomial in C[X1, X2, . . . , Xn]:\nfT = ΠjΠi<i′(XTij −XTi′j).\nLet Sλ be the subspace of C[X1, X2, . . . , Xn] spanned by fT ’s, where T\nranges over all tableaux of shape λ. It is a representation of Sn. Here Sn\nacts on C[X1, X2, . . . , Xn] as:\n(σ.f)(X1, X2, . . . , Xn) = f(Xσ(1), Xσ(2), . . . , Xσ(n))\nTheorem 4.1.\n1. Sλ is irreducible.\n2. Sλ ̸≈Sλ′ if λ ̸= λ′\n3. The set {fT }, where T ranges over standard tableau of shape λ, is a\nbasis of Sλ.\n4.1.2\nSecond Construction\nLet T be a numbering of a Young diagram with distinct numbers from\n{1, . . . , n}. An element σ in Sn acts on T in the usual way by permuting\nthe numbers. Let R(T), C(T) ⊂Sn be the sets of permutations that fix\nthe rows and columns of T, respectively. We have R(σT) = σR(T)σ−1 and\nC(σT) = σR(T)σ−1. We say T ≡T ′ if the rows of T and T ′ are the same\nup to ordering. The equivalence class of T, called the tabloid, is denoted by\n{T}. Its orbit is isomorphic to Sn/R(T).\nLet C[Sn] be the group algebra of Sn. Representations of Sn are the same\nas the representations of C[Sn]. The element aT = P\np∈R(T) p in C[Sn] is\ncalled the row symmetrizer, bT = P\nq∈C(T) sign(q)q the column symmetrizer,\nand cT = aT bT the Young symmetrizer.\nLet\nVT = bT .{T} =\nX\nq∈C(T)\nsign(q){qT}.\nThen σ.VT = VσT . Let Sλ be the span of all VT ’s, where T ranges over\nall numberings of shape λ.\nTheorem 4.2.\n1. Sλ is irreducible\n31"},{"page":33,"text":"2. Sλ ̸≈Sλ′ if λ ̸= λ′.\n3. The set {VT |T standard} forms a basis for Sλ.\n4.1.3\nThird Construction\nLet T be a canonical numbering of shape λ = (λ1, . . . , λk). By this, we mean\nthe first row is numbered by 1, . . . , λ1, the second row by λ1 + 1, . . . λ1 + λ2,\nand so on, and the rows are increasing. Let aλ = aT , bλ = bT , and cλ =\ncT = aT .bT .\nThen Sλ = C[Sn].cλ is a representation of Sn from the left.\nTheorem 4.3.\n1. Sλ is irreducible\n2. Sλ ̸∼= Sλ′ if λ ̸= λ′.\n3. The basis: an exercise.\n4.1.4\nCharacter of Sλ [Frobenius character formula]\nLet i = (i1, i2, . . . , ik) be such that P\nj jij = n. Let Ci be the conjugacy\nclass consisting of permutations with ij cycles of length j. Let χλ be the\ncharacter of Sλ. The goal is to find χλ(Ci).\nLet λ : λ1 ≥· · · ≥λk be a partition of length k. Given k variables\nX1, X2, . . . , Xk, let\nPj(X) =\nX\nXj\ni ,\nbe the power sum, and\n∆(X) = Πi<j(Xj −Xi)\nthe discriminant. Let f(X) be a formal power series on Xi’s. Let [f(X)]l1,l2,...,lk\ndenote the coefficient of Xl1\n1 Xl2\n2 . . . Xlk\nk in f(X). Let li = λi + k −i.\nTheorem 4.4 (Frobenius Character Formula).\nχλ(Ci) =\n \n∆(X) · ΠjPj(X)ij \nl1,l2,...,lk\n32"},{"page":34,"text":"4.2\nThe first decision problem in GCT\nNow we can state the first hard decision problem in representation theory\nthat arises in the context of the flip. Let Sα and Sβ be two Specht modules\nof Sn. Since Sn is reductive, Sα ⊗Sβ decomposes as\nSα ⊗Sβ =\nM\nkλ\nαβSλ\nHere kλ\nαβ is called the Kronecker coefficient.\nProblem 4.1. (Kronecker problem) Given λ, α and β decide if kλ\nαβ > 0.\nConjecture 4.1 (GCT6). This can be done in polynomial time; i.e. in time\npolynomial in the bit lengths of the inputs λ, α and β.\n33"},{"page":35,"text":"55\n34"},{"page":36,"text":"Chapter 5\nRepresentations of GLn(C)\nScribe: Joshua A. Grochow\nGoal: To determine the irreducible representations of GLn(C) and their\ncharacters.\nReferences: [FH, F]\nThe goal of today’s lecture is to classify all irreducible representations\nof GLn(C) and compute their characters. We will go over two approaches,\nthe first due to Deruyts and the second due to Weyl.\nA polynomial representation of GLn(C) is a representation ρ : GLn(C) →\nGL(V ) such that each entry in the matrix ρ(g) is a polynomial in the entries\nof the matrix g ∈GLn(C).\nThe main result is that the polynomial irreducible representations of\nGLn(C) are in bijective correspondence with Young diagrams λ of height\nat most n, i.e. λ1 ≥λ2 ≥· · · ≥λn ≥0. Because of the importance of\nWeyl’s construction (similar constructions can be used on many other Lie\ngroups besides GLn(C)), the irreducible representation corresponding to λ\nis known as the Weyl module Vλ.\n5.1\nFirst Approach [Deruyts]\nLet X = (xij) be a generic n × n matrix with variable entries xij. Consider\nthe polynomial ring C[X] = C[x11, x12, . . . , xnn].\nThen GLn(C) acts on\nC[X] by (A ◦f)(X) = f(ATX) (it is easily checked that this is in fact a left\naction).\nLet T be a tableau of shape λ. To each column C of T of length r, we\nassociate an r × r minor of X as follows: if C has the entries i1, . . . , ir, then\n35"},{"page":37,"text":"take from the first r columns of X the rows i1, . . . , ir. Visually:\nC =\n \n \n \ni1\n...\nir\n \n \n −→eC =\n1\n· · ·\nr\n↓\n↓\ni1 →\ni2 →\n...\nir →\n \n \n \n \n \n \n \n \n \n \n \nxi1,1\n· · ·\nxi1,r\n· · ·\nxi1,n\nxi2,1\n· · ·\nxi2,r\n· · ·\nxi2,n\n...\n...\n...\nxir,1\n· · ·\nxir,r\n· · ·\nxir,n\n \n \n \n \n \n \n \n \n \n \n \n(Thus if there is a repeated number in the column C, eC = 0, since\nthe same row will get chosen twice.) Using these monomials eC for each\ncolumn C of the tableau T, we associate a monomial to the entire tableau,\neT = Q\nC eC. (Thus, if in any column of T there is a repeated number,\neT = 0. Furthermore, the numbers must all come from {1, . . . , n} if they\nare to specify rows of an n × n matrix.\nSo we restrict our attention to\nnumberings of T from {1, . . . , n} in which the numbers in any given column\nare all distinct.)\nLet Vλ be the vector space generated by the set {eT }, where T ranges over\nall such numberings of shape λ. Then GLn(C) acts on Vλ: for g ∈GLn(C),\neach row of gX is a linear combination of the rows of X, and since eC is a\nminor of X, g · eC is a linear combination of minors of X of the same size,\ni.e. g(eC) = P\nD ag\nC,DeD (this follows from standard linear algebra). Then\ng(eT )\n=\ng(eC1eC2 · · · eCk)\n=\n X\nD\nag\nC1,DeD\n!\n· · ·\n X\nD\nag\nCk,DeD\n!\nIf we expand this product out, we find that each term is in fact eT ′ for some\nT ′ of the appropriate shape. We then have the following theorem:\nTheorem 5.1.\n1. Vλ is an irreducible representation of GLn(C).\n2. The set {eT |T is a semistandard tableau of shape λ} is a basis for Vλ.\n(Recall that a semistandard tableau is one whose numbering is weakly\nincreasing across each row and strictly increasing down each column.)\n3. Every polynomial irreducible representation of GLn(C) of degree d is\nisomorphic to Vλ for some partition λ of d of height at most n.\n36"},{"page":38,"text":"4. Every rational irreducible representation of GLn(C) (each entry of ρ(g)\nis a rational function in the entries of g ∈GLn(C)) is isomorphic to\nVλ⊗detk for some partition λ of height at most n and for some integer\nk (where det is the determinant representation).\n5. (Weyl’s character formula) Define the character χλ of Vλ by χλ(g) =\nTr(ρ(g)), where ρ : GLn(C) →GL(Vλ) is the representation map.\nThen, for g ∈GLn(C) with eigenvalues x1, . . . , xn,\nχλ(g) = Sλ(x1, . . . , xn) :=\n xλi+n−i\nj\n \n xn−i\nj\n \n(where |yi\nj| is the determinant of the n × n matrix whose entries are\nyij = yi\nj, so, e.g., the determinant in the denominator is the usual van\nder Monde determinant, which is equal to Q\ni<j(xi −xj)). Here Sλ is\na polynomial, called the Schur polynomial.\nNB: It turns out that all holomorphic representations of GLn(C) are\nrational, and, by part (4) of the theorem, the Weyl modules classify all such\nrepresentations up to scalar multiplication by powers of the determinant.\nWe’ll give here a very brief introduction to the Schur polynomial in-\ntroduced in the above theorem, and explain why the Schur polynomial Sλ\nassociated to λ gives the character of Vλ.\nLet λ be a partition, and T a semistandard tableau of shape λ. Define\nx(T) = Q\ni xμi(T)\ni\n∈C[x1, . . . , xn], where μi(T) is the number of times i\nappears in T. Then it can be shown [F] that\nSλ(x1, . . . , xn) =\nX\nT\nx(T),\nwhere the sum is taken over all semistandard tableau of shape λ.\nProposition 5.1. Sλ(x1, . . . , xn) is the character of Vλ, where x1, . . . , xn\ndenote the eigenvalues of an element of GLn(C).\nProof. It suffices to show this diagonalizable g ∈GLn(C), since the diago-\nnalizable matrices are dense in GLn(C).\nSo let g ∈GLn(C) be diagonalizable with eigenvalues x1, . . . , xn. We\ncan assume that g is diagonal. If not, let A be a matrix that diagonalizes\ng.\nSo AgA−1 is diagonal with x1, . . . , xn as its diagonal entries.\nIf ρ :\nGLn(C) →GL(Vλ) is the representation corresponding to the module Vλ,\n37"},{"page":39,"text":"then conjugate ρ by A to get ρ′ : GLn(C) →GL(Vλ) defined by ρ′(h) =\nAρ(h)A−1. In particular, since trace is invariant under conjugation, ρ and\nρ′ have the same character. The module corresponding to ρ′ is simply A·Vλ,\nwhich is clearly isomorphic to Vλ since A is invertible. Thus to compute the\ncharacter χλ(g), it suffices to compute the character of g under ρ′, i.e., when\ng is diagonal, as we shall assume now.\nWe will show that eT is an eigenvector of g with eigenvalue x(T), i.e.\ng(eT ) = x(T)eT . Then since {eT |T is a semistandard tableau of shape λ} is\na basis for Vλ, the trace of g on Vλ will just be P\nT x(T), where the sum is\nover semistandard T of shape λ; this is exactly Sλ(x1, . . . , xn).\nWe reduce to the case where T is a single column. Suppose the claim\nis true for all columns C. Then since eT is a product of eC where C is\na column, the corresponding eigenvalue of eT will be Q\nC x(C) (where the\nproduct is taken over the columns C of T), which is exactly x(T).\nSo assume T is a single column, say with entries i1, . . . , ir. Then eT is\nsimply the above-mentioned r×r minor of the generic n×n matrix X = (xij)\n(do not confuse the double-indexed entries of the matrix X with the single-\nindexed eigenvalues of g). Since g is diagonal, gt = g. So (g ◦eT )(X) =\neT (gtX) = eT (gX). Thus g multiplies the ij-th column by xij. Thus its\neffect on eT is simply to multiply it by Qr\nj=1 xij, which is exactly x(T).\n5.1.1\nHighest weight vectors\nThe subgroup B ⊂GLn(C) of lower triangular invertible matrices, called\nthe Borel subgroup, is solvable.\nSo every irreducible representation of B\nis one-dimensional. A weight vector for GLn(C) is a vector v which is an\neigenvector for every matrix b ∈B.\nIn other words, there is a function\nλ : B →C such that b · v = λ(b)v for all b ∈B. The restriction of λ to the\nsubgroup of diagonal matrices in B is known as the weight of v.\nAs we showed in the proof of the above proposition,\n \n \n \nx1\n...\nxn\n \n \n eT = x(T)eT = xλ1\n1 · · · xλn\nn eT .\nSo eT is a weight vector with weight x(T). Thus Theorem 5.1 (2) gives a\nbasis consisting entirely of weight vectors. We abbreviate the weight x(T)\nby the sequence of exponents (λ1, . . . , λn). We say eT is a highest weight\nvector if its weight is the highest in the lexicographic ordering (using the\nabove sequence notation for the weight).\n38"},{"page":40,"text":"Each Vλ has a unique (up to scalars) B-invariant vector, which turns\nout to be the highest weight vector: namely eT , where T is canonical. For\nexample, for λ = (5, 3, 2, 2, 1), the canonical T is:\nT =\n1 1 1 1 1\n2 2 2\n3 3\n4 4\n5\nNote that the weight of such eT is (λ1, . . . , λn), so that the highest weight\nvector uniquely determines λ, and thus the entire representation Vλ. (This\nis a general feature of highest weight vectors in the representation theory of\nLie algebras and Lie groups.) Thus the irreducible representations GLn(C)\nare in bijective correspondence wih the highest weights of GLn(C), i.e. the\nsequences of exponents of the eigenvalues of the B-invariant eigenvectors.\n5.2\nSecond Approach [Weyl]\nLet V = Cn and consider the d-th tensor power V ⊗d. The group GLn(C)\nacts on V ⊗d on the left by the diagonal action\ng(v1 ⊗· · · ⊗vd) = gv1 ⊗· · · ⊗gvd (g ∈GL(V ))\nwhile the symmetric group Sd acts on the right by\n(v1 ⊗· · · ⊗vd)τ = v1τ ⊗· · · ⊗vdτ (τ ∈Sd) .\nThese two actions commute, so V ⊗d is a representation of H = GLn(C) ×\nSd. Every irreducible representation of H is of the form U ⊗W for some\nirreducible representation U of GLn(C) and some irreducible representation\nW of Sd. Since both GLn(C) and Sd are reductive (every finite-dimensional\nrepresentation is a direct sum of irreducible representations), their product\nH is reductive as well. So there are some partitions α and β and integers\nmαβ such that\nV ⊗d =\nM\n(Vα ⊗Sβ)mαβ,\nwhere Vα are Weyl modules and Sβ are Specht modules (irreducible repre-\nsentations of the symmetric group Sd).\nTheorem 5.2. V ⊗d = L\nλ Vλ⊗Sλ, where the sum is taken over partitions λ\nof d of height at most n. (Note that each summand appears with multiplicity\none, so this is a “multiplicity-free” decomposition.)\n39"},{"page":41,"text":"Now, let T be any standard tableau of shape λ, and recall the Young\nsymmetrizer cT from our discussion of the irreducible representations of Sd.\nThen V ⊗dcT is a representation of GLn(C) from the left (since cT ∈C[Sd]\nacts on the right, and the left action of GLn(C) and the right action of Sd\ncommute.)\nTheorem 5.3. V ⊗dcT ∼= Vλ\nThus\nV ⊗d =\nM\nλ:|λ|=d\nM\nstd. tableau\nT of shape λ\nV ⊗dcT ,\nwhere |λ| = P λi denotes the size of λ. In particular, Vλ occurs in V ⊗d with\nmultiplicity dim(Sλ).\nFinally, we construct a basis for V ⊗dcT . A bitableau of shape λ is a pair\n(U, T) where U is a semistandard tableau of shape λ and T is a standard\ntableau of shape λ. (Recall that the the semistandard tableau of shape λ\nare in natural bijective correspondence with a basis for the Weyl module Vλ,\nwhile the standard tableau of shape λ are in natural bijective correspondence\nwith a basis for the Specht module Sλ.)\nTo each bitableau we associate a vector e(U,T) = ei1 ⊗· · · ⊗eid where\nij is defined as follows. Each number 1, . . . , n appears in T exactly once.\nThe number ij is the entry of U in the same location as the number j in T;\npictorially:\nU\nT\nij\nj\nThen:\nTheorem 5.4. The set {e(U,T)cT } is a basis for V ⊗dcT .\n40"},{"page":42,"text":"Chapter 6\nDeciding nonvanishing of\nLittlewood-Richardson\ncoefficients\nScribe: Hariharan Narayanan\nGoal:\nTo show that nonvanishing of Littlewood-Richardson coefficients\ncan be decided in polynomial time.\nReferences: [DM2, GCT3, KT2]\n6.1\nLittlewood-Richardson coefficients\nFirst we define Littlewood-Richardson coefficients, which are basic quanti-\nties encountered in representation theory. Recall that the irreducible rep-\nresentations Vλ of GLn(C), the Weyl modules, are indexed by partitions λ,\nand:\nTheorem 6.1 (Weyl). Every finite dimensional representation of GLn(C)\nis completely reducible.\nLet G = GLn(C). Consider the diagonal embedding of G ֒→G × G.\nThis is a group homomorphism. Any G × G module, in particular, Vα ⊗Vβ\ncan also be viewed as a G module via this homomorphism. It then splits\ninto irreducible G-submodules:\n41"},{"page":43,"text":"Vα ⊗Vβ = ⊕γcγ\nαβVλ.\n(6.1)\nHere cγ\nαβ is the multiplicity of Vγ in Vα ⊗Vβ and is known as the Littlewood-\nRichardson coefficient.\nThe character of Vλ is the Schur polynomial Sλ. Hence, it follows from\n(6.1) that the Schur polynomials satisfy the following relation:\nSαSβ = ⊕γcγ\nαβSλ.\n(6.2)\nTheorem 6.2. cγ\nαβ is in PSPACE.\nProof: This easily follows from eq.(6.2) and the definition of Schur poly-\nnomials.\nAs a matter of fact, a stronger result holds:\nTheorem 6.3. cγ\nαβ is in #P.\nRecall that #P ⫅PSPACE.\nProof:\nThis is an immediate consequence of the following Littlewood-\nRichardson rule (formula) for cγ\nαβ. To state it, we need a few definitions.\nGiven partitions γ and α, a skew Young diagram of shape γ/α is the dif-\nference between the Young diagrams for γ and α, with their top-left corners\naligned; cf. Figure 6.2. A skew tableau of shape γ/α is a numbering of the\nboxes in this diagram. It is called semi-standard (SST) if the entries in each\ncolumn are strictly increasing top to bottom and the entries in each row\nare weakly increasing left to right; see Figures 6.1 and 6.2. The row word\nrow(T) of a skew-tableau T is the sequence of numbers obtained by reading\nT left to right, bottom to top; e.g. row(T) for Figure 6.2 is 13312211. It is\ncalled a reverse lattice word, if when read right to left, for each i, the number\nof i’s encountered at any point is at least the number of i + 1’s encountered\ntill that point; thus the row word for Figure 6.2 is a reverse lattice word.\nWe say that T is an LR tableau for given α, β, γ of shape γ/α and content\nβ if\n1. T is an SST,\n2. row(T) is a reverse lattice word,\n3. T has shape γ/α, and\n4. the content of T is β, i. e. the number of i’s in T is βi.\n42"},{"page":44,"text":"1 1 2 5\n2 2 3\n4\nFigure 6.1: semi-standard Young tableau\n:\n:\n:\n:\n:\n: 1 1\n:\n:\n: 1 2 2\n:\n: 3\n1 3\nFigure 6.2: Littlewood–Richardson skew tableau\nFor example, Figure 6.2 shows an LR tableau with α = (6, 3, 2), β =\n(4, 2, 2) and γ = (8, 6, 3, 2).\nThe Littlewood-Richardson rule [F, FH]: cγ\nαβ is equal to the number\nof LR skew tableaux of shape γ/α and content β.\nRemark: It may be noticed that the Littlewood-Richardson rule depends\nonly on the partitions α, β and γ and not on n, the rank of GLn(C) (as long\nas it is greater than or equal to the maximum height of α, β or γ). For this\nreason, we can assume without loss of generalitity that n is the maximum\nof the heights of α, β and γ, as we shall henceforth.\nNow we express cγ\nαβ as the number of integer points in some polytope\nP γ\nαβ using the Littlewood-Richardson rule:\nLemma 6.1. There exists a polytope P = P γ\nαβ of dimension polynomial in\nn such that the number of integer points in it is cγ\nαβ.\nProof: Let ri\nj(T), i ≤n, j ≤n, denote the number of j’s in the i-th row of\nT. If T is an LR-tableau of shape γ/α with content β then these integers\nsatisfy the following constraints:\n1. Nonnegativity: ri\nj ≥0.\n2. Shape constraints: For i ≤n,\nαi +\nX\nj\nri\nj = γi.\n43"},{"page":45,"text":"3. Content constraints: For j ≤n:\nX\ni\nri\nj = βj.\n4. Tableau constraints:\nαi+1 +\nX\nk≤j\nri+1\nk\n≤αi +\nX\nk′<j\nri\nk′.\n5. Reverse lattice word constraints: ri\nj = 0 for i < j, and for i ≤n,\n1 < j ≤n:\nX\ni′≤i\nri′\nj ≤\nX\ni′<i\nri′\nj−1.\nLet P γ\nαβ be the polytope defined by these constraints. Then cγ\nαβ is the\nnumber of integer points in this polytope. This proves the lemma.\n□\nThe membership function of the polytope P γ\nαβ is clearly computable in\ntime that is polynomial in the bitlengths of α, β and γ. Hence cγ\nαβ belongs\nto #P. This proves the theorem.\n□\nThe complexity-theoretic content of the Littlewood-Richardson rule is\nthat it puts a quantity, which is a priori only in PSPACE, in #P. We also\nhave:\nTheorem 6.4 ([H]). cγ\nαβ is #P-complete.\nFinally, the main complexity-theoretic result that we are interested in:\nTheorem 6.5 (GCT3, Knutson-Tao, De Loera-McAllister). The problem\nof deciding nonvanishing of cγ\nαβ is in P, i. e. , it can be solved in time that is\npolynomial in the bitlengths of α, β and γ. In fact, it can solved in strongly\npolynomial time [GCT3].\nHere, by a strongly polynomial time algorithm, we mean that the num-\nber of arithmetic steps +, −, ∗, ≤, . . . in the algorithm is polynomial in the\nnumber of parts of α, β and γ regardless of their bitlengths, and the bit-\nlength of each intermediate operand is polynomial in the bitlengths of α, β\nand γ.\nProof: Let P = P γ\nαβ be the polytope as in Lemma 6.1.\nAll vertices of\nP have rational coefficients. Hence, for some positive integer q, the scaled\npolytope qP has an integer point. It follows that, for this q, cqγ\nqα,qβ is positive.\nThe saturation Theorem [KT] says that, in this case, cγ\nα,β is positive. Hence,\nP contains an integer point. This implies:\n44"},{"page":46,"text":"Lemma 6.2. If P ̸= ∅then cγ\nαβ > 0.\nBy this lemma, to decide if cγ\nαβ > 0, it suffices to test if P is nonempty.\nThe polytope P is given by Ax ≤b where the entries of A are 0 or 1–\nsuch linear programs are called combinatorial. Hence, this can be done in\nstrongly polynomial time using Tardos’ algorithm [GLS] for combinatorial\nlinear programming. This proves the theorem.\n□\nThe integer programming problem is NP-complete, in general. However,\nlinear programming works for the specific integer programming problem here\nbecause of the saturation property [KT].\nProblem: Find a genuinely combinatorial poly-time algorithm for deciding\nnon-vanishing of cγ\nαβ.\n45"},{"page":47,"text":"Chapter 7\nLittlewood-Richardson\ncoefficients (cont)\nScribe: Paolo Codenotti\nGoal: We continue our study of Littlewood-Richardson coefficients and\ndefine Littlewood-Richardson coefficients for the orthogonal group On(C).\nReferences: [FH, F]\nRecall\nLet us first recall some definitions and results from the last class. Let cγ\nα,β\ndenote the Littlewood-Richardson coefficient for GLn(C).\nTheorem 7.1 (last class). Non-vanishing of cγ\nα,β can be decided in poly(⟨α⟩, ⟨β⟩, ⟨γ⟩)\ntime, where ⟨⟩denotes the bit length.\nThe positivity hypotheses which hold here are:\n• cγ\nα,β ∈#P, and more strongly,\n• Positivity Hypothesis 1 (PH1): There exists a polytope P γ\nα,β of\ndimension polynomial in the heights of α, β and γ such that cγ\nα,β =\nφ(P γ\nα,β), where φ indicates the number of integer points.\n• Saturation Hypothesis (SH): If ckγ\nkα,kβ ̸= 0 for some k ≥1, then\ncγ\nα,β ̸= 0 [Saturation Theorem].\nProof. (of theorem)\nPH1 + SH + Linear programming.\n46"},{"page":48,"text":"This is the general form of algorithms in GCT. The main principle is\nthat linear programming works for integer programming when PH1 and SH\nhold.\n7.1\nThe stretching function\nWe define ecγ\nα,β(k) = ckγ\nkα,kβ.\nTheorem 7.2 (Kirillov, Derkesen Weyman [Der, Ki]). ecγ\nα,β(k) is a polyno-\nmial in k.\nHere we prove a weaker result. For its statement, we will quickly review\nthe theory of Ehrhart quasipolynomials (cf. Stanley [S]).\nDefinition 7.1. (Quasipolynomial) A function f(k) is called a quasipoly-\nnomial if there exist polynomials fi, 1 ≤i ≤l, for some lsuch that\nf(k) = fi(k) if k ≡i mod l.\nWe denote such a quasipolynomial f by f = (fi). Here lis called the period\nof f(k) (we can assume it is the smallest such period).\nThe degree of a\nquasipolynomial f is the max of the degrees of the fi’s.\nNow let P ⊆Rm be a polytope given by Ax ≤b. Let φ(P) be the number\nof integer points inside P. We define the stretching function fP(k) = φ(kP),\nwhere kP is the dilated polytope defined by Ax ≤kb.\nTheorem 7.3. (Ehrhart) The stretching function fP(k) is a quasipolyno-\nmial. Furthermore, fP (k) is a polynomial if P is an integral polytope (i.e.\nall vertices of P are integral).\nIn view of this result, fP(k) is called the Ehrhart quasi-polynomial of\nP. Now ecγ\nα,β(k) is just the Ehrhart quasipolynomial of P γ\nα,β, and cγ\nα,β =\nφ(P γ\nα,β), the number of integer points in P γ\nα,β. Moreover P γ\nα,β is defined by\nthe inequality Ax ≤b, where A is constant, and b is a homogeneous linear\nform in the coefficients of α, β, and γ.\nHowever, P γ\nα,β need not be integral. Therefore Theorem (7.2) does not\nfollow from Ehrhart’s result. Its proof needs representation theory.\nDefinition 7.2. A quasipolynomial f(k) is said to be positive if all the\ncoefficients of fi(k) are nonnegative. In particular, if f(k) is a polynomial,\nthen it’s positive if all its coefficients are nonnegative.\n47"},{"page":49,"text":"The Ehrhart quasipolynomial of a polytope is positive only in exceptional\ncases. In this context:\nPH2 (positivity hypothesis 2) [KTT]: The polynomial ecγ\nα,β(k) is positive.\nThere is considerable computer evidence for this.\nProposition 7.1. PH2 implies SH.\nProof. Look at:\nc(k) = ecγ\nα,β(k) =\nX\naiki.\nIf all the coefficients ai are nonnegative (by PH2), and c(k) ̸= 0, then c(1) ̸=\n0.\nSH has a proof involving algebraic geometry [B]. Therefore we suspect\nthat the stronger PH2 is a deep phenomenon related to algebraic geometry.\n7.2\nOn(C)\nSo far we have talked about GLn(C).\nNow we move on to the orthogo-\nnal group On(C). Fix Q, a symmetric bilinear form on Cn; for example,\nQ(V, W) = V T W.\nDefinition 7.3. The orthogonal group On(C) ⊆GLn(C) is the group con-\nsisting of all A ∈GLn(C) s.t.\nQ(AV, AW) = Q(V, W) for all V and\nW ∈Cn. The subgroup SOn(C) ⊆SLn(C), where SLn(C) is the set of\nmatrices with determinant 1, is defined similarly.\nTheorem 7.4 (Weyl). The group On(C) is reductive\nProof. The proof is similar to the reductivity of GLn(C), based on Weyl’s\nunitary trick.\nThe next step is to classify all irreducible polynomial representations of\nOn(C).\nFix a partition λ = (λ1 ≥λ2 ≥. . . ) of length at most n.\nLet\n|λ| = d = P λi be its size. Let V = Cn, V ⊗d = V ⊗· · · ⊗V d times, and\nembed the Weyl module Vλ of GLn(C) in V ⊗d as per Theorem 5.3. Define\na contraction map\nφp,q : V ⊗d →V ⊗(d−2)\nfor 1 ≤p ≤q ≤d by:\nφp,q(vi1 ⊗· · · ⊗vid) = Q(vip, viq)(vi1 ⊗· · · ⊗c\nvip ⊗· · · ⊗c\nviq ⊗· · · ⊗vid),\n48"},{"page":50,"text":"λ\nFigure 7.1: The first two columns of the partition λ are highlighted.\nwhere c\nvip means omit vip.\nIt is On(C)-equivariant, i.e. the following diagram commutes:\nV ⊗d\nφp,q\n−−−−→V ⊗d−2\n yσ∈On(C)\n yσ∈On(C)\nV ⊗d\nφp,q\n−−−−→V ⊗d−2\nLet\nV [d] =\n\\\npq\nker(φp,q).\nBecause the maps are equivariant, each kernel is an On(C)-module, and V [d]\nis an On(C)-module. Let V[λ] = V [d] T Vλ, where Vλ ⊆V ⊗d is the embedded\nWeyl module as above. Then V[λ] is an On(C)-module.\nTheorem 7.5 (Weyl). V[λ] is an irreducible representation of On(C). More-\nover, the following two conditions hold:\n1. If n is odd, then V[λ] is non-zero if and only if the sum of the lengths\nof the first two columns of λ is ≤n (see figure 7.1).\n2. If n is odd, then each polynomial irreducible representation is isomor-\nphic to V[λ] for some λ.\nLet\nV[λ] ⊗V[μ] = ⊗γdγ\nλ,μV[γ]\n49"},{"page":51,"text":"be the decomposition of V[λ]⊗V[μ] into irreducibles. Here dγ\nλ,μ is called the\nLittlewood-Richardson coefficient of type B. The types of various connected\nreductive groups are defined as follows:\n• GLn(C): type A\n• On(C), n odd: type B\n• Spn(C): type C\n• On(C), n even: type D\nThe Littlewood-Richardson coefficient can be defined for any type in a sim-\nilar fashion.\nTheorem 7.6 (Generalized Littlewood-Richardson rule). The Littlewood-\nRichardson coefficient dγ\nλ,μ ∈#P. This also holds for any type.\nProof. The most transparent proof of this theorem comes through the theory\nof quantum groups [K]; cf. Chapter 20.\nAs in type A this leads to:\nHypothesis 7.1 (PH1). There exists a polytope P γ\nλ,μ of dimension polyno-\nmial in the heights of λ, μ and γ such that:\n1. dγ\nλ,μ = φ(P γ\nλ,μ), the number of integer points in P γ\nλ,μ, and\n2. edγ\nλ,μ(k) = dkγ\nkλ,kμ is the Ehrhart quasipolynomial of P γ\nλ,μ.\nThere are several choices for such polytopes; e.g. the BZ-polytope [BZ].\nTheorem 7.7 (De Loera, McAllister [DM2]). The stretching function edγ\nλ,μ(k)\nis a quasipolynomial of degree at most 2; so also for types C and D.\nA verbatim translation of the saturation property fails here [Z]): there\nexist λ, μ and γ such that d2γ\n2λ,2μ ̸= 0 but dγ\nλ,μ = 0. Therefore we change the\ndefinition of saturation:\nDefinition 7.4. Given a quasipolynomial f(k) = (fi), index(f) is the\nsmallest i such that fi(k) is not an identically zero polynomial.\nIf f(k)\nis identically zero, index(f) = 0.\nDefinition 7.5. A quasipolynomial f(k) is saturated if f(index(f)) ̸= 0.\nIn particular, if index(f) = 1, then f(k) is saturated if f(1) ̸= 0.\n50"},{"page":52,"text":"A positive quasi-polynomial is clearly saturated.\nPositivity Hypothesis 2 (PH2) [DM2]: The stretching quasipolyomial\nedγ\nλ,μ(k) is positive.\nThere is considerable evidence for this.\nSaturation Hypothesis (SH): The stretching quasipolynomial edγ\nλ,μ(k) is\nsaturated.\nPH2 implies SH.\nTheorem 7.8. [GCT5] Assuming SH (or PH2), positivity of the Littlewood-\nRichardson coefficient dγ\nλ,μ of type B can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨γ⟩)\ntime.\nThis is also true for all types.\nProof. next class.\n51"},{"page":53,"text":"Chapter 8\nDeciding nonvanishing of\nLittlewood-Richardson\ncoefficients for On(C)\nScribe: Hariharan Narayanan\nGoal: A polynomial time algorithm for deciding nonvanishing of Littlewood-\nRichardson coefficients for the orthogonal group assuming SH.\nReference: [GCT5]\nLet dν\nλ,μ denote the Littlewood-Richardson coefficient of type B (i.e. for\nthe orthogonal group On(C), n odd) as defined in the earlier lecture. In this\nlecture we describe a polynomial time algorithm for deciding nonvanishing\nof dν\nλ,μ assuming the following positivity hypothesis PH2.\nSimilar result\nalso holds for all types, though we shall only concentrate on type B in this\nlecture.\nLet ̃dν\nλ,μ(k) = dkν\nkλ,kμ denote the associated stretching function.\nIt is\nknown to be a quasi-polynomial of period at most two [DM2]. This means\nthere are polynomials f1(k) and f2(k) such that\ndkν\nkλ,kμ =\n f1(k),\nif k is odd;\nf2(k),\nif k is even.\nPositivity Hypothesis (PH2) [DM2]: The stretching quasi-polynomial\n ̃dν\nλ,μ(k) is positive.\nThis means the coefficients of f1 and f2 are all non-\nnegative.\nThe main result in this lecture is:\n52"},{"page":54,"text":"Theorem 8.1. [GCT5] If PH2 holds, then the problem of deciding the posi-\ntivity (nonvanishing) of dν\nλμ belongs to P. That is, this problem can be solved\nin time polynomial in the bitlengths of λ, μ and ν.\nWe need a few lemmas for the proof.\nLemma 8.1. If PH2 holds, the following are equivalent:\n(1) dν\nλμ ≥1.\n(2) There exists an odd integer k such that dkν\nkλ kμ ≥1.\nProof: Clearly (1) implies (2). By PH2, there exists a polynomial f1\nwith non-negative coefficients such that\n∀odd k, f1(k) = dkν\nkλ kμ.\nSuppose that for some odd k, dkν\nkλ kμ ≥1. Then f1(k) ≥1. Therefore f1 has\nat least one non-zero coefficient. Since all coefficients of f1 are nonnegative,\ndν\nλμ = f1(1) > 0. Since dν\nλμ is an integer, (1) follows.\n□\nDefinition 8.1. Let Z<2> be the subring of Q obtained by localizing Z at\n2:\nZ<2> :=\n p\nq | p, q −1\n2\n∈Z\n \n.\nThis ring consists of all fractions whose denominators are odd.\nLemma 8.2. Let P ∈Rd be a convex polytope specified by Ax ≤B, xi ≥0\nfor all i, where A and B are integral. Let Aff(P) denote its affine span. The\nfollowing are equivalent:\n(1) P contains a point in Zd\n<2>.\n(2) Aff(P) contains a point in Zd\n<2>.\nProof: Since P ⊆Aff(P), (1) implies (2). Now suppose (2) holds. We\nhave to show (1). Let z ∈Zd\n<2> ∩Aff(P).\nFirst, consider the case when Aff(P) is one dimensional. In this case, P\nis the line segment joining two points x and y in Qd. The point z can be\nexpressed as an affine linear combination, z = ax + (1−a)y for some a ∈Q.\nThere exists q ∈Z such that qx ∈Zd\n<2> and qy ∈Zd\n<2>. Note that\n{z + λ(qx −qy) | λ ∈Z<2>} ⊆Aff(P) ∩Zd\n<2>.\n53"},{"page":55,"text":"Since Z<2> is a dense subset of Q, the l.h.s. and hence the r.h.s. is a dense\nsubset of Aff(P). Consequently, P ∩Zd\n<2> ̸= ∅.\nNow consider the general case. Let u be any point in the interior of P\nwith rational coordinates, and L the line through u and z. By restricting to\nL, the lemma reduces to the preceding one dimensional case.\n□\nLemma 8.3. Let\nP = {x | Ax ≤B, (∀i)xi ≥0} ⊆Rd\nbe a convex polytope where A and B are integral. Then, it is possible to\ndetermine in polynomial time whether or not Aff(P) ∩Zd\n<2> = ∅.\nProof: Using Linear Programming [Kha79, Kar84], a presentation of\nthe form Cx = D can be obtained for Aff(P) in polynomial time, where C\nis an integer matrix and D is a vector with integer coordinates. We may\nassume that C is square since this can be achieved by padding it with 0’s\nif necessary, and extending D. The Smith Normal Form over Z of C is a\nmatrix S such that C = USV where U and V are unimodular and S has\nthe form\n \n \n \n \n \ns11\n0\n. . .\n0\n0\ns22\n. . .\n0\n...\n...\n...\n0\n0\n0\n. . .\nsdd\n \n \n \n \n \nwhere for 1 ≤i ≤d−1, sii divides si+1 i+1. It can be computed in polynomial\ntime [KB79].\nThe question now reduces to whether USV x = D has a\nsolution x ∈Zd\n<2>. Since V is unimodular, its inverse has integer entries\ntoo, and y := V x ∈Zd\n<2> ⇔x ∈Zd\n<2>.\nThis is equivalent to whether\nSy = U−1D has a solution y ∈Zd\n<2>.\nSince S is diagonal, this can be\nanswered in polynomial time simply by checking each coordinate.\n□\nProof of Theorem 8.1: By [BZ], there exists a polytope P = P ν\nλ,μ\nsuch that the Littlewood-Richardson coefficient dν\nλμ is equal to the number\nof integer points in P. This polytope is such that the number of integer\npoints in the dilated polytope kP is dkν\nkλ kμ. Assuming PH2, we know from\nLemma 8.1 that\nP ∩Zd ̸= ∅⇔(∃odd k), kP ∩Zd ̸= ∅.\nThe latter is equivalent to\nP ∩Zd\n<2> ̸= ∅.\n54"},{"page":56,"text":"The theorem now follows from Lemma 8.2 and Lemma 8.3.\n□\nIn combinatorial optimization, LP works if the polytope is integral. In\nour setting, this is not necessarily the case [DM1]: the denominators of the\ncoordinates of the vertices of P can be Ω(l), where l is the total height of\nλ, μ and ν. LP works here nevertheless because of PH2; it can be checked\nthat SH is also sufficient.\n55"},{"page":57,"text":"Chapter 9\nThe plethysm problem\nScribe: Joshua A. Grochow\nGoal: In this lecture we describe the general plethysm problem, state anal-\nogous positivity and saturation hypotheses for it, and state the results from\nGCT 6 which imply a polynomial time algorithm for deciding positivity of\na plethysm constant assuming these hypotheses.\nReference: [GCT6]\nRecall\nRecall that a function f(k) is quasipolynomial if there are functions fi(k)\nfor i = 1, . . . , lsuch that f(k) = fi(k) whenever k ≡i mod l. The number\nlis then the period of f. The index of f is the least i such that fi(k) is\nnot identically zero.\nIf f is identically zero, then the index of f is zero\nby convention. We say f is positive if all the coefficients of each fi(k) are\nnonnegative. We say f is saturated if f(index(f)) ̸= 0. If f is positive, then\nit is saturated.\nGiven any function f(k), we associate to it the rational series F(t) =\nP\nk≥0 f(k)tk.\nProposition 9.1. [S] The following are equivalent:\n1. f(k) is a quasipolynomial of period l.\n2. F(t) is a rational function of the form A(t)\nB(t) where deg A < deg B and\nevery root of B(t) is an l-th root of unity.\n56"},{"page":58,"text":"9.1\nLittlewood-Richardson Problem [GCT 3,5]\nLet G = GLn(C) and cγ\nα,β the Littlewood-Richardson coefficient – i.e. the\nmultiplicity of the Weyl module Vγ in Vα ⊗Vβ. We saw that the positivity\nof cγ\nα,β can be decided in poly(⟨α⟩, ⟨β⟩, ⟨γ⟩) time, where ⟨·⟩denotes the bit-\nlength. Furthermore, we saw that the stretching function ecγ\nα,β(k) = ckγ\nkα,kβ\nis a polynomial, and the analogous stretching function for type B is a\nquasipolynomial of period at most 2.\n9.2\nKronecker Problem [GCT 4,6]\nNow we study the analogous problem for the representations of the symmet-\nric group (the Specht modules), called the Kronecker problem.\nLet Sα be the Specht module of the symmetric group Sm associated to\nthe partition α. Define the Kronecker coefficient κπ\nλ,μ to be the multiplicity\nof Sπ in Sλ ⊗Sμ (considered as an Sm-module via the diagonal action). In\nother words, write Sλ ⊗Sμ = L\nπ κπ\nλ,μSπ. We have κπ\nλ,μ = (χλχμ, χπ), where\nχλ denotes the character of Sλ. By the Frobenius character formula, this\ncan be computed in PSPACE. More strongly, analogous to the Littlewood-\nRichardson problem:\nConjecture 9.1. [GCT4, GCT6] The Kronecker coefficient κπ\nλ,μ ∈# P.\nIn other words, there is a positive #P-formula for κπ\nλ,μ.\nThis is a fundamental problem in representation theory. More concretely,\nit can be phrased as asking for a set of combinatorial objects I and a char-\nacteristic function χ : {I} →{0, 1} such that χ ∈FP and κπ\nλ,μ = P\nI χ(I).\nContinuing our analogy:\nConjecture 9.2. [GCT6] The problem of deciding positivity of κπ\nλ,μ belongs\nto P.\nTheorem 9.1. [GCT6] The stretching function eκπ\nλ,μ(k) = κkπ\nkλ,kμ is a quasipoly-\nnomial.\nNote that κkπ\nkλ,kμ is a Kronecker coefficient for Skm.\nThere is also a dual definition of the Kronecker coefficients. Namely,\nconsider the embedding\nH = GLn(C) × GLn(C) ֒→G = GL(Cn ⊗Cn),\nwhere (g, h)(v ⊗w) = (gv ⊗hw). Then\n57"},{"page":59,"text":"Proposition 9.2. [FH] The Kronecker coefficient κπ\nλ,μ is the multiplicity of\nthe tensor product of Weyl modules Vλ(GLn(C)) ⊗Vμ(GLn(C)) (this is an\nirreducible H-module) in the Weyl module Vπ(G) considered as an H-module\nvia the embedding above.\n9.3\nPlethysm Problem [GCT 6,7]\nNext we consider the more general plethysm problem.\nLet H = GLn(C), V = Vμ(H) the Weyl module of H corresponding to\na partition μ, and ρ : H →G = GL(V ) the corresponding representation\nmap. Then the Weyl module Vλ(G) of G for a given partition λ can be\nconsidered an H-module via the map ρ. By complete reducibility, we may\ndecompose this H-representation as\nVλ(G) =\nM\nπ\naπ\nλ,μVπ(H).\nThe coefficients aπ\nλ,μ are known as plethsym constants (this definition can\neasily be generalized to any reductive group H). The Kronecker coefficient\nis a special case of the plethsym constant [Ki].\nTheorem 9.2 (GCT 6). The plethysm constant aπ\nλ,μ ∈PSPACE.\nThis is based on a parallel algorithm to compute the plethysm constant\nusing Weyl’s character formula. Continuing in our previous trend:\nConjecture 9.3. [GCT6] aπ\nλ,μ ∈# P and the problem of deciding positivity\nof aπ\nλ,μ belongs to P.\nFor the stretching function, we need to be a bit careful. Define eaπ\nλ,μ =\nakπ\nkλ,μ. Here the subscript μ is not stretched, since that would change G,\nwhile stretching λ and π only alters the representations of G.\nAs in the beginning of the lecture, we can associate a function Aπ\nλ,μ(t) =\nP\nk≥0 eaπ\nλ,μ(k)tk to the plethysm constant. Kirillov conjectured that Aπ\nλ,μ(t) is\nrational. In view of Proposition 9.1, this follows from the following stronger\nresult:\nTheorem 9.3 (GCT 6). The stretching function eaπ\nλ,μ(k) is a quasipolyno-\nmial.\nThis is the main result of GCT 6, which in some sense allows GCT to\ngo forward.\nWithout it, there would be little hope for proving that the\n58"},{"page":60,"text":"positivity of plethysm constants can be decided in polynomial time.\nIts\nproof is essentially algebro-geometric. The basic idea is to show that the\nstretching function is the Hilbert function of some algebraic variety with nice\n(i.e. “rational”) singularities. Similar results are shown for the stretching\nfunctions in the algebro-geometric problems arising in GCT.\nThe main complexity-theoretic result in [GCT6] shows that, under the\nfollowing positivity and saturation hypotheses (for which there is much ex-\nperimental evidence), the positivity of the plethysm constants can indeed\nbe decided in polynomial time (cf. Conjecture 9.3).\nThe first positivity hypothesis is suggested by Theorem 9.3: since the\nstretching function is a quasipolynomial, we may suspect that it is captured\nby some polytope:\nPositivity Hypothesis 1 (PH1). There exists a polytope P = P π\nλ,μ\nsuch that:\n1. aπ\nλ,μ = φ(P), where φ denotes the number of integer points inside the\npolytope,\n2. The stretching quasipolynomial (cf. Thm. 9.3) eaπ\nλ,μ(k) is equal to the\nEhrhart quasipolynomial fP(k) of P,\n3. The dimension of P is polynomial in ⟨λ⟩, ⟨μ⟩, and ⟨π⟩,\n4. the membership in P π\nλ,μ can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) time, and\nthere is a polynomial time separation oracle [GLS] for P.\nHere (4) does not imply that the polytope P has only polynomially\nmany constraints. In fact, in the plethysm problem there may be a super-\npolynomial number of constraints.\nPositivity Hypothesis 2 (PH2).\nThe stretching quasipolynomial\neaπ\nλ,μ(k) is positive.\nThis implies:\nSaturation Hypothesis (SH). The stretching quasipolynomial is sat-\nurated.\nTheorem 9.3 is essential to state these hypotheses, since positivity and\nsaturation are properties that only apply to quasipolynomials. Evidence for\nPH1, PH2, and SH can be found in GCT 6.\nTheorem 9.4. [GCT6] Assuming PH1 and SH (or PH2), positivity of the\nplethysm constant aπ\nλ,μ can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) time.\n59"},{"page":61,"text":"This follows from the polynomial time algorithm for saturated integer\nprogramming described in the next class. As with Theorem 9.3, this also\nholds for more general problems in algebraic geometry.\n60"},{"page":62,"text":"Chapter 10\nSaturated and positive\ninteger programming\nScribe: Sourav Chakraborty\nGoal :\nA polynomial time algorithm for saturated integer programming\nand its application to the plethysm problem.\nReference: [GCT6]\nNotation :\nIn this class we denote by ⟨a⟩the bit-length of the a.\n10.1\nSaturated, positive integer programming\nLet Ax ≤b be a set of inequalities.\nThe number of constraints can be\nexponential. Let P ⊂Rn be the polytope defined by these inequalities. The\nbit length of P is defined to be ⟨P⟩= n + ψ, where ψ is the maximum\nbit-length of a constraint in the set of inequalities. We assume that P is\ngiven by a separating oracle. This means membership in P can be decided\nin poly(⟨P⟩) time, and if x ̸∈P then a separating hyperplane is given as a\nproof as in [GLS].\nLet fP(k) be the Ehrhart quasi-polynomial of P. Quasi-polynomiality\nmeans there exist polynomials fi(k), 1 ≤i ≤l, l the period, so that fP(k) =\nfi(k) if k = i modulo l. Then\nIndex(fP) = min{i|fi(k)not identically 0 as a polynomial}\nThe integer programming problem is called positive if fP(k) is positive\nwhenever P is non-empty, and saturated if fP(k) is saturated whenever P is\nnon-empty.\n61"},{"page":63,"text":"Theorem 10.1 (GCT6).\n1. Index(fP ) can be computed in time polyno-\nmial in the bit length ⟨P⟩of P assuming that the separation oracle\nworks in poly-⟨P⟩-time.\n2. Saturated and hence positive integer programming problem can be solved\nin poly-⟨P⟩-time.\nThe second statement follow from the first.\nProof. Let Aff(P) denote the affine span of P. By [GLS] we can compute\nthe specifications Cx = d, C and d integral, of Aff(P) in poly(⟨P⟩) time.\nWithout loss of generality, by padding, we can assume that C is square. By\n[KB79] we find the Smith-normal form of C in polynomial time. Let it be\n ̄C. So,\n ̄C = ACB\nwhere A and B are unimodular, and ̄C is a diagonal matrix, where the\ndiagonal entries c1, c2, . . . are such that with ci|ci+1.\nClearly Cx = d iff ̄Cz = ̄d where z = B−1x and ̄d = Ad.\nSo all equations here are of form\n ̄cizi = di\n(10.1)\nWithout loss of generality we can assume that ci and di are relatively\nprime. Let ̃c = lcm(ci).\nClaim 10.1. Index(fP ) = ̃c.\nFrom this claim the theorem clearly follows.\nProof of the claim. Let fP(t) = P\nk≥0 fP(k)tk be the Ehrhart Series of P.\nNow kP will not have an integer point unless ̃c divides k because of\n(10.1).\nHence fP(t) = f ̄P(t ̃c) where ̄P is the stretched polytope ̃cP and f ̄P(s) is\nthe Ehrhart series of ̄P. From this it follows that\nIndex(fP ) = ̃cIndex(f ̄P )\nNow we show that Index(f ̄P) = 1.\nThe equations of ̄P are of the form\nzi = ̃c\nci\ndi\n62"},{"page":64,"text":"where each\n ̃c\nci is an integer.\nTherefore without loss of generality we can\nignore these equations and assume the ̄P is full dimensional.\nThen it suffices to show that ̄P contains a rational point whose denomi-\nnators are all 1 modulo l( ̄P), the period of the quasi-polynomial f ̄P (s).\nThis follows from a simple density argument that we saw earlier (cf. the\nproof of Lemma 8.2).\nFrom this the claim follows.\n10.2\nApplication to the plethysm problem\nNow we can prove the result stated in the last class:\nTheorem 10.2. Assuming PH1 and SH, positivity of the plethysm constant\naπ\nλ,μ can be decided in time polynomial in ⟨λ⟩, ⟨μ⟩and ⟨π⟩.\nProof. Let P = P π\nλ,μ be the polytope as in PH1 such that aπ\nλ,μ is the number\nof integer points in P. The goal is to decide if P contains an integer point.\nThis integer programming problem is saturated because of SH. Hence the\nresult follows from Theorem 10.1.\n63"},{"page":65,"text":"Chapter 11\nBasic algebraic geometry\nScribe: Paolo Codenotti\nGoal: So far we have focussed on purely representation-theoretic aspects of\nGCT. Now we have to bring in algebraic geometry. In this lecture we review\nthe basic definitions and results in algebraic geometry that will be needed\nfor this purpose. The proofs will be omitted or only sketched. For details,\nsee the books by Mumford [Mm] and Fulton [F].\n11.1\nAlgebraic geometry definitions\nLet V = Cn, and v1, . . . , vn the coordinates of V .\nDefinition 11.1.\n• Y is an affine algebraic set in V if Y is the set of\nsimultaneous zeros of a set of polynomials in vi’s.\n• An algebraic set that cannot be written as the union of two proper\nalgebraic sets Y1 and Y2 is called irreducible.\n• An irreducible affine algebraic set is called an affine variety.\n• The ideal of an affine algebraic set Y is I(Y ), the set of all polynomials\nthat vanish on Y .\nFor example, Y = (v1 −v2\n2 + v3, v2\n3 −v2 + 4v1) is an irreducible affine\nalgebraic set (and therefore an affine variety).\nTheorem 11.1 (Hilbert). I(Y ) is finitely generated, i.e. there exist poly-\nnomials g1, . . . , gk that generate I(Y ). This means every f ∈I(Y ) can be\nwritten as f = P figi for some polynomials fi.\n64"},{"page":66,"text":"Let C[V ], the coordinate ring of V , be the ring of polynomials over\nthe variables v1, . . . , vn. The coordinate ring of Y is defined to be C[Y ] =\nC[V ]/I(Y ). It is the set of polynomial functions over Y .\nDefinition 11.2.\n• P(V ) is the projective space associated with V , i.e.\nthe set of lines through the origin in V .\n• V is called the cone of P(V ).\n• C[V ] is called the homogeneous coordinate ring of P(V ).\n• Y ⊆P(V ) is a projective algebraic set if it is the set of simultaneous\nzeros of a set of homogeneous forms (polynomials) in the variables\nv1, . . . , vn. It is necessary that the polynomials be homogeneous because\na point in P(V ) is a line in V .\n• A projective algebraic set Y is irreducible if it can not be expressed as\nthe union of two proper algebraic sets in P(V ).\n• An irreducible projective algebraic set is called a projective variety.\nLet Y ⊆P(V ) be a projective variety, and define I(Y ), the ideal of Y\nto be the set of all homogeneous forms that vanish on Y . Hilbert’s result\nimplies that I(Y ) is finitely generated.\nDefinition 11.3. The cone C(Y ) ⊆V of a projective variety Y ⊆P(V ) is\ndefined to be the set of all points on the lines in Y .\nDefinition 11.4. We define the homogeneous coordinate ring of Y as\nR(Y ) = C[V ]/I(Y ), the set of homogeneous polynomial forms on the cone\nof Y .\nDefinition 11.5. A Zariski open subset of Y is the complement of a pro-\njective algebraic subset of Y . It is called a quasi-projective variety.\nLet G = GLn(C), and V a finite dimensional representation of G. Then\nC[V ] is a G-module, with the action of σ ∈G defined by:\n(σ · f)(v) = f(σ−1v), v ∈V.\nDefinition 11.6. Let Y ⊆P(V ) be a projective variety with ideal I(Y ). We\nsay that Y is a G-variety if I(Y ) is a G-module, i.e., I(Y ) is a G-submodule\nof C[V ].\n65"},{"page":67,"text":"If Y is a projective variety, then R(Y ) = C[V ]/I(Y ) is also a G-module.\nTherefore Y is G-invariant, i.e.\ny ∈Y ⇒σy ∈Y, ∀σ ∈G.\nThe algebraic geometry of G-varieties is called geometric invariant theory\n(GIT).\n11.2\nOrbit closures\nWe now define special classes of G-varieties called orbit closures. Let v ∈\nP(V ) be a point, and Gv the orbit of v:\nGv = {gv|g ∈G}.\nLet the stabilizer of v be\nH = Gv = {g ∈G|gv = v}.\nThe orbit Gv is isomorphic to the space G/H of cosets, called the ho-\nmogeneous space. This is a very special kind of algebraic variety.\nDefinition 11.7. The orbit closure of v is defined by:\n∆V [v] = Gv ⊆P(V ).\nHere Gv is the closure of the orbit Gv in the complex topology on P(V ) (see\nfigure 11.1).\nA basic fact of algebraic geometry:\nTheorem 11.2. The orbit closure ∆V [v] is a projective G-variety\nIt is also called an almost homogeneous space.\nLet IV [v] be the ideal of ∆V [v], and RV [v] the homogeneous coordinate\nring of ∆V [v]. The algebraic geometry of general orbit closures is hopeless,\nsince the closures can be horrendous (see figure 11.1). Fortunately we shall\nonly be interested in very special kinds of orbit closures with good algebraic\ngeometry.\nWe now define the simplest kind of orbit closure, which is obtained when\nthe orbit itself is closed. Let Vλ be an irreducible Weyl module of GLn(C),\nwhere λ = (λ1 ≥λ2 ≥· · · ≥λn ≥0) is a partition. Let vλ be the highest\nweight point in P(Vλ), i.e., the point corresponding to the highest weight\n66"},{"page":68,"text":"v\nGv\n∆V [v]\nLimit points of Gv\nFigure 11.1: The limit points of Gv in ∆V [v] can be horrendous.\nvector in Vλ. This means bvλ = vλ for all b ∈B, where B ⊆GLn(C) is the\nBorel subgroup of lower triangular matrices. Recall that the highest weight\nvector is unique.\nConsider the orbit Gvλ of vλ. Basic fact:\nProposition 11.1. The orbit Gvλ is already closed in P(V ).\nIt can be shown that the stabilizer Pλ = Gvλ is a group of block lower\ntriangular matrices, where the block lengths only depend on λ (see figure\n11.2). Such subgroups of GLn(C) are called parabolic subgroups, and will be\ndenoted by P. Clearly Gvλ ∼= G/Pλ = G/P.\n11.3\nGrassmanians\nThe simplest examples of G/P are Grassmanians.\nDefinition 11.8. Let G = Gln(C), and V = Cn. The Grassmanian Grn\nd is\nthe space of d-dimensional subspaces (containing the origin) of V .\nExamples:\n1. Gr2\n1 is the set of lines in C2 (see figure 11.3).\n2. More generally, P(V ) = Grn\n1 .\nProposition 11.2. The Grassmanian Grn\nd is a projective variety (just like\nP(V ) = Grn\n1 ).\n67"},{"page":69,"text":"*\nA1\nA2\nA3\nA4\nA5\nm1\nm2\nm3\nm4\nm5\nFigure 11.2: The parabolic subgroup of block lower triangular matrices. The sizes\nmi only depend on λ.\nFigure 11.3: Gr2\n1 is the set of lines in C2.\n68"},{"page":70,"text":"It is easy to see that Grn\nd is closed (since the limit of a sequence of d-\ndimensional subspaces of V is a d-dimensional subspace). Hence this follows\nfrom:\nProposition 11.3. Let λ = (1, . . . , 1) be the partition of d, whose all parts\nare 1. Then Grn\nd ∼= Gvλ ⊆P(Vλ).\nProof. For the given λ, Vλ can be identified with the dth wedge product\nΛd(V ) = span{(vi1 ∧· · ·∧vid)|i1, . . . , id are distinct} ⊆V ⊗· · ·⊗V (d times),\nwhere\n(vi1 ∧· · · ∧vid) = 1\nd!\nX\nσ∈Sd\nsgn(σ)(vσ(i1) ⊗· · · ⊗vσ(id)).\nLet Z be a variable d × n matrix.\nThen C[Z] is a G-module: given\nf ∈C[Z] and σ ∈GLn(C), we define the action of σ by\n(σ · f)(Z) = f(Zσ).\nNow Λd(V ), as a G-module, is isomorphic to the span in C[Z] of all d × d\nminors of Z.\nLet A ∈Grn\nd be a d-dimensional subspace of V . Take any basis {v1, . . . , vd}\nof A. The point v1 ∧· · · ∧vd ∈Λd(V ) depends only on the subspace A, and\nnot on the basis, since the change of basis does not change the wedge prod-\nuct. Let ZA be the d × n complex matrix whose rows are the basis vectors\nv1, . . . , vd of A. The Plucker map associates with A the tuple of all d × d\nminors Aj1,...,jd of ZA, where Aj1,...,jd denotes the minor of ZA formed by\nthe columns j1, . . . , jd. This depends only on A, and not on the choice of\nbasis for A.\nThe proposition follows from:\nClaim 11.1. The Plucker map is a G-equivariant map from Grn\nd to Gvλ ⊆\nP(Vλ) and Grn\nd ≈Gvλ ⊆P(Vλ).\nProof. Exercise. Hint: take the usual basis, and note that the highest weight\npoint vλ corresponds to v1 ∧· · · ∧vd.\n69"},{"page":71,"text":"Chapter 12\nThe class varieties\nScribe: Hariharan Narayanan\nGoal: Associate class varieties with the complexity classes #P and NC\nand reduce the NC ̸= P #P conjecture over C to a conjecture that the class\nvariety for #P cannot be embedded in the class variety for NC.\nreference: [GCT1]\nThe NC ̸= P #P conjecture over C says that the permanent of an n × n\ncomplex matrix X cannot be expressed as a determinant of an m × m com-\nplex matrix Y , m = poly(n), whose entries are (possibly nonhomogeneous)\nlinear forms in the entries of X. This obviously implies the NC ̸= P #P\nconjecture over Z, since multivariate polynomials over Cn are determined\nby the values that they take over the subset Zn. The conjecture over Z is\nimplied by the usual NC ̸= P #P conjecture over a finite field Fp, p ̸= 2,\nand hence, has to be proved first anyway.\nFor this reason, we concentrate on the NC ̸= P #P conjecture over C in\nthis lecture. The goal is to reduce this conjecture to a statement in geometric\ninvariant theory.\n12.1\nClass Varieties in GCT\nTowards that end, we associate with the complexity classes #P and NC\ncertain projective algebraic varieties, which we call class varieties. For this,\nwe need a few definitions.\nLet G = GLl(C), V a finite dimensional representation of G. Let P(V )\nbe the associated projective space, which inherits the group action. Given a\npoint v ∈P(V ), let ∆V [v] = Gv ⊆P(V ) be its orbit closure. Here Gv is the\n70"},{"page":72,"text":"closure of the orbit Gv in the complex topology on P(v). It is a projective\nG-variety; i.e., a projective variety with the action of G.\nAll class varieties in GCT are orbit closures (or their slight generaliza-\ntions), where v ∈P(V ) corresponds to a complete function for the class in\nquestion. The choice of the complete function is crucial, since it determines\nthe algebraic geometry of ∆V [v].\nWe now associate a class variety with NC. Let g = det(Y ), Y an m × m\nvariable matrix. This is a complete function for NC. Let V = symm(Y )\nbe the space of homogeneous forms in the entries of Y of degree m. It is a\nG-module, G = GLm2(C), with the action of σ ∈G given by:\nσ : f(Y ) 7−→f(σ−1Y ).\nHere σ−1Y is defined thinking of Y as an m2-vector.\nLet ∆V [g] = ∆V [g, m] = Gg, where we think of g as an element of\nP(V ).\nThis is the class variety associated with NC.\nIf g is a different\nfunction instead of det(Y ), the algebraic geometry of ∆V [g] would have been\nunmanageable. The main point is that the algebraic geometry of ∆V [g] is\nnice, because of the very special nature of the determinant function.\nWe next associate a class variety with #P. Let h = perm(X), X an\nn × n variable matrix. Let W = symn(X). It is similarly an H-module,\nH = GLk(C), k = n2. Think of h as an element of P(W), and let ∆W[h] =\nHh be its orbit closure. It is called the class variety associated with #P.\nNow assume that m > n, and think of X as a submatrix of Y , say the\nlower principal submatrix. Fix a variable entry y of Y outside of X. Define\nthe map φ : W →V which takes w(x) ∈W to ym−nw(x) ∈V . This induces\na map from P(V ) to P(W) which we call φ as well. Let φ(h) = f ∈P(V )\nand ∆V [f, m, n] = Gf its orbit closure. It is called the extended class variety\nassociated with #P.\nProposition 12.1 (GCT 1).\n1. If h(X) ∈W can be computed by a cir-\ncuit (over C) of depth ≤logc(n), c a constant, then f = φ(h) ∈\n∆V [g, m], for m = O(2logc n).\n2. Conversely if f ∈∆V [g, m] for m = 2logc n, then h(X) can be approx-\nimated infinitesimally closely by a circuit of depth log2c m. That is,\n∀ǫ > 0, there exists a function ̃h(X) that can be computed by a circuit\nof depth ≤log2c m such that ∥ ̃h −h∥< ǫ in the usual norm on P(V ).\nIf the permanent h(X) can be approximated infinitesimally closely by\nsmall depth circuits, then every function in #P can be approximated in-\nfinitesimally closely by small depth circuits. This is not expected. Hence:\n71"},{"page":73,"text":"Conjecture 12.1 (GCT 1). Let h(X) = perm(X), X an n × n variable\nmatrix. Then f = φ(h) ̸∈∆V [g; m] if m = 2polylog(n) and n is sufficiently\nlarge.\nThis is equivalent to:\nConjecture 12.2 (GCT 1). The G-variety ∆V [f; m, n] cannot be embedded\nas a G-subvariety of ∆V [g, m], symbolically\n∆V [f; m, n] ̸֒→∆V [g, m],\nif m = 2polylog(n) and n →∞.\nThis is the statement in geometric invariant theory (GIT) that we sought.\n72"},{"page":74,"text":"Chapter 13\nObstructions\nScribe: Paolo Codenotti\nGoal: Define an obstruction to the embedding of the #P-class variety in\nthe NC-class-variety and describe why it should exist.\nReferences: [GCT1, GCT2]\nRecall\nLet us first recall some definitions and results from the last class. Let Y be\na generic m × m variable matrix, and X an n × n minor of Y (see figure\n13.1).\nLet g = det(Y ), h = perm(X), f = φ(h) = ym−nperm(X), and\nV = Symm[Y ] the set of homogeneous forms of degree m in the entries of Y .\nThen V is a G-module for G = GL(Y ) = GLl(C), l = m2, with the action\nY\nX\nn\nn\nm\nm\nFigure 13.1: Here Y is a generic m by m matrix, and X is an n by n minor.\n73"},{"page":75,"text":"of σ ∈G given by\nσ : f(Y ) →f(σ−1Y ),\nwhere Y is thought of as an l-vector, and P(V ) a G-variety. Let\n∆V [f; m, n] = Gf ⊆P(V ),\nand\n∆V [g; m] = Gg ⊆P(V )\nbe the class varieties associated with #P and NC.\n13.1\nObstructions\nConjecture 13.1. [GCT1] There does not exist an embedding ∆V [f; m, n] ֒→\n∆V [g; m] with m = 2polylog(n), n →∞.\nThis implies Valiant’s conjecture that the permanent cannot be com-\nputed by circuits of polylog depth. Now we discuss how to go about proving\nthe conjecture.\nSuppose to the contrary,\n∆[f; m, n] ֒→∆V [g; m].\n(13.1)\nWe denote ∆V [f; m, n] by ∆V [f], and ∆V [g; m] by ∆V [g]. Let RV [g] be\nthe homogeneous coordinate ring of ∆V [g]. The embedding (13.1) implies\nexistence of a surjection:\nRV [f] եւ RV [g]\n(13.2)\nThis is a basic fact from algebraic geometry. The reason is that RV [g] is the\nset of homogeneous polynomial functions on the cone C of ∆V [g], and any\nsuch function τ can be restricted to ∆V [f] (see figure 13.2). Conversely, any\npolynomial function on ∆V [f] can be extended to a homogeneous polynomial\nfunction on the cone C.\nLet RV [f]d and RV [g]d be the degree d components of RV [f] and RV [g].\nThese are G-modules since ∆V [f] and ∆V [g] are G-varieties. The surjection\n(13.2) is degree preserving. So there is a surjection\nRV [f]d եւ RV [g]d\n(13.3)\nfor every d. Since G is reductive, both RV [f]d and RV [g]d are direct sums\nof irreducible G-modules. Hence the surjection (13.3) implies that RV [f]d\ncan be embedded as a G submodule of RV [g]d.\n74"},{"page":76,"text":"C\nτ\n∆V [f]\nFigure 13.2: C denotes the cone of ∆V [g].\nDefinition 13.1. We say that a Weyl-module S = Vλ(G) is an obstruction\nfor the embedding (13.1) (or, equivalently, for the pair (f, g)) if Vλ(G) occurs\nin RV [f; m, n]d, but not in RV [g; m]d, for some d. Here occurs means the\nmultiplicity of Vλ(G) in the decomposition of RV [f; m, n]d is nonzero.\nIf an obstruction exists for given m, n, then the embedding (13.1) does\nnot exist.\nConjecture 13.2 (GCT2). An obstruction exists for the pair (f, g) for all\nlarge enough n if m = 2polylog(n).\nThis implies Conjecture 13.1. In essence, this turns a nonexistence prob-\nlem (of polylog depth circuit for the permanent) into an existence problem\n(of an obstruction).\nIf we replace the determinant here by any other complete function in\nNC, an obstruction need not exist. Because, as we shall see in the next\nlecture, the existence of an obstruction crucially depends on the exceptional\nnature of the class variety constructed from the determinant.\nThe main\ngoals of GCT in this context are:\n1. understand the exceptional nature of the class varieties for NC and\n#P, and\n2. use it to prove the existence of obstructions.\n13.1.1\nWhy are the class varieties exceptional?\nWe now elaborate on the exceptional nature of the class varieties. Its signif-\nicance for the existence of obstructions will be discussed in the next lecture.\nLet V be a G-module, G = GLn(C). Let P(V ) be a projective variety\nover V . Let v ∈P(V ), and recall ∆V [v] = Gv. Let H = Gv be the stabilizer\nof v, that is, Gv = {σ ∈G|σv = v}.\n75"},{"page":77,"text":"Definition 13.2. We say that v is characterized by its stabilizer H = Gv\nif v is the only point in P(V ) such that hv = v, ∀h ∈H.\nIf v is characterized by its stabilizer, then ∆V [v] is completely determined\nby the group triple H ֒→G ֒→K = GL(V ).\nDefinition 13.3. The orbit closure ∆V [v], when v is characterized by its\nstabilizer, is called a group-theoretic variety.\nProposition 13.1. [GCT1]\n1. The determinant g = det(Y ) ∈P(V ) is characterized by its stabilizer.\nTherefore ∆V [g] is group theoretic.\n2. The permanent h = perm(X) ∈P(W), where W = Symn(X), is also\ncharacterized by its stabilizer. Therefore ∆W[h] is also group theoretic.\n3. Finally, f = φ(h) ∈P(V ) is also characterized by its stabilizer. Hence\n∆V [f] is also group theoretic.\nProof. (1) It is a fact in classical representation theory that the stabilizer\nof det(Y ) in G = GL(Y ) = GLm2(C) is the subgroup Gdet that consists of\nlinear transformations of the form Y →AY ∗B, where Y ∗= Y or Y t, for any\nA, B ∈GLm(C). It is clear that linear transformation of this form stabilize\nthe determinant since:\n1. det(AY B) = det(A)det(B)det(Y ) = c det(Y ), where c = det(A) det(B).\nNote that the constant c doesn’t matter because we get the same point\nin the projective space.\n2. det(Y ∗) = det(Y ).\nIt is a basic fact in classical invariant theory that det(Y ) is the only point in\nP(V ) stabilized by Gdet. Furthermore, the stabilizer Gdet is reductive, since\nits connected part is (Gdet)◦≈GLm × GLm with the natural embedding\n(Gdet)◦= GLm × GLm ֒→GL(Cm ⊗Cm) = GLm2(C) = G.\n(2) The stabilizer of perm(x) is the subgroup Gperm ⊆GL(X) = GLn2(C)\ngenerated by linear transformations of the form X →λX∗μ, where X∗=\nXorXt, and λ and μ are diagonal (which change the permanent by a con-\nstant factor) or permutation matrices (which do not change the permanent).\nFinally, the discrete component of Gperm is isomorphic to S2 ⋊Sn ×Sn,\nwhere ⋊denotes semidirect product. The continuous part is (C∗)n × (C∗)n.\nSo Gperm is reductive.\n76"},{"page":78,"text":"(3) Similar.\nThe main significance of this proposition is the following.\nBecause\n∆V [g], ∆V [f], and ∆W[h] are group theoretic, the algebraic geometric prob-\nlems concerning these varieties can be “reduced” to problems in the theory\nof quantum groups. So the plan is:\n1. Use the theory of quantum groups to understand the structure of the\ngroup triple associated with the algebraic variety.\n2. Translate this understanding to the structure of the algebraic variety.\n3. Use this to show the existence of obstructions.\n77"},{"page":79,"text":"Chapter 14\nGroup theoretic varieties\nScribe: Joshua A. Grochow\nGoal: In this lecture we continue our discussion of group-theoretic varieties.\nWe describe why obstructions should exist, and why the exceptional group-\ntheoretic nature of the class varieties is crucial for this existence.\nRecall\nLet G = GLn(C), V a G-module, and P(V ) the associated projective space.\nLet v ∈P(V ) be a point characterized by its stabilizer H = Gv ⊂G. In\nother words, v is the only point in P(V ) stabilized by H. Then ∆V [v] = Gv\nis called a group-theoretic variety because it is completely determined by the\ngroup triple\nH ֒→G ֒→GL(V ).\nThe simplest example of a group-theoretic variety is a variety of the form\nG/P that we described in the earlier lecture. Let V = Vλ(G) be a Weyl\nmodule of G and vλ ∈P(V ) the highest weight point (recall: the unique\npoint stabilized by the Borel subgroup B ⊂G of lower triangular matrices).\nThen the stabilizer of vλ consists of block-upper triangular matrices, where\n78"},{"page":80,"text":"the block sizes are determined by λ:\nPλ := Gvλ =\n \n \n \n \n \n \n \n \n \n \n \n \n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n0\n0\n0\n∗\n∗\n∗\n∗\n∗\n0\n0\n0\n∗\n∗\n∗\n∗\n∗\n0\n0\n0\n0\n0\n∗\n∗\n∗\n0\n0\n0\n0\n0\n∗\n∗\n∗\n0\n0\n0\n0\n0\n∗\n∗\n∗\n \n \n \n \n \n \n \n \n \n \n \n \nThe orbit ∆V [vλ] = Gvλ ∼= G/Pλ is a group-theoretic variety determined\nentirely by the triple\nPλ = Gvλ ֒→G ֒→K = GL(V ).\nThe group-theoretic varieties of main interest in GCT are the class va-\nrieties associated with the various complexity classes.\n14.1\nRepresentation theoretic data\nThe main principle guiding GCT is that the algebraic geometry of a group-\ntheoretic variety ought to be completely determined by the representation\ntheory of the corresponding group triple.\nThis is a natural extension of\nwork already pursued in mathematics by Deligne and Milne on Tannakien\ncategories [DeM], showing that an algebraic group is completely determined\nby its representation theory. So the goal is to associate to a group-theoretic\nvariety some representation-theoretic data that will analogously capture the\ninformation in the variety completely. We shall now illustrate this for the\nclass variety for NC. First a few definitions.\nLet v ∈P(V ) be the point as above characterized by its stabilizer Gv.\nThis means the line Cv ⊆V corresponding to v is a one-dimensional repre-\nsentation of Gv. Thus (Cv)⊗d is a one-dimensional degree d representation,\ni.e. the representation ρ : G →GL(Cv) ∼= C∗is polynomial of degree d\nin the entries of the matrix of an element in G. Recall that C[V ] is the\ncoordinate ring of V , and C[V ]d is its degree d homogeneous component, so\n(Cv)⊗d ⊆C[V ]d.\nTo each v ∈P(V ) that is characterized by its stabilizer, we associate a\nrepresentation-theoretic data, which is the set of G-modules\nΠv =\n[\nd\nΠv(d),\n79"},{"page":81,"text":"where Πv(d) is the set of all irreducible G-submodules S of C[V ]d whose\nduals S∗do not contain a Gv-submodule isomorphic to (Cv)⊗d∗(the dual of\n(Cv)⊗d). The following proposition elucidates the importance of this data:\nProposition 14.1. [GCT2] Πv ⊆IV [v] (where IV [v] is the ideal of the\nprojective variety ∆V [v] ⊆P(V )).\nProof. Fix S ∈Πv(d). Suppose, for the sake of contradiction, that S ⊈\nIV [v]. Since S ⊆C[V ], S consists of “functions” on the variety P(V ) (ac-\ntually homogeneous polynomials on V ). The coordinate ring of ∆V [v] is\nC[V ]/IV [v], and since S ⊈IV [v], S must not vanish identically on ∆V [v].\nSince the orbit Gv is dense in ∆V [v], S must not vanish identically on this\nsingle orbit Gv. Since S is a G-module, if S were to vanish identically on the\nline Cv, then it would vanish on the entire orbit Gv, so S does not vanish\nidentically on Cv.\nNow S consists of functions of degree d. Restrict them to the line Cv.\nThe dual of this restriction gives an injection of (Cv)⊗d∗as a Gv-submodule\nof S∗, contradicting the definition of Πv(d).\n14.2\nThe second fundamental theorem\nWe now ask essentially the reverse question: when does the representation\ntheoretic data Πv generate the ideal IV [v]? For if Πv generates IV [v], then Πv\ncompletely captures the coordinate ring C[V ]/IV [v], and hence the variety\n∆V [v].\nTheorem 14.1 (Second fundamental theorem of invariant theory for G/P).\nThe G-modules in Πvλ(2) generate the ideal IV [vλ] of the orbit Gvλ ∼= G/Pλ,\nwhen V = Vλ(G).\nThis theorem justifies the main principle for G/P, so we can hope that\nsimilar results hold for the class varieties in GCT (though not always exactly\nin the same form).\nNow, let ∆V [g] be the class variety for NC (in other words, take g =\ndet(Y ) for a matrix Y of indeterminates).\nBased on the main principle,\nwe have the following conjecture, which essentially generalizes the second\nfundamental theorem of invariant theory for G/P to the class variety for\nNC:\nConjecture 14.1 (GCT 2). ∆V [g] = X(Πg) where X(Πg) is the zero-set\nof all forms in the G-modules contained in Πg.\n80"},{"page":82,"text":"Theorem 14.2 (GCT 2). A weaker version of the above conjecture holds.\nSpecifically, assuming that the Kronecker coefficients satisfy a certain sep-\naration property, there exists a G-invariant (Zariski) open neighbourhood\nU ⊆P(V ) of the orbit Gg such that X(Πg) ∩U = ∆V [g] ∩U.\nThere is a notion of algebro-geometric complexity called Luna-Vust com-\nplexity which quantifies the gap between G/P and class varieties. The Luna-\nVust complexity of G/P is 0. The Luna-Vust complexity of the NC class\nvariety is Ω(dim(Y )). This is analogous to the difference between circuits\nof constant depth and circuits of superpolynomial depth. This is why the\nprevious conjecture and theorem turn out to be far harder than the corre-\nsponding facts for G/P.\n14.3\nWhy should obstructions exist?\nThe following proposition explains why obstructions should exist to separate\nNC from P #P.\nProposition 14.2 (GCT 2). Let g = det(Y ), h = perm(X), f = φ(h),\nn = dim(X), m = dim(Y ). If Conjecture 14.1 holds and the permanent can-\nnot be approximated arbitrarily closely by circuits of poly-logarithmic depth\n(hardness assumption), then an obstruction for the pair (f, g) exists for all\nlarge enough n, when m = 2logc n for some constant c. Hence, under these\nconditions, NC ̸= P #P over C.\nThis proposition may seem a bit circular at first, since it relies on a hard-\nness assumption. But we do not plan to prove the existence of obstructions\nby proving the assumptions of this proposition. Rather, this proposition\nshould be taken as evidence that obstructions exist (since we expect the\nhardness assumption therein to hold, given that the permanent is # P-\ncomplete), and we will develop other methods to prove their existence.\nProof. The hardness assumption implies that f /∈∆V [g] if m = 2logc n [GCT\n1].\nConjecture 14.1 says that X(Πg) = ∆V [g]. So there exists an irreducible\nG-module S ∈Πg such that S does not vanish on f. So S occurs in RV [f]\nas a G-submodule.\nOn the other hand, since S ∈Πg, S ⊆IV [g] by Proposition 14.1. So\nS does not occur in RV [g] = C[V ]/IV [g]. Thus S is not a G-submodule of\nRV [g], but it is a G-submodule of RV [f], i.e., S is an obstruction.\n81"},{"page":83,"text":"Chapter 15\nThe flip\nScribe: Hariharan Narayanan\nGoal: Describe the basic principle of GCT, called the flip, in the context of\nthe NC vs. P #P problem over C.\nreferences: [GCTflip1, GCT1, GCT2, GCT6]\nRecall\nAs in the previous lectures, let g = det(Y ) ∈P(V ), Y an m × m vari-\nable matrix, G = GLm2(C), and ∆V (g) = ∆V [g; m] = Gg ⊆P(V ) the\nclass variety for NC. Let h = perm(X), X an n × n variable matrix,\nf = φ(h) = ym−nh ∈P(V ), and ∆V (f) = ∆V [f; m, n] = Gf ⊆P(V ) the\nclass variety for P #P. Let RV [f; m, n] denote the homogeneous coordinate\nring of ∆V [f; m, n], RV [g; m] the homogeneous coordinate ring of ∆V [g; m],\nand RV [f; m, n]d and RV [g; m]d their degree d-components. A Weyl module\nS = Vλ(G) of G is an obstruction of degree d for the pair (f, g) if Vλ occurs\nin RV [f; m, n]d but not RV [g; m]d.\nConjecture 15.1. [GCT2] An obstruction (of degree polynomial in m) ex-\nists if m = 2polylog(n) as n →∞.\nThis implies NC ̸= P #P over C.\n82"},{"page":84,"text":"15.1\nThe flip\nIn this lecture we describe an approach to prove the existence of such ob-\nstructions. It is based on the following complexity theoretic positivity hy-\npothesis:\nPHflip [GCTflip1]:\n1. Given n, m and d, whether an obstruction of degree d for m and n\nexists can be decided in poly(n, m, ⟨d⟩) time, and if it exists, the label\nλ of such an obstruction can be constructed in poly(n, m, ⟨d⟩) time.\nHere ⟨d⟩denotes the bitlength of d.\n2.\n(a) Whether Vλ occurs in RV [f; m, n]d can be decided in poly(n, m, ⟨d⟩, ⟨λ⟩)\ntime.\n(b) Whether Vλ occurs in RV [g; m]d can be decided in poly(n, m, ⟨d⟩, ⟨λ⟩)\ntime.\nThis suggests the following approach for proving Conjecture 15.1:\n1. Find polynomial time algorithms sought in PHflip-2 for the basic de-\ncision problems (a) and (b) therein.\n2. Using these find a polynomial time algorithm sought in PHflip-1 for\ndeciding if an obstruction exists.\n3. Transform (the techniques underlying) this “easy” (polynomial time)\nalgorithm for deciding if an obstruction exists for given n and m into\nan “easy” (i.e., feasible) proof of existence of an obstruction for every\nn →∞when d is large enough and m = 2polylog(n).\nThe first step here is the crux of the matter. The main results of [GCT6]\nsay that the polynomial time algorithms for the basic decision problems as\nsought in PHflip-2 indeed exist assuming natural analogues of PH1 and SH\n(PH2) that we have seen earlier in the context of the plethysm problem. To\nstate them, we need some definitions.\nLet Sλ\nd [f] = Sλ\nd [f; m, n] be the multiplicity of Vλ = Vλ(G) in RV [f; m, n].\nThe stretching function ̃Sd[f] = ̃Sλ\nd [f; m, n] is defined by\n ̃Sλ\nd [f](k) := Skλ\nkd[f].\nThe stretching function for g, ̃Sλ\nd [g] = ̃Sλ\nd [g; m], is defined analogously.\nThe main mathematical result of [GCT6] is:\n83"},{"page":85,"text":"Theorem 15.1. [GCT6] The stretching functions ̃Sλ\nd [g] and ̃Sλ\nd [f] are quasipoly-\nnomials assuming that the singularities of ∆V [f; m, n] and ∆V [g; m] are ra-\ntional.\nHere rational means “nice”; we shall not worry about the exact defini-\ntion.\nThe main complexity-theoretic result is:\nTheorem 15.2. [GCT6] Assuming the following mathematical positivity\nhypothesis PH1 and the saturation hypothesis SH (or the stronger positivity\nhypothesis PH2), PHflip-2 holds.\nPH1: There exists a polytope P = P λ\nd [f] such that\n1. The Ehrhart quasi-polynomial of P, fP(k), is ̃Sλ\nd [f](k).\n2. dim(P) = poly(n, m, ⟨d⟩).\n3. Membership in P can be answered in polynomial time.\n4. There is a polynomial time separation oracle [GLS] for P.\nSimilarly, there exists a polytope Q = Qλ\nd[g] such that\n1. The Ehrhart quasi-polynomial of Q, fQ(k), is ̃Sλ\nd [g](k).\n2. dim(Q) = poly(m, ⟨d⟩).\n3. Membership in Q can be answered in polynomial time.\n4. There is a polynomial time separation oracle for Q.\nPH2: The quasi-polynomials ̃Sλ\nd [g] and ̃Sλ\nd [f] are positive.\nThis implies:\nSH: The quasi-polynomials ̃Sλ\nd [g] and ̃Sλ\nd [f] are saturated.\nPH1 and SH imply that the decision problems in PHflip-2 can be trans-\nformed into saturated positive integer programming problems. Hence The-\norem 15.2 follows from the polynomial time algorithm for saturated linear\nprogramming that we described in an earlier class.\nThe decision problems in PHflip-2 are “hyped” up versions of the plethysm\nproblem discussed earlier. The article [GCT6] provides evidence for PH1\n84"},{"page":86,"text":"and PH2 for the plethysm problem. This constitutes the main evidence for\nPH1 and PH2 for the class varieties in view of their group-theoretic nature;\ncf. [GCTflip1].\nThe following problem is important in the context of PHflip-2:\nProblem 15.1. Understand the G-module structure of the homogeneous\ncoordinate rings RV [f]d and RV [g]d.\nThis is an instance of the following abstract:\nProblem 15.2. Let X be a projective group-theoretic G-variety. Let R =\nL∞\nd=0 Rd be its homogeneous coordinate ring.\nUnderstand the G-module\nstructure of Rd.\nThe simplest group-theoretic variety is G/P. For it, a solution to this\nabstract problem is given by the following results:\n1. The Borel-Weil theorem.\n2. The Second Fundamental theorem of invariant theory [SFT].\nThese will be covered in the next class for the simplest case of G/P, the\nGrassmanian.\n85"},{"page":87,"text":"Chapter 16\nThe Grassmanian\nScribe: Hariharan Narayanan\nGoal: The Borel-Weil and the second fundamental theorem of invariant\ntheory for the Grassmanian.\nReference: [F]\nRecall\nLet V = Vλ(G) be a Weyl module of G = GLn(C) and vλ ∈P(V ) the point\ncorresponding to its highest weight vector. The orbit ∆V [vλ] := Gvλ, which\nis already closed, is of the form G/P, where P is the parabolic stabilizer of\nvλ. When λ is a single column, it is called the Grassmannian.\nAn alternative description of the Grassmanian is as follows. Assume that\nλ is a single column of length d. Let Z be a d × n matrix of variables zij.\nThen V = Vλ(G) can be identified with the span of d × d minors of Z with\nthe action of σ ∈G given by:\nσ : f(z) 7→f(zσ).\nLet Grn\nd be the space of all d-dimensional subspaces of Cn. Let W be a\nd-dimensional subspace of Cn. Let B = B(W) be a basis of W. Construct\nthe d × n matrix zB, whose rows are vectors in B. Consider the Pl ̈ucker\nmap from Grn\nd to P(V ) which maps any W ∈Grn\nd to the tuple of d × d\nminors of ZB. Here the choice of B = B(W) does not matter, since any\nchoice gives the same point in P(V ). Then the image of Grn\nd is precisely the\nGrassmanian Gvλ ⊆P(V ).\n86"},{"page":88,"text":"16.1\nThe second fundamental theorem\nNow we ask:\nQuestion 16.1. What is the ideal of Grn\nd ≈Gvλ ⊆P(V )?\nThe homogeneous coordinate ring of P(V ) is C[V ].\nWe want an ex-\nplicit set of generators of this ideal in C[V ]. This is given by the second\nfundamental theorem of invariant theory, which we describe next.\nThe coordinates of P(V ) are in one-to-one correspondence with the d×d\nminors of the matrix Z. Let each minor of Z be indexed by its columns. Thus\nfor 1 ≤i1 < · · · < id ≤n, Zi1,...,id is a coordinate of P(V ) corresponding\nto the minor of Z formed by the columns i1, i2, . . ..\nLet Λ(n, d) be the\nset of ordered d-tuples of {1, . . . , n}. The tuple [i1, . . . , id] in this set will\nbe identified with the coordinate Zi1,...,id of P(V ).\nThere is a bijection\nbetween the elements of Λ(n, d) and of Λ(n, n −d) obtained by associating\ncomplementary sets:\nΛ(n, d) ∋λ ↭λ∗∈Λ(n, n −d).\nWe define sgn(λ, λ∗) to be the sign of the permutation that takes [1, . . . , n]\nto [λ1, . . . , λd, λ∗\n1, . . . , λ∗\nn−d].\nGiven s ∈{1. . . . , d}, α ∈Λ(n, s−1), β ∈Λ(n, d+1), and γ ∈Λ(n, d−s),\nwe now define the Van der Waerden Syzygy [[α, β, γ]], which is an element\nof the degree two component C[V ]2 of C[V ], as follows:\n[[α, β, γ]] =\nP\nτ∈Λ(d+1,s) sgn(τ, τ ∗)[α1, . . . , αs−1, βτ ∗\n1 , . . . , βτ ∗\nd+1−s][βτ1, . . . , βτs, γ1, . . . , γd−s].\nIt is easy to show that this syzygy vanishes on the Grassmanian Grn\nd :\nbecause it is an alternating (d+1)-multilinear-form, and hence has to vanish\non any d-dimensional space W ∈Grn\nd . Thus it belongs to the ideal of the\nGrassmanian. Moreover:\nTheorem 16.1 (Second fundamental theorem). The ideal of the Grassma-\nnian Grn\nd is generated by the Van-der-Waerden syzygies.\nAn alternative formulation of this result is as follows. Let Pλ ⊆G be the\nstabilizer of vλ. Let Πvλ(2) be the set of irreducible G-submodules of C[V ]2\nwhose duals do not contain a Pλ-submodule isomorphic to Cv⊗2∗\nλ\n(the dual\nof Cv⊗2\nλ ). Here Cvλ denotes the line in P(V ) corresponding to vλ, which is\na one-dimensional representation of Pλ since it stabilizes vλ ∈P(V ). It can\n87"},{"page":89,"text":"be shown that the span of the G-modules in Πvλ(2) is equal to the span of\nthe Van-der-Waerden syzygies. Hence, Theorem 16.1 is equivalent to:\nTheorem 16.2 (Second Fundamental Theorem(SFT)). The G-modules in\nΠvλ(2) generate the ideal of Grn\nd .\nThis formulation of SFT for the Grassmanian looks very similar to the\ngeneralized conjectural SFT for the NC-class variety described in the ear-\nlier class. This indicates that the class varieties in GCT are “qualitatively\nsimilar” to G/P.\n16.2\nThe Borel-Weil theorem\nWe now describe the G-module structure of the homogeneous coordinate\nring R of the Grassmannian Gvλ ⊆P(V ), where λ is a single column of\nheight d. The goal is to give an explicit basis for R. Let Rs be the degree\ns component of R. Corresponding to any numbering T of the shape sλ,\nwhich is a d × s rectangle, whose columns have strictly increasing elements\ntop to bottom, we have a monomial mT = Q\nc Zc ∈C[V ]s, were Zc is the\ncoordinate of P(V ) indexed by the d-tuple c, and c ranges over the s columns\nof T. We say that mT is (semi)-standard if the rows of T are nondecreasing,\nwhen read left to right. It is called nonstandard otherwise.\nLemma 16.1 (Straightening Lemma). Each non-standard mT can be straight-\nened to a normal form, as a linear combination of standard monomials, by\nusing Van der Waerden Syzygies as straightening relations (rewriting rules).\nFor any numbering T as above, express mT in a normal form as per the\nlemma:\nmT =\nX\n(Semi)-Standard Tableau S\nα(S, T), mS\nwhere α(S, T) ∈C.\nTheorem 16.3 (Borel-Weil Theorem for Grassmannians). Standard mono-\nmials {mT } form a basis of Rs, where T ranges over all semi-standard\ntableaux of rectangular shape sλ. Hence, Rs ∼= V ∗\nsλ, the dual of the Weyl\nmodule Vsλ.\nThis gives the G-module structure of R completely. It follows that the\nproblem of deciding if Vβ(G) occurs in Rs can be solved in polynomial time:\nthis is so if and only if (sλ)∗= β, where (sλ)∗denotes the dual partition,\nwhose description is left as an exercise.\n88"},{"page":90,"text":"The second fundamental theorem as well as the Borel-Weil theorem easily\nfollow from the straightening lemma and linear independence of the standard\nmonomials (as functions on the Grassmanian).\n89"},{"page":91,"text":"Chapter 17\nQuantum group: basic\ndefinitions\nScribe: Paolo Codenotti\nGoal: The basic plan to implement the flip in [GCT6] is to prove PH1 and\nSH via the theory of quantum groups. We introduce the basic concepts in\nthis theory in this and the next two lectures, and briefly show their relevance\nin the context of PH1 in the final lecture.\nReference: [KS]\n17.1\nHopf Algebras\nLet G be a group, and K[G] the ring of functions on G with values in the\nfield K, which will be C in our applications. The group G is defined by the\nfollowing operations:\n• multiplication: G × G →G,\n• identity e: e →G,\n• inverse: G →G.\nIn order for G to be a group, the following properties have to hold:\n• eg = ge = g,\n• g1(g2g3) = (g1g2)g3,\n• g−1g = gg−1 = e.\n90"},{"page":92,"text":"We now want to translate these properties to properties of K[G]. This\nshould be possible since K[G] contains all the information that G has. In\nother words, we want to translate the notion of a group in terms of K[G].\nThis translate is called a Hopf algebra. Thus if G is a group, K[G] is a Hopf\nalgebra. Let us first define the dual operations.\n• Multiplication is a map:\n· : G × G →G.\nSo co-multiplication ∆will be a map as follows:\nK[G × G] = K[G] ⊗K[G] ←K[G].\nWe want ∆to be the pullback of multiplication. So for a given f ∈\nK[G] we define ∆(f) ∈K[G] ⊗K[G] by:\n∆(f)(g1, g2) = f(g1g2).\nPictorially:\nG × G\n·\n−−−−→G\n∆(f)\n y\n yf\nk\nk\n• The unit is a map:\ne →G.\nTherefore we want the co-unit ǫ to be a map:\nK ǫ←−K[G],\ndefined by: for f ∈K[G], ǫ(f) = f(e).\n• Inverse is a map:\n( )−1 : G →G.\nWe want the dual antipode S to be the map:\nK[G] ←K[G]\ndefined by: for f ∈K[G], S(f)(g) = f(g−1).\nThe following are the abstract axioms satisfied by ∆, ǫ and S.\n91"},{"page":93,"text":"1. ∆and ǫ are algebra homomorphisms.\n∆: K[G] →K[G] ⊗K[G]\nǫ : K[G] →K.\n2. co-associativity: Associativity is defined so that the following diagram\ncommutes:\nG × G×G\nG×G × G\n·\n y\nid\n y\n yid\n y·\nG\n×G\nG×\nG\n·\n y\n y·\nG\nG\nSimilarly, we define co-associativity so that the following dual diagram\ncommutes:\nK[G] ⊗K[G] ⊗K[G]\nK[G] ⊗K[G] ⊗K[G]\n∆\nx \nid\nx \nx id\nx ∆\nK[G]\n⊗K[G]\nK[G] ⊗\nK[G]\n∆\nx \nx ∆\nK[G]\nK[G]\nTherefore co-associativity says:\n(∆⊗id) ◦∆= (id ⊗∆) ◦∆.\n3. The property ge = g is defined so that the following diagram com-\nmutes:\ne ×G\nG\ne\n y\n yid\n y\nG×G\nid\n y·\n y\nG\nG\n92"},{"page":94,"text":"We define the co of this property so that the following diagram com-\nmutes:\nK\n× K[G]\nK[G]\nǫ\nx \nx id\nx \nK[G] × K[G]\nid\nx ∆\nx \nK[G]\nK[G]\nThat is, id = (ǫ⊗id)◦∆. Similarly, ge = g translates to: id = (id⊗ǫ)◦∆.\nTherefore we get\nid = (ǫ ⊗id) ◦∆= (id ⊗ǫ) ◦∆.\n4. The last property is gg−1 = e = g−1g. The first equality is equivalent\nto requiring that the following diagram commute:\nG\nG\ndiag\n y\n y\nG×G\n y\n()−1\n y\n yid\ne\nG×G\n y\n y·\n y\nG\nG\nWhere diag : G →G× G is the diagonal embedding. The co of diag is\nm : K[G] ←K[G] ⊗K[G] defined by m(f1, f2)(g) = f1(g) · f2(g). So\nthe co of this property will hold when the following diagram commutes:\n93"},{"page":95,"text":"K[G]\nK[G]\nm\nx \nx \nK[G] ⊗\nk[G]\nν\nx \nS\nx \nx id\nK\nK[G] ⊗K[G]\nx \nx ∆\nǫ\nx \nK[G]\nK[G]\nWhere ν is the embedding of K into K[G]. Therefore the last property\nwe want to be satisfied is:\nm ◦(S ⊗id) ◦∆= ν ◦ǫ.\nFor e = g−1g, we similarly get:\nm ◦(id ⊗S) ◦∆= ν ◦ǫ.\nDefinition 17.1 (Hopf algebra). A K-algebra A is called a Hopf algebra if\nthere exist homomorphisms ∆: A ⊗A →A, S : A →A, ǫ : A →K, and\nν : A →K that satisfy (1) −(4) above, with A in place of K[G].\nWe have shown that if G is a group, the ring K[G] of functions on\nG is a (commutative) Hopf algebra, which is non-co-commutative if G is\nnon-commutative. Thus for every usual group, we get a commutative Hopf\nalgebra. However, in general, Hopf algebras may be non-commutative.\nDefinition 17.2. A quantum group is a (non-commutative and non-co-\ncommutative) Hopf algebra.\nA nontrivial example of a quantum group will be constructed in the next\nlecture.\nNext we want to look at what happens to group theoretic notions such\nas representations, actions, and homomorphisms, in the context of Hopf\nalgebras. These will correspond to co-representations, co-actions, and co-\nhomomorphisms.\nLet us look closely at the notion of co-representation. A representation\nis a map · : G × V →V , such that\n• (h1h2) · v = h1 · (h2 · v), and\n94"},{"page":96,"text":"• e · v = v.\nTherefore a (right) co-representation of A will be a linear mapping φ : V →\nV ⊗A, where V is a K-vector space, and φ satisfies the following:\n• The following diagram commutes:\nV ⊗A ⊗A\nid⊗∆\n←−−−−V ⊗A\nφ⊗id\nx \nx φ\nV ⊗A\n←−−−−\nφ\nV\nThat is, the following equality holds:\n(φ ⊗id) ◦φ = (id ⊗∆) ◦φ.\n• The following diagram commutes:\nV ⊗K\nid\n←−−−−V ⊗K\nid⊗ǫ\nx \n\nV ⊗A ←−−−−\nφ\nV\nThat is, the following equality holds:\n(id ⊗ǫ) ◦φ = id\nIn fact all usual group theoretic notions can be “Hopfified” in this sense\n[exercise].\nLet us look now at an example. Let\nG = GLn(C) = GL(Cn) = GL(V ),\nwhere V = Cn. Let Mn be the matrix space of n×n C-matrices, and O(Mn)\nthe coordinate ring of Mn,\nO(Mn) = C[U] = C[{ui\nj}],\nwhere U is an n × n variable matrix with entries ui\nj. Let C[G] = O(G) be\nthe coordinate ring of G obtained by adjoining det(U)−1 to O(Mn). That\nis, C[G] = O(G) = C[U][det(U)−1], which is the C algebra generated by ui\nj’s\nand det(U)−1.\n95"},{"page":97,"text":"Proposition 17.1. C[G] is a Hopf algebra, with ∆, ǫ, and S as follows.\n• Recall that the axioms of a Hopf algebra require that\n∆: C[G] →C[G] ⊗C[G],\n∆(f)(g1, g2) = f(g1g2).\nTherefore we define\n∆(ui\nj) =\nX\nk\nui\nk ⊗uk\nj,\nwhere U denotes the generic matrix in Mn as above.\n• Again, it is required that\nǫ(f) = f(e).\nTherefore we define\nǫ(ui\nj) = δij,\nwhere δij is the Kronecker delta function.\n• Finally, the antipode is required to satisfy S(f)(g) = f(g−1). Let eU\nbe the cofactor matrix of U, U−1 =\n1\ndet(U) eU, and eui\nj the entries of eU.\nThen we define S by:\nS(ui\nj) =\n1\ndet(U) eui\nj = (U−1)i\nj.\n96"},{"page":98,"text":"Chapter 18\nStandard quantum group\nScribe: Paolo Codenotti\nGoal: In this lecture we construct the standard (Drinfeld-Jimbo) quantum\ngroup, which is a q-deformation of the general linear group GLn(C) with\nremarkable properties.\nReference: [KS]\nLet G = GL(V ) = GL(Cn), and V = Cn. In the earlier lecture, we\nconstructed the commutative and non co-commutative Hopf algebra C[G].\nIn this lecture we quantize C[G] to get a non-commutative and non-co-\ncommutative Hopf algebra Cq[G], and then define the standard quantum\ngroup Gq = GLq(V ) = GLq(n) as the virtual object whose coordinate ring\nis Cq[G].\nWe start by defining GLq(2) and SLq(2), for n = 2.\nThen we will\ngeneralize this construction to arbitrary n. Let O(M2) be the coordinate\nring of M2, the set of 2 × 2 complex matrices, C[V ] the coordinate ring of\nV generated by the coordinates x1 and x2 of V which satisfy x1x2 = x2x1.\nLet\nU =\n a\nb\nc\nd\n \nbe the generic (variable) matrix in M2. It acts on V = C2 from the left and\nfrom the right. Let\nx =\n x1\nx2\n \n.\nThe left action is defined by\nx →x′ := Ux.\n97"},{"page":99,"text":"Let\nx′ =\n x′\n1\nx′\n2\n \n.\nSimilarly, the right action is defined by\nxT →(x′′)T := xT U.\nLet\nx′′ =\n x′′\n1\nx′′\n2\n \n.\nThe action of M2 on V satisfies\nx′\n1x′\n2 = x′\n2x′\n1, and\nx′′\n1x′′\n2 = x′′\n2x′′\n1.\nNow instead of V , we take its q-deformation Vq, a quantum space, whose\ncoordinates x1 and x2 satisfy\nx1x2 = qx2x1,\n(18.1)\nwhere q ∈C is a parameter. Intuitively, in quantum physics if x1 and x2\nare position and momentum, then q = eiħwhen ħis Planck’s constant. Let\nCq[V ] be the ring generated by x1 and x2 with the relation (18.1). That is,\nCq[V ] = C[x1, x2]/ < x1x2 −qx2x1 > .\nIt is the coordinate ring of the quantum space Vq. Now we want to quantize\nM(2) to get Mq(2), the space of quantum 2 × 2 matrices, and GL(2) to\nGLq(2), the space of quantum 2×2 nonsingular matrices. Intuitively, Mq(2)\nis the space of linear transformations of the quantum space Vq which preserve\nthe equation (18.1) under the left and right actions, and similarly, GLq(2) is\nthe space of non-singular linear transformation that preserve the equation\n(18.1) under the left and right actions. We now formalize this intuition.\nLet U =\n a\nb\nc\nd\n \nbe a quantum matrix whose coordinates do not\ncommute. The left and right actions of U must preserve 18.1.\n[Left action:] Let the left action be φL : x →Ux, and Ux = x′. Then we\nmust have:\n a\nb\nc\nd\n x1\nx2\n \n=\n ax1 + bx2\ncx1 + dx2\n \n=\n x′\n1\nx′\n2\n \n.\n98"},{"page":100,"text":"[Right action:] Let the right action be φR : xT →xT U, and let x′′ =\n(xT U)T = UT x. Then we must have:\n x1\nx2\n a\nb\nc\nd\n \n=\n ax1 + cx2\nbx1 + dx2\n \n=\n x′′\n1\nx′′\n2\n \n.\nThe preservation of x1x2 = qx2x1 under left multiplication means\nx′\n1x′\n2 = qx′\n2x′\n1.\nThat is,\n(ax1 + bx2)(cx1 + dx2) = q(cx1 + dx2)(ax1 + bx2).\n(18.2)\nThe left hand side of (18.2) is\nacx2\n1 + bcx2x1 + adx1x2 + bdx2\n2 = acx2\n1 + (bc + adq)x2x1 + bdx2\n2.\nSimilarly, the right hand side of (18.2) is\nq(cax2\n1 + (da + cbq)x2x1 + bdx2\n2).\nTherefore equation (18.2) implies:\nac = qca\nbd = qdb\nbc + adq = da + qcb.\nThat is,\nac = qca\nbd = qdb\nad −da −qcb + q−1bc = 0.\nSimilarly, since x′′\n1x′′\n2 = qx′′\n2x′′\n1, we get:\nab = qba\ncd = qdc\nad −da −qbc + q−1cb = 0.\nThe last equations from each of these sets imply bc = cb.\nSo we define O(Mq(2)), the coordinate ring of the space of 2×2 quantum\nmatrices Mq(2), to be the C-algebra with generators a, b, c, and d, satisfying\nthe relations:\nab = qba,\nac = qca,\nbd = qdb,\ncd = qdc,\nbc = cb,\nad −da = (q −q−1)bc.\n99"},{"page":101,"text":"Let\nU =\n a\nb\nc\nd\n \n=\n u1\n1\nu1\n2\nu2\n1\nu2\n2\n \n.\nDefine the quantum determinant of U to be\nDq = det(U) = ad −qbc = da −q−1bc.\nDefine Cq[G] = O(GLq(2)), the coordinate ring of the virtual quantum group\nGLq(2) of invertible 2 × 2 quantum matrices, to be\nO(GLq(2)) = O(Mq(2))[D−1\nq ],\nwhere the square brackets indicate adjoining.\nProposition 18.1. The coordinate ring O(GLq(2)) is a Hopf algebra, with\n∆(ui\nj) =\nX\nk\nui\nk ⊗uk\nj ,\nS(ui\nj) = 1\nDq\neui\nj = (U−1)i\nj,\nǫ(ui\nj) = δij,\nwhere eU = [ ̃ui\nj] is the cofactor matrix\neU =\n d\n−q−1b\n−qc\na\n \n.\n(defined so that U eU = DqI) and U−1 = ̃U/Dq is the inverse of U.\nThis is a non-commutative and non-co-commutative Hopf algebra.\nNow we go to the general n. Let Vq be the n-dimensional quantum space,\nthe q-deformation of V , with coordinates xi’s which satisfy\nxixj = qxjxi\n∀i < j.\n(18.3)\nLet Cq[V ] be the coordinate ring of Vq defined by\nCq[V ] = C[x1, . . . , xn]/ < xixj −qxjxi > .\n100"},{"page":102,"text":"Let Mq(n) be the space of quantum n × n matrices, that is the set of linear\ntransformations on Vq which preserve (18.3) under the left as well as the\nright action. The left action is given by:\n \n \nx1\n...\nxn\n \n = x →Ux = x′,\nwhere U is the n × n generic quantum matrix. Similarly, the right action is\ngiven by:\nxT →xT U = (x′′)T .\nPreservation of (18.3) under the left and right actions means:\nx′\niy′\nj = qx′\njx′\ni,\nfor\ni < j\nx′′\ni y′′\nj = qx′′\njx′′\ni ,\nfor\ni < j.\nAfter straightforward calculations, these yield the following relations on\nthe entries uij = ui\nj of U:\nujkuik = q−1uikujk\n(i < j)\nukjuki = q−1ukiukj\n(i < j)\nujkuil= uilujk\n(i < j, k < l)\nujluik = uikujl−(q −q−1)ujkuil\n(i < j, k < l).\n(18.4)\nThe quantum determinant is defined as\nDq =\nX\nσ∈Sn\n(−q)l(σ)u\niσ(1)\nj1\n. . . u\niσ(n)\njn\n,\nwhere l(σ) denotes the length of the permutation σ, that is, the number of\ninversions in σ. This determinant formula is the same as the usual formula\nsubstituting (−q) for (−1).\nWe define the coordinate ring of the space Mq(n) of quantum n × n\nmatrices by\nO(Mq(n)) = C[U]/ < (18.4) >, and\nand the coordinate ring of the virtual quantum group GLq(n) by\nCq[G] = O(GLq(n)) = O(Mq(n))[D−1\nq ].\nWe define the quantum minors and, using these, the quantum co-factor\nmatrix eU and the quantum inverse matrix U−1 = eU/Dq in a straightforward\nfashion (these constructions are left as exercises).\n101"},{"page":103,"text":"Theorem 18.1. The algebra O(GLq(n)) is a Hopf algebra, with\n∆(ui\nj) =\nX\nk\nui\nk ⊗uk\nj\nǫ(ui\nj) = δij\nS(ui\nj) = 1\nDq\neui\nj = (U−1)i\nj\nS(D−1\nq ) = Dq.\nWe also denote the quantum group GLq(n) by Gq, GLq(Cn) or GLq(V ).\nIt has to be emphasized that this is only a virtual object. Only its coordinate\nring Cq[G] is real. Henceforth, whenever we say representation or action of\nGq, we actually mean corepresentation or coaction of Cq[G], and so forth.\n102"},{"page":104,"text":"Chapter 19\nQuantum unitary group\nScribe: Joshua A. Grochow\nGoal:\nDefine the quantum unitary subgroup of the standard quantum\ngroup.\nReference: [KS]\nRecall\nLet V = Cn, G = GLn(C) = GL(V ) = GL(Cn), and O(G) the coordinate\nring of G. The quantum group Gq = GLq(V ) is the virtual object whose\ncoordinate ring is\nO(Gq) = C[U]/⟨relations ⟩,\nwhere U is the generic n × n matrix of indeterminates, and the relations\nare the quadratic relations on the coordinates uj\ni defined in the last class so\nas to preserve the non-commuting relations among the coordinates of the\nquantum vector space Vq on which Gq acts. This coordinate ring is a Hopf\nalgebra.\n19.1\nA q-analogue of the unitary group\nIn this lecture we define a q-analogue of the unitary subgroup U = Un(C) =\nU(V ) ⊆GLn(C) = GL(V ) = G. This is a q-deformation Uq = Uq(V ) ⊆Gq\nof U(V ). Since Gq is only a virtual object, Uq will also be virtual. To define\nUq, we must determine how to capture the notion of unitarity in the setting\nof Hopf algebras. As we shall see, it is captured by the notion of a Hopf\n∗-algebra.\n103"},{"page":105,"text":"Definition 19.1. A ∗-vector space is a vector space V with an involution\n∗: V →V satisfying\n(αv + βw)∗= αv∗+ βw∗\n(v∗)∗= v\nfor all v, w ∈V , and α, β ∈C.\nWe think of ∗as a generalization of complex conjugation; and in fact\nevery complex vector space is a ∗-vector space, where ∗is exactly complex\nconjugation.\nDefinition 19.2. A Hopf ∗-algebra is a Hopf algebra (A, ∆, ǫ, S) with an\ninvolution ∗: A →A such that (A, ∗) is a ∗-vector space, and:\n1. (ab)∗= b∗a∗, 1∗= 1\n2. ∆(a∗) = ∆(a)∗(where ∗acts diagonally on the tensor product A ⊗A:\n(v ⊗w)∗= (v∗⊗w∗))\n3. ǫ(a∗) = ǫ(a)\nThere is no explicit condition here on how ∗interacts with the antipode\nS.\nLet O(G) = C[G] be the coordinate ring of G as defined earlier.\nProposition 19.1. Then O(G) is a Hopf ∗-algebra.\nProof. We think of the elements in O(G) as C-valued functions on G and\ndefine ∗: O(G) →O(G) so that it satisfies the three conditions for a Hopf\n∗-algebra, and\n(4) For all f ∈O(G) and g ∈U ⊆G, f ∗(g) = f(g)\nLet uj\ni be the coordinate functions which, together with D−1, D = det(U),\ngenerate O(G). Because of the first condition on a Hopf ∗-algebra (relating\nthe involution ∗to multiplication), specifying (uj\ni)∗and D∗suffices to define\n∗completely. We define\n(uj\ni)∗= S(ui\nj) = (U−1)i\nj\nand D∗= D−1. We can check that this satifies (1)-(4). Here we will only\ncheck (4), and leave the remaining verification as an exercise. Let g be an\nelement of the unitary group U. Then (uj\ni)∗(g) = S(ui\nj)(g) = (g−1)i\nj = (g)j\ni,\nwhere the last equality follows from the fact that g is unitary (i.e. g−1 = g†,\nwhere † denotes conjugate transpose).\n104"},{"page":106,"text":"Thus, we have defined a map f 7→f ∗purely algebraically in such a way\nthat the restriction of f ∗to the unitary group U is the same as taking the\ncomplex conjugate f on U.\nProposition 19.2. The coordinate ring Cq[G] = O(Gq) of the quantum\ngroup Gq = GLq(V ) is also a Hopf ∗-algebra.\nProof. The proof is syntactically identical to the proof for O(G), except\nthat the coordinate function uj\ni now lives in O(Gq) and the determinant D\nbecomes the q-determinant Dq. The definition of ∗is: (uj\ni)∗= S(ui\nj) and\nD∗\nq = D−1\nq , essentially the same as in the classical case.\nIntuitively, the “quantum subgroup” Uq of Gq is the virtual object such\nthat the restriction to Uq of the involution ∗just defined coincides with the\ncomplex conjugate.\n19.2\nProperties of Uq\nWe would like the nice properties of the classical unitary group to transfer\nover to the quantum unitary group, and this is indeed the case. Some of the\nnice properties of U are:\n1. It is compact, so we can integrate over U.\n2. we can do harmonic analysis on U (viz.\nthe Peter-Weyl Theorem,\nwhich is an analogue for U of the Fourier analysis on the circle U1).\n3. Every finite dimensional representation of U has a G-invariant Hermi-\ntian form, and thus a unitary basis – we say that every finite dimen-\nsional representation of U is unitarizable.\n4. Every finite dimensional representation X of U is completely reducible;\nthis follows from (3), since any subrepresentation W ⊆X has a per-\npendicular subrepresentation W ⊥under the G-invariant Hermitian\nform.\nCompactness is in some sense the key here. The question is how to define\nit in the quantum setting. Following Woronowicz, we define compactness to\nmean that every finite dimensional representation of Uq is unitarizable. Let\nus see what this means formally.\nLet A be a Hopf ∗-algebra, and W a corepresentation of A. Let ρ : W →\nW ⊗A be the corepresentation map. Let {bi} be a basis of W. Then, under\n105"},{"page":107,"text":"ρ, bi 7→P\nj bj ⊗mj\ni for some mj\ni ∈A. We can thus define the matrix of\nthe (co)representation M = (mj\ni) in the basis {bi}. We define M∗such that\n(M∗)j\ni = (Mi\nj)∗. Thus, in the classical case (i.e. when q = 1), M∗= M†.\nWe say that the corepresentation W is unitarizable if it has a basis B =\n{bi} such that the corresponding matrix MB of corepresentation satisfies the\nunitarity condition: MBM∗\nB = I. In this case, we say B is a unitary basis\nof the corepresentation W.\nDefinition 19.3. A Hopf ∗-algebra A is compact if every finite dimensional\ncorepresentation of A is unitarizable.\nTheorem 19.1 (Woronowicz). The coordinte ring Cq[G] = O(Gq) is a\ncompact Hopf ∗-algebra. This implies that every finite dimensional repre-\nsentation of Gq, by which we mean a finite dimensional coorepresentation\nof Cq[G], is completely reducible.\nWoronowicz goes further to show that we can q-integrate on Uq, and that\nwe can do quantized harmonic analysis on Uq; i.e., a quantum analogue of\nthe Peter-Weyl theorem holds.\nNow that we know the finite dimensional representations of Gq are com-\npletely reducible, we can ask what the irreducible representations are.\n19.3\nIrreducible Representations of Gq\nWe proceed by analogy with the Weyl modules Vλ(G) for G. Recall that\nevery polynomial irreducible representation of G = GLn(C) is of this form.\nTheorem 19.2.\n1. For all partitions λ of length at most n, there exists\na q-Weyl module Vq,λ(Gq) which is an irreducible representation of Gq\nsuch that\nlim\nq→1 Vq,λ(Gq) = Vλ(G).\n2. The q-Weyl modules give all polynomial irreducible representations of\nGq.\n19.4\nGelfand-Tsetlin basis\nTo understand the q-Weyl modules better, we wish to get an explicit basis for\neach module Vq,λ. We begin by defining a very useful basis – the Gel’fand-\nTestlin basis – in the classical case for Vλ(G), and then describe the q-\nanalogue of this basis.\n106"},{"page":108,"text":"By Pieri’s rule [FH]\nVλ(GLn(C)) =\nM\nλ′\nVλ′(GLn−1(C))\nwhere the sum is taken over all λ′ obtained from λ by removing any number\nof boxes (in a legal way) such that no two removed boxes come from the\nsame column. This is an orthogonal decomposition (relative to the GLn(C)-\ninvariant Hermitian form on Vλ) and it is also multiplicity-free, i.e., each Vλ′\nappears only once.\nFix a G-invariant Hermitian form on Vλ. Then the Gel’fand-Tsetlin basis\nfor Vλ(GLn(C)), denoted GT n\nλ , is the unique orthonormal basis for Vλ such\nthat\nGT n\nλ =\n[\nλ′\nGT n−1\nλ′\n,\nwhere the disjoint union is over the λ′ as in Pieri’s rule, and GT n−1\nλ′\nis defined\nrecursively, the case n = 1 being trivial.\nThe dimension of Vλ is the number of semistandard tableau of shape λ.\nWith any tableau T of this shape, one can also explicitly associate a basis\nelement GT(T) ∈GT n\nλ ; we shall not worry about how.\nWe can define the Gel’fand-Tsetlin basis GT n\nq,λ for Vq,λ(Gq(Cn)) analo-\ngously. We have the q-analogue of Pieri’s rule:\nVq,λ(Gq(Cn)) =\nM\nλ′\nVq,λ′(Gq(Cn−1))\nwhere the decomposition is orthogonal and multiplicity-free, and the sum\nranges over the same λ′ as above. So we can define GT n\nq,λ to be the unique\nunitary basis of Vq,λ such that\nGT n\nq,λ =\n[\nλ′\nGT n−1\nq,λ′ .\nWith any semistandard tableau T, one can also explicitly associate a basis\nelement GTq(T) ∈GT n\nq,λ′; details omitted.\n107"},{"page":109,"text":"Chapter 20\nTowards positivity\nhypotheses via quantum\ngroups\nScribe: Joshua A. Grochow\nGoal: In this final brisk lecture, we indicate the role of quantum groups in\nthe context of the positivity hypothesis PH1. Specifically, we sketch how the\nLittlewood-Richardson rule – the gist of PH1 in the Littlewood-Richardson\nproblem – follows from the theory of standard quantum groups. We then\nbriefly mention analogous (nonstandard) quantum groups for the Kronecker\nand plethysm problems defined in [GCT4, GCT7], and the theorems and\nconjectures for them that would imply PH1 for these problems.\nReferences: [KS, K, Lu2, GCT4, GCT6, GCT7, GCT8]\nLet V = Cn, G = GLn(C) = GL(V ), Vλ = Vλ(G) a Weyl module of\nG, Gq = GLq(V ) the standard quantum group, Vq the q-deformation of V\non which GLq(V ) acts, Vq,λ = Vq,λ(Gq) the q-deformation of Vλ(G), and\nGTq,λ = GT n\nq,λ the Gel’fand-Tsetlin basis for Vq,λ.\n20.1\nLittlewood-Richardson rule via standard quan-\ntum groups\nWe now sketch how the Littlewood-Richardson rule falls out of the standard\nquantum group machinery, specifically the properties of the Gelfand-Tsetlin\nbasis.\n108"},{"page":110,"text":"20.1.1\nAn embedding of the Weyl module\nFor this, we have to embed the q-Weyl module Vq,λ in V ⊗d\nq\n, where d = |λ| =\nP λi is the size of λ. We first describe how to embed the Weyl module Vλ\nof G in V ⊗d in a standard way that can be quantized.\nIf d = 1, then Vλ(G) = V = V ⊗1. Otherwise, obtain a Young diagram\nμ from λ by removing its top-rightmost box that can be removed to get a\nvalid Young diagram, e.g.:\nx\n⇝\nλ\nμ\nIn the following, the box must be removed from the second row, since\nremoving from the first row would result in an illegal Young diagram:\nx\n⇝\nλ\nμ\nBy induction on d, we have a standard embedding Vμ(G) ֒→V ⊗d−1. This\ngives us an embedding Vμ(G) ⊗V ֒→V ⊗d. By Pieri’s rule [FH]\nVμ(G) ⊗V =\nM\nβ\nVβ(G),\nwhere the sum is over all β obtained from μ by adding one box in a legal way.\nIn particular, Vλ(G) ⊂Vμ(G) ⊗V . By restricting the above embedding, we\nget a standard embedding Vλ(G) ֒→V ⊗d.\nNow Pieri’s rule also holds in a quantized setting:\nVq,μ ⊗Vq =\nM\nβ\nVq,β(G),\nwhere β is as above.\nHence, the standard embedding Vλ ֒→V ⊗d above\ncan be quantized in a straightforward fashion to get a standard embedding\nVq,λ ֒→V ⊗d\nq\n. We shall denote it by ρ. Here the tensor product is meant to\nbe over Q(q). Actually, Q(q) doesn’t quite work. We have to allow square\nroots of elements of Q(q), but we won’t worry about this. For a semistandard\ntableau b of shape λ, we denote the image of a Gelfand-Tsetlin basis element\nGTq,λ(b) ∈GTq,λ under ρ by GT ρ\nq,λ(b) = ρ(GTq,λ(b)) ∈V ⊗d\nq\n.\n109"},{"page":111,"text":"20.1.2\nCrystal operators and crystal bases\nTheorem 20.1 (Crystallization). [DJM] The Gelfand-Tsetlin basis ele-\nments crystallize at q = 0. This means:\nlim\nq→0 GT ρ\nq,λ(b) = vi1(b) ⊗· · · ⊗vid(b),\n(20.1)\nfor some integer functions i1(b), . . . , id(b), and\nlim\nq→∞GT ρ\nq,λ(b) = vj1(b) ⊗· · · ⊗vjd(b),\n(20.2)\nfor some integer functions j1(b), . . . , jd(b).\nThe phenomenon that these limits consists of monomials, i.e., simple\ntensors is known as crystallization. It is related to the physical phenomenon\nof crystallization, hence the name. The maps b 7→i(b) = (i1(b), . . . , id(b))\nand b 7→j(b) = (j1(b), . . . , jd(b)) are computable in poly(⟨b⟩) time (where\n⟨b⟩is the bit-length of b).\nNow we want to define a special crystal basis of Vq,λ based on this phe-\nnomenon of crystallization. Towards that end, consider the following family\nof n × n matrices:\nEi =\n \n \n \n \n \n \n \n \n \n \n0\n0\n· · ·\n0\n...\n...\n...\n0\n1\n· · ·\n0\n0\n· · ·\n0\n...\n...\n0\n \n \n \n \n \n \n \n \n \n \n,\nwhere the only nonzero entry is a 1 in the i-th row and (i + 1)-st column.\nLet Fi = ET\ni . Corresponding to Ei and Fi, Kashiwara associates certain\noperators ˆEi and ˆFi on Vq,λ(Gq). We shall not worry about their actual\nconstruction here (for the readers familiar with Lie algebras: these are closely\nrelated to the usual operators in the Lie algebra of G associated with Ei and\nFi).\nIf we let ˆEi act on GT ρ\nq,λ(b), we get some linear combination\nˆEi(GT ρ\nq,λ(b)) =\nX\nb′\nab\nb′(q)GT ρ\nq,λ(b′),\nwhere ab\nb′(q) ∈Q(q) (actually an algebraic extension of Q(q) as mentioned\nabove). Essentially because of crystallization (Theorem 20.1), it turns out\n110"},{"page":112,"text":"that limq→0 ab\nb′(q) is always either 0 or 1, and for a given b, this limit is 1 for\nat most one b′, if any. A similar result holds for ˆFi(GT ρ\nq,λ(b)). This allows\nus to define the crystal operators (due to Kashiwara):\neei · b =\n b′\nif limq→0 ab\nb′(q) = 1,\n0\nif no such b′ exists,\nand similarly for efi. Although these operators are defined according to a\nparticular embedding Vq,λ ֒→V ⊗d\nq\nand a basis, they can be defined intrinsi-\ncally, i.e., without reference to the embedding or the Gel’fand-Tsetlin basis.\nNow, let W be a finite-dimensional representation of Gq, and R the\nsubring of functions in Q(q) regular at q = 0 (i.e. without a pole at q = 0).\nA lattice within W is an R-submodule of W such that Q(q) ⊗R L = W.\n(Intuition behind this definition: R ⊂Q(q) is analogous to Z ⊂Q. A lattice\nin Rn is a Z-submodule L of Rn such that R ⊗Z L = Rn.)\nDefinition 20.1. An (upper) crystal basis of a representation W of Gq is\na pair (L, B) such that\n• L is a lattice in W preserved by the Kashiwara operators ˆEi and ˆFi,\ni.e. ˆEi(L) ⊆L and ˆFi(L) ⊆L.\n• B is a basis of L/qL preserved by the crystal operators eei and efi, i.e.,\neei(B) ⊆B ∪{0} and efi(B) ⊆B ∪{0}.\n• The crystal operators eei and efi are inverse to each other wherever\npossible, i.e., for all b, b′ ∈B, if eei(b) = b′ ̸= 0 then efi(b′) = b, and\nsimilarly, if efi(b) = b′ ̸= 0 then eei(b′) = b.\nIt can be shown that if W = Vq,λ(Gq), then there exists a unique b ∈B\nsuch that eei(b) = 0 for all i; this corresponds to the highest weight vector\nof Vq,λ (the weight vectors in Vq,λ are analogous to the weight vectors in Vλ;\nwe do not give their exact definition here). By the work of Kashiwara and\nDate et al [K, DJM] above, the Gel’fand-Tsetlin basis (after appropriate\nrescaling) is in fact a crystal basis: just let\nL = LGT\n=\nthe R-module generated by GTq,λ, and\nBGT\n=\nGTq,λ(b),\nwhere GTq,λ(b) is the image under the projection L 7→L/qL of the set of\nbasis vectors in GTq,λ(b).\n111"},{"page":113,"text":"Theorem 20.2 (Kashiwara).\n1. Every finite-dimensional Gq-module has\na unique crystal basis (up to isomorphism).\n2. Let (Lλ, Bλ) be the unique crystal basis corresponding to Vq,λ. Then\n(Lα, Bα) ⊗(Lβ, Bβ) = (Lα ⊗Lβ, Bα ⊗Bβ) is the unique crystal basis\nof Vq,α ⊗Vq,β, where Bα ⊗Bβ denotes {ba ⊗bb|ba ∈Bα, bb ∈Bβ}.\nIt can be shown that every b ∈Bλ has a weight; i.e., it is the image of a\nweight vector in Lλ under the projection Lλ →Lλ/qLλ.\nNow let us see how the Littlewood-Richardson rule falls out of the prop-\nerties of the crystal bases. Recall that the specialization of Vq,α at q = 1 is\nthe Weyl module Vα of G = GLn(C), and\nVα ⊗Vβ =\nM\nγ\ncγ\nα,βVγ\n(20.3)\nwhere cγ\nα,β are the Littlewood-Richardson coefficients.\nThe Littlewood-\nRichardson rule now follows from the following fact:\ncγ\nα,β = #{b ⊗b′ ∈Bα ⊗Bβ|∀i, eei(b ⊗b′) = 0 and b ⊗b′ has weight γ}.\nIntuitively, b ⊗b′ here correspond to the highest weight vectors of the G-\nsubmodules of Vα ⊗Vβ isomorphic to Vγ.\n20.2\nExplicit decomposition of the tensor product\nThe decomposition (20.3) is only an abstract decomposition of Vα ⊗Vβ as a\nG-module. Next we consider the explicit decomposition problem. The goal\nis to find an explicit basis B = Bα⊗β of Vα ⊗Vβ that is compatible with this\nabstract decomposition. Specifically, we want to construct an explicit basis\nB of Vα ⊗Vβ in terms of suitable explicit bases of Vα and Vβ such that B\nhas a filtration\nB = B0 ⊇B1 ⊇· · · ⊇∅\nwhere each ⟨Bi⟩/⟨Bi+1⟩is an irreducible representation of G and ⟨Bi⟩denotes\nthe linear span of Bi.\nFurthermore, each element b ∈B should have a\nsufficiently explicit representation in terms of the basis Bα ⊗Bβ of Vα ⊗Vβ.\nThe explicit decomposition problem for the q-analogue Vq,α ⊗Vq,β is similar.\nFor example, we have already constructed explicit Gelfand-Tsetlin bases\nof Weyl modules. But it is not known how to construct an explicit basis B\n112"},{"page":114,"text":"with filtration as above in terms of the Gelfand-Tsetlin bases of Vα and Vβ\n(except when the Young diagram of either α or β is a single row).\nKashiwara and Lusztig [K, Lu2] construct certain canonical bases Bq,α\nand Bq,β of Vq,α and Vq,β, and Lusztig furthermore constructs a canonical\nbasis Bq = Bq,α⊗β of Vq,α ⊗Vq,β such that:\n1. Bq has a filtration as above,\n2. Each b ∈Bq has an expansion of the form\nb =\nX\nbα∈Bq,α,bβ∈Bq,β\nabα,bβ\nb\nbα ⊗bβ,\nwhere each abα,bβ\nb\nis a polynomial in q and q−1 with nonnegative inte-\ngral coefficients,\n3. Crystallization: For each b, as q →0, exactly one coefficient abα,bβ\nb\n→1,\nand the remaining all vanish.\nThe proof of nonnegativity of the coefficients of abα,bβ\nb\nis based on the Rie-\nmann hypothesis (theorem) over finite fields [Dl2], and explicit formulae for\nthese coefficients are known in terms of perverse sheaves [BBD] (which are\ncertain types of algebro-geometric objects).\nThis then provides a satisfactory solution to the explicit decomposition\nproblem, which is far harder and deeper than the abstract decomposition\nprovided by the Littlewood-Richardson rule. By specializing at q = 1, we\nalso get a solution to the explicit decomposition problem for Vα ⊗Vβ. This\n(i.e. via quantum groups) is the only known solution to the explicit decom-\nposition problem even at q = 1. This may give some idea of the power of\nthe quantum group machinery.\n20.3\nTowards nonstandard quantum groups for the\nKronecker and plethysm problems\nNow the goal is to construct quantum groups which can be used to de-\nrive PH1 and explicit decomposition for the Kronecker and plethysm prob-\nlems just as the standard quantum group can be used for the same in the\nLittlewood-Richardson problem.\nIn the Kronecker problem, we let H = GL(Cn) and G = GL(Cn ⊗Cn).\nThe Kronecker coefficient κγ\nα,β is the multiplicity of Vα(H)⊗Vβ(H) in Vγ(G):\nVγ(G) =\nM\nα,β\nκγ\nα,βVα(H) ⊗Vβ(H).\n113"},{"page":115,"text":"The goal is to get a positive # P-formula for κγ\nα,β; this is the gist of PH1\nfor the Kronecker problem.\nIn the plethysm problem, we let H = GL(Cn) and G = GL(Vμ(H)).\nThe plethysm constant aπ\nλ,μ is the multiplicity of Vπ(H) in Vλ(G):\nVλ(G) =\nM\nπ\naπ\nλ,μVπ(H).\nAgain, the goal is to get a positive # P-formula for the plethysm constant;\nthis is the gist of PH1 for the plethysm problem.\nTo apply the quantum group approach, we need a q-analogue of the\nembedding H ֒→G. Unfortunately, there is no such q-analogue in the theory\nof standard quantum groups. Because there is no nontrivial quantum group\nhomomorphism from the standard quantum group Hq = GLq(Cn) and to\nthe standard quantum group Gq.\nTheorem 20.3. (1) [GCT4]: Let H and G be as in the Kronecker problem.\nThen there exists a quantum group ˆGq such that the homomorphism H →G\ncan be quantized in the form Hq ֒→ˆGq. Furthermore, ˆGq has a unitary\nquantum subgroup ˆUq which corresponds to the maximal unitary subgroup\nU ⊆G, and a q-analogue of the Peter-Weyl theorem holds for ˆGq. The\nlatter implies that every finite dimensional representation of ˆGq is completely\ndecomposible into irreducibles.\n(2) [GCT7] There is an analogous (possibly singular) quantum group ˆGq\nwhen H and G are as in the plethysm problem. This also holds for general\nconnected reductive (classical) H.\nSince the Kronecker problem is a special case of the (generalized) plethysm\nproblem, the quantum group in GCT 4 is a special case of the quantum group\nin GCT 7. The quantum group in the plethysm problem can be singular, i.e.,\nits determinant can vanish and hence the antipode need not exist. We still\ncall it a quantum group because its properties are very similar to those of the\nstandard quantum group; e.g. q-analogue of the Peter-Weyl theorem, which\nallows q-harmonic analysis on these groups. We call the quantum group\nˆGq nonstandard, because though it is qualitatively similar to the standard\n(Drinfeld-Jimbo) quantum group Gq, it is also, as expected, fundamentally\ndifferent.\nThe article [GCT8] gives a conjecturally correct algorithm to construct a\ncanonical basis of an irreducible polynomial representation of ˆGq which gen-\neralizes the canonical basis for a polynomial representation of the standard\nquantum group as per Kashiwara and Lusztig. It also gives a conjecturally\n114"},{"page":116,"text":"correct algorithm to construct a canonical basis of a certain q-deformation of\nthe symmetric group algebra C[Sr] which generalizes the Kazhdan-Lusztig\nbasis [KL] of the Hecke algebra (a standard q-deformation of C[Sr]). It is\nshown in [GCT7, GCT8] that PH1 for the Kronecker and plethysm problems\nfollows assuming that these canonical bases in the nonstandard setting have\nproperties akin to the ones in the standard setting. For a discussion on SH,\nsee [GCT6].\n115"},{"page":117,"text":"Part II\nInvariant theory with a view\ntowards GCT\nBy Milind Sohoni\n116"},{"page":118,"text":"Chapter 21\nFinite Groups\nReferences: [FH, N]\n21.1\nGeneralities\nLet V be a vector space over C, and let GL(V ) denote the group of all\nisomorphisms on V . For a fixed basis of V , GL(V ) is isomorphic to the\ngroup GLn(C), the group of all n × n invertible matrices.\nLet G be a group and ρ : G →GL(V ) be a representation. We also\ndenote this by the tuple (ρ, V ) or say that V is a G-module. Let Z ⊆V be\na subspace such that ρ(g)(Z) ⊆Z for all g ∈G. Then, we say that Z is\nan invariant subspace. We say that (ρ, V ) is irreducible if there is no\nproper subspace W ⊂V such that ρ(g)(W) ⊆W for all g ∈G. We say that\n(ρ, V ) is indecomposable is there is no expression V = W1 ⊕W2 such that\nρ(g)(Wi) ⊆Wi, for all g ∈G.\nFor a point v ∈V , the orbit O(v), and the stabilizer Stab(v) are\ndefined as:\nO(v)\n=\n{v′ ∈V |∃g ∈G with ρ(g)(v) = v′}\nStab(v)\n=\n{g ∈G|ρ(g)(v) = v}\nOne may also define v ∼v′ if there is a g ∈G such that ρ(g)(v) = v′. It is\nthen easy to show that [v]∼= O(v).\nLet V ∗be the dual-space of V . The representation (ρ, V ) induces the\ndual representation (ρ∗, V ∗) defined as ρ∗(v∗)(v) = v∗(ρ(g−1)(v)). It will\nbe convenient for ρ∗to act on the right, i.e., ((v∗)(ρ∗))(v) = v∗(ρ(g−1)(v)).\nWhen ρ is fixed, we abbrieviate ρ(g)(v) as just g · v. Along with this,\nthere are the standard constructions of the tensor T d(V ), the symmetric\npower Symd(V ) and the alternating power ∧d(V ) representations.\n117"},{"page":119,"text":"Of special significance is Symd(V ∗), the space of homogeneous polyno-\nmial functions on V of degree d. Let dim(V ) = n and let X1, . . . , Xn be a\nbasis of V ∗. We define as follows:\nR = C[X1, . . . , Xn] = ⊕∞\nd=0Rd = ⊕∞\nd=0Symd(V ∗)\nThus R is the ring of all polynomial functions on V and is isomorphic\nto the algebra (over C) of n indeterminates. Since G acts on the domain V ,\nG also acts on all functions f : V →C as follows:\n(f · g)(v) = f(g−1 · v)\nThis action of G on all functions extends the action of G on polynomial\nfunctions above.\nIndeed, for any g ∈G, the map tg : R →R given by\nf →f · g is an algebra isomorphism. This is called the translation map.\nFor an f ∈R, we say that f is an invariant if f · g = f for all g ∈G.\nThe following are equivalent:\n• f ∈R is an invariant.\n• Stab(f) = G.\n• f(g · v) = f(v) for all g ∈G and v ∈V .\n• For all v, v′ such that v′ ∈Orbit(v), we have f(v) = f(v′).\nIf W1 and W2 are two modules of G and φ : W1 →W2 is a linear map\nsuch that g · φ(w1) = φ(g · w1) for all g ∈G and w1 ∈W1 then we say that\nφ is G-equivariant or that φ is a morphism of G-modules.\n21.2\nThe finite group action\nLet G be a finite group and (μ, W) be a representation.\nRecall that a complex inner product on W is a map h : W × W →C\nsuch that:\n• h(αw + βw′, w′′) = αh(w, w′′) + βh(w′, w′′) for all α, β ∈C and all\nw, w′, w′′ ∈W.\n• h(w′′, αw + βw′) = αh(w′′, w) + βh(w′′, w′) for all α, β ∈C and all\nw, w′, w′′ ∈W.\n• h(w, w) > 0 for all w ̸= 0.\n118"},{"page":120,"text":"Also recall that if Z ⊆W is a subspace, then Z⊥is defined as:\nZ⊥= {w ∈W|h(w, z) = 0 ∀z ∈Z}\nAlso recall that W = Z ⊕Z⊥.\nWe say that an inner product h is G-invariant if h(g·w, g·w′) = h(w, w′)\nfor all w, w′ ∈W and g ∈G.\nProposition 21.1. Let W be as above, and Z be an invariant subspace of\nW. Then Z⊥is also an invariant subspace. Thus every reducible represen-\ntation of G is also decomposable.\nProof: Let x ∈Z⊥, z ∈Z and let us examine (g · x, z). Applying g−1 to\nboth sides, we see that:\nh(g · x, z) = h(g−1 · g · x, g−1 · z) = h(x, g−1 · z) = 0\nThus, G preserves Z⊥as claimed. □\nLet h be a complex inner product on W. We define the inner product\nhG as follows:\nhG(w, w′) =\n1\n|G|\nX\ng′∈G\nh(g′ · w, g′ · w′)\nLemma 21.1. hG is a G-invariant inner product.\nProof: First we see that\nhG(w, w) =\n1\n|G|\nX\ng′∈G\nh(w′, w′)\nwhere w′ = g′ · w. Thus hG(w, w) > 0 unless w = 0. Secondly, by the\nlinearity of the action of G, we see that hG is indeed an inner product.\nFinally, we see that:\nhG(g · w, g · w′) =\n1\n|G|\nX\ng′∈G\nh(g′ · g · w, g′ · g · w′)\nSince as g′ ranges over G, so will g′ · g for any fixed g, we have that hG is\nG-invariant. □\nTheorem 1.\n• Let G be a finite group and (ρ, V ) be an indecomposable\nrepresentation, then it is also irreducible.\n119"},{"page":121,"text":"• Every representation (ρ, V ) may be decomposed into irreducible repre-\nsentations Vi. Thus V = ⊕iVi, where (ρi, Vi) is an irreducible repre-\nsentation.\nProof: Suppose that Z ⊆V is an invariant subspace, then V = Z ⊕Z⊥is\na non-trivial decomposition of V contradicting the hypothesis. The second\npart is proved by applying the first, recursively. □\nWe have seen the operation of averaging over the group in going\nfrom the inner product h to the G-invariant inner product hG. A similar\napproach may be used for constructing invariant polynomials functions. So\nlet p(X) ∈R = C[X1, . . . , Xn] be a polynomial function. We define the\nfunction pG : V →C as:\npG(v) =\n1\n|G|\nX\ng∈G\np(g · v)\nThe transition from p to pG is called the Reynold’s operator.\nProposition 21.2. Let p ∈R be of degree atmost d, then pG is also a\npolynomial function of degree atmost d. Next, pG is an invariant.\nLet RG denote the set of all invariant polynomial functions on the space\nV . It is easy to see that RG ⊆R is actually a subring of R.\nLet Z ⊆V be an arbitrary subset of V . We say that Z is G-closed if\ng ·z ∈Z for all g ∈G and z ∈Z. Thus Z is a union of orbits of points in V .\nLemma 21.2. Let p ∈RG be an invariant and let Z = V (p) be the variety\nof p. Then Z is G-closed.\nWe have already seen that O(v), the orbit of v arises from the equivalence\nclass ∼on V . Since the group is finite, |O(v)| ≤|G| for any v. Let O1 and\nO2 be disjoint orbits.\nIt is essential to determine if elements of RG can\nseparate O1 and O2.\nLemma 21.3. Let O1 and O2 be as above, and I1 and I2 be their ideals in\nR. Then there are p1 ∈I1 and p2 ∈I2 so that p1 + p2 = 1.\nProof: This follows from the Hilbert Nullstellensatz. Since the point sets\nare finite, there is an explicit construction based on Lagrange interpolation.\n□\nLet G be a finite group and (ρ, V ) be a representation as above. We\nhave see that this induces an action on C[X1, . . . , Xn]. Also note that this\naction is homogeneous: for a g ∈G and p ∈Rd, we have that p · g ∈Rd\n120"},{"page":122,"text":"as well. Thus RG, the ring of invariants, is a homogeneous subring of R. In\nother words:\nRG = ⊕∞\nd=0RG\nd\nwhere RG\nd are invariants which are homogeneous of degree d. The existence\nof the above decomposition implies that every invariant is a sum of homo-\ngeneous invariants. Now, since RG\nd ⊆Rd as a vector space over C. Thus\ndimC(RG\nd ) ≤dimC(Rd) ≤\n n + d −1\nn −1\n \nWe define the hilbert function h(RG) of RG (or for that matter, of\nany homogeneous ring) as:\nh(RG) =\n∞\nX\nd=0\ndimC(RG\nd )zd\nWe will see now that h(RG) is actually a rational function which is easily\ncomputed. We need a lemma.\nLet (ρ, W) be a representation of the finite group G. Let\nW G = {w ∈W|g · w = w}\nbe the set of all vector invariants in W, and this is a subspace of W. .\nLemma 21.4. Let (ρ, W) be as above. We have:\ndimC(W G) =\n1\n|G|\nX\ng∈G\ntrace(ρ(g))\nProof: Define P =\n1\n|G|\nP\ng∈G ρ(g), as the average of the representation\nmatrices. We see that ρ(g) · P = P · ρ(g) and that P 2 = P. Thus P is\ndiagonalizable and the eigenvalues of P are in the set {1, 0}. Let W1 and\nW0 be the corresponding eigen-spaces. It is clear that W G ⊆W1 and that\nW1 is fixed by each g ∈G. We now argue that every w ∈W1 is actually an\ninvariant. For that, let wg = g · w. We then have that Pw = w implies that\nw =\n1\n|G|\nX\ng∈G\nwg\nNote that a change-of-basis does not affect the hypothesis nor the assertion.\nWe may thus assume that each ρ(g) is unitary, we have that wg = w for all\ng ∈G. Now, the claim follows by computing trace(P). □\nWe are now ready to state Molien’s Theorem:\n121"},{"page":123,"text":"Theorem 2. Let (ρ, W) be as above. We have:\nh(RG) =\n1\n|G|\nX\ng∈G\n1\ndet(I −zρ(g))\nProof: Let dimC(W) = n and let {X1, . . . , Xn} be a basis of W ∗. Since\nRG = P\nd RG\nd and each RG\nd ⊆C[X1, . . . , Xn]d. Note that each C[X1, . . . , Xn]d\nis also a representation ρd of G.\nFurthermore, it is easy to see that if\n{λ1, . . . , λn} are the eigenvalues of ρ(g), then the eigen-values of the matrix\nρd(g) are precisely (including multiplicity)\n{\nY\ni\nλdi\ni |\nX\ni\ndi = d}\nThus\ntrace(ρd(g)) =\nX\nd:|d|=d\nY\ni\nλdi\ni\nWe then have:\nh(RG)\n=\nP\nd zddimC(RG\nd )\n=\nP\nd zd[ 1\n|G|\nP\ng trace(ρd(g))]\n=\n1\n|G|\nP\ng\n1\n(1−λ1(g)z)...(1−λn(g)z)\n=\n1\n|G|\nP\ng\n1\ndet(I−zρ(g))\nThis proves the theorem. □\n21.3\nThe Symmetric Group\nSn will denote the symmetric group of all bijections on the set [n]. The\nstandard representation of Sn is obviously on V = Cn with\nσ · (v1, . . . , vn) = (vσ(1), . . . , vσ(n))\nThus, regarding V as column vectors, and Sn as the group of n × n-\npermutation matrices, we see that the action of permutation P on vector v\nis given by the matrix multiplication P · v.\nLet X1, . . . , Xn be a basis of V ∗.\nSn acts on R = C[X1, . . . , Xn] by\nXi · σ = Xσ(i). The orbit of any point v = (v1, . . . , vn) is the collection of\nall permutation of the entries of the vector v and thus the size of the orbit\nis bounded by n!.\n122"},{"page":124,"text":"The invariants for this action are given by the elementary symmetric\npolynomials ek(X), for k = 1, . . . , n, where\nek(X) =\nX\ni1<i2<...<ik\nXi1Xi2 . . . Xik\nGiven two vector v and w, if w ̸∈O(v), then there is a k such that\nek(v) ̸= ek(w). This follows from the theory of equations in one variable.\nThe ring RG equals C[e1, . . . , en] and has no algebraic dependencies. The\nhilbert function of RG may then be expressed as:\nh(RG) =\n1\n(1 −z)(1 −z2) . . . (1 −zn)\nIt is an exercise to verify that Molien’s expression agrees with the above.\nA related action of Sn is the diagonal action: Let X = {X1, . . . , Xn},\nY = {Y1, . . . , Yn}, and so on upto W = {W1, . . . , Wn} be a family of r\n(disjoint) variables. Let B = C[X, Y, . . . , W] be the ring of polynomials in\nthe variables of the disjoint union.\nWe define the action of Sn on X ∪Y ∪. . . ∪W as Xi · σ = Xσ(i),\nYi · σ = Yσ(i), and so on.\nThe matrix equivalence of this action is the action of the permutation\nmatrices on n × r matrices A, where the action of P on A is given by P · A.\nThe invariants BG is obtained from the r = 1 case by a curious operation:\nLet DXY denote the operator:\nDXY = Y1\n∂\n∂X1\n+ . . . + Yn\n∂\n∂Xn\nThe ring BG is obtained from RG by applying the operators DXY , DXW , DW X\nand so on, to elements of RG. As an example, we have\ne2(X) = X1X2 + X1X3 + . . . + Xn−1Xn\nWe have DXY (e2) as:\nDXY (e2(X)) =\nX\ni̸=j\nXiYj\nThis is clearly an element of BG.\n123"},{"page":125,"text":"Chapter 22\nThe Group SLn\nReferences: [FH, N]\n22.1\nThe Canonical Representation\nLet V be a vector space of dimension n, and let x1, . . . , xn be a basis for V .\nLet X1, . . . , Xn be the dual basis of V ∗.\nSL(V ) will denote the group of all unimodular linear transformations on\nV . In the above basis, this group is isomorphic to that of all n × n matrices\nof determinant 1, or in other words SLn(C).\nThe standard representation of SL(V ) is obviously V itself: Given φ ∈\nSL(V ) and v ∈V , we have φ · v = φ(v) is the action of φ on v.\nIn terms of the basis x above we may write v = [x1, . . . , xn][α1, . . . , αn]T\nand thus φ·v as [φ·x1, . . . , φ·xn][α1, . . . , αn]T . If φ·xi = [x1, . . . , xn][a1i, . . . , ani]T ,\nthen we have\nφ · v = [x1, . . . , xn]\n \n \na11\n. . .\na1n\n...\n...\nan1\n. . .\nann\n \n \n \n \nα1\n...\nαn\n \n \nWe denote this matrix as Aφ.\nWe may now work with SLn(C) or simply SLn. Given a vector a =\n[α1, . . . , αn]T , we see that the matrix multiplication A · a is the action of A\non the column vector a.\nLet us now understand the orbits of typical elements in the column space\nCn. The typical v ∈Cn is a non-zero column vector. For any non-zero vector\nw, we see that there is an element A ∈SLn such that w = Av. Furthermore,\n124"},{"page":126,"text":"for any B ∈SLn, clearly Bv ̸= 0. Thus we see that Cn has exactly two\norbits:\nO0 = {0} O1{v ∈Cn|v ̸= 0}\nNote that O1 is dense in V = Cn and its closure includes the orbit O0 and\ntherefore, the whole of V .\nLet R = C[X1, . . . Xn] be the ring of polynomial functions on Cn. We\nexamine the action of SLn on C[X]. Recall that the action of A on X should\nbe such that the evaluations Xi(xj) = δij must be conserved. Thus if the\ncolumn vectors xi = [0, . . . , 0, 1, 0 . . . , 0]T are the basis vectors of V and the\nrow vectors Xi = [0, . . . , 0, 1, 0 . . . , 0] that of V ∗, then a matrix A ∈SLn\ntransforms xi to Axi and Xj to XjA−1. Thus Xj(xi) = Xj/cdotxi goes to\nXjA−1Axi = Xj · xi.\nNext, we examine R = C[X] for invariants. First note that the action of\nSLn is homogeneous and thus we may assume that an invariant p is actually\nhomogeneous of degree d. Next we see that p must be constant on O1 and\nO0. In this case, if p(O1) = α then by the density of O1 in V , we see that p\nis actually constant on V . Thus RG = R0 = C.\n22.2\nThe Diagonal Representation\nLet us now consider the diagonal representation of the above representation.\nIn other words, let V r be the space of all complex n × r matrices x. The\naction of A on x is obvious A · x. Let X be the r × n-matrix dual to x. It\ntransforms according to X →XA−1.\nLet us examine the case when r < n and compute the orbits in V r. Let\nx ∈V r be a matrix and y be a column vector such that x · y = 0. We see\nthat (A · x) · y = 0 as well. Thus if ann(x) = {y ∈Cr|x · y = 0} is the\nannihilator space of x, then we see that ann(x) = ann(A · x).\nWe show that the converse is also true:\nProposition 22.1. Let r < n and x, x′ ∈V r be such that ann(x) = ann(x′).\nThen there is an A ∈SLn such that x = A · x′.\nProof: Use the row-echelon form construction. Make the pivots as 1 using\nappropriate diagonal matrices. Since r < n these can be chosen from SLn.\n□\nThis decides the orbit structure of V r for r < n. The largest orbit Or is\nof course when ann(x) = 0. Thus, when x ∈Vr is a mtrix of rank r, we see\nthat ann(x) = 0 is trivial. The generic element of V r is of this form. Thus\nOr is dense in V r. For this reason, there are no non-constant invariants.\n125"},{"page":127,"text":"Another calculation is the computation of the closure of orbits. Let O, O′\nbe two arbitrary orbits of V r. We have:\nProposition 22.2. O′ lies in the closure of O if and only if ann(O′) ⊇\nann(O).\nProof: One direction is clear. We prove the other direction when ann(O) ⊆\nann(O′) and dim(ann(O)) = dim(ann(O′)) −1.\nThus, we may assume\nthat O = [x] and O′ = [x′] with both x and x′ such that rowspace(x) ⊇\nrowspace(x′) with rank(x′) = rank(x)−1. Then, upto SLn, we may assume\nthat the rows x′[1], . . . , x′[k] match the first k rows of x, and that x′[k+1] =\nx′[k + 2] = . . . = x′[n] = 0. Note that x[k + 1] is non-zero and x[k + 2] exists\nand is zero. We then construct the matrix A(t) as follows:\nA(t) =\n \n \nIk\n0\n0\n0\n0\nt\n0\n0\n0\n0\nt−1\n0\n0\n0\n0\nIn−k−2\n \n \nWe see that A(t) ∈SLn for all t ̸= 0. Next, if we let x(t) = A(t) · x, then\nwe see that:\nlim\nt→0 x(t) = x′\nThis shows that x′ lies in the closure of O. □\nWe now look at the case when r ≥n. We have:\nProposition 22.3. Let r ≥n and x, x′ ∈V r be such that ann(x) = ann(x′).\nIf (i) rank(x) < n then there is an A ∈SLn such that x′ = Ax, (ii) if\nrank(x) = n there is a unique A ∈SLn and a λ ∈C∗such that if z = A · x,\nthen the first n −1 rows of z equal those of x′ and z[n] = λx′[n].\nThe proof is easy. Let x be matrix in V r of rank n, and C be a subset\nof [r] = {1, 2, . . . , r} such that det(x[C]) ̸= 0.\nProposition 22.4. Let x be as above. Then O(x), the orbit of x, equals\nall points x′ ∈V r such that (i) ann(x) = ann(x′) and (ii) det(X′[C]) =\ndet(x[C]). The set of all rank n points in V r is dense in V r.\nThe proof is easy.\nProposition 22.5. The orbit O(x) as above is closed.\n126"},{"page":128,"text":"Proof: Notice that if A ∈SLn and z = Ax then det(x[C]) = det(z[C]).\nWe may rewrite condition (i) above as ann(x′) ⊇ann(x). Condition (ii)\nensures that rank(x′) = n and that condition (i) holds with equality. Thus\nis y1, . . . , yr−n are column vectors generating ann(x), then the equations\nx′ys = 0 and det(x′[C]) = det(x[C]) determines the orbit O(x). Thus O(x)\nis the zero-set of some algebraic equations and thus is closed. □\nProposition 22.6. Let x ∈V r be such that rank(x) < n. Then (i) O(x)\nequals those x′ such that ann(x) = ann(x′), (ii) O(x) is not closed and its\nclosure O(x) equals points x′ such that ann(x′) ⊇ann(x).\nWe now move to the computation of invariants. The space C[V r] equals\nthe space of all polynomials in the variable matrix X = (Xij) where i =\n1, . . . , n and j = 1, . . . , r. It is clear that for any set C of n columns of X,\nwe see that det(X[C]) is an invariant. We will denote C as C = c1 < c2 <\n. . . < cn and det(X[C]) as pC, the C-th Plucker coordinate. We aim to\nshow now that these are the only invariants.\nLet C0 = {1 < 2 < . . . < n} and W ⊆V r be the space of all matrices\nx ∈V r such that x[C0] = diag(1, . . . , 1, λ). Let W ′ be those elements of W\nfor which λ ̸= 0.\nLemma 22.1. Let W ′ be as above. (i) If x ∈W ′, then O(x) ∩W ′ = x, (ii)\nfor any x ∈V r such that det(x[C0]) ̸= 0, there is a unique A ∈SLn such\nthat Ax ∈W ′.\nLet us call Z′ ⊆V r as those x such that det(x[C0]) ̸= 0. We then have\nthe projection map\nπ : Z′ →W ′\ngiven by the above lemma. Note that Z′ is SLn-invariant: if x ∈Z′ and\nA ∈SLn then A · x ∈Z′ as well.\nThe ring C[Z′] of regular functions on Z′ is precisely C[X]det([X[C0]), the\nlocalization of C[X] at det(X[C0]).\nWe may parametrize W ′ as:\nW =\n \n \n1\n. . .\n0\nw1,n+1\n. . .\nw1,r\n...\n...\n...\n...\n0\n. . .\nwn,n\nwn,n+1\n. . .\nwn,r\n \n \nLet\nW = {Wi,j|i = 1, . . . , n, j = n + 1, . . . , r} ∪{Wn,n}\nbe a set of (n −r) · r + 1 variables. The ring C[W ′] of regular function on\nW ′ is precisely C[W]Wn,n.\n127"},{"page":129,"text":"Let x ∈Z′ be an arbitrary point and A = x[C0]. Let Ci,j be the set\nC0 −i + j. The map π is given by:\nπ(x)n,n\n=\ndet(A)\nπ(x)i,j\n=\n det(x[Ci,j])/det(A)\nfor i ̸= n\ndet(x[Ci,j])\notherwise\nThe map π causes the map:\nπ∗: C[W ′] →C[Z′]\ngiven by:\nπ∗(Wn,n)\n=\ndet(X[C0])\nπ∗(Wi,j)\n=\n det(X[Ci,j])/det(X[C0])\nfor i ̸= n\ndet(X[Ci,j])\notherwise\nNow we note that Z′ is dense in V r. Let p ∈C[X]SLn be an invariant.\nClearly p restricted to W ′ defines an element pW ′ of C[W ′].\nThis then\nextends to Z′ via π∗. Clearly π∗(pW ′) must match p on Z′. Thus we have\nthat p is a polynomial in det(X[Ci,j]) possibly localized at det(X[C0]).\nNote that for a general C, det(X[C]) is already expressible in det(X[C0])\nand det(X[Ci,j]).\n22.3\nOther Representations\nWe discuss two other representations:\nThe Conjugate Action: Let M be the space of all n × n matrices with\ncomplex entries. we define the action of A ∈SLn on M as follows. For an\nM ∈M, A acts on it by conjugation.\nA · M = AMA−1\nNote that dimC(M) = n2. Let X = {Xij|1 ≤i, j ≤n} be the dual space\nto M. The invariants for this action are Tr(X), . . . , Tr(Xi), . . . , Tr(Xn).\nThat these are invariants ic clear, for Tr(AMiA−1) = Tr(M) for any A, M.\nAlso note that Tr(X) = X11 + . . . + Xnn is linear in the X’s.\nThis\nindicates an invariant hyperplane in M.\nThis is precisely the trace zero\nmatrices. Thus M = M0 ⊕M1 where M0 are all matrices M such that\nTr(M) = 0. The one-dimensional complementary space M1 is composed of\nmultiples of the identity matrix.\n128"},{"page":130,"text":"The orbits are parametrized by the Jordan canonical form JCF(M).\nIn other words, if JCF(M) = JCF(M′) them M′ ∈O(M). Furthermore,\nif JCF(M) is diagonal then the orbit of M is closed in M.\nThe Space of Forms: Let V be a complex vector space of dimension n\nandV ∗its dual. Let X1, . . . , Xn be the dual basis. The space Symd(V ∗)\nconsists of degree d forms are formal linear combinations of the monomials\nXi1\n1 . . . Xin\nn with i1 + . . . + in = d. The typical form may be written as:\nf(X) =\nX\ni1+...+in=d\nBi1...inXi1\n1 . . . Xin\nn\nLet A ∈SLn be such that\nA−1 =\n \n \na11\n. . .\nan1\n...\n...\na1n\n. . .\nann\n \n \nThe space Symd(V ∗) is the formal linear combination of the generaic coef-\nficients {Bi||i| = d}. The action of A is obtained by substituting\nXi →\nX\nj\naijXi\nin f(X) and recoomputing the coefficients. we illustrate this for n = 2 and\nd = 2. Then, the generic form is given by:\nf(X1, X2) = B20X2\n1 + B11X1\n1X1\n2 + B02X2\n2\nUpon substitution, we get f(a11X1 + a12X2, a21X1 + a22X2): Thus the new\ncoefficients are:\nB20\n→\nB20a2\n11+\nB11a11a21+\nB02a2\n21\nB11\n→\nB202a11a12+\nB11(a11a22 + a12a21)+\nB022a21a22\nB02\n→\nB20a2\n12+\nB11a12a22+\nB02a2\n22\nWe thus see that the variables B move linearly, with coefficients as homo-\ngeneous polynomials of degree 2 in the coefficients of the group elements. In\nthe above case, we know that the discriminant B2\n11−4B20B02 is an invariant,\nThese spaces have been the subject of intense analysis and their study by\nGordan, Hilbert and other heralded the beginning of commutative algebra\nand invariant theory.\n129"},{"page":131,"text":"22.4\nFull Reducibility\nLet W be a representation of SLn and let Z ⊆W be an invariant subspace.\nThe reducibility question is whether there exists a complement Z′ such that\nW = Z ⊕Z′ and Z′ is SLn-invariant as well.\nThe above result is indeed true although we will not prove it here. There\nare many proofs known, each with a specific objective in a specific situation,\nand each extremely instructive.\nThe simplest is possibly through the Weyl Unitary trick.\nIn this, a\nsuitable compact subgroup U ⊆SLn is chosen. The theory of compact\ngroups is much like that of finite groups and a complement Z′ may easily\nbe found. It is then shown that Z′ is SLn-invariant.\nThe second attack is through showing the fill reducibility of the module\n⊗dV , the d-th tensor product representation of V . This goes through the\nconstruction of the commutator of the same module regarded as a right\nSd-module, with the symmetric group permutating the contents of the d\npositions. The full reducibility then follows from Maschke’s theorem and\nthat of the finite group Sd.\nThe oldest approach was through the construction of a symbolic reynold’s\noperator, which is the Cayley Ω-process. Let C[W] be the ring of polyno-\nmial functions on the space W. The operator Ωis a map:\nΩ: C[W] →C[W]SLn\nsuch that if p is an invariant then Ω(p · p′) = p · Ω(p′). The definition of Ω\nis nothing but curious.\n130"},{"page":132,"text":"Chapter 23\nInvariant Theory\nReferences: [FH, N]\n23.1\nAlgebraic Groups and affine actions\nAn algebraic group (over C) is an affine variety G equipped with (i) the\ngroup product, i.e., a morphism of algebraic varieties · : G×G →G which is\nassociative, i.e. (g1·g2)·g3 = g1·(g2·g3) for all g1, g2, g3 ∈G, (ii) and algebraic\ninverse i : G →G, i.e, g · i(g) = i(g) · g = 1G, where 1G is (iii) a special\nelement 1G ∈G, which functions as the identity, i.e., 1G · g = g · 1G = g for\nall g ∈G. Let C[G] be the ring of regular functions on G. The requirements\n(i)-(iii) above are via morphisms of algebraic varities, and thus the group\nproduct and the inverse must be defined algebraically. Thus the product\n· : G × G →G results in a morphism of C-algebras ·∗: C[G] →C[G] ⊗C[G]\nand the inverse into another morphism i∗: C[G] →C[G].\nThe essential example is obviously SLn. Clearly for G = SLn, we have\nC[G] = C[X]/(det(X)−1), where X is the n×n indeterminate matrix. The\nmorphism ·∗and i∗are clearly:\n·∗(Xij)\n=\nP\nk Xik ⊗Xkj\ni∗(Xij)\n=\ndet(Mji)\nwhere Mji is the corresponding minor of X.\nNext, let Z be another affine variety with C[Z] as its ring of regular\nfunctions. We say that Z is a G-variety if there is a morphism μ : G×Z →Z\nwhich is a group action. Thus, not only must G act on Z, it must do so\nalgebraically. This μ induces the map μ∗:\nμ∗: C[Z] →C[G] ⊗C[Z]\n131"},{"page":133,"text":"Thus every function f ∈C[Z] goes to a finite sum:\nμ∗(f) =\nk\nX\ni=1\nhi ⊗fi\nwhere fi ∈C[Z] and hi ∈C[G] for all i.\nTo continue with our example, consider SL2 and Z = Symd(V ∗). C[Z] =\nC[B20, B11, B02] and C[G] = C[A11, A12, A21, A22]/(A11A22 −A21A12 −1).\nWe have for example:\nμ∗(B20) = A2\n11 ⊗B20 + A11A21 ⊗B11 + A2\n21 ⊗B02\nLet μ : G × Z →Z and μ′ : G × Z′ →Z′ be two G-varities and let\nφ : Z →Z′ be a morphism. We say that φ is G-equivariant if φ(μ(g, z)) =\nμ′(g, φ(z)) for all g ∈G and z ∈Z. Thus φ commutes with the action of G.\n23.2\nOrbits and Invariants\nEvery g ∈G induces an algebraic bijection on Z by restricting the map\nμ : G × Z →Z to g. We denote this map by μ(g) and call it translation\nby g. The map:\nμ(g) : Z →Z\ninduces the isomorphism of C-algebras:\nμ(g)∗: C[Z] →C[Z]\nGiven any function f ∈C[Z], μ(g)∗(f) ∈C[Z] is the translated function\nand denotes the action of G on f. This makes C[Z] into a G-module. If\nφ : Z →Z′ is a G-equivariant map, then the map φ∗: C[Z′] →C[Z] is also\nG-equivariant, for the G action on C[Z] and C[Z′] as above.\nWe next examine the equation:\nμ∗(f) =\nk\nX\ni=1\nhi ⊗fi\nwhere fi ∈C[Z] and hi ∈C[G] for all i. For a fixed g, we see that\nμ(g)∗(f) =\nk\nX\ni=1\nhi(g) ⊗fi\n132"},{"page":134,"text":"Thus we see that every translate of f lies in the k-dimensional vector space\nC · {f1, . . . , fk}. Let\nM(f) = C · {μ(g)∗(f)|g ∈G} ⊆C · {f1, . . . , fk}\nClearly, M(f) is a G-invariant subspace of C[Z]. This may be generalized:\nProposition 23.1. Let S = {s1, . . . , sm} be a finite subset of C[Z]. Then\nthere is a finite-dimensional G-invariant subspace M(S) of C[Z] containing\ns1, . . . , sm.\nNext, let us consider μ : G × Z →Z and fix a z ∈Z. We get the map\nμz : G →X. Since G is an affine variety, we see that the image μz(G) is a\nconstructible set, whose closure is an affine variety. The image is precisely\nthe orbit O(z) ⊆Z. The closure of O(z) will be denoted by ∆(z).\nIf O(z) = ∆(z), then we have the G-equivariant embedding iz : O(z) →\nZ. This gives us the map i∗\nz : C[Z] →C[O(z)] which is a surjection. Thus\nthere is an ideal I(z) = ker(i∗\nz) such that C[O(z)] ∼= C[Z]/I(z). Since the\nmap iz is G-equivariant, I(z) is a G-submodule of C[Z] and is the ideal of\ndefinition of the orbit O(z).\nIn general, if I ⊆C[Z] is an ideal which is G-invariant, then the variety\nof I is also G-invariant and is the union of orbits.\nThe second construction that we make is that of the quotient Z/G.\nSince C[Z] is a G-module, we examine C[Z]G, the subring of G-invariant\nfunctions in C[Z]. We define Z/G as the spectrum Spec(C[Z]G). The inclu-\nsion C[Z]G →C[Z] gives us the quotient map:\nπ : Z →Z/G\nExercise 23.1. Let us consider Z = Sym2(V ∗) ∼= C3 where V is the stan-\ndard representation of G = SL2. As we have seen, C[Z] = C[B20, B11, B02].\nThere is only one invariant δ = B2\n11 −4B02B20. Thus C[Z]G = C[δ]. Thus\nZ/G is precisely Spec(C[δ]) = C, the complex plane. The map π is executed\nas follows: given a form aX2\n1 + bX1X2 + cX2\n2 ≡(a, b, c) ∈Z, we eveluate\nthe invariant δ at the point (a, b, c). Thus π : C3 →C is given by:\nπ(a, b, c) = b2 −4ac\nClearly, if f, f ′ ∈Z such that f = g · f ′ for g ∈SL2 then δ(f) = δ(f ′). We\nlook at the converse: if f, f ′ are such that δ(f) = δ(f ′) then is it that f ∈\nO(f ′)? We begin with f = aX2\n1 +bX1X2 +cX2\n2. We assume for the moment\nthat a ̸= 0. If that is the case, we make the substitution X1 →X1 −αX2\n133"},{"page":135,"text":"and X2 →X2. Note that this transformation is unimodular for all α. This\ntransforms f to:\na(X1 −αX2)2 + b(X1 −αX2)X2 + cX2\n2\nThe coefficient of X1X2 is −2aα+b. Thus by choosing α as b/2a we see that\nf is transformed into a′′X2\n1 −c′′X2\n2 for some a′′, c′′. By a similar token, even\nif a = 0 one may do a similar transform. Thus in general, if f is not the\nzero form, there is a point in O(f) which is of the form aX2\n1 −cX2\n2. Thus,\nwe may assume that both f and f ′ are in this form. We can simplify the\nform further by a diagonal element of SL2 to put both f and f ′ as X2\n1 −cX2\n2\nand X2\n1 −c′X2\n2. It is now clear that δ(f) = δ(f ′) implies that c = c′. Thus\nthe general answer is that if f, f ′ ̸= 0 and δ(f) = δ(f ′) then f ∈O(f ′).\nNext, let us examine the form 0. we see that δ(0) = 0. Thus we see that\n(i) for any point d ∈C, if d ̸= 0, then π−1(d) consists of a single orbit, (ii)\nπ−1(0) consists of two orbits, O(0) and O(X2\n1), the perfect square, (iii) the\norbits O(f) are closed when δ(f) ̸= 0, (iv) O(X2\n1) is not closed. Its closure\nincludes the closed orbit 0.\nThe above example illustrates the utility of contructing the quotient Z/G\nas a variety parametrizing closed orbits albeit with some deviant points. In\nthe example above, the discrepancy was at the point 0, wherein the pre-\nimage is closed but decomposes into two orbits.\nWe state the all-important theorem linking a space and its quotient in\nthe restricted case when G is finite and Z is a finite-dimensional G-module.\nThus C[Z] is a polynomial ring and the action of G is homogeneous.\nTheorem 3. Let G be a finite group and act on the space Z. Let R = C[Z]\nand RG = C[Z]G be the ring of invariants. Let π : Z →Z/G be the quotient\nmap. Then\n(i) For any ideal J ⊆RG, we have (J · R) ∩RG = J.\n(ii) The map π is surjective. Further, for any x ∈Z/G, π−1(x) is a single\norbit in Z.\nProof: Let f1, . . . , fk be elements of RG and f ∈RG be such that:\nf = r1f1 + . . . + rkfk\nwhere ri are elements of R. Since G is finite, we apply the Reynolds operator\n134"},{"page":136,"text":"p →pG. In other words, we have:\nf\n=\n1\n|G|\nX\ng∈G\nf\n=\n1\n|G|\nX\ng∈G\nX\ni\nrifi\n=\n1\n|G|\nX\ni\nfi\nX\ng∈G\nri\n=\nX\ni\nfi\n1\n|G|\nX\ng∈G\nri\n=\nX\ni\nfirG\ni\nNote that we have used the fact that if h ∈RG and p ∈R then (h · p)G =\nh · pG. Thus we see that any element f of RG which is expressible as an\nR-linear combination of elements fi’s of RG is already expressible as an\nRG-linear combination of the same elements. This proves (i) above.\nNow we prove (ii). Firstly, let J be a maximal ideal of RG. By part (i)\nabove, J · R is a proper ideal of R and thus π is surjective. Let x ∈Z/G\nand Jx ⊆RG be the maximal ideal for the point x. Let z be such that\nπ(z) = x and let IO(x) ⊆R be the ideal of all functions in R vanishing at\nall points of the orbit O(z) of z. We show that rad(Jx · R) = IO(z) which\nproves (ii). Towards that, it is clear that (a) JxR ⊆IO(z) and (b) the variety\nof JxR is G-invariant. Let O(z′) is another orbit in the variety of Jx · R.\nIf O(z′) ̸= O(z) then we already have that there is a p ∈RG such that\np(O(z)) = 0 and p(O(z′)) = 1. Since this p ∈Jx · R, the variety of Jx · R\nexcludes the orbit O(z′) proving our claim. □\nTheorem 4. Let Z be a finite-dimesional G-module for a finite group G.\nThen C[Z]G, the ring of G-invariants, is finitely generated as a C-algebra.\nProof: Since the action of G is homogeneous, every invariant f ∈C[Z]G is\nthe sum of homogeneous invariants. Let IG ⊆C[Z] be the ideal generated\nby the positive degree invariants.\nBy the Hilbert Basis theorem, IG =\n(f1, . . . , fk)·C[Z], where each fi is itself an invariant. we claim that C[Z]G =\nC[f1, . . . , fk] ⊆C[Z], or in other words, every invariant is a polynomial over\nC in the terms f1, . . . , fk. To this end, let f be a homogeneous invariant.\nWe prove this by induction over the degree d of f. Since f ∈IG, we have\nf =\nX\ni\nrifi where for all i, ri ∈C[Z]\n135"},{"page":137,"text":"By applying the reynolds operator, we have:\nf =\nX\ni\nhifi where for all i, hi ∈C[Z]G\nNow since each hi has degree less than d, by our induction hypothesis, each\nis an element of C[f1, . . . , fk]. This then shows that f too is an element of\nC[f1, . . . , fk]. □\n23.3\nThe Nagata Hypothesis\nWe now generalize the above two theorems to the case of more general\ngroups. This generlization is possible if the group G satisfies what we call\nthe Nagata Hypothesis.\nDefinition 23.1. Let G be an algebraic group. We say that G satisfies the\nNagata hypothesis if for every finite-dimensional module M over C and a\nG-invariant hyperplane N ⊆M, there is a decomposition M = H ⊕P as\nG-modules, where H is a hyperplane and P is an invariant (i.e., P is the\ntrivial representation.\nThe group SLn satisfies the hypothesis, and so do the so-called reductive\ngroups over C.\nTheorem 5. Let G satisfy the Nagata hypothesis and let Z be an affine\nG-variety. The ring C[Z]G is finitely generated as a C-algebra.\nThe proof will go through several steps.\nLet f ∈C[Z] be an arbitrary element. We define two modules M(f) and\nN(f).\nM(f)\n=\nC · {s · f|s ∈G}\nN(f)\n=\nC · {s · f −t · f|s, t ∈G}\nThus M(f) ⊇N(f) are finite-dimensional submodules of C[Z].\nLemma 23.1. Let f ∈C[Z] be an arbitrary element. Then there exists an\nf ∗∈C[Z]G ∩M(f) such that f −f ∗∈N(f).\nProof: We prove this by induction over dim(M(f)).\nNote that s · f =\nf+(s·f−1·f) and thus M(f) = N(f)⊕C·f as vector spaces. Thus N(f) is a\nG-invariant hyperplane of M(f). By the Nagata hypothesis, M(f) = f ′⊕H,\nwhere f ′ is an invariant. If f ′ ̸∈N(f) then M(f) = f ′⊕N(f) and the lemma\nfollows. However, if f ′ ∈N(f), then f = f ′ + h where h ∈H. Since H is\n136"},{"page":138,"text":"a G-module, we see that M(h) ⊆H which is of dimension less than that of\nM(f). Thus there is an invariant h∗such that h −h∗∈N(h). Note that\nf −f ′ = h with f ′ invariant, implies that s · h −t · h = s · f −t · f. Thus\nN(h) ⊆N(f). We take f ∗= h∗. Examining f −f ∗, we see that\nf −h∗= f ′ + h −h∗∈N(f)\nThis proves the lemma. □\nLemma 23.2. Let f1, . . . , fr ∈C[Z]G.\nThen C[Z]G ∩(P\ni C[Z] · fi) =\nP\ni C[Z]G · fi. Thus if an invariant f is a C[Z]linear combination of invari-\nants, then it is already a C[Z]G linear combination.\nProof: This is proved by induction on r. Say f = Pr\ni=1 hifi with hi ∈C[Z].\nApplying the above lemma to hr, there is an h′′ ∈C[Z]G and an h′ ∈N(hk)\nsuch that hr = h′ + h′′. We tackle h′ as follows. Since f is an invariant, we\nhave for all s, t ∈G:\nr\nX\ni=1\n(s · hi −t · hi)fi = s · f −t · f = 0\nHence:\n(s · hr −t · hr)fr =\nr−1\nX\ni=1\n(s · hi −t · hi)fi\nIt follows from this that h′fr = Pr−1\ni=1 h′\nifi for some h′\ni ∈C[Z]. Substituting\nthis in the expression\nf =\nr−1\nX\ni=1\nhifi + (h′ + h′′)fr\nwe get:\nf −h′′fr =\nr−1\nX\ni=1\n(hi + h′\ni)fi\nThus the invariant f −h′′fr is C[Z]-linear combination of r −1 invariants,\nand thus the induction hypothesis applies. This then results in an expression\nfor f as a C[Z]G-linear combination for f. □\nThe above lemma proves part (i) of Theorem 3 for groups G for which\nthe Nagata hypothesis holds. The route to Theorem 4 is now a straight-\nforward adaptation of its proof for finite groups. In the case when Z is a\nG-module, C[Z]G would then be homogeneous and the proof of Theorem 4\nholds. For general Z here is a trick which converts it to the homogeneous\ncase:\n137"},{"page":139,"text":"Proposition 23.2. Let Z be an affine G-variety. Then there is a G-module\nW and an equivariant embedding φ : Z →W.\nProof: Since C[Z] is a finitely generated C-algebra, C[Z] = C[f1, . . . , fk]\nwhere f1, . . . , fk are some specific elements of C[Z]. By expanding the list\nof generators, we may assume that the vector space C · {f1, . . . , fk} ⊆C[Z]\nis a finite-dimensional G-module.\nLet us construct W as an isomorphic\ncopy of this module with the dual basis W1, . . . , Wk. We construct the map\nφ∗: C[W] →C[Z] by defining φ(Wi) = fi. Since C[W] is a free algebra, we\nindeed have the surjection:\nφ∗: C[W] →C[Z]\nThis proves the proposition. □\nWe are now ready to prove:\nTheorem 6. Let G satisfy the Nagata hypothesis. Let Z be an affine G-\nvariety with coordinate ring C[Z]. Then C[Z]G is finitely generated as a\nC-algebra.\nProof: We construct the equivariant surjection φ∗: C[W] →C[Z]. Note\nthat C[W]G is already finitely generated over C, say C[W]G = C[h1, . . . , hk].\nWe claim that φ∗(h1), . . . , φ∗(hk) generate C[Z]G.\nNow, let f ∈C[Z]G. By the surjectivity of φ∗, there is an h ∈C[W] such\nthat φ∗(h) = f. Consider the space N(h). A typical generator of N(h) is\ns · h −t · h. Applying φ∗to this, we see that:\nφ∗(s · h −t · h)\n=\ns · φ∗(h) −t · φ∗(h)\n=\nf −f = 0\nBy an earlier lemma there is an invariant h∗such that h −h∗∈N(h). thus\napplying φ∗we see that φ∗(h∗) = φ(h) = f. Thus there is an invariant h∗\nsuch that φ∗(h∗) = f. Now h∗∈C[h1, . . . , hk] implies that f = φ∗(h∗) ∈\nC[φ∗(h1), . . . , φ∗(hk)]. □\n138"},{"page":140,"text":"Chapter 24\nOrbit-closures\nReference: [Ke, N]\nIn this chapter we will analyse the validity of Theorem 3 for general\ngroups G with the Nagata hypothesis. The objective is to analyse the map\nπ : Z →Z/G.\nProposition 24.1. Let G satisfy the Nagata hypothesis and act on the affine\nvariety Z. Then the map π : Z →Z/G is surjective.\nProof: Let J ⊆C[Z]G be a maximal ideal. By lemma 23.2 J · C[Z] is a\nproper ideal of C[Z]. This implies that π is surjective. □\nTheorem 7. Let G satisfy the Nagata hypothesis and act on the affine\nvariety Z. Let W1 and W2 be G-invariant (Zariski-)closed subsets of Z such\nthat W1 ∩W2 is empty. Then there is an invariant f ∈C[Z]G such that\nf(W1) = 0 and f(W2) = 1.\nProof: Let I1 and I2 be the ideals of W1 and W2 in C[Z]. Since their inter-\nsection is empty, by Hilbert Nullstellensatz, we have I1 + I2 = 1. Whence\nthere are functions f1 ∈I1 and f2 ∈I2 such that f1 + f2 = 1. For arbitrary\ns, t ∈G we have:\n(s · f1 −t · f1) + (s · f2 −t · f2) = 1 −1 = 0\nNote that M(fi) and N(fi) are submodules of the G-invariant ideal Ii. Now\napplying lemma 23.1 to f1, we see that there are elements f ′\ni ∈N(fi) such\nthat f1 + f ′\n1 ∈C[Z]G. Whence we see that:\n(f1 + f ′\n1) + (f2 + f ′\n2) = 1\n139"},{"page":141,"text":"where f = f1 + f ′\n1 is an invariant. Since f1 + f ′\n1 ∈I1 we see that f(W1) = 0.\nOn the other hand, f = 1 −(f2 + f ′\n2) ∈1 + I2 and thus f(W2) = 1. □\nRecall that ∆(z) denotes the closure of the orbit O(z) in Z.\nTheorem 8. Let G satisfy the Nagata hypothesis and act on an affine va-\nriety Z. We define the relation ≈on Z as follows: z1 ≈z2 if and only if\n∆(z1) ∩∆(z2) is non-empty. Then\n(i) ≈is an equivalence relation.\n(ii) z1 ≈z2 ifff(z1) = f(z2) for all f ∈C[Z]G.\n(iii) Within each ∆(z) there is a unique closed orbit, and this is of minimum\ndimension among all orbits in ∆(z).\nProof: It is clear that (ii) proves (i). Towards (ii), if ∆(z1)∩∆(z2) is empty,\nthen by Theorem 7, there is an invariant separating the two points. Thus it\nremains to show that if ∆(z10 ∩∆(z2) is non-empty, and f is any invariant,\nthen f(z1) = f(z2). So let z be an element of the intersection. Since f is\nan invariant, we have f(O(z1)) is a constant, say α. Since O(z1) ⊆∆(z1) is\ndense, and f is continuous, we have f(z) = α. Thus f(z1) = f(z2) = f(z) =\nα.\nNow (iii) is easy. Clearly ∆(z) cannot have two closed distinct orbits,\nfor otherwise they woulbe separated by an invariant. But this must take the\nsame value on all points of ∆(z). That it is of minimum dimension follows\nalgebraic arguments. □\nDefinition 24.1. Let Z be a G-variety and z ∈Z. We say that z is stable\nif the orbit O(z) ⊆Z is closed in Z.\nBy the above theorem, every point x of Spec(C[Z]G) corresponds to\nexactly one stable point: the point whose orbit is of minimum dimension in\nπ−1(x).\nExercise 24.1. Consider the action of G = SLnon M, the space of n × n-\nmatrices by conjugation. Thus, given A ∈SLn and M ∈M, we have:\nA · M = AMA−1\nLet R = C[M] = C[X11, . . . , Xnn] be the ring of functions on M.\nThe\ninvariants RG is generated as a C-algebra by the forms ei(X) = Tr(Xi), for\ni = 1, . . . , n. The forms {ei|1 ≤i ≤n} are algebraically independent and\n140"},{"page":142,"text":"thus RG is the polynomial ring C[e1, . . . , en]. Clearly then Spec(RG) ∼= Cn\nand we have:\nπ : M ∼= Cn2 →Cn\nGiven a matrix M with eigenvalues {λ1, . . . , λn}, we have:\nei(M) = λi\n1 + . . . + λi\nn\nThus, by the fundamental theorem of algebra, the image π(M) determines\nthe set {λ1, . . . , λn} (with multiplicities).\nOn the other hand, given any\ntuple μ = (μ1, . . . , μn) there is a unique set λμ = {λ1, . . . , λn} such that\nP\nr λi\nr = μi. Clearly, for the diagonal matrix D(λμ) = diag(λ1, . . . , λn), we\nhave that π(D(λ)) = μ. This verifies that π is surjective.\nFor a given μ, the set π−1(μ) are all matrices M with Spec(M) = λ =\nλμ. By the Jordan canonical form (JCF), this set may be stratified by the\nvarious Jordan canonical blocks of spectrum λ. If λ has no multiplicities\nthen π−1(μ) consists of just one orbit: matrices M such that JCF(M) =\nD(λ). For a general λ, the orbit of M with JCF(M) = D(λ) is the unique\nclosed orbit of minimum dimension. All other orbits contain this orbit in its\nclosure. Thus stable points M ∈M are the diagonalizable matrices.\nAs an example, consider the case when n = 2 and the matrix:\nN =\n λ\n1\n0\nλ\n \nConsider the family A(t) = diag(t, t−1) ∈SL2. We see that:\nN(t) = A(t)NA(t)−1 =\n λ\nt2\n0\nλ\n \nThus limt→0 N(t) = diag(λ, λ), the diagonal matrix.\nThus, we see that the invariant ring C[Z]G puts a different equivalence\nrelation ≈on points in Z which is coarser than ∼=, the orbit equivalence\nrelation. The relation ≈is more ‘topological’ than group-theoretic and cor-\nrectly classifies orbits by their separability by invariants. The special case\nof Theorem 8 when Z is a representation was analysed by Hilbert in 1893.\nThe point 0 ∈Z is then the smallest closed orbit, and the equivalence class\n[0]≈is termed as the null-cone of Z. We see that the null-cone consists\nof all points z ∈Z such that 0 lies in the orbit-closure ∆(z) of z. It was\nHilbert who discovered that if 0 ∈∆(z) then 0 lies in the orbit-closure of a 1-\nparameter diagonal subgroup of SLn. To understand the intricay of Hilbert’s\nconstructions, it is essential that we understand diagonal subgroups of SLn.\n141"},{"page":143,"text":"Chapter 25\nTori in SLn\nReference: [Ke, N]\nLet C∗denote the multiplicative group of non-zero complex numbers. A\ntorus is the abstract group (C∗)m for some m. Note that C∗is an abelian\nalgebraic group with C[G] = C[T, T −1]. Furthermore, C∗has a compact\nsubroup S1 = {z ∈C, |z| = 1}, the unit circle.\nNext, let us look at representations of tori. For C∗, the simplest repre-\nsentations are indexed by integers k ∈Z. So let k ∈Z. The representation\nC[k] corresponds to the 1-dimensional vector space C with the action:\nt · z = tkz\nThus a non-zero t ∈C∗acts on z by multiplication of the k-th power. Next,\nfor (C∗)m, let χ = (χ[1], . . . , χ[m]) be a sequence of integers. For such a χ,\nwe define the representation Cχ as follows: Let t = (t1, . . . , tm) ∈(C∗)m be\na general element and z ∈C. The action is given by:\nt · z = tχ[1]\n1\n. . . tχ[m]\nm\nz\nSuch a χ is called a character of (C∗)m.\nThese 1-dimensional representations of tori are crucial in the analysis of\nalgebraic group actions.\nLet us begin by understanding the structure of algebraic homomorphisms\nfrom C∗to SLn(C). So let\nλ : C∗→SLn(C)\nbe such a map such that λ(t) = [aij(t)] where aij(T) ∈C[T, T −1].\nAn\nimportant substitution is for t = eiθ, and we obtain a 2π-periodic map\nλ : C →SLn\n142"},{"page":144,"text":"We see that λ(0) = I. Let the derivative at 0 for λ be X.\nWe have the following general lemma:\nLemma 25.1. Let f : C →SLn be a smooth map such that f(0) = I and\nf ′(0) = X, where X is an n × n-matrix. Then for θ ∈C,\nlim\nk→∞\n \nf\n θ\nk\n k\n= eθX\nThe proof follows from the local diffeomorphism of the exponential map\nin the neighborhood of the identity matrix.\nApplying this lemma to λ we see that eθX is in the image of λ for all θ.\nNow, since λ is 2π-periodic, we must have e(n2π+θ)X = eθX. This forces (i)\nX to be diagonalizable, and (ii) with integer eigenvalues. This proves:\nProposition 25.1. Let λ : C∗→SL(V ) be an algebraic homomorphism.\nThen the image of λ is closed and V ∼= C[m1] ⊕. . .⊕C[mn], for some integers\nm1, . . . , mn, where n = dimC(V ).\nBased on this, we have the generalization:\nProposition 25.2. Let λ : (C∗)r →SL(V ) be an algebraic homomorphism.\nThen the image of λ is closed and V ∼= Cχ1 ⊕. . . ⊕Cχn, for some integers\nχ1, . . . , χn, where n = dimC(V ).\nThus, in effect, for every homomorphism λ : (C∗)r →SLn, there is a\nfixed invertible matrix A such that for all t ∈(C∗)r, the conjugate Aλ(t)A−1\nis diagonal.\nA torus in SLn is defined as an abstract subgroup H of SLn which is\nisomorphic to (C∗)r for some r. The standard maximal torus D of SLn\nis the diagonal matrices diag(t1, . . . , tn) where ti ∈C∗and t1t2 . . . tn = 1.\nThis clears the way for the important theorem:\nTheorem 9.\n(i) Every torus is contained in a maximal torus. All maxi-\nmal tori in SLn are isomorphic to (C∗)n−1.\n(ii) If T and T ′ are two maximal tori then there is an A ∈SLn such that\nT ′ = ATA−1. Thus all maximal tori are conjugate to D above.\n(iii) Let N(D) be the normalizer of D and N(D)o be the connected compo-\nnent of N(D). Then N(D)o = D and N(D)/D is the Weyl group\nW, isomorphic to the symmetric group Sn.\n143"},{"page":145,"text":"Definition 25.1. Let G be an algebraic group. Γ(G) will denote the col-\nlection of all 1-parameter subgroups of G, i.e., morphisms λ : C∗→G.\nX(G) will denote the collection of all characters of G, i.e., homomorphisms\nχ : G →C∗.\nWe consider the case when G = (C∗)r. Clearly, for a given λ : C∗→G,\nthere are integers m1, . . . , mr ∈Z such that:\nλ(t) = (tm1, . . . , tmr)\nIn the same vein, for the character χ : G →C∗, we have integers a1, . . . , ar\nsuch that\nχ(t1, . . . , tr) =\nY\ni\ntai\ni\nWe also have the composition λ ◦χ : C∗→C∗, where by:\nλ ◦χ(t) = tm1a1+...+mrar\nConsolidating all this, we have:\nTheorem 10. Let G = (C∗)r. Then Γ(G) ∼= Zr and X(G) ∼= Zr. Further-\nmore, there is the pairing\n< , >: Γ(G) × X(G) →Z\nwhich is a unimodular pairing on lattices.\nExercise 25.1. Let G = (C∗)3 and λ and χ be as follows:\nλ(t)\n=\n(t3, t−1, t2)\nχ(t1, t2, t3)\n=\nt−1\n1 t2t2\n3\nThen, λ ∼= [3, −1, 2] and χ ∼= [−1, 1, 2]. We evaluate the pairing:\n< λ, χ >= 3 · −1 + (−1) · 1 + 2 · 2 = 0\nWe now turn to the special case of D ⊆SLn, the maximal torus which\nis isomorphic to (C∗)n−1.\nBy the above theorem, Γ(D), X(D) ∼= Zn−1.\nHowever, it will more convenient to identify this space as a subset of Zn. So\nlet:\nYn = {[m1, . . . , mn] ∈Zn|m1 + . . . + mn = 0}\nIt is easy to see that Yn ∼= Zn−1. In fact, we will set up a special bijection\nθ : Yn →Zn−1 defined as:\nθ([m1, m2, . . . , mn]) = [m1, m1 + m2, . . . , m1 + . . . + mn−1]\n144"},{"page":146,"text":"The inverse θ−1 is also easily computed:\nθ−1[a1, . . . , an−1] = [a1, a2 −a1, a3 −a2, . . . , an−1 −an−2, −an−1]\nThis θ corresponds to the Z-basis of Yn consisting of the vectors e1 −\ne2, . . . , en−1 −en where ei is the standard basis of Zn. This is also equivalent\nto the embedding θ∗: (C∗)n−1 →D as follows:\n(t1, . . . , tn−1) →\n \n \nt1\n0\n. . .\n0\n0\nt−1\n1 t2\n0\n. . .\n0\n...\n0\n. . .\n0\nt−1\nn−2tn−1\n0\n0\n. . .\n0\nt−1\nn−1\n \n \nA useful computation is to consider the inclusion D ⊆D∗, where D∗⊆GLn\nis subgroup of all diagonal matrices. Clearly Γ(D) ⊆Γ(D∗), however there\nis a surjection X(D∗) →X(D). It will be useful to work out this surjection\nexplicitly via θ and θ∗. If [m1, . . . .mn] ∈Zn ∼= X(D∗), then it maps to\n[m1 −m2, . . . , mn−1 −mn] ∈Zn−1 ∼= X((C∗)n−1) via θ∗. If we push this\nback into Yn via θ−1, we get:\n[m1, . . . , mn] →[m1−m2, 2m2−m1−m3, , . . . , 2mn−1−mn−2−mn, mn−mn−1]\nWe are now ready to define the weight spaces of an SLn-module W.\nSo let W be such a module.\nBy restricting this module to D ⊆G, via\nProposition 25.2, we see that W is a direct sum W = Cχ1 ⊕. . . ⊕CχN,\nwhere N = dimC(W). Collecting identical characters, we see that:\nW = ⊕χ∈X(D)Cmχ\nχ\nThus W is a sum of mχ copies of the module Cχ. Clearly mχ = 0 for all but\na finite number, and is called the multiplicity of χ. For a given module\nW, computing mχ is an intricate combinatorial exercise and is given by the\ncelebrated Weyl Character Formula.\nExercise 25.2. Let us look at SL3 and the weight-spaces for some modules\nof SL3.\nAll modules that we discuss will also be GL3-modules and thus\nD∗modules. The formula for converting D∗-modules to D-modules will be\nuseful. This map is Z3 →Y3 and is given by:\n[m1, m2, m3] →[m1 −m2, 2m2 −m1 −m3, m3 −m2]\n145"},{"page":147,"text":"The simplest SLn module is C3 with the basis {X1, X2, X3} with D∗\nweights [1, 0, 0], [0, 1, 0] and [0, 0, 1].\nThis converted to D-weights give us\n{[1, −1, 0], [−1, 2, −1], [0, −1, 1]}, with C[1,−1,0] ∼= C · X1 and so on.\nThe next module is Sym2(C3) with the basis X2\ni and XiXj. The six D∗\nand D-weights with the weight-spaces are given below:\nD∗-wieghts\nD-weights\nweight-space\n[2, 0, 0]\n[2, −2, 0]\nX2\n1\n[0, 2, 0]\n[−2, 4, −2]\nX2\n2\n[0, 0, 2]\n[0, −2, 2]\nX2\n3\n[0, 1, 1]\n[−1, 1, 0]\nX2X3\n[1, 0, 1]\n[1, −2, 1]\nX1X3\n[1, 1, 0]\n[0, 1, −1]\nX1X2\nThe final example is the space of 3× 3-matrices M acted upon by conju-\ngation. We see at once that M = M0 ⊕C·I where M0 is the 8-dimensional\nspace of trace-zero matrices, and C · I is 1-dimensional space of multiples\nof the idenity matrix. Weight vectors are Eij, with 1 ≤i, j ≤3. The D∗\nweights are [1, −1, 0], [1, 0, −1], [0, 1, −1], [−1, 0, 1], [0, −1, 1], [−1, 1, 0] and [0, 0, 0].\nThe multiplicity of [0, 0, 0] in M is 3 and in M0 is 2. Note that Eii ̸∈M0.\nThe D-weights are [2, −3, 1], [1, 0, −1], [−1, 3, −2] and its negatives, and ob-\nviously [0, 0, 0].\nThe normalizer N(D) gives us an action of N(D) on the weight spaces.\nIf w is a weight-vector of weight χ, t ∈D and g ∈N(D), then g · w is also\na weight vector. Afterall t · (g · w) = g · t′ · w where t′ = g−1tg. Thus\nt · (g · w) = χ(t′)(g · w)\nwhence g · w must also be a weight-vector with some weight χ′. This χ′ is\neasily computed via the action of D∗. Here the action of N(D∗) is clear: if\nχ = [m1, . . . , mn], then χ′ = [mσ(1), . . . , mσ(n)] for some permutation σ ∈Sn\ndetermined by the component of N(D∗) containing g. Thus the map χ to\nχ′ for D-weights in the case of SL3 is as follows:\n[m1−m2, 2m2−m1−m3, m3−m2] →[mσ(1)−mσ(2), 2mσ(2)−mσ(1)−mσ(3), mσ(3)−mσ(2)]\nCaution: Note that though Y3 ⊆Z3 is an S3-invariant subset, the action\nof S3 on χ ∈Y3 is different. Note that, e.g., in the last example above,\n[2, −3, 1] is a weight but not the ‘permuted’ vector [−3, 2, 1]. This is because\nof our peculiar embedding of Zn−1 →Yn.\n146"},{"page":148,"text":"Chapter 26\nThe Null-cone and the\nDestabilizing flag\nReference: [Ke, N]\nThe fundamental result of Hilbert states:\nTheorem 11. Let W be an SLn-module, and let w ∈W be an element of\nthe null-cone. Then there is a 1-parameter subgroup λ : C∗→SLn such that\nlim\nt→0 λ(t) · w = 0W\nIn other words, if the zero-vector 0W lies in the orbit-closure of w, then\nthere is a 1-parameter subgroup taking it there, in the limit. We will not\nprove this statement here. Our objective for this chapter is to interpret the\ngeometric content of the theorem. We will show that there is a standard\nform for an element of the null-cone. For well-known representations, this\nstandard form is easily identified by geometric concepts.\n26.1\nCharacters and the half-space criterion\nTo begin, let D be the fixed maximal torus. For any w ∈W, we may express:\nw = w1 + w2 + . . . + wr\nwhere wi ∈Wχi, the weight-space for character χi. Note the the above ex-\npression is unique if we insist that each wi be non-zero. The set of characters\n{χ1, . . . , χr} will be called the support of w and denoted as supp(w). Let\n147"},{"page":149,"text":"λ : C∗→SLn be such that Im(λ) ⊆D. In this case, the action of t ∈C∗\nvia λ is easily described:\nt · w = t(λ,χ1)w1 + . . . + t(λ,χr)wr\nThus, if limt→0 t · w exists (and is 0W ), then for all χ ∈supp(w), we have\n(λ, χ) ≥0 (and further (λ, χ) > 0).\nNote that (λ, χ) is implemented as a linear functional on Yn. Thus, if\nlimt→0 t · w exists (and is )W ) then there is a hyperplane in Yn such that\nthe support of w is on one side of the hyperplane (strictly on one side of\nthe hyperplane). The normal to this hyperplane is given by the conversion\nof λ into Yn notation.\nOn the other hand if the support supp(w) enjoys the geometric/combinatorial\nproperty, then by the approximability of reals by rationals, we see that there\nis a λ such that limt→0 t · w exists (and is zero).\nThus for 1-parameter subgroups of D, Hilbert’s theorem translates into\na combinatorial statement on the lattice subset supp(w) ⊂Yn.\nWe call\nthis the (strict) half-space property. In the general case, we know that\ngiven any λ : C∗→SLn, there is a maximal torus T containing Im(λ). By\nthe conjugacy result on maximal tori, we know that T = ADA−1 for some\nA ∈SLn. Thus, we may say that w is in the null-cone iffthere is a translate\nA · w such that supp(A · w) satisfies the strict half-space property.\nExercise 26.1. Let us consider SL3 acting of the space of forms of degree\n2. For the standard torus D, the weight-spaces are C · X2\ni and C · XiXj.\nConsider the form f = (X1 + X2 + X3)2. We see that supp(f) is set of all\ncharacters of Sym2(C3) and does not satisfy the combinatorial property.\nHowever, under a basis change A:\nX1\n→\nX1 + X2 + X3\nX2\n→\nX2\nX3\n→\nX3\nwe see that A·f = X2\n1. Thus A·f does satisfy the strict half-space property.\nIndeed consider the λ\nλ(t) =\n \n \nt\n0\n0\n0\nt−1\n0\n0\n0\n1\n \n \nWe see that\nlim\nt→0 t · (A · f) = t2X2\n1 = 0\n148"},{"page":150,"text":"Thus we see that every form in the null-cone has a standard form with a\nvery limited sets of possible supports.\nLet us look at the module M of 3 × 3-matrices under conjugation. Let\nus fix a λ:\nλ(t) =\n \n \ntn1\n0\n0\n0\ntn2\n0\n0\n0\ntn3\n \n \nsuch that n1 + n2 + n3 = 0. We may assume that n1 ≥n2 ≥n3. Looking at\nthe action of λ(t) on a general matrix X, we see that:\nt · X = (tni−njxij)\nThus if limt→0 t · X is to be 0 then xij = 0 for all i > j. In other words,\nX is strictly upper-triangular. Considering the general 1-parameter group\ntells us that X is in the null-cone iffthere is an A such that AXA−1 is\nstrictly upper-triangular. In other words, X is nilpotent. The 1-parameter\nsubgroup identifies this transformation and thus the flag of nilpotency.\n26.2\nThe destabilizing flag\nIn this section we do a more refined analysis of elements of the null-cone. The\nbasic motivation is to identify a unique set of 1-parameter subgroups which\ndrive a null-point to zero. The simplest example is given by X2\n1 ∈Sym2(C3).\nLet λ, λ′ and λ′′ be as below:\nλ(t) =\n \n \nt\n0\n0\n0\nt−1\n0\n0\n0\n1\n \n \nλ′(t) =\n \n \nt\n0\n0\n0\n1\n0\n0\n0\nt−1\n \n \nλ′′(t) =\n \n \nt\n0\n0\n0\n0\n−1\n0\nt−1\n0\n \n \nWe see that all the three λ, λ′ and λ′′ drive X2\n1 to zero.\nThe question\nis whether these are related, and to classify such 1-parameter subgroups.\nAlternately, one may view this to a more refined classification of points in\nthe null-cone, such as the stratification of the nilpotent matrices by their\nJordan canonical form.\nThere are two aspects to this analysis. Firstly, to identify a metric by\nwhich to choose the ’best’ 1-parameter subgroup driving a null-point to\nzero. Next, to show that there is a unique equivalence class of such ’best’\nsubgroups.\n149"},{"page":151,"text":"Towards the first objective, let λ : C∗→SLn be a 1-parameter subgroup.\nWithout loss of generality, we may assume that Im(λ) ⊆D. If w is a null-\npoint then we have:\nt · w = tn1w1 + . . . + tnkwk\nwhere ni > 0 for all i. Clearly, a measure of how fast λ drives w to zero\nis m(λ) = min{n1, . . . , nk}. Verify that this really does not depend on the\nchoice of the maximal torus at all, and thus is well-defined.\nNext, we see that for a λ as above, we consider λ2 : C∗→SLn such that\nλ2(t) = λ(t2). It is easy to see that m(λ2) = 2 · m(λ). Clearly, λ and λ2 are\nintrinsically identical and we would like to have a measure invariant under\nsuch scaling. This comes about by associating a length to each λ. Let λ be\nas above and let Im(λ) ⊆D. Then, there are integers a1, . . . , an such that\nλ(t) =\n \n \nta1\n0\n0\n0\n0\nta2\n0\n0\n...\n0\n0\n0\ntan\n \n \nWe define ∥λ∥as\n∥λ∥=\nq\na2\n1 + . . . + a2n\nWe must show that this does not depend on the choice of the maximal\ntorus D. Let T (SLn) denote the collection of all maximal tori of SLn as\nabstract subgroups. For every A ∈SLn, we may define the map φA : T →T\ndefined by T →ATA−1. The stabilizer of a torus T for this action of SLn\nis clearly N(T), the normalizer of T.\nAlso recall that N(T)/T = W is\nthe (discrete) weyl group.\nLet Im(λ) ⊆D ∩D′ for some two maximal\ntori D and D′.\nSince there is an A such that AD′A−1 = D, it is clear\nthat ∥λ∥= ∥AλA−1∥. Thus, we are left to check if ∥λ′∥= ∥λ∥when (i)\nIm(λ), Im(λ′) ⊆D, and (ii) λ′ = AλA−1 for some A ∈SLn. This throws\nthe question to invariance of ∥λ∥under N(D), or in other words, symmetry\nunder the weyl group.\nSince W ∼= Sn, the symmetric group, and since\np\na2\n1 + . . . + a2n is a symmetric function on a1, . . . , an, we have that ∥λ∥is\nwell defined.\nWe now define the efficiency of λ on a null-point w to be\ne(λ) = m(λ)\n∥λ∥\nWe immediately see that e(λ) = e(λ2).\n150"},{"page":152,"text":"Lemma 26.1. Let W be a representation of SLn and let w ∈W be a null-\npoint. Let N(w, D) be the collection of all λ : C∗→D such that limt→0 t ·\nw = 0W . If N(w, D) is non-empty then there is a unique λ′ ∈N(w, D)\nwhich maximizes the efficiency, i.e., e(λ′) > e(λ) for all λ ∈N(w, D) and\nλ ̸= (λ′)k for any k ∈Z. This 1-parameter subgroup will be denoted by\nλ(w, D).\nProof: Suppose that N(w, D) is non-empty.\nThen in the weight-space\nexpansion of w for the maximal torus D, we see that supp(w) staisfies the\nhalf-space property for some λ ∈Yn.\nNote that the λ ∈N(w, D) are\nparametrized by lattice points λ ∈Yn such that (λ, χ) > 0 for all χ ∈\nsupp(w).\nLet Cone(w) be the conical combination (over R) of all χ ∈\nsupp(w) and Cone(w)◦its polar. Thus, in other words, N(w, D) is precisely\nthe collection of lattice points in the cone Cone(w)◦. Next, we see that e(λ)\nis a convex function of Cone(w)◦which is constant over rays R+ · λ for all\nλ ∈Cone(w)◦. By a routine analysis, the maximum of such a function must\nbe a unique ray with rational entries. This proves the lamma. □\nThis covers one important part in our task of identifying the ’best’ 1-\nparameter subgroup driving a null-point to zero. The next part is to relate\nD to other maximal tori.\nLet λ : C∗→SLn and let P(λ) be defined as follows:\nP(λ) = {A ∈SLn| lim\nt→0 λ(t)Aλ(t−1) = I ∈SLn}\nHaving fixed a maximal torus D containing IM(λ), we easily identify\nP(λ) as a parabolic subgroup, i.e., block upper-triangular. Indeed, let\nλ(t) =\n \n \nta1\n0\n0\n0\n0\nta2\n0\n0\n...\n0\n0\n0\ntan\n \n \nwith a1 ≥a2 ≥. . . ≥an (obviously with a1 + . . . + an = 0). Then\nP(λ) = {(xij|xij = 0 for all i, j such that ai < aj}\nThe unipotent radical U(λ) is a normal subgroup of P(λ) defined as:\nU(λ) = (xij) where =\n \nxij = 0\nif ai < aj\nxij = δij\nif ai = aj\n151"},{"page":153,"text":"Lemma 26.2. Let λ ∈N(w, D) and let g ∈P(λ), then (i) gλg−1 ⊆\nP(λ) and P(gλg−1) = P(λ), (ii) gλg−1 ∈N(w, gDg−1), and (iii) e(λ) =\ne(gλg−1).\nThis actually follows from the construction of the explicit SLn-modules\nand is left to the reader. We now come to the unique object that we will\ndefine for each w ∈W in the null-cone. This is the parabolic subgroup P(λ)\nfor any ’best’ λ. We have already seen above that if λ′ is a P(λ)-conjugate\nof a best λ then λ′ is ’equally best’ and P(λ) = P(λ′).\nWe now relate two general equally best λ and λ′. For this we need a\npreliminary definition and a lemma:\nDefinition 26.1. Let V be a vector space over C.\nA flag F of V is a\nsequence (V0, . . . , Vr) of nested subspaces 0 = V0 ⊂V1 ⊂. . . ⊂Vr = V .\nLemma 26.3. Let dimC(V ) = r and let F = (V0, . . . , Vr) and F′ =\n(V ′\n0, . . . , V ′\nr) be two (complete) flags for V . Then there is a basis b1, . . . , br\nof V and a permutation σ ∈Sr such that Vi = {b1, . . . , bi} and V ′\ni =\n{bσ(1), . . . , bσ(i)} for all i.\nThis is proved by induction on r.\nCorollary 26.1. Let λ and λ′ be two 1-parameter subgroups and P(λ) and\nP(λ′) be their corresponding parabolic subgroups. Then there is a maximal\ntorus T of SLn such that T ⊆P(λ) ∩P(λ′).\nProof: It is clear that there is a correspondence between parabolic sub-\ngroups of SLn and flags. We refine the flags associated to the parabolic\nsubgroups P(λ) and P(λ′) to complete flags and apply the above lemma. □\nWe are now prepared to prove Kempf’d theorem:\nTheorem 12. Let W be a representation of SLn and w ∈W a null-point.\nThen there is a 1-parameter subgroup λ ∈Γ(SLn) such that (i) for all λ′ ∈\nΓ(SLn), we have e(λ) ≥e(λ′), and (ii) for all λ′ such that e(λ) = e(λ′) we\nhave P(λ) = P(λ′) and that there is a g ∈P(λ) such that λ′ = gλg−1.\nProof: Let N(w) be all elements of Γ(SLn) which drive w to zero. Let Ξ(W)\nbe the (finite) collection of D-characters appearing in the representation W.\nFor every λ(w, T) such that Im(λ) ⊆D, we may consider an A ∈SLn such\nthat AλA−1 ∈N(A · w, D) and e(λ) = e(AλA−1). Since the ’best’ element\nof N(A · w, D) is determined by supp(A · w) ⊆Ξ, we see that there are only\nfinitely many possibilities for e(A · w, AλA−1) and therefore for e(λ) for the\n’best’ λ driving w to zero.\n152"},{"page":154,"text":"Thus the length k of any sequence λ(w, T1), . . . , λ(w, Tk) such that e(λ(w, T1)) <\n. . . < e(λ(w, Tk)) must be bounded by the number 2Ξ. This proves (i).\nNext, let λ1 = λ(w, T1) and λ2 = λ(w, T2) be two ’best’ elements of\nN(w, T1) and N(w, T2) respectively. By corollary 26.1, we have a torus ,\nsay D, and P(λ(w, Ti))-conjugates λi such that (i) e(λi) = e(λ(w, Ti)) and\n(ii) Im(λi) ⊆D. By lemma 26.1, we have λ1 = λ2 and thus P(λ1) = P(λ2).\nOn the other hand, P(λ(w, Ti)) = P(λi) and this proves (ii). □\nThus 12 associates a unique parabolic subgroup P(w) to every point in\nthe null-cone. This subgroup is called the destabilizing flag of w. Clearly,\nif w is in the null-cone then so is A · w, where A ∈SLn. Furthermore, it is\nclear that P(A · w) = AP(w)A−1.\nCorollary 26.2. Let w ∈W be in the null-cone and let Gw ⊆SLn stabilize\nw. Then Gw ⊆P(w).\nProof: Let g ∈Gw. Since g · w = w, we see that gP(w)g−1 = P(w), and\nthat g normalizes P(w). Since the normalizer of any parabolic subgroup is\nitself, we see that g ∈P(w). □\n153"},{"page":155,"text":"Chapter 27\nStability\nReference: [Ke, GCT1]\nRecall that z ∈W is stable iffits orbit O(z) is closed in W. In the last\nchapter, we tackled the points in the null-cone, i.e., points in the set [0W ]≈,\nor in other words, points which close onto the stable point 0W. A similar\nanalysis may be done for arbitrary stable points.\nFollowing kempf, let S ⊆W be a closed SLn-invariant subset.\nLet\nz ∈W be arbitrary. If the orbit-closure ∆(z) intersects S, then we associate\na unique parabolic subgroup Pz,S ⊆SLn as a witness to this fact.\nThe\nconstruction of this parabolic subgroup is in several steps.\nAs the first step, we construct a representation X of SLn and a closed\nSLn-invariant embedding φ : W →X such that φ−1(0X) = S, scheme-\ntheoretically. This may be done as follows: since S is a closed sub-variety\nof W, there is an ideal Is = (f1, . . . , fk) of definition for S. We may further\nassume that the vector space {f1, . . . , fk} is itself an SLn-module, say X.\nWe assume that X is k-dimensional.\nWe now construct the map φ : W →X as follows:\nφ(w) = (f1(x), . . . , fk(x))\nNote that φ(S) = 0X and that IS = (f1, . . . , fk) ensure that the requirements\non our φ do hold.\nNext, there is an adaptation of (Hilbert’s) Theorem 11 which we do not\nprove:\nTheorem 13. Let W be an SLn-module and let y ∈W be a stable point.\nLet z ∈[y]≈be an element which closes onto y. Then there is a 1-parameter\nsubgroup λ : C∗→SLn such that\nlim\nt→0 λ(t) · w ∈O(y)\n154"},{"page":156,"text":"Thus the limit exists and lies in the closed orbit of y.\nNow suppose that ∆(z) ∩S is non-empty. Then there must be stable\ny ∈∆(z). We apply the theorem to O(y) and obtain the λ as above. This\nshows that there is indeed a 1-parameter subgroup driving z into S. Next,\nit is easy to see that\nlim\nt→0[λ(t) · φ(z)] = 0X\nThus φ(z) actually lies in the null-cone of X. We may now be tempted to\napply the techniques of the previous chapter to come up with the ’best’ λ\nand its parabolic, now called P(z, S). This is almost the technique to be\nadopted , except that this ’best’ λ drives φ(z) into 0X but limt→0[λ(t) · z]\n(which is supposed to be in S) may not exist! This is because we are using the\nunproved (and untrue) converse of the assertion that 1-parameter subgroups\nwhich drive z into S drive φ(z) into 0X.\nThis above argument is rectified by limiting the domain of allowed 1-\nparameter subgroups to (i) Cone(supp(φ(z))◦as before, and (ii) those λ\nsuch that limt→0[λ(t) · z] exists. This second condition is also a ’convex’\ncondition and then the ’best’ λ does exist. This completes the construction\nof P(z, S).\nAs before, if Gz ⊆SLn stabilizes z then it normalizes P(z, S) thus must\nbe contained in it:\nProposition 27.1. If Gz stabilizes z then Gz ⊆P(z, S).\nLet us now consider the permanent and the determinant. Let M\nbe the n2-dimensional space of all n × n-matrices.\nSince det and perm\nare homogeneous n-forms on M, we consider the SL(M)-module W =\nSymn(M∗). We recall now certain stabilizing groups of the det and the\nperm. We will need the definition of a certain group L′. This is defined as\nthe group generated by the permutation and diagonal matrices in GLn. In\nother words, L′ is the normalizer of the complete standard torus D∗⊆GLn.\nL is defined as that subgroup of L′ which is contained in SLn.\nProposition 27.2.\n(A) Consider the group K = SLn × SLn. We define\nthe action μK of typical element (A, B) ∈K on X ∈M as given by:\nX →AXB−1\nThen (i) M is an irreducible representation of K and Im(K) ⊆SL(M),\nand (ii) K stabilizes the determinant.\n155"},{"page":157,"text":"(B) Consider the group H = L × L. We define the action μH of typical\nelement (A, B) ∈H on X ∈M as given by:\nX →AXB−1\nThen (i) M is an irreducible representation of H and Im(H) ⊆SL(M),\nand (ii) H stabilizes the permanent.\nWe are now ready to show:\nTheorem 14. The points det and perm in the SL(M-module W = Symn(M∗)\nare stable.\nProof: Lets look at det, the perm being similar. If det were not stable,\nthen there would be a closed SL(M)-invariant subset S ⊂W such that\ndet ̸∈S but closes onto S: just take S to be the unique closed orbit in\n[det]≈. Whence there is a parabolic P(det, S) which, by Proposition 27.1,\nwould contain K. This would mean that there is a K-invariant flag in M\ncorresponding to P(det, S). This contradicts the irreducibility of M as a\nK-module. □\n156"},{"page":158,"text":"Bibliography\n[BBD] A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Ast ́erisque\n100, (1982), Soc. Math. France.\n[B]\nP. Belkale, Geometric proofs of Horn and saturation conjectures,\nmath.AG/0208107.\n[BZ] A. 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Fulton, Young Tableaux: With Applications to Representation The-\nory and Geometry. Cambridge University Press, 1997.\n[FH] W. Fulton and J. Harris, Representation Theory: A First Course.\nSpringer-Verlang, 1991.\n157"},{"page":159,"text":"[GCTabs] K.\nMulmuley,\nGeometric\ncomplexity\ntheory:\nabstract,\ntechnical\nreport\nTR-2007-12,\nComputer\nscience\ndepartment,\nThe\nUniversity\nof\nChicago,\nSeptember,\n2007.\navailable\nat\nhttp://ramakrishnadas.cs.uchicago.edu.\n[GCTflip1] K. Mulmuley, On P. vs. NP, geometric complexity theory, and\nthe flip I: a high-level view, Technical Report TR-2007-13, Computer\nScience Department, The University of Chicago, September 2007. Avail-\nable at: http://ramakrishnadas.cs.uchicago.edu\n[GCTflip2] K. Mulmuley, On P vs. NP, geometric complexity theory, and\nthe flip II, under preparation.\n[GCTconf] K. Mulmuley, M. Sohoni, Geometric complexity theory, P vs.\nNP and explicit obstructions, in “Advances in Algebra and Geometry”,\nEdited by C. Musili, the proceedings of the International Conference\non Algebra and Geometry, Hyderabad, 2001.\n[GCT1] K. Mulmuley, M. Sohoni, Geometric complexity theory I: an ap-\nproach to the P vs. NP and related problems, SIAM J. Comput., vol\n31, no 2, pp 496-526, 2001.\n[GCT2] K. Mulmuley, M. Sohoni, Geometric complexity theory II: towards\nexplicit obstructions for embeddings among class varieties, to appear in\nSIAM J. Comput., cs. ArXiv preprint cs. CC/0612134, December 25,\n2006.\n[GCT3] K. Mulmuley, M. Sohoni, Geometric complexity theory III, on\ndeciding positivity of Littlewood-Richardson coefficients, cs. ArXiv\npreprint cs. CC/0501076 v1 26 Jan 2005.\n[GCT4] K. Mulmuley,\nM. Sohoni,\nGeometric complexity theory IV:\nquantum group for the Kronecker problem, cs. ArXiv preprint cs.\nCC/0703110, March, 2007.\n[GCT5] K. Mulmuley, H. Narayanan, Geometric complexity theory V: on\ndeciding nonvanishing of a generalized Littlewood-Richardson coeffi-\ncient, Technical report TR-2007-05, Comp. Sci. Dept. The university of\nchicago, May, 2007.\n[GCT6] K. Mulmuley,\nGeometric complexity theory VI: the flip via\nsaturated and positive integer programming in representation the-\nory and algebraic geometry, Technical report TR 2007-04, Comp.\n158"},{"page":160,"text":"Sci. Dept., The University of Chicago, May, 2007. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu. Revised version to be available\nhere.\n[GCT7] K. Mulmuley, Geometric complexity theory VII: nonstandard quan-\ntum group for the plethysm problem (Extended Abstract), Technical\nreport TR-2007-14, Comp. Sci. Dept., The University of Chicago, Sept.\n2007. Available at: http://ramakrishnadas.cs.uchicago.edu.\n[GCT8] K. Mulmuley, Geometric complexity theory VIII: On canonical\nbases for the nonstandard quantum groups (Extended Abstract), Tech-\nnical report TR-2007-15, Comp. Sci. Dept., The University of Chicago,\nSept. 2007. Available at: http://ramakrishnadas.cs.uchicago.edu.\n[GCT9] B. Adsul, M. Sohoni, K. Subrahmanyam, Geometric complexity\ntheory IX: algbraic and combinatorial aspects of the Kronecker prob-\nlem, under preparation.\n[GCT10] K. Mulmuley, Geometric complexity theory X: On class varieties,\nand the natural proof barrier, under preparation.\n[GCT11] K. Mulmuley, Geometric complexity theory XI: on the flip over fi-\nnite or algebraically closed fields of positive characteristic, under prepa-\nration.\n[GLS] M. Gr ̈otschel, L. Lov ́asz, A. Schrijver, Geometric algorithms and com-\nbinatorial optimzation, Springer-Verlag, 1993.\n[H] H. Narayanan, On the complexity of computing Kostka numbers and\nLittlewood-Richardson coefficients Journal of Algebraic Combinatorics,\nVolume 24 , Issue 3 (November 2006) 347 - 354, 2006\n[KB79] R. Kannan and A. Bachem. Polynomial algorithms for computing\nthe Smith and Hermite normal forms of an integer matrix, SIAM J.\nComput., 8(4), 1979.\n[Kar84] N. Karmarkar. A new polynomial-time algorithm for linear pro-\ngramming. Combinatorica, 4(4):373–395, 1984.\n[KL] D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke\nalgebras, Invent. Math. 53 (1979), 165-184.\n[KL2] D. Kazhdan, G. Lusztig, Schubert varieties and Poincare duality,\nProc. Symp. Pure Math., AMS, 36 (1980), 185-203.\n159"},{"page":161,"text":"[Kha79] L. G. Khachian. A polynomial algorithm for linear programming.\nDoklady Akedamii Nauk SSSR, 244:1093–1096, 1979. In Russian.\n[K] M. Kashiwara, On crystal bases of the q-analogue of universal envelop-\ning algebras, Duke Math. J. 63 (1991), 465-516.\n[Ke] G. Kempf: Instability in invariant theory, Annals of Mathematics, 108\n(1978), 299-316.\n[KTT] R. King, C. Tollu, F. Toumazet, Tretched Littlewood-Richardson\ncoefficients and Kostak coefficients. In, Winternitz, P. Harnard, J. Lam,\nC.S. and Patera, J. (eds.) Symmetry in Physics: In Memory of Robert\nT. Sharp. Providence, USA, AMS OUP, 99-112, CRM Proceedings and\nLecture Notes 34, 2004.\n[Ki] A. Kirillov, An invitation to the generalized saturation conjecture,\nmath. CO/0404353, 20 Apr. 2004.\n[KS] A. Klimyck, and K. Schm ̈udgen, Quantum groups and their represen-\ntations, Springer, 1997.\n[KT] A. Knutson, T. Tao, The honeycomb model of GLn(C) tensor products\nI: proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999)\n1055-1090.\n[KT2] A. Knutson, T. Tao: Honeycombs and sums of Hermitian matrices,\nNotices Amer. Math. Soc. 48 (2001) No. 2, 175-186.\n[LV] D. Luna and T. Vust, Plongements d’espaces homogenes, Comment.\nMath. Helv. 58, 186(1983).\n[Lu2] G. Lusztig, Introduction to quantum groups, Birkh ̈auser, 1993.\n[Ml] K. Mulmuley, Lower bounds in a parallel model without bit operations.\nSIAM J. Comput. 28, 1460–1509, 1999.\n[Mm] D. Mumford, Algebraic Geometry I, Springer-Verlang, 1995.\n[N] M. Nagata, Polynomial Rings and Affine Spaces. CBMS Regional Con-\nference no. 37, American Mathematical Society, 1978.\n[S]\nR. Stanley,\nEnumerative combinatorics,\nvol. 1,\nWadsworth and\nBrooks/Cole, Advanced Books and Software, 1986.\n[Z]\nA.\nZelevinsky,\nLittlewood-Richardson\nsemigroups,\narXiv:math.CO/9704228 v1 30 Apr 1997.\n160"},{"page":162,"text":"Gv\nClosure\nDv"},{"page":163,"text":"d\nZ\nn"}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"plexity Theory (GCT), which is an approach to proving P ̸= NP via al-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"of the group G = GLn(C) on them. This action induces an action on the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"homogeneous coordinate ring of the variety, given by (σf)(x) = f(σ−1x)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"Weyl’s theorem–that all finite-dimensional representations of G = GLn(C)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"all n →∞, assuming m = nlog n (say), then NP ̸= P in characteristic zero.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"when m = nlog n. But the conjecture is not just that there is an algorithm,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"n if m = nlog n. The main mathematical work in GCT takes steps towards","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"The map ρ induces a natural action of G on V , defined by g·v = (ρ(g))(v).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"That is, if φ(g·v) = g·φ(v). A G-equivariant map is also called G-invariant","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"• Tensor product: V ⊗W. g · (v ⊗w) = (g · v) ⊗(g · w).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"(v1 ⊗· · · ⊗vn) · σ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"sgn(σ)(v1 ⊗· · · ⊗vn) · σ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"(g · φ)(v) = g · (φ(g−1 · v)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"K = C), with a basis vector associated to each element in S. More","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"αses =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"representation V , then there exists a representation W ⊥s.t. V = W ⊕W ⊥.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"H(v, w) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"H(v, w) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"Ho(g · v, g · w) = H(h · v, h · w)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"1. Either φ is an isomorphism or φ = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"2. If V = W, φ = λI for some λ ∈C.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"1. Since Ker(φ), and Imφ are G-submodules, either Im(φ) = V","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"or Im(φ) = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"λ of φ. Look at the map φ−λI : V →V . By (1), φ−λI = 0 (it can’t","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"be an isomorphism because something maps to 0). So φ = λI.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"V =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"H(v, w) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"Un(C) ⊆GLn(C) ֒→V = ⊕iVi,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"representation V (denoted χV ) by χV (g) = Tr(ρ(g)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"Since Tr(A−1BA) = Tr(B), χV (hgh−1) = χV (g). This means charac-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"Lemma 3.1. χV ⊕W = χV + χW","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"eigenvalues of σ(g). Then (ρ ⊕σ)(g) = (ρ(g), σ(g)), so the eigenvalues of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"Then χV (g) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"i λi, χW (g) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"i μi, and χV ⊕W = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"Lemma 3.2. χV ⊗W = χV χW","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"Then, Tr((ρ ⊗σ)(g)) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"i,j λiμj = (P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"representation. Given a G-module V , let V G = {v|∀g ∈G, g · v = v}. We","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"HomG(V, V ) = (Hom(V, V ))G.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"h · π = π for all h ∈G.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"h · φ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"hgh−1 =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"g = φ.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"h · φ = φ, ∀h ∈G,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"v = φ(w) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"h · v = h · φ(w) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"hg · w = v, for any h ∈G.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"φ(v) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"g · v =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"|G||G|v = v.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"dim(V G) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"Proof. We have: dim(V G) = Tr(φ), because φ is a projection (φ = φ|V G ⊕","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"Tr(φ) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"TrV (g) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"Let Cclass(G) be the space of class functions on (G), and let (α, β) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"(χV , χW ) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"χV (g)χW (g) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"if V ∼= W","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"Proof. Since Hom(V, W) ∼= V ∗⊗W, χHom(V,W ) = χV ∗χW = χV χW . Now","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"If V ,W are irreducible, then ⟨χV , χW ⟩is 0 if V ̸= W and 1 otherwise.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"Proof. Let V = L","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":". So χV = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"i aiχVi, and ai = (χV , χVi). This","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq65","equation_number":null,"raw_text":"R =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq66","equation_number":null,"raw_text":"(χR, χV ) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq67","equation_number":null,"raw_text":"χR(g)χV (g) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq68","equation_number":null,"raw_text":"|G||G|χV (e) = χV (e) = dim(V )","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq69","equation_number":null,"raw_text":"Let α : G →C. For any G-module V , let φα,V = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq70","equation_number":null,"raw_text":"Proposition 3.1. Suppose α : G →C is a class function, and (α, χV ) = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq71","equation_number":null,"raw_text":"and V is irreducible, φα,V = λId, where λ = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq72","equation_number":null,"raw_text":"nTr(φα,V ), n = dim(V ). We","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq73","equation_number":null,"raw_text":"α(g)χV (g) = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq74","equation_number":null,"raw_text":"Now V is irreducible iffV ∗is irreducible. So λ = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq75","equation_number":null,"raw_text":"n|G|0 = 0. Therefore,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq76","equation_number":null,"raw_text":"φα,V = 0 for any irreducible representation, and hence for any representa-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq77","equation_number":null,"raw_text":"V are linearly independent, φα,V = 0 implies that α(g) = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq78","equation_number":null,"raw_text":"If V = L","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq79","equation_number":null,"raw_text":"a formula π =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq80","equation_number":null,"raw_text":"Exercise 3.2. πi = dimVi","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq81","equation_number":null,"raw_text":"G dμ = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq82","equation_number":null,"raw_text":"Let χV (g) = Tr(ρ(g)). Let V = L","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq83","equation_number":null,"raw_text":"dim(V G) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq84","equation_number":null,"raw_text":"(χV , χW ) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq85","equation_number":null,"raw_text":"χV χW dμ = dim(HomG(V, W)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq86","equation_number":null,"raw_text":"Lemma 3.11. If V, W are irreducible, (χV , χW) = 1 if V and W are iso-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq87","equation_number":null,"raw_text":"morphic, and (χV , χW ) = 0 otherwise.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq88","equation_number":null,"raw_text":"If V is reducible, V = L","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq89","equation_number":null,"raw_text":"ai = (χV , χVi) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq90","equation_number":null,"raw_text":"ρ(g). For G = U1(C), (2) gives the Fourier series expansion on the circle.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq91","equation_number":null,"raw_text":"Number of irreducible representations of Sn = Number of partitions of n","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq92","equation_number":null,"raw_text":"Let λ = {λ1 ≥λ2 ≥. . . } be a partition of n; i.e., the size |λ| = P λi is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq93","equation_number":null,"raw_text":"fT = ΠjΠi<i′(XTij −XTi′j).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq94","equation_number":null,"raw_text":"(σ.f)(X1, X2, . . . , Xn) = f(Xσ(1), Xσ(2), . . . , Xσ(n))","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq95","equation_number":null,"raw_text":"2. Sλ ̸≈Sλ′ if λ ̸= λ′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq96","equation_number":null,"raw_text":"the rows and columns of T, respectively. We have R(σT) = σR(T)σ−1 and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq97","equation_number":null,"raw_text":"C(σT) = σR(T)σ−1. We say T ≡T ′ if the rows of T and T ′ are the same","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq98","equation_number":null,"raw_text":"as the representations of C[Sn]. The element aT = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq99","equation_number":null,"raw_text":"called the row symmetrizer, bT = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq100","equation_number":null,"raw_text":"and cT = aT bT the Young symmetrizer.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq101","equation_number":null,"raw_text":"VT = bT .{T} =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq102","equation_number":null,"raw_text":"Then σ.VT = VσT . Let Sλ be the span of all VT ’s, where T ranges over","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq103","equation_number":null,"raw_text":"2. Sλ ̸≈Sλ′ if λ ̸= λ′.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq104","equation_number":null,"raw_text":"Let T be a canonical numbering of shape λ = (λ1, . . . , λk). By this, we mean","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq105","equation_number":null,"raw_text":"and so on, and the rows are increasing. Let aλ = aT , bλ = bT , and cλ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq106","equation_number":null,"raw_text":"cT = aT .bT .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq107","equation_number":null,"raw_text":"Then Sλ = C[Sn].cλ is a representation of Sn from the left.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq108","equation_number":null,"raw_text":"2. Sλ ̸∼= Sλ′ if λ ̸= λ′.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq109","equation_number":null,"raw_text":"Let i = (i1, i2, . . . , ik) be such that P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq110","equation_number":null,"raw_text":"j jij = n. Let Ci be the conjugacy","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq111","equation_number":null,"raw_text":"Pj(X) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq112","equation_number":null,"raw_text":"∆(X) = Πi<j(Xj −Xi)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq113","equation_number":null,"raw_text":"k in f(X). Let li = λi + k −i.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq114","equation_number":null,"raw_text":"χλ(Ci) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq115","equation_number":null,"raw_text":"Sα ⊗Sβ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq116","equation_number":null,"raw_text":"Let X = (xij) be a generic n × n matrix with variable entries xij. Consider","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq117","equation_number":null,"raw_text":"the polynomial ring C[X] = C[x11, x12, . . . , xnn].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq118","equation_number":null,"raw_text":"C[X] by (A ◦f)(X) = f(ATX) (it is easily checked that this is in fact a left","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq119","equation_number":null,"raw_text":"C =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq120","equation_number":null,"raw_text":"−→eC =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq121","equation_number":null,"raw_text":"(Thus if there is a repeated number in the column C, eC = 0, since","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq122","equation_number":null,"raw_text":"eT = Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq123","equation_number":null,"raw_text":"eT = 0. Furthermore, the numbers must all come from {1, . . . , n} if they","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq124","equation_number":null,"raw_text":"i.e. g(eC) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq125","equation_number":null,"raw_text":"5. (Weyl’s character formula) Define the character χλ of Vλ by χλ(g) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq126","equation_number":null,"raw_text":"χλ(g) = Sλ(x1, . . . , xn) :=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq127","equation_number":null,"raw_text":"yij = yi","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq128","equation_number":null,"raw_text":"x(T) = Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq129","equation_number":null,"raw_text":"Sλ(x1, . . . , xn) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq130","equation_number":null,"raw_text":"then conjugate ρ by A to get ρ′ : GLn(C) →GL(Vλ) defined by ρ′(h) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq131","equation_number":null,"raw_text":"g(eT ) = x(T)eT . Then since {eT |T is a semistandard tableau of shape λ} is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq132","equation_number":null,"raw_text":"simply the above-mentioned r×r minor of the generic n×n matrix X = (xij)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq133","equation_number":null,"raw_text":"indexed eigenvalues of g). Since g is diagonal, gt = g. So (g ◦eT )(X) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq134","equation_number":null,"raw_text":"eT (gtX) = eT (gX). Thus g multiplies the ij-th column by xij. Thus its","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq135","equation_number":null,"raw_text":"j=1 xij, which is exactly x(T).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq136","equation_number":null,"raw_text":"λ : B →C such that b · v = λ(b)v for all b ∈B. The restriction of λ to the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq137","equation_number":null,"raw_text":"eT = x(T)eT = xλ1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq138","equation_number":null,"raw_text":"example, for λ = (5, 3, 2, 2, 1), the canonical T is:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq139","equation_number":null,"raw_text":"T =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq140","equation_number":null,"raw_text":"Let V = Cn and consider the d-th tensor power V ⊗d. The group GLn(C)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq141","equation_number":null,"raw_text":"g(v1 ⊗· · · ⊗vd) = gv1 ⊗· · · ⊗gvd (g ∈GL(V ))","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq142","equation_number":null,"raw_text":"(v1 ⊗· · · ⊗vd)τ = v1τ ⊗· · · ⊗vdτ (τ ∈Sd) .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq143","equation_number":null,"raw_text":"These two actions commute, so V ⊗d is a representation of H = GLn(C) ×","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq144","equation_number":null,"raw_text":"V ⊗d =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq145","equation_number":null,"raw_text":"Theorem 5.2. V ⊗d = L","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq146","equation_number":null,"raw_text":"Theorem 5.3. V ⊗dcT ∼= Vλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq147","equation_number":null,"raw_text":"V ⊗d =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq148","equation_number":null,"raw_text":"λ:|λ|=d","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq149","equation_number":null,"raw_text":"where |λ| = P λi denotes the size of λ. In particular, Vλ occurs in V ⊗d with","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq150","equation_number":null,"raw_text":"To each bitableau we associate a vector e(U,T) = ei1 ⊗· · · ⊗eid where","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq151","equation_number":null,"raw_text":"Let G = GLn(C). Consider the diagonal embedding of G ֒→G × G.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq152","equation_number":null,"raw_text":"Vα ⊗Vβ = ⊕γcγ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq153","equation_number":null,"raw_text":"SαSβ = ⊕γcγ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq154","equation_number":null,"raw_text":"For example, Figure 6.2 shows an LR tableau with α = (6, 3, 2), β =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq155","equation_number":null,"raw_text":"(4, 2, 2) and γ = (8, 6, 3, 2).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq156","equation_number":null,"raw_text":"Lemma 6.1. There exists a polytope P = P γ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq157","equation_number":null,"raw_text":"j = γi.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq158","equation_number":null,"raw_text":"j = βj.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq159","equation_number":null,"raw_text":"j = 0 for i < j, and for i ≤n,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq160","equation_number":null,"raw_text":"Proof: Let P = P γ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq161","equation_number":null,"raw_text":"Lemma 6.2. If P ̸= ∅then cγ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq162","equation_number":null,"raw_text":"kα,kβ ̸= 0 for some k ≥1, then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq163","equation_number":null,"raw_text":"α,β ̸= 0 [Saturation Theorem].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq164","equation_number":null,"raw_text":"α,β(k) = ckγ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq165","equation_number":null,"raw_text":"f(k) = fi(k) if k ≡i mod l.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq166","equation_number":null,"raw_text":"We denote such a quasipolynomial f by f = (fi). Here lis called the period","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq167","equation_number":null,"raw_text":"of integer points inside P. We define the stretching function fP(k) = φ(kP),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq168","equation_number":null,"raw_text":"c(k) = ecγ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq169","equation_number":null,"raw_text":"α,β(k) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq170","equation_number":null,"raw_text":"If all the coefficients ai are nonnegative (by PH2), and c(k) ̸= 0, then c(1) ̸=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq171","equation_number":null,"raw_text":"Q(V, W) = V T W.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq172","equation_number":null,"raw_text":"Q(AV, AW) = Q(V, W) for all V and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq173","equation_number":null,"raw_text":"Fix a partition λ = (λ1 ≥λ2 ≥. . . ) of length at most n.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq174","equation_number":null,"raw_text":"|λ| = d = P λi be its size. Let V = Cn, V ⊗d = V ⊗· · · ⊗V d times, and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq175","equation_number":null,"raw_text":"φp,q(vi1 ⊗· · · ⊗vid) = Q(vip, viq)(vi1 ⊗· · · ⊗c","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq176","equation_number":null,"raw_text":"V [d] =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq177","equation_number":null,"raw_text":"is an On(C)-module. Let V[λ] = V [d] T Vλ, where Vλ ⊆V ⊗d is the embedded","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq178","equation_number":null,"raw_text":"V[λ] ⊗V[μ] = ⊗γdγ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq179","equation_number":null,"raw_text":"λ,μ = φ(P γ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq180","equation_number":null,"raw_text":"λ,μ(k) = dkγ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq181","equation_number":null,"raw_text":"2λ,2μ ̸= 0 but dγ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq182","equation_number":null,"raw_text":"λ,μ = 0. Therefore we change the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq183","equation_number":null,"raw_text":"Definition 7.4. Given a quasipolynomial f(k) = (fi), index(f) is the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq184","equation_number":null,"raw_text":"is identically zero, index(f) = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq185","equation_number":null,"raw_text":"Definition 7.5. A quasipolynomial f(k) is saturated if f(index(f)) ̸= 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq186","equation_number":null,"raw_text":"In particular, if index(f) = 1, then f(k) is saturated if f(1) ̸= 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq187","equation_number":null,"raw_text":"λ,μ(k) = dkν","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq188","equation_number":null,"raw_text":"kλ,kμ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq189","equation_number":null,"raw_text":"∀odd k, f1(k) = dkν","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq190","equation_number":null,"raw_text":"λμ = f1(1) > 0. Since dν","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq191","equation_number":null,"raw_text":"Z<2> :=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq192","equation_number":null,"raw_text":"expressed as an affine linear combination, z = ax + (1−a)y for some a ∈Q.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq193","equation_number":null,"raw_text":"P = {x | Ax ≤B, (∀i)xi ≥0} ⊆Rd","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq194","equation_number":null,"raw_text":"the form Cx = D can be obtained for Aff(P) in polynomial time, where C","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq195","equation_number":null,"raw_text":"matrix S such that C = USV where U and V are unimodular and S has","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq196","equation_number":null,"raw_text":"The question now reduces to whether USV x = D has a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq197","equation_number":null,"raw_text":"too, and y := V x ∈Zd","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq198","equation_number":null,"raw_text":"Sy = U−1D has a solution y ∈Zd","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq199","equation_number":null,"raw_text":"Proof of Theorem 8.1: By [BZ], there exists a polytope P = P ν","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq200","equation_number":null,"raw_text":"P ∩Zd ̸= ∅⇔(∃odd k), kP ∩Zd ̸= ∅.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq201","equation_number":null,"raw_text":"for i = 1, . . . , lsuch that f(k) = fi(k) whenever k ≡i mod l. The number","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq202","equation_number":null,"raw_text":"nonnegative. We say f is saturated if f(index(f)) ̸= 0. If f is positive, then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq203","equation_number":null,"raw_text":"Given any function f(k), we associate to it the rational series F(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq204","equation_number":null,"raw_text":"Let G = GLn(C) and cγ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq205","equation_number":null,"raw_text":"α,β(k) = ckγ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq206","equation_number":null,"raw_text":"other words, write Sλ ⊗Sμ = L","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq207","equation_number":null,"raw_text":"λ,μ = (χλχμ, χπ), where","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq208","equation_number":null,"raw_text":"λ,μ = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq209","equation_number":null,"raw_text":"λ,μ(k) = κkπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq210","equation_number":null,"raw_text":"H = GLn(C) × GLn(C) ֒→G = GL(Cn ⊗Cn),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq211","equation_number":null,"raw_text":"where (g, h)(v ⊗w) = (gv ⊗hw). Then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq212","equation_number":null,"raw_text":"Let H = GLn(C), V = Vμ(H) the Weyl module of H corresponding to","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq213","equation_number":null,"raw_text":"a partition μ, and ρ : H →G = GL(V ) the corresponding representation","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq214","equation_number":null,"raw_text":"Vλ(G) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq215","equation_number":null,"raw_text":"λ,μ(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq216","equation_number":null,"raw_text":"Positivity Hypothesis 1 (PH1). There exists a polytope P = P π","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq217","equation_number":null,"raw_text":"λ,μ = φ(P), where φ denotes the number of integer points inside the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq218","equation_number":null,"raw_text":"bit length of P is defined to be ⟨P⟩= n + ψ, where ψ is the maximum","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq219","equation_number":null,"raw_text":"means there exist polynomials fi(k), 1 ≤i ≤l, l the period, so that fP(k) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq220","equation_number":null,"raw_text":"fi(k) if k = i modulo l. Then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq221","equation_number":null,"raw_text":"Index(fP) = min{i|fi(k)not identically 0 as a polynomial}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq222","equation_number":null,"raw_text":"the specifications Cx = d, C and d integral, of Aff(P) in poly(⟨P⟩) time.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq223","equation_number":null,"raw_text":"̄C = ACB","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq224","equation_number":null,"raw_text":"Clearly Cx = d iff ̄Cz = ̄d where z = B−1x and ̄d = Ad.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq225","equation_number":null,"raw_text":"̄cizi = di","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq226","equation_number":null,"raw_text":"prime. Let ̃c = lcm(ci).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq227","equation_number":null,"raw_text":"Claim 10.1. Index(fP ) = ̃c.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq228","equation_number":null,"raw_text":"Proof of the claim. Let fP(t) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq229","equation_number":null,"raw_text":"Hence fP(t) = f ̄P(t ̃c) where ̄P is the stretched polytope ̃cP and f ̄P(s) is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq230","equation_number":null,"raw_text":"Index(fP ) = ̃cIndex(f ̄P )","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq231","equation_number":null,"raw_text":"Now we show that Index(f ̄P) = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq232","equation_number":null,"raw_text":"zi = ̃c","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq233","equation_number":null,"raw_text":"Proof. Let P = P π","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq234","equation_number":null,"raw_text":"Let V = Cn, and v1, . . . , vn the coordinates of V .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq235","equation_number":null,"raw_text":"For example, Y = (v1 −v2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq236","equation_number":null,"raw_text":"written as f = P figi for some polynomials fi.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq237","equation_number":null,"raw_text":"the variables v1, . . . , vn. The coordinate ring of Y is defined to be C[Y ] =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq238","equation_number":null,"raw_text":"R(Y ) = C[V ]/I(Y ), the set of homogeneous polynomial forms on the cone","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq239","equation_number":null,"raw_text":"Let G = GLn(C), and V a finite dimensional representation of G. Then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq240","equation_number":null,"raw_text":"(σ · f)(v) = f(σ−1v), v ∈V.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq241","equation_number":null,"raw_text":"If Y is a projective variety, then R(Y ) = C[V ]/I(Y ) is also a G-module.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq242","equation_number":null,"raw_text":"Gv = {gv|g ∈G}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq243","equation_number":null,"raw_text":"H = Gv = {g ∈G|gv = v}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq244","equation_number":null,"raw_text":"∆V [v] = Gv ⊆P(V ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq245","equation_number":null,"raw_text":"where λ = (λ1 ≥λ2 ≥· · · ≥λn ≥0) is a partition. Let vλ be the highest","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq246","equation_number":null,"raw_text":"vector in Vλ. This means bvλ = vλ for all b ∈B, where B ⊆GLn(C) is the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq247","equation_number":null,"raw_text":"It can be shown that the stabilizer Pλ = Gvλ is a group of block lower","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq248","equation_number":null,"raw_text":"denoted by P. Clearly Gvλ ∼= G/Pλ = G/P.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq249","equation_number":null,"raw_text":"Definition 11.8. Let G = Gln(C), and V = Cn. The Grassmanian Grn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq250","equation_number":null,"raw_text":"2. More generally, P(V ) = Grn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq251","equation_number":null,"raw_text":"P(V ) = Grn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq252","equation_number":null,"raw_text":"Proposition 11.3. Let λ = (1, . . . , 1) be the partition of d, whose all parts","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq253","equation_number":null,"raw_text":"d ∼= Gvλ ⊆P(Vλ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq254","equation_number":null,"raw_text":"Λd(V ) = span{(vi1 ∧· · ·∧vid)|i1, . . . , id are distinct} ⊆V ⊗· · ·⊗V (d times),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq255","equation_number":null,"raw_text":"(vi1 ∧· · · ∧vid) = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq256","equation_number":null,"raw_text":"(σ · f)(Z) = f(Zσ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq257","equation_number":null,"raw_text":"and reduce the NC ̸= P #P conjecture over C to a conjecture that the class","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq258","equation_number":null,"raw_text":"The NC ̸= P #P conjecture over C says that the permanent of an n × n","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq259","equation_number":null,"raw_text":"plex matrix Y , m = poly(n), whose entries are (possibly nonhomogeneous)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq260","equation_number":null,"raw_text":"linear forms in the entries of X. This obviously implies the NC ̸= P #P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq261","equation_number":null,"raw_text":"implied by the usual NC ̸= P #P conjecture over a finite field Fp, p ̸= 2,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq262","equation_number":null,"raw_text":"For this reason, we concentrate on the NC ̸= P #P conjecture over C in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq263","equation_number":null,"raw_text":"Let G = GLl(C), V a finite dimensional representation of G. Let P(V )","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq264","equation_number":null,"raw_text":"point v ∈P(V ), let ∆V [v] = Gv ⊆P(V ) be its orbit closure. Here Gv is the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq265","equation_number":null,"raw_text":"We now associate a class variety with NC. Let g = det(Y ), Y an m × m","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq266","equation_number":null,"raw_text":"variable matrix. This is a complete function for NC. Let V = symm(Y )","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq267","equation_number":null,"raw_text":"G-module, G = GLm2(C), with the action of σ ∈G given by:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq268","equation_number":null,"raw_text":"Let ∆V [g] = ∆V [g, m] = Gg, where we think of g as an element of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq269","equation_number":null,"raw_text":"We next associate a class variety with #P. Let h = perm(X), X an","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq270","equation_number":null,"raw_text":"n × n variable matrix. Let W = symn(X). It is similarly an H-module,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq271","equation_number":null,"raw_text":"H = GLk(C), k = n2. Think of h as an element of P(W), and let ∆W[h] =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq272","equation_number":null,"raw_text":"a map from P(V ) to P(W) which we call φ as well. Let φ(h) = f ∈P(V )","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq273","equation_number":null,"raw_text":"and ∆V [f, m, n] = Gf its orbit closure. It is called the extended class variety","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq274","equation_number":null,"raw_text":"cuit (over C) of depth ≤logc(n), c a constant, then f = φ(h) ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq275","equation_number":null,"raw_text":"∆V [g, m], for m = O(2logc n).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq276","equation_number":null,"raw_text":"2. Conversely if f ∈∆V [g, m] for m = 2logc n, then h(X) can be approx-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq277","equation_number":null,"raw_text":"Conjecture 12.1 (GCT 1). Let h(X) = perm(X), X an n × n variable","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq278","equation_number":null,"raw_text":"matrix. Then f = φ(h) ̸∈∆V [g; m] if m = 2polylog(n) and n is sufficiently","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq279","equation_number":null,"raw_text":"if m = 2polylog(n) and n →∞.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq280","equation_number":null,"raw_text":"Let g = det(Y ), h = perm(X), f = φ(h) = ym−nperm(X), and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq281","equation_number":null,"raw_text":"V = Symm[Y ] the set of homogeneous forms of degree m in the entries of Y .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq282","equation_number":null,"raw_text":"Then V is a G-module for G = GL(Y ) = GLl(C), l = m2, with the action","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq283","equation_number":null,"raw_text":"∆V [f; m, n] = Gf ⊆P(V ),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq284","equation_number":null,"raw_text":"∆V [g; m] = Gg ⊆P(V )","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq285","equation_number":null,"raw_text":"∆V [g; m] with m = 2polylog(n), n →∞.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq286","equation_number":null,"raw_text":"Definition 13.1. We say that a Weyl-module S = Vλ(G) is an obstruction","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq287","equation_number":null,"raw_text":"large enough n if m = 2polylog(n).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq288","equation_number":null,"raw_text":"Let V be a G-module, G = GLn(C). Let P(V ) be a projective variety","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq289","equation_number":null,"raw_text":"over V . Let v ∈P(V ), and recall ∆V [v] = Gv. Let H = Gv be the stabilizer","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq290","equation_number":null,"raw_text":"of v, that is, Gv = {σ ∈G|σv = v}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq291","equation_number":null,"raw_text":"Definition 13.2. We say that v is characterized by its stabilizer H = Gv","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq292","equation_number":null,"raw_text":"if v is the only point in P(V ) such that hv = v, ∀h ∈H.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq293","equation_number":null,"raw_text":"by the group triple H ֒→G ֒→K = GL(V ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq294","equation_number":null,"raw_text":"1. The determinant g = det(Y ) ∈P(V ) is characterized by its stabilizer.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq295","equation_number":null,"raw_text":"2. The permanent h = perm(X) ∈P(W), where W = Symn(X), is also","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq296","equation_number":null,"raw_text":"3. Finally, f = φ(h) ∈P(V ) is also characterized by its stabilizer. Hence","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq297","equation_number":null,"raw_text":"of det(Y ) in G = GL(Y ) = GLm2(C) is the subgroup Gdet that consists of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq298","equation_number":null,"raw_text":"linear transformations of the form Y →AY ∗B, where Y ∗= Y or Y t, for any","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq299","equation_number":null,"raw_text":"1. det(AY B) = det(A)det(B)det(Y ) = c det(Y ), where c = det(A) det(B).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq300","equation_number":null,"raw_text":"2. det(Y ∗) = det(Y ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq301","equation_number":null,"raw_text":"(Gdet)◦= GLm × GLm ֒→GL(Cm ⊗Cm) = GLm2(C) = G.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq302","equation_number":null,"raw_text":"(2) The stabilizer of perm(x) is the subgroup Gperm ⊆GL(X) = GLn2(C)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq303","equation_number":null,"raw_text":"generated by linear transformations of the form X →λX∗μ, where X∗=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq304","equation_number":null,"raw_text":"Let G = GLn(C), V a G-module, and P(V ) the associated projective space.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq305","equation_number":null,"raw_text":"Let v ∈P(V ) be a point characterized by its stabilizer H = Gv ⊂G. In","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq306","equation_number":null,"raw_text":"other words, v is the only point in P(V ) stabilized by H. Then ∆V [v] = Gv","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq307","equation_number":null,"raw_text":"G/P that we described in the earlier lecture. Let V = Vλ(G) be a Weyl","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq308","equation_number":null,"raw_text":"Pλ := Gvλ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq309","equation_number":null,"raw_text":"The orbit ∆V [vλ] = Gvλ ∼= G/Pλ is a group-theoretic variety determined","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq310","equation_number":null,"raw_text":"Pλ = Gvλ ֒→G ֒→K = GL(V ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq311","equation_number":null,"raw_text":"i.e. the representation ρ : G →GL(Cv) ∼= C∗is polynomial of degree d","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq312","equation_number":null,"raw_text":"Πv =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq313","equation_number":null,"raw_text":"The G-modules in Πvλ(2) generate the ideal IV [vλ] of the orbit Gvλ ∼= G/Pλ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq314","equation_number":null,"raw_text":"when V = Vλ(G).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq315","equation_number":null,"raw_text":"Now, let ∆V [g] be the class variety for NC (in other words, take g =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq316","equation_number":null,"raw_text":"Conjecture 14.1 (GCT 2). ∆V [g] = X(Πg) where X(Πg) is the zero-set","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq317","equation_number":null,"raw_text":"U ⊆P(V ) of the orbit Gg such that X(Πg) ∩U = ∆V [g] ∩U.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq318","equation_number":null,"raw_text":"Proposition 14.2 (GCT 2). Let g = det(Y ), h = perm(X), f = φ(h),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq319","equation_number":null,"raw_text":"n = dim(X), m = dim(Y ). If Conjecture 14.1 holds and the permanent can-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq320","equation_number":null,"raw_text":"large enough n, when m = 2logc n for some constant c. Hence, under these","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq321","equation_number":null,"raw_text":"conditions, NC ̸= P #P over C.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq322","equation_number":null,"raw_text":"Proof. The hardness assumption implies that f /∈∆V [g] if m = 2logc n [GCT","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq323","equation_number":null,"raw_text":"Conjecture 14.1 says that X(Πg) = ∆V [g]. So there exists an irreducible","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq324","equation_number":null,"raw_text":"S does not occur in RV [g] = C[V ]/IV [g]. Thus S is not a G-submodule of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq325","equation_number":null,"raw_text":"As in the previous lectures, let g = det(Y ) ∈P(V ), Y an m × m vari-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq326","equation_number":null,"raw_text":"able matrix, G = GLm2(C), and ∆V (g) = ∆V [g; m] = Gg ⊆P(V ) the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq327","equation_number":null,"raw_text":"class variety for NC. Let h = perm(X), X an n × n variable matrix,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq328","equation_number":null,"raw_text":"f = φ(h) = ym−nh ∈P(V ), and ∆V (f) = ∆V [f; m, n] = Gf ⊆P(V ) the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq329","equation_number":null,"raw_text":"S = Vλ(G) of G is an obstruction of degree d for the pair (f, g) if Vλ occurs","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq330","equation_number":null,"raw_text":"ists if m = 2polylog(n) as n →∞.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq331","equation_number":null,"raw_text":"This implies NC ̸= P #P over C.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq332","equation_number":null,"raw_text":"n →∞when d is large enough and m = 2polylog(n).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq333","equation_number":null,"raw_text":"d [f] = Sλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq334","equation_number":null,"raw_text":"d [f; m, n] be the multiplicity of Vλ = Vλ(G) in RV [f; m, n].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq335","equation_number":null,"raw_text":"The stretching function ̃Sd[f] = ̃Sλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq336","equation_number":null,"raw_text":"d [f](k) := Skλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq337","equation_number":null,"raw_text":"d [g] = ̃Sλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq338","equation_number":null,"raw_text":"PH1: There exists a polytope P = P λ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq339","equation_number":null,"raw_text":"2. dim(P) = poly(n, m, ⟨d⟩).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq340","equation_number":null,"raw_text":"Similarly, there exists a polytope Q = Qλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq341","equation_number":null,"raw_text":"2. dim(Q) = poly(m, ⟨d⟩).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq342","equation_number":null,"raw_text":"Problem 15.2. Let X be a projective group-theoretic G-variety. Let R =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq343","equation_number":null,"raw_text":"d=0 Rd be its homogeneous coordinate ring.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq344","equation_number":null,"raw_text":"Let V = Vλ(G) be a Weyl module of G = GLn(C) and vλ ∈P(V ) the point","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq345","equation_number":null,"raw_text":"corresponding to its highest weight vector. The orbit ∆V [vλ] := Gvλ, which","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq346","equation_number":null,"raw_text":"Then V = Vλ(G) can be identified with the span of d × d minors of Z with","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq347","equation_number":null,"raw_text":"d-dimensional subspace of Cn. Let B = B(W) be a basis of W. Construct","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq348","equation_number":null,"raw_text":"minors of ZB. Here the choice of B = B(W) does not matter, since any","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq349","equation_number":null,"raw_text":"top to bottom, we have a monomial mT = Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq350","equation_number":null,"raw_text":"mT =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq351","equation_number":null,"raw_text":"tableaux of rectangular shape sλ. Hence, Rs ∼= V ∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq352","equation_number":null,"raw_text":"this is so if and only if (sλ)∗= β, where (sλ)∗denotes the dual partition,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq353","equation_number":null,"raw_text":"• eg = ge = g,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq354","equation_number":null,"raw_text":"• g1(g2g3) = (g1g2)g3,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq355","equation_number":null,"raw_text":"• g−1g = gg−1 = e.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq356","equation_number":null,"raw_text":"K[G × G] = K[G] ⊗K[G] ←K[G].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq357","equation_number":null,"raw_text":"∆(f)(g1, g2) = f(g1g2).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq358","equation_number":null,"raw_text":"defined by: for f ∈K[G], ǫ(f) = f(e).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq359","equation_number":null,"raw_text":"defined by: for f ∈K[G], S(f)(g) = f(g−1).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq360","equation_number":null,"raw_text":"(∆⊗id) ◦∆= (id ⊗∆) ◦∆.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq361","equation_number":null,"raw_text":"3. The property ge = g is defined so that the following diagram com-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq362","equation_number":null,"raw_text":"That is, id = (ǫ⊗id)◦∆. Similarly, ge = g translates to: id = (id⊗ǫ)◦∆.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq363","equation_number":null,"raw_text":"id = (ǫ ⊗id) ◦∆= (id ⊗ǫ) ◦∆.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq364","equation_number":null,"raw_text":"4. The last property is gg−1 = e = g−1g. The first equality is equivalent","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq365","equation_number":null,"raw_text":"m : K[G] ←K[G] ⊗K[G] defined by m(f1, f2)(g) = f1(g) · f2(g). So","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq366","equation_number":null,"raw_text":"m ◦(S ⊗id) ◦∆= ν ◦ǫ.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq367","equation_number":null,"raw_text":"For e = g−1g, we similarly get:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq368","equation_number":null,"raw_text":"m ◦(id ⊗S) ◦∆= ν ◦ǫ.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq369","equation_number":null,"raw_text":"• (h1h2) · v = h1 · (h2 · v), and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq370","equation_number":null,"raw_text":"• e · v = v.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq371","equation_number":null,"raw_text":"(φ ⊗id) ◦φ = (id ⊗∆) ◦φ.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq372","equation_number":null,"raw_text":"(id ⊗ǫ) ◦φ = id","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq373","equation_number":null,"raw_text":"G = GLn(C) = GL(Cn) = GL(V ),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq374","equation_number":null,"raw_text":"where V = Cn. Let Mn be the matrix space of n×n C-matrices, and O(Mn)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq375","equation_number":null,"raw_text":"O(Mn) = C[U] = C[{ui","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq376","equation_number":null,"raw_text":"j. Let C[G] = O(G) be","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq377","equation_number":null,"raw_text":"is, C[G] = O(G) = C[U][det(U)−1], which is the C algebra generated by ui","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq378","equation_number":null,"raw_text":"∆(f)(g1, g2) = f(g1g2).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq379","equation_number":null,"raw_text":"j) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq380","equation_number":null,"raw_text":"ǫ(f) = f(e).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq381","equation_number":null,"raw_text":"j) = δij,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq382","equation_number":null,"raw_text":"• Finally, the antipode is required to satisfy S(f)(g) = f(g−1). Let eU","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq383","equation_number":null,"raw_text":"be the cofactor matrix of U, U−1 =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq384","equation_number":null,"raw_text":"j) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq385","equation_number":null,"raw_text":"j = (U−1)i","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq386","equation_number":null,"raw_text":"Let G = GL(V ) = GL(Cn), and V = Cn. In the earlier lecture, we","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq387","equation_number":null,"raw_text":"group Gq = GLq(V ) = GLq(n) as the virtual object whose coordinate ring","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq388","equation_number":null,"raw_text":"We start by defining GLq(2) and SLq(2), for n = 2.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq389","equation_number":null,"raw_text":"V generated by the coordinates x1 and x2 of V which satisfy x1x2 = x2x1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq390","equation_number":null,"raw_text":"U =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq391","equation_number":null,"raw_text":"be the generic (variable) matrix in M2. It acts on V = C2 from the left and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq392","equation_number":null,"raw_text":"x =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq393","equation_number":null,"raw_text":"x →x′ := Ux.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq394","equation_number":null,"raw_text":"x′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq395","equation_number":null,"raw_text":"xT →(x′′)T := xT U.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq396","equation_number":null,"raw_text":"x′′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq397","equation_number":null,"raw_text":"2 = x′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq398","equation_number":null,"raw_text":"2 = x′′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq399","equation_number":null,"raw_text":"x1x2 = qx2x1,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq400","equation_number":null,"raw_text":"are position and momentum, then q = eiħwhen ħis Planck’s constant. Let","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq401","equation_number":null,"raw_text":"Cq[V ] = C[x1, x2]/ < x1x2 −qx2x1 > .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq402","equation_number":null,"raw_text":"Let U =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq403","equation_number":null,"raw_text":"[Left action:] Let the left action be φL : x →Ux, and Ux = x′. Then we","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq404","equation_number":null,"raw_text":"[Right action:] Let the right action be φR : xT →xT U, and let x′′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq405","equation_number":null,"raw_text":"(xT U)T = UT x. Then we must have:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq406","equation_number":null,"raw_text":"The preservation of x1x2 = qx2x1 under left multiplication means","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq407","equation_number":null,"raw_text":"2 = qx′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq408","equation_number":null,"raw_text":"(ax1 + bx2)(cx1 + dx2) = q(cx1 + dx2)(ax1 + bx2).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq409","equation_number":null,"raw_text":"2 = acx2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq410","equation_number":null,"raw_text":"ac = qca","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq411","equation_number":null,"raw_text":"bd = qdb","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq412","equation_number":null,"raw_text":"bc + adq = da + qcb.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq413","equation_number":null,"raw_text":"ac = qca","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq414","equation_number":null,"raw_text":"bd = qdb","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq415","equation_number":null,"raw_text":"ad −da −qcb + q−1bc = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq416","equation_number":null,"raw_text":"2 = qx′′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq417","equation_number":null,"raw_text":"ab = qba","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq418","equation_number":null,"raw_text":"cd = qdc","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq419","equation_number":null,"raw_text":"ad −da −qbc + q−1cb = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq420","equation_number":null,"raw_text":"The last equations from each of these sets imply bc = cb.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq421","equation_number":null,"raw_text":"ab = qba,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq422","equation_number":null,"raw_text":"ac = qca,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq423","equation_number":null,"raw_text":"bd = qdb,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq424","equation_number":null,"raw_text":"cd = qdc,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq425","equation_number":null,"raw_text":"bc = cb,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq426","equation_number":null,"raw_text":"ad −da = (q −q−1)bc.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq427","equation_number":null,"raw_text":"U =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq428","equation_number":null,"raw_text":"Dq = det(U) = ad −qbc = da −q−1bc.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq429","equation_number":null,"raw_text":"Define Cq[G] = O(GLq(2)), the coordinate ring of the virtual quantum group","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq430","equation_number":null,"raw_text":"O(GLq(2)) = O(Mq(2))[D−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq431","equation_number":null,"raw_text":"j) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq432","equation_number":null,"raw_text":"j) = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq433","equation_number":null,"raw_text":"j = (U−1)i","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq434","equation_number":null,"raw_text":"j) = δij,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq435","equation_number":null,"raw_text":"where eU = [ ̃ui","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq436","equation_number":null,"raw_text":"eU =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq437","equation_number":null,"raw_text":"(defined so that U eU = DqI) and U−1 = ̃U/Dq is the inverse of U.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq438","equation_number":null,"raw_text":"xixj = qxjxi","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq439","equation_number":null,"raw_text":"Cq[V ] = C[x1, . . . , xn]/ < xixj −qxjxi > .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq440","equation_number":null,"raw_text":"= x →Ux = x′,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq441","equation_number":null,"raw_text":"xT →xT U = (x′′)T .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq442","equation_number":null,"raw_text":"j = qx′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq443","equation_number":null,"raw_text":"j = qx′′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq444","equation_number":null,"raw_text":"the entries uij = ui","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq445","equation_number":null,"raw_text":"ujkuik = q−1uikujk","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq446","equation_number":null,"raw_text":"ukjuki = q−1ukiukj","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq447","equation_number":null,"raw_text":"ujkuil= uilujk","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq448","equation_number":null,"raw_text":"ujluik = uikujl−(q −q−1)ujkuil","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq449","equation_number":null,"raw_text":"Dq =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq450","equation_number":null,"raw_text":"O(Mq(n)) = C[U]/ < (18.4) >, and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq451","equation_number":null,"raw_text":"Cq[G] = O(GLq(n)) = O(Mq(n))[D−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq452","equation_number":null,"raw_text":"matrix eU and the quantum inverse matrix U−1 = eU/Dq in a straightforward","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq453","equation_number":null,"raw_text":"j) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq454","equation_number":null,"raw_text":"j) = δij","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq455","equation_number":null,"raw_text":"j) = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq456","equation_number":null,"raw_text":"j = (U−1)i","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq457","equation_number":null,"raw_text":"q ) = Dq.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq458","equation_number":null,"raw_text":"Let V = Cn, G = GLn(C) = GL(V ) = GL(Cn), and O(G) the coordinate","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq459","equation_number":null,"raw_text":"ring of G. The quantum group Gq = GLq(V ) is the virtual object whose","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq460","equation_number":null,"raw_text":"O(Gq) = C[U]/⟨relations ⟩,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq461","equation_number":null,"raw_text":"In this lecture we define a q-analogue of the unitary subgroup U = Un(C) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq462","equation_number":null,"raw_text":"U(V ) ⊆GLn(C) = GL(V ) = G. This is a q-deformation Uq = Uq(V ) ⊆Gq","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq463","equation_number":null,"raw_text":"(αv + βw)∗= αv∗+ βw∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq464","equation_number":null,"raw_text":"(v∗)∗= v","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq465","equation_number":null,"raw_text":"1. (ab)∗= b∗a∗, 1∗= 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq466","equation_number":null,"raw_text":"2. ∆(a∗) = ∆(a)∗(where ∗acts diagonally on the tensor product A ⊗A:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq467","equation_number":null,"raw_text":"(v ⊗w)∗= (v∗⊗w∗))","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq468","equation_number":null,"raw_text":"3. ǫ(a∗) = ǫ(a)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq469","equation_number":null,"raw_text":"Let O(G) = C[G] be the coordinate ring of G as defined earlier.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq470","equation_number":null,"raw_text":"(4) For all f ∈O(G) and g ∈U ⊆G, f ∗(g) = f(g)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq471","equation_number":null,"raw_text":"i be the coordinate functions which, together with D−1, D = det(U),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq472","equation_number":null,"raw_text":"i)∗= S(ui","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq473","equation_number":null,"raw_text":"j) = (U−1)i","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq474","equation_number":null,"raw_text":"and D∗= D−1. We can check that this satifies (1)-(4). Here we will only","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq475","equation_number":null,"raw_text":"i)∗(g) = S(ui","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq476","equation_number":null,"raw_text":"j)(g) = (g−1)i","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq477","equation_number":null,"raw_text":"j = (g)j","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq478","equation_number":null,"raw_text":"where the last equality follows from the fact that g is unitary (i.e. g−1 = g†,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq479","equation_number":null,"raw_text":"Proposition 19.2. The coordinate ring Cq[G] = O(Gq) of the quantum","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq480","equation_number":null,"raw_text":"group Gq = GLq(V ) is also a Hopf ∗-algebra.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq481","equation_number":null,"raw_text":"i)∗= S(ui","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq482","equation_number":null,"raw_text":"q = D−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq483","equation_number":null,"raw_text":"the (co)representation M = (mj","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq484","equation_number":null,"raw_text":"i = (Mi","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq485","equation_number":null,"raw_text":"j)∗. Thus, in the classical case (i.e. when q = 1), M∗= M†.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq486","equation_number":null,"raw_text":"We say that the corepresentation W is unitarizable if it has a basis B =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq487","equation_number":null,"raw_text":"B = I. In this case, we say B is a unitary basis","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq488","equation_number":null,"raw_text":"Theorem 19.1 (Woronowicz). The coordinte ring Cq[G] = O(Gq) is a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq489","equation_number":null,"raw_text":"every polynomial irreducible representation of G = GLn(C) is of this form.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq490","equation_number":null,"raw_text":"q→1 Vq,λ(Gq) = Vλ(G).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq491","equation_number":null,"raw_text":"Vλ(GLn(C)) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq492","equation_number":null,"raw_text":"recursively, the case n = 1 being trivial.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq493","equation_number":null,"raw_text":"Vq,λ(Gq(Cn)) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq494","equation_number":null,"raw_text":"q,λ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq495","equation_number":null,"raw_text":"Let V = Cn, G = GLn(C) = GL(V ), Vλ = Vλ(G) a Weyl module of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq496","equation_number":null,"raw_text":"G, Gq = GLq(V ) the standard quantum group, Vq the q-deformation of V","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq497","equation_number":null,"raw_text":"on which GLq(V ) acts, Vq,λ = Vq,λ(Gq) the q-deformation of Vλ(G), and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq498","equation_number":null,"raw_text":"GTq,λ = GT n","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq499","equation_number":null,"raw_text":", where d = |λ| =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq500","equation_number":null,"raw_text":"If d = 1, then Vλ(G) = V = V ⊗1. Otherwise, obtain a Young diagram","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq501","equation_number":null,"raw_text":"Vμ(G) ⊗V =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq502","equation_number":null,"raw_text":"Vq,μ ⊗Vq =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq503","equation_number":null,"raw_text":"q,λ(b) = ρ(GTq,λ(b)) ∈V ⊗d","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq504","equation_number":null,"raw_text":"ments crystallize at q = 0. This means:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq505","equation_number":null,"raw_text":"q,λ(b) = vi1(b) ⊗· · · ⊗vid(b),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq506","equation_number":null,"raw_text":"q,λ(b) = vj1(b) ⊗· · · ⊗vjd(b),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq507","equation_number":null,"raw_text":"of crystallization, hence the name. The maps b 7→i(b) = (i1(b), . . . , id(b))","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq508","equation_number":null,"raw_text":"and b 7→j(b) = (j1(b), . . . , jd(b)) are computable in poly(⟨b⟩) time (where","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq509","equation_number":null,"raw_text":"Ei =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq510","equation_number":null,"raw_text":"Let Fi = ET","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq511","equation_number":null,"raw_text":"q,λ(b)) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq512","equation_number":null,"raw_text":"eei · b =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq513","equation_number":null,"raw_text":"b′(q) = 1,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq514","equation_number":null,"raw_text":"subring of functions in Q(q) regular at q = 0 (i.e. without a pole at q = 0).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq515","equation_number":null,"raw_text":"A lattice within W is an R-submodule of W such that Q(q) ⊗R L = W.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq516","equation_number":null,"raw_text":"in Rn is a Z-submodule L of Rn such that R ⊗Z L = Rn.)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq517","equation_number":null,"raw_text":"possible, i.e., for all b, b′ ∈B, if eei(b) = b′ ̸= 0 then efi(b′) = b, and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq518","equation_number":null,"raw_text":"similarly, if efi(b) = b′ ̸= 0 then eei(b′) = b.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq519","equation_number":null,"raw_text":"It can be shown that if W = Vq,λ(Gq), then there exists a unique b ∈B","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq520","equation_number":null,"raw_text":"such that eei(b) = 0 for all i; this corresponds to the highest weight vector","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq521","equation_number":null,"raw_text":"L = LGT","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq522","equation_number":null,"raw_text":"(Lα, Bα) ⊗(Lβ, Bβ) = (Lα ⊗Lβ, Bα ⊗Bβ) is the unique crystal basis","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq523","equation_number":null,"raw_text":"erties of the crystal bases. Recall that the specialization of Vq,α at q = 1 is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq524","equation_number":null,"raw_text":"the Weyl module Vα of G = GLn(C), and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq525","equation_number":null,"raw_text":"Vα ⊗Vβ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq526","equation_number":null,"raw_text":"α,β = #{b ⊗b′ ∈Bα ⊗Bβ|∀i, eei(b ⊗b′) = 0 and b ⊗b′ has weight γ}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq527","equation_number":null,"raw_text":"is to find an explicit basis B = Bα⊗β of Vα ⊗Vβ that is compatible with this","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq528","equation_number":null,"raw_text":"B = B0 ⊇B1 ⊇· · · ⊇∅","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq529","equation_number":null,"raw_text":"basis Bq = Bq,α⊗β of Vq,α ⊗Vq,β such that:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq530","equation_number":null,"raw_text":"b =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq531","equation_number":null,"raw_text":"provided by the Littlewood-Richardson rule. By specializing at q = 1, we","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq532","equation_number":null,"raw_text":"position problem even at q = 1. This may give some idea of the power of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq533","equation_number":null,"raw_text":"In the Kronecker problem, we let H = GL(Cn) and G = GL(Cn ⊗Cn).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq534","equation_number":null,"raw_text":"Vγ(G) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq535","equation_number":null,"raw_text":"In the plethysm problem, we let H = GL(Cn) and G = GL(Vμ(H)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq536","equation_number":null,"raw_text":"Vλ(G) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq537","equation_number":null,"raw_text":"homomorphism from the standard quantum group Hq = GLq(Cn) and to","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq538","equation_number":null,"raw_text":"(ρ, V ) is indecomposable is there is no expression V = W1 ⊕W2 such that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq539","equation_number":null,"raw_text":"{v′ ∈V |∃g ∈G with ρ(g)(v) = v′}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq540","equation_number":null,"raw_text":"{g ∈G|ρ(g)(v) = v}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq541","equation_number":null,"raw_text":"One may also define v ∼v′ if there is a g ∈G such that ρ(g)(v) = v′. It is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq542","equation_number":null,"raw_text":"then easy to show that [v]∼= O(v).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq543","equation_number":null,"raw_text":"dual representation (ρ∗, V ∗) defined as ρ∗(v∗)(v) = v∗(ρ(g−1)(v)). It will","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq544","equation_number":null,"raw_text":"be convenient for ρ∗to act on the right, i.e., ((v∗)(ρ∗))(v) = v∗(ρ(g−1)(v)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq545","equation_number":null,"raw_text":"mial functions on V of degree d. Let dim(V ) = n and let X1, . . . , Xn be a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq546","equation_number":null,"raw_text":"R = C[X1, . . . , Xn] = ⊕∞","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq547","equation_number":null,"raw_text":"d=0Rd = ⊕∞","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq548","equation_number":null,"raw_text":"d=0Symd(V ∗)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq549","equation_number":null,"raw_text":"(f · g)(v) = f(g−1 · v)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq550","equation_number":null,"raw_text":"For an f ∈R, we say that f is an invariant if f · g = f for all g ∈G.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq551","equation_number":null,"raw_text":"• Stab(f) = G.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq552","equation_number":null,"raw_text":"• f(g · v) = f(v) for all g ∈G and v ∈V .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq553","equation_number":null,"raw_text":"• For all v, v′ such that v′ ∈Orbit(v), we have f(v) = f(v′).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq554","equation_number":null,"raw_text":"such that g · φ(w1) = φ(g · w1) for all g ∈G and w1 ∈W1 then we say that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq555","equation_number":null,"raw_text":"• h(αw + βw′, w′′) = αh(w, w′′) + βh(w′, w′′) for all α, β ∈C and all","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq556","equation_number":null,"raw_text":"• h(w′′, αw + βw′) = αh(w′′, w) + βh(w′′, w′) for all α, β ∈C and all","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq557","equation_number":null,"raw_text":"• h(w, w) > 0 for all w ̸= 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq558","equation_number":null,"raw_text":"Z⊥= {w ∈W|h(w, z) = 0 ∀z ∈Z}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq559","equation_number":null,"raw_text":"Also recall that W = Z ⊕Z⊥.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq560","equation_number":null,"raw_text":"We say that an inner product h is G-invariant if h(g·w, g·w′) = h(w, w′)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq561","equation_number":null,"raw_text":"h(g · x, z) = h(g−1 · g · x, g−1 · z) = h(x, g−1 · z) = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq562","equation_number":null,"raw_text":"hG(w, w′) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq563","equation_number":null,"raw_text":"hG(w, w) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq564","equation_number":null,"raw_text":"where w′ = g′ · w. Thus hG(w, w) > 0 unless w = 0. Secondly, by the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq565","equation_number":null,"raw_text":"hG(g · w, g · w′) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq566","equation_number":null,"raw_text":"sentations Vi. Thus V = ⊕iVi, where (ρi, Vi) is an irreducible repre-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq567","equation_number":null,"raw_text":"Proof: Suppose that Z ⊆V is an invariant subspace, then V = Z ⊕Z⊥is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq568","equation_number":null,"raw_text":"let p(X) ∈R = C[X1, . . . , Xn] be a polynomial function. We define the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq569","equation_number":null,"raw_text":"pG(v) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq570","equation_number":null,"raw_text":"Lemma 21.2. Let p ∈RG be an invariant and let Z = V (p) be the variety","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq571","equation_number":null,"raw_text":"R. Then there are p1 ∈I1 and p2 ∈I2 so that p1 + p2 = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq572","equation_number":null,"raw_text":"RG = ⊕∞","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq573","equation_number":null,"raw_text":"d=0RG","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq574","equation_number":null,"raw_text":"h(RG) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq575","equation_number":null,"raw_text":"d=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq576","equation_number":null,"raw_text":"W G = {w ∈W|g · w = w}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq577","equation_number":null,"raw_text":"dimC(W G) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq578","equation_number":null,"raw_text":"Proof: Define P =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq579","equation_number":null,"raw_text":"matrices. We see that ρ(g) · P = P · ρ(g) and that P 2 = P. Thus P is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq580","equation_number":null,"raw_text":"invariant. For that, let wg = g · w. We then have that Pw = w implies that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq581","equation_number":null,"raw_text":"w =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq582","equation_number":null,"raw_text":"We may thus assume that each ρ(g) is unitary, we have that wg = w for all","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq583","equation_number":null,"raw_text":"h(RG) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq584","equation_number":null,"raw_text":"Proof: Let dimC(W) = n and let {X1, . . . , Xn} be a basis of W ∗. Since","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq585","equation_number":null,"raw_text":"RG = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq586","equation_number":null,"raw_text":"di = d}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq587","equation_number":null,"raw_text":"trace(ρd(g)) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq588","equation_number":null,"raw_text":"d:|d|=d","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq589","equation_number":null,"raw_text":"standard representation of Sn is obviously on V = Cn with","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq590","equation_number":null,"raw_text":"σ · (v1, . . . , vn) = (vσ(1), . . . , vσ(n))","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq591","equation_number":null,"raw_text":"Sn acts on R = C[X1, . . . , Xn] by","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq592","equation_number":null,"raw_text":"Xi · σ = Xσ(i). The orbit of any point v = (v1, . . . , vn) is the collection of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq593","equation_number":null,"raw_text":"polynomials ek(X), for k = 1, . . . , n, where","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq594","equation_number":null,"raw_text":"ek(X) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq595","equation_number":null,"raw_text":"ek(v) ̸= ek(w). This follows from the theory of equations in one variable.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq596","equation_number":null,"raw_text":"h(RG) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq597","equation_number":null,"raw_text":"A related action of Sn is the diagonal action: Let X = {X1, . . . , Xn},","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq598","equation_number":null,"raw_text":"Y = {Y1, . . . , Yn}, and so on upto W = {W1, . . . , Wn} be a family of r","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq599","equation_number":null,"raw_text":"(disjoint) variables. Let B = C[X, Y, . . . , W] be the ring of polynomials in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq600","equation_number":null,"raw_text":"We define the action of Sn on X ∪Y ∪. . . ∪W as Xi · σ = Xσ(i),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq601","equation_number":null,"raw_text":"Yi · σ = Yσ(i), and so on.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq602","equation_number":null,"raw_text":"The invariants BG is obtained from the r = 1 case by a curious operation:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq603","equation_number":null,"raw_text":"DXY = Y1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq604","equation_number":null,"raw_text":"e2(X) = X1X2 + X1X3 + . . . + Xn−1Xn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq605","equation_number":null,"raw_text":"DXY (e2(X)) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq606","equation_number":null,"raw_text":"i̸=j","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq607","equation_number":null,"raw_text":"SL(V ) and v ∈V , we have φ · v = φ(v) is the action of φ on v.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq608","equation_number":null,"raw_text":"In terms of the basis x above we may write v = [x1, . . . , xn][α1, . . . , αn]T","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq609","equation_number":null,"raw_text":"and thus φ·v as [φ·x1, . . . , φ·xn][α1, . . . , αn]T . If φ·xi = [x1, . . . , xn][a1i, . . . , ani]T ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq610","equation_number":null,"raw_text":"φ · v = [x1, . . . , xn]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq611","equation_number":null,"raw_text":"We may now work with SLn(C) or simply SLn. Given a vector a =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq612","equation_number":null,"raw_text":"w, we see that there is an element A ∈SLn such that w = Av. Furthermore,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq613","equation_number":null,"raw_text":"for any B ∈SLn, clearly Bv ̸= 0. Thus we see that Cn has exactly two","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq614","equation_number":null,"raw_text":"O0 = {0} O1{v ∈Cn|v ̸= 0}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq615","equation_number":null,"raw_text":"Note that O1 is dense in V = Cn and its closure includes the orbit O0 and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq616","equation_number":null,"raw_text":"Let R = C[X1, . . . Xn] be the ring of polynomial functions on Cn. We","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq617","equation_number":null,"raw_text":"be such that the evaluations Xi(xj) = δij must be conserved. Thus if the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq618","equation_number":null,"raw_text":"column vectors xi = [0, . . . , 0, 1, 0 . . . , 0]T are the basis vectors of V and the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq619","equation_number":null,"raw_text":"row vectors Xi = [0, . . . , 0, 1, 0 . . . , 0] that of V ∗, then a matrix A ∈SLn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq620","equation_number":null,"raw_text":"transforms xi to Axi and Xj to XjA−1. Thus Xj(xi) = Xj/cdotxi goes to","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq621","equation_number":null,"raw_text":"XjA−1Axi = Xj · xi.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq622","equation_number":null,"raw_text":"Next, we examine R = C[X] for invariants. First note that the action of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq623","equation_number":null,"raw_text":"O0. In this case, if p(O1) = α then by the density of O1 in V , we see that p","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq624","equation_number":null,"raw_text":"is actually constant on V . Thus RG = R0 = C.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq625","equation_number":null,"raw_text":"x ∈V r be a matrix and y be a column vector such that x · y = 0. We see","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq626","equation_number":null,"raw_text":"that (A · x) · y = 0 as well. Thus if ann(x) = {y ∈Cr|x · y = 0} is the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq627","equation_number":null,"raw_text":"annihilator space of x, then we see that ann(x) = ann(A · x).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq628","equation_number":null,"raw_text":"Proposition 22.1. Let r < n and x, x′ ∈V r be such that ann(x) = ann(x′).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq629","equation_number":null,"raw_text":"Then there is an A ∈SLn such that x = A · x′.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq630","equation_number":null,"raw_text":"of course when ann(x) = 0. Thus, when x ∈Vr is a mtrix of rank r, we see","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq631","equation_number":null,"raw_text":"that ann(x) = 0 is trivial. The generic element of V r is of this form. Thus","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq632","equation_number":null,"raw_text":"ann(O′) and dim(ann(O)) = dim(ann(O′)) −1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq633","equation_number":null,"raw_text":"that O = [x] and O′ = [x′] with both x and x′ such that rowspace(x) ⊇","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq634","equation_number":null,"raw_text":"rowspace(x′) with rank(x′) = rank(x)−1. Then, upto SLn, we may assume","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq635","equation_number":null,"raw_text":"that the rows x′[1], . . . , x′[k] match the first k rows of x, and that x′[k+1] =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq636","equation_number":null,"raw_text":"x′[k + 2] = . . . = x′[n] = 0. Note that x[k + 1] is non-zero and x[k + 2] exists","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq637","equation_number":null,"raw_text":"A(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq638","equation_number":null,"raw_text":"We see that A(t) ∈SLn for all t ̸= 0. Next, if we let x(t) = A(t) · x, then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq639","equation_number":null,"raw_text":"t→0 x(t) = x′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq640","equation_number":null,"raw_text":"Proposition 22.3. Let r ≥n and x, x′ ∈V r be such that ann(x) = ann(x′).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq641","equation_number":null,"raw_text":"If (i) rank(x) < n then there is an A ∈SLn such that x′ = Ax, (ii) if","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq642","equation_number":null,"raw_text":"rank(x) = n there is a unique A ∈SLn and a λ ∈C∗such that if z = A · x,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq643","equation_number":null,"raw_text":"then the first n −1 rows of z equal those of x′ and z[n] = λx′[n].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq644","equation_number":null,"raw_text":"of [r] = {1, 2, . . . , r} such that det(x[C]) ̸= 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq645","equation_number":null,"raw_text":"all points x′ ∈V r such that (i) ann(x) = ann(x′) and (ii) det(X′[C]) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq646","equation_number":null,"raw_text":"Proof: Notice that if A ∈SLn and z = Ax then det(x[C]) = det(z[C]).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq647","equation_number":null,"raw_text":"ensures that rank(x′) = n and that condition (i) holds with equality. Thus","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq648","equation_number":null,"raw_text":"x′ys = 0 and det(x′[C]) = det(x[C]) determines the orbit O(x). Thus O(x)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq649","equation_number":null,"raw_text":"equals those x′ such that ann(x) = ann(x′), (ii) O(x) is not closed and its","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq650","equation_number":null,"raw_text":"the space of all polynomials in the variable matrix X = (Xij) where i =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq651","equation_number":null,"raw_text":"1, . . . , n and j = 1, . . . , r. It is clear that for any set C of n columns of X,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq652","equation_number":null,"raw_text":"we see that det(X[C]) is an invariant. We will denote C as C = c1 < c2 <","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq653","equation_number":null,"raw_text":"Let C0 = {1 < 2 < . . . < n} and W ⊆V r be the space of all matrices","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq654","equation_number":null,"raw_text":"x ∈V r such that x[C0] = diag(1, . . . , 1, λ). Let W ′ be those elements of W","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq655","equation_number":null,"raw_text":"for which λ ̸= 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq656","equation_number":null,"raw_text":"Lemma 22.1. Let W ′ be as above. (i) If x ∈W ′, then O(x) ∩W ′ = x, (ii)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq657","equation_number":null,"raw_text":"for any x ∈V r such that det(x[C0]) ̸= 0, there is a unique A ∈SLn such","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq658","equation_number":null,"raw_text":"Let us call Z′ ⊆V r as those x such that det(x[C0]) ̸= 0. We then have","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq659","equation_number":null,"raw_text":"W =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq660","equation_number":null,"raw_text":"W = {Wi,j|i = 1, . . . , n, j = n + 1, . . . , r} ∪{Wn,n}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq661","equation_number":null,"raw_text":"Let x ∈Z′ be an arbitrary point and A = x[C0]. Let Ci,j be the set","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq662","equation_number":null,"raw_text":"for i ̸= n","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq663","equation_number":null,"raw_text":"for i ̸= n","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq664","equation_number":null,"raw_text":"A · M = AMA−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq665","equation_number":null,"raw_text":"Note that dimC(M) = n2. Let X = {Xij|1 ≤i, j ≤n} be the dual space","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq666","equation_number":null,"raw_text":"That these are invariants ic clear, for Tr(AMiA−1) = Tr(M) for any A, M.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq667","equation_number":null,"raw_text":"Also note that Tr(X) = X11 + . . . + Xnn is linear in the X’s.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq668","equation_number":null,"raw_text":"matrices. Thus M = M0 ⊕M1 where M0 are all matrices M such that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq669","equation_number":null,"raw_text":"Tr(M) = 0. The one-dimensional complementary space M1 is composed of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq670","equation_number":null,"raw_text":"In other words, if JCF(M) = JCF(M′) them M′ ∈O(M). Furthermore,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq671","equation_number":null,"raw_text":"n with i1 + . . . + in = d. The typical form may be written as:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq672","equation_number":null,"raw_text":"f(X) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq673","equation_number":null,"raw_text":"i1+...+in=d","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq674","equation_number":null,"raw_text":"A−1 =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq675","equation_number":null,"raw_text":"ficients {Bi||i| = d}. The action of A is obtained by substituting","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq676","equation_number":null,"raw_text":"in f(X) and recoomputing the coefficients. we illustrate this for n = 2 and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq677","equation_number":null,"raw_text":"d = 2. Then, the generic form is given by:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq678","equation_number":null,"raw_text":"f(X1, X2) = B20X2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq679","equation_number":null,"raw_text":"W = Z ⊕Z′ and Z′ is SLn-invariant as well.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq680","equation_number":null,"raw_text":"such that if p is an invariant then Ω(p · p′) = p · Ω(p′). The definition of Ω","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq681","equation_number":null,"raw_text":"associative, i.e. (g1·g2)·g3 = g1·(g2·g3) for all g1, g2, g3 ∈G, (ii) and algebraic","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq682","equation_number":null,"raw_text":"inverse i : G →G, i.e, g · i(g) = i(g) · g = 1G, where 1G is (iii) a special","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq683","equation_number":null,"raw_text":"element 1G ∈G, which functions as the identity, i.e., 1G · g = g · 1G = g for","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq684","equation_number":null,"raw_text":"The essential example is obviously SLn. Clearly for G = SLn, we have","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq685","equation_number":null,"raw_text":"C[G] = C[X]/(det(X)−1), where X is the n×n indeterminate matrix. The","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq686","equation_number":null,"raw_text":"μ∗(f) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq687","equation_number":null,"raw_text":"i=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq688","equation_number":null,"raw_text":"To continue with our example, consider SL2 and Z = Symd(V ∗). C[Z] =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq689","equation_number":null,"raw_text":"C[B20, B11, B02] and C[G] = C[A11, A12, A21, A22]/(A11A22 −A21A12 −1).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq690","equation_number":null,"raw_text":"μ∗(B20) = A2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq691","equation_number":null,"raw_text":"φ : Z →Z′ be a morphism. We say that φ is G-equivariant if φ(μ(g, z)) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq692","equation_number":null,"raw_text":"μ∗(f) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq693","equation_number":null,"raw_text":"i=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq694","equation_number":null,"raw_text":"μ(g)∗(f) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq695","equation_number":null,"raw_text":"i=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq696","equation_number":null,"raw_text":"M(f) = C · {μ(g)∗(f)|g ∈G} ⊆C · {f1, . . . , fk}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq697","equation_number":null,"raw_text":"Proposition 23.1. Let S = {s1, . . . , sm} be a finite subset of C[Z]. Then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq698","equation_number":null,"raw_text":"If O(z) = ∆(z), then we have the G-equivariant embedding iz : O(z) →","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq699","equation_number":null,"raw_text":"there is an ideal I(z) = ker(i∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq700","equation_number":null,"raw_text":"z) such that C[O(z)] ∼= C[Z]/I(z). Since the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq701","equation_number":null,"raw_text":"Exercise 23.1. Let us consider Z = Sym2(V ∗) ∼= C3 where V is the stan-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq702","equation_number":null,"raw_text":"dard representation of G = SL2. As we have seen, C[Z] = C[B20, B11, B02].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq703","equation_number":null,"raw_text":"There is only one invariant δ = B2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq704","equation_number":null,"raw_text":"11 −4B02B20. Thus C[Z]G = C[δ]. Thus","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq705","equation_number":null,"raw_text":"Z/G is precisely Spec(C[δ]) = C, the complex plane. The map π is executed","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq706","equation_number":null,"raw_text":"π(a, b, c) = b2 −4ac","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq707","equation_number":null,"raw_text":"Clearly, if f, f ′ ∈Z such that f = g · f ′ for g ∈SL2 then δ(f) = δ(f ′). We","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq708","equation_number":null,"raw_text":"look at the converse: if f, f ′ are such that δ(f) = δ(f ′) then is it that f ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq709","equation_number":null,"raw_text":"O(f ′)? We begin with f = aX2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq710","equation_number":null,"raw_text":"that a ̸= 0. If that is the case, we make the substitution X1 →X1 −αX2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq711","equation_number":null,"raw_text":"if a = 0 one may do a similar transform. Thus in general, if f is not the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq712","equation_number":null,"raw_text":"2. It is now clear that δ(f) = δ(f ′) implies that c = c′. Thus","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq713","equation_number":null,"raw_text":"the general answer is that if f, f ′ ̸= 0 and δ(f) = δ(f ′) then f ∈O(f ′).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq714","equation_number":null,"raw_text":"Next, let us examine the form 0. we see that δ(0) = 0. Thus we see that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq715","equation_number":null,"raw_text":"(i) for any point d ∈C, if d ̸= 0, then π−1(d) consists of a single orbit, (ii)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq716","equation_number":null,"raw_text":"orbits O(f) are closed when δ(f) ̸= 0, (iv) O(X2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq717","equation_number":null,"raw_text":"Theorem 3. Let G be a finite group and act on the space Z. Let R = C[Z]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq718","equation_number":null,"raw_text":"and RG = C[Z]G be the ring of invariants. Let π : Z →Z/G be the quotient","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq719","equation_number":null,"raw_text":"(i) For any ideal J ⊆RG, we have (J · R) ∩RG = J.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq720","equation_number":null,"raw_text":"f = r1f1 + . . . + rkfk","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq721","equation_number":null,"raw_text":"Note that we have used the fact that if h ∈RG and p ∈R then (h · p)G =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq722","equation_number":null,"raw_text":"π(z) = x and let IO(x) ⊆R be the ideal of all functions in R vanishing at","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq723","equation_number":null,"raw_text":"all points of the orbit O(z) of z. We show that rad(Jx · R) = IO(z) which","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq724","equation_number":null,"raw_text":"If O(z′) ̸= O(z) then we already have that there is a p ∈RG such that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq725","equation_number":null,"raw_text":"p(O(z)) = 0 and p(O(z′)) = 1. Since this p ∈Jx · R, the variety of Jx · R","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq726","equation_number":null,"raw_text":"By the Hilbert Basis theorem, IG =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq727","equation_number":null,"raw_text":"(f1, . . . , fk)·C[Z], where each fi is itself an invariant. we claim that C[Z]G =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq728","equation_number":null,"raw_text":"f =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq729","equation_number":null,"raw_text":"f =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq730","equation_number":null,"raw_text":"G-invariant hyperplane N ⊆M, there is a decomposition M = H ⊕P as","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq731","equation_number":null,"raw_text":"Note that s · f =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq732","equation_number":null,"raw_text":"f+(s·f−1·f) and thus M(f) = N(f)⊕C·f as vector spaces. Thus N(f) is a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq733","equation_number":null,"raw_text":"G-invariant hyperplane of M(f). By the Nagata hypothesis, M(f) = f ′⊕H,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq734","equation_number":null,"raw_text":"where f ′ is an invariant. If f ′ ̸∈N(f) then M(f) = f ′⊕N(f) and the lemma","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq735","equation_number":null,"raw_text":"follows. However, if f ′ ∈N(f), then f = f ′ + h where h ∈H. Since H is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq736","equation_number":null,"raw_text":"f −f ′ = h with f ′ invariant, implies that s · h −t · h = s · f −t · f. Thus","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq737","equation_number":null,"raw_text":"N(h) ⊆N(f). We take f ∗= h∗. Examining f −f ∗, we see that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq738","equation_number":null,"raw_text":"f −h∗= f ′ + h −h∗∈N(f)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq739","equation_number":null,"raw_text":"i C[Z] · fi) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq740","equation_number":null,"raw_text":"Proof: This is proved by induction on r. Say f = Pr","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq741","equation_number":null,"raw_text":"i=1 hifi with hi ∈C[Z].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq742","equation_number":null,"raw_text":"such that hr = h′ + h′′. We tackle h′ as follows. Since f is an invariant, we","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq743","equation_number":null,"raw_text":"i=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq744","equation_number":null,"raw_text":"(s · hi −t · hi)fi = s · f −t · f = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq745","equation_number":null,"raw_text":"(s · hr −t · hr)fr =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq746","equation_number":null,"raw_text":"i=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq747","equation_number":null,"raw_text":"It follows from this that h′fr = Pr−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq748","equation_number":null,"raw_text":"i=1 h′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq749","equation_number":null,"raw_text":"f =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq750","equation_number":null,"raw_text":"i=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq751","equation_number":null,"raw_text":"f −h′′fr =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq752","equation_number":null,"raw_text":"i=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq753","equation_number":null,"raw_text":"Proof: Since C[Z] is a finitely generated C-algebra, C[Z] = C[f1, . . . , fk]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq754","equation_number":null,"raw_text":"φ∗: C[W] →C[Z] by defining φ(Wi) = fi. Since C[W] is a free algebra, we","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq755","equation_number":null,"raw_text":"that C[W]G is already finitely generated over C, say C[W]G = C[h1, . . . , hk].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq756","equation_number":null,"raw_text":"that φ∗(h) = f. Consider the space N(h). A typical generator of N(h) is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq757","equation_number":null,"raw_text":"f −f = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq758","equation_number":null,"raw_text":"applying φ∗we see that φ∗(h∗) = φ(h) = f. Thus there is an invariant h∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq759","equation_number":null,"raw_text":"such that φ∗(h∗) = f. Now h∗∈C[h1, . . . , hk] implies that f = φ∗(h∗) ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq760","equation_number":null,"raw_text":"f(W1) = 0 and f(W2) = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq761","equation_number":null,"raw_text":"section is empty, by Hilbert Nullstellensatz, we have I1 + I2 = 1. Whence","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq762","equation_number":null,"raw_text":"there are functions f1 ∈I1 and f2 ∈I2 such that f1 + f2 = 1. For arbitrary","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq763","equation_number":null,"raw_text":"(s · f1 −t · f1) + (s · f2 −t · f2) = 1 −1 = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq764","equation_number":null,"raw_text":"where f = f1 + f ′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq765","equation_number":null,"raw_text":"1 ∈I1 we see that f(W1) = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq766","equation_number":null,"raw_text":"On the other hand, f = 1 −(f2 + f ′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq767","equation_number":null,"raw_text":"2) ∈1 + I2 and thus f(W2) = 1. □","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq768","equation_number":null,"raw_text":"(ii) z1 ≈z2 ifff(z1) = f(z2) for all f ∈C[Z]G.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq769","equation_number":null,"raw_text":"then f(z1) = f(z2). So let z be an element of the intersection. Since f is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq770","equation_number":null,"raw_text":"dense, and f is continuous, we have f(z) = α. Thus f(z1) = f(z2) = f(z) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq771","equation_number":null,"raw_text":"Exercise 24.1. Consider the action of G = SLnon M, the space of n × n-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq772","equation_number":null,"raw_text":"A · M = AMA−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq773","equation_number":null,"raw_text":"Let R = C[M] = C[X11, . . . , Xnn] be the ring of functions on M.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq774","equation_number":null,"raw_text":"invariants RG is generated as a C-algebra by the forms ei(X) = Tr(Xi), for","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq775","equation_number":null,"raw_text":"i = 1, . . . , n. The forms {ei|1 ≤i ≤n} are algebraically independent and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq776","equation_number":null,"raw_text":"thus RG is the polynomial ring C[e1, . . . , en]. Clearly then Spec(RG) ∼= Cn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq777","equation_number":null,"raw_text":"π : M ∼= Cn2 →Cn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq778","equation_number":null,"raw_text":"ei(M) = λi","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq779","equation_number":null,"raw_text":"tuple μ = (μ1, . . . , μn) there is a unique set λμ = {λ1, . . . , λn} such that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq780","equation_number":null,"raw_text":"r = μi. Clearly, for the diagonal matrix D(λμ) = diag(λ1, . . . , λn), we","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq781","equation_number":null,"raw_text":"have that π(D(λ)) = μ. This verifies that π is surjective.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq782","equation_number":null,"raw_text":"For a given μ, the set π−1(μ) are all matrices M with Spec(M) = λ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq783","equation_number":null,"raw_text":"then π−1(μ) consists of just one orbit: matrices M such that JCF(M) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq784","equation_number":null,"raw_text":"D(λ). For a general λ, the orbit of M with JCF(M) = D(λ) is the unique","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq785","equation_number":null,"raw_text":"As an example, consider the case when n = 2 and the matrix:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq786","equation_number":null,"raw_text":"N =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq787","equation_number":null,"raw_text":"Consider the family A(t) = diag(t, t−1) ∈SL2. We see that:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq788","equation_number":null,"raw_text":"N(t) = A(t)NA(t)−1 =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq789","equation_number":null,"raw_text":"Thus limt→0 N(t) = diag(λ, λ), the diagonal matrix.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq790","equation_number":null,"raw_text":"relation ≈on points in Z which is coarser than ∼=, the orbit equivalence","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq791","equation_number":null,"raw_text":"algebraic group with C[G] = C[T, T −1]. Furthermore, C∗has a compact","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq792","equation_number":null,"raw_text":"subroup S1 = {z ∈C, |z| = 1}, the unit circle.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq793","equation_number":null,"raw_text":"t · z = tkz","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq794","equation_number":null,"raw_text":"for (C∗)m, let χ = (χ[1], . . . , χ[m]) be a sequence of integers. For such a χ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq795","equation_number":null,"raw_text":"we define the representation Cχ as follows: Let t = (t1, . . . , tm) ∈(C∗)m be","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq796","equation_number":null,"raw_text":"t · z = tχ[1]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq797","equation_number":null,"raw_text":"be such a map such that λ(t) = [aij(t)] where aij(T) ∈C[T, T −1].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq798","equation_number":null,"raw_text":"important substitution is for t = eiθ, and we obtain a 2π-periodic map","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq799","equation_number":null,"raw_text":"We see that λ(0) = I. Let the derivative at 0 for λ be X.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq800","equation_number":null,"raw_text":"Lemma 25.1. Let f : C →SLn be a smooth map such that f(0) = I and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq801","equation_number":null,"raw_text":"f ′(0) = X, where X is an n × n-matrix. Then for θ ∈C,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq802","equation_number":null,"raw_text":"= eθX","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq803","equation_number":null,"raw_text":"Now, since λ is 2π-periodic, we must have e(n2π+θ)X = eθX. This forces (i)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq804","equation_number":null,"raw_text":"Then the image of λ is closed and V ∼= C[m1] ⊕. . .⊕C[mn], for some integers","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq805","equation_number":null,"raw_text":"m1, . . . , mn, where n = dimC(V ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq806","equation_number":null,"raw_text":"Then the image of λ is closed and V ∼= Cχ1 ⊕. . . ⊕Cχn, for some integers","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq807","equation_number":null,"raw_text":"χ1, . . . , χn, where n = dimC(V ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq808","equation_number":null,"raw_text":"is the diagonal matrices diag(t1, . . . , tn) where ti ∈C∗and t1t2 . . . tn = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq809","equation_number":null,"raw_text":"T ′ = ATA−1. Thus all maximal tori are conjugate to D above.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq810","equation_number":null,"raw_text":"nent of N(D). Then N(D)o = D and N(D)/D is the Weyl group","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq811","equation_number":null,"raw_text":"We consider the case when G = (C∗)r. Clearly, for a given λ : C∗→G,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq812","equation_number":null,"raw_text":"λ(t) = (tm1, . . . , tmr)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq813","equation_number":null,"raw_text":"χ(t1, . . . , tr) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq814","equation_number":null,"raw_text":"λ ◦χ(t) = tm1a1+...+mrar","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq815","equation_number":null,"raw_text":"Theorem 10. Let G = (C∗)r. Then Γ(G) ∼= Zr and X(G) ∼= Zr. Further-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq816","equation_number":null,"raw_text":"Exercise 25.1. Let G = (C∗)3 and λ and χ be as follows:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq817","equation_number":null,"raw_text":"Then, λ ∼= [3, −1, 2] and χ ∼= [−1, 1, 2]. We evaluate the pairing:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq818","equation_number":null,"raw_text":"By the above theorem, Γ(D), X(D) ∼= Zn−1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq819","equation_number":null,"raw_text":"Yn = {[m1, . . . , mn] ∈Zn|m1 + . . . + mn = 0}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq820","equation_number":null,"raw_text":"It is easy to see that Yn ∼= Zn−1. In fact, we will set up a special bijection","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq821","equation_number":null,"raw_text":"θ([m1, m2, . . . , mn]) = [m1, m1 + m2, . . . , m1 + . . . + mn−1]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq822","equation_number":null,"raw_text":"θ−1[a1, . . . , an−1] = [a1, a2 −a1, a3 −a2, . . . , an−1 −an−2, −an−1]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq823","equation_number":null,"raw_text":"explicitly via θ and θ∗. If [m1, . . . .mn] ∈Zn ∼= X(D∗), then it maps to","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq824","equation_number":null,"raw_text":"[m1 −m2, . . . , mn−1 −mn] ∈Zn−1 ∼= X((C∗)n−1) via θ∗. If we push this","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq825","equation_number":null,"raw_text":"Proposition 25.2, we see that W is a direct sum W = Cχ1 ⊕. . . ⊕CχN,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq826","equation_number":null,"raw_text":"where N = dimC(W). Collecting identical characters, we see that:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq827","equation_number":null,"raw_text":"W = ⊕χ∈X(D)Cmχ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq828","equation_number":null,"raw_text":"Thus W is a sum of mχ copies of the module Cχ. Clearly mχ = 0 for all but","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq829","equation_number":null,"raw_text":"{[1, −1, 0], [−1, 2, −1], [0, −1, 1]}, with C[1,−1,0] ∼= C · X1 and so on.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq830","equation_number":null,"raw_text":"gation. We see at once that M = M0 ⊕C·I where M0 is the 8-dimensional","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq831","equation_number":null,"raw_text":"a weight vector. Afterall t · (g · w) = g · t′ · w where t′ = g−1tg. Thus","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq832","equation_number":null,"raw_text":"t · (g · w) = χ(t′)(g · w)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq833","equation_number":null,"raw_text":"χ = [m1, . . . , mn], then χ′ = [mσ(1), . . . , mσ(n)] for some permutation σ ∈Sn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq834","equation_number":null,"raw_text":"t→0 λ(t) · w = 0W","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq835","equation_number":null,"raw_text":"w = w1 + w2 + . . . + wr","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq836","equation_number":null,"raw_text":"t · w = t(λ,χ1)w1 + . . . + t(λ,χr)wr","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq837","equation_number":null,"raw_text":"the conjugacy result on maximal tori, we know that T = ADA−1 for some","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq838","equation_number":null,"raw_text":"Consider the form f = (X1 + X2 + X3)2. We see that supp(f) is set of all","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq839","equation_number":null,"raw_text":"we see that A·f = X2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq840","equation_number":null,"raw_text":"λ(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq841","equation_number":null,"raw_text":"t→0 t · (A · f) = t2X2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq842","equation_number":null,"raw_text":"λ(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq843","equation_number":null,"raw_text":"such that n1 + n2 + n3 = 0. We may assume that n1 ≥n2 ≥n3. Looking at","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq844","equation_number":null,"raw_text":"t · X = (tni−njxij)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq845","equation_number":null,"raw_text":"Thus if limt→0 t · X is to be 0 then xij = 0 for all i > j. In other words,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq846","equation_number":null,"raw_text":"λ(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq847","equation_number":null,"raw_text":"λ′(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq848","equation_number":null,"raw_text":"λ′′(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq849","equation_number":null,"raw_text":"t · w = tn1w1 + . . . + tnkwk","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq850","equation_number":null,"raw_text":"is m(λ) = min{n1, . . . , nk}. Verify that this really does not depend on the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq851","equation_number":null,"raw_text":"λ2(t) = λ(t2). It is easy to see that m(λ2) = 2 · m(λ). Clearly, λ and λ2 are","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq852","equation_number":null,"raw_text":"λ(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq853","equation_number":null,"raw_text":"Also recall that N(T)/T = W is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq854","equation_number":null,"raw_text":"Since there is an A such that AD′A−1 = D, it is clear","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq855","equation_number":null,"raw_text":"that ∥λ∥= ∥AλA−1∥. Thus, we are left to check if ∥λ′∥= ∥λ∥when (i)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq856","equation_number":null,"raw_text":"Im(λ), Im(λ′) ⊆D, and (ii) λ′ = AλA−1 for some A ∈SLn. This throws","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq857","equation_number":null,"raw_text":"Since W ∼= Sn, the symmetric group, and since","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq858","equation_number":null,"raw_text":"e(λ) = m(λ)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq859","equation_number":null,"raw_text":"We immediately see that e(λ) = e(λ2).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq860","equation_number":null,"raw_text":"w = 0W . If N(w, D) is non-empty then there is a unique λ′ ∈N(w, D)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq861","equation_number":null,"raw_text":"λ ̸= (λ′)k for any k ∈Z. This 1-parameter subgroup will be denoted by","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq862","equation_number":null,"raw_text":"P(λ) = {A ∈SLn| lim","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq863","equation_number":null,"raw_text":"t→0 λ(t)Aλ(t−1) = I ∈SLn}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq864","equation_number":null,"raw_text":"λ(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq865","equation_number":null,"raw_text":"with a1 ≥a2 ≥. . . ≥an (obviously with a1 + . . . + an = 0). Then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq866","equation_number":null,"raw_text":"P(λ) = {(xij|xij = 0 for all i, j such that ai < aj}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq867","equation_number":null,"raw_text":"U(λ) = (xij) where =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq868","equation_number":null,"raw_text":"xij = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq869","equation_number":null,"raw_text":"xij = δij","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq870","equation_number":null,"raw_text":"if ai = aj","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq871","equation_number":null,"raw_text":"P(λ) and P(gλg−1) = P(λ), (ii) gλg−1 ∈N(w, gDg−1), and (iii) e(λ) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq872","equation_number":null,"raw_text":"of a best λ then λ′ is ’equally best’ and P(λ) = P(λ′).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq873","equation_number":null,"raw_text":"sequence (V0, . . . , Vr) of nested subspaces 0 = V0 ⊂V1 ⊂. . . ⊂Vr = V .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq874","equation_number":null,"raw_text":"Lemma 26.3. Let dimC(V ) = r and let F = (V0, . . . , Vr) and F′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq875","equation_number":null,"raw_text":"of V and a permutation σ ∈Sr such that Vi = {b1, . . . , bi} and V ′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq876","equation_number":null,"raw_text":"i =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq877","equation_number":null,"raw_text":"Γ(SLn), we have e(λ) ≥e(λ′), and (ii) for all λ′ such that e(λ) = e(λ′) we","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq878","equation_number":null,"raw_text":"have P(λ) = P(λ′) and that there is a g ∈P(λ) such that λ′ = gλg−1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq879","equation_number":null,"raw_text":"that AλA−1 ∈N(A · w, D) and e(λ) = e(AλA−1). Since the ’best’ element","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq880","equation_number":null,"raw_text":"Next, let λ1 = λ(w, T1) and λ2 = λ(w, T2) be two ’best’ elements of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq881","equation_number":null,"raw_text":"say D, and P(λ(w, Ti))-conjugates λi such that (i) e(λi) = e(λ(w, Ti)) and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq882","equation_number":null,"raw_text":"(ii) Im(λi) ⊆D. By lemma 26.1, we have λ1 = λ2 and thus P(λ1) = P(λ2).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq883","equation_number":null,"raw_text":"On the other hand, P(λ(w, Ti)) = P(λi) and this proves (ii). □","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq884","equation_number":null,"raw_text":"clear that P(A · w) = AP(w)A−1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq885","equation_number":null,"raw_text":"Proof: Let g ∈Gw. Since g · w = w, we see that gP(w)g−1 = P(w), and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq886","equation_number":null,"raw_text":"SLn-invariant embedding φ : W →X such that φ−1(0X) = S, scheme-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq887","equation_number":null,"raw_text":"of W, there is an ideal Is = (f1, . . . , fk) of definition for S. We may further","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq888","equation_number":null,"raw_text":"φ(w) = (f1(x), . . . , fk(x))","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq889","equation_number":null,"raw_text":"Note that φ(S) = 0X and that IS = (f1, . . . , fk) ensure that the requirements","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq890","equation_number":null,"raw_text":"t→0[λ(t) · φ(z)] = 0X","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq891","equation_number":null,"raw_text":"are homogeneous n-forms on M, we consider the SL(M)-module W =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq892","equation_number":null,"raw_text":"(A) Consider the group K = SLn × SLn. We define","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq893","equation_number":null,"raw_text":"(B) Consider the group H = L × L. We define the action μH of typical","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq894","equation_number":null,"raw_text":"Theorem 14. The points det and perm in the SL(M-module W = Symn(M∗)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq895","equation_number":null,"raw_text":"q = 0 and the Robinson-Schensted correspondence, in Physics and","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":221311,"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}}