{"paper_meta":{"paper_id":"arxiv:0709.0748","title":"0709.0748","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:0709.0748v1 [cs.CC] 5 Sep 2007\nOn P vs. NP, Geometric Complexity Theory, and\nThe Flip I: a high-level view\nDedicated to Sri Ramakrishna\nKetan D. Mulmuley ∗\nThe University of Chicago\nhttp://ramakrishnadas.cs.uchicago.edu\nTechnical Report TR-2007-13, Computer Science Department,\nThe University of Chicago\nSeptember, 2007\nNovember 7, 2018\nAbstract\nGeometric complexity theory (GCT) is an approach to the P vs.\nNP and related problems through algebraic geometry and representa-\ntion theory. This article gives a high-level exposition of the basic plan\nof GCT based on the principle, called the flip, without assuming any\nbackground in algebraic geometry or representation theory.\nContents\n1\nIntroduction\n4\n1.1\nThe flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n7\n1.1.1\nFrom nonexistence to existence . . . . . . . . . . . . .\n7\n1.1.2\nFrom hard to easy . . . . . . . . . . . . . . . . . . . .\n10\n1.2\nThe P-barrier and its crossing . . . . . . . . . . . . . . . . . .\n11\n∗Part of this work was done while the author was visiting I.I.T. Mumbai\n1\n\n1.3\nWhy should PH1 and PH2 hold? . . . . . . . . . . . . . . . .\n13\n1.4\nThe reduction . . . . . . . . . . . . . . . . . . . . . . . . . . .\n15\n1.5\nTowards PH1 and SH via PH0\n. . . . . . . . . . . . . . . . .\n16\n1.6\nNonstandard quantum groups . . . . . . . . . . . . . . . . . .\n18\n1.7\nNonstandard Riemann hypotheses? . . . . . . . . . . . . . . .\n19\n1.8\nObstructions vs. expanders\n. . . . . . . . . . . . . . . . . . .\n22\n1.9\nIs there a simpler proof technique? . . . . . . . . . . . . . . .\n23\n1.10 Organization of the paper . . . . . . . . . . . . . . . . . . . .\n24\n2\nBasics in algebraic geometry and representation theory\n24\n2.1\nRepresentation theory . . . . . . . . . . . . . . . . . . . . . .\n26\n2.1.1\nIrreducible representations of GLn(C) . . . . . . . . .\n27\n2.1.2\nIrreducible representations of the symmetric group . .\n29\n2.1.3\nTensor products\n. . . . . . . . . . . . . . . . . . . . .\n30\n2.2\nAlgebraic geometry . . . . . . . . . . . . . . . . . . . . . . . .\n32\n3\nGroup-theoretic varieties\n34\n4\nClass varieties\n35\n4.1\nNC vs. P #P\n. . . . . . . . . . . . . . . . . . . . . . . . . . .\n36\n4.2\nP vs. NP problem over C . . . . . . . . . . . . . . . . . . . .\n39\n5\nObstructions\n40\n6\nWhy should obstructions exist?\n41\n7\nThe flip\n44\n8\nWhy should the flip work?: the P-barrier\n45\n9\nOn crossing the P-barrier\n46\n9.1\nA basic prototype with constant depth complexity\n. . . . . .\n46\n9.2\nFrom constant to superpolynomial depth . . . . . . . . . . . .\n48\n9.3\nSaturated and positive integer programming . . . . . . . . . .\n49\n2\n\n10 Why should PH1 and PH2 hold?\n51\n11 Decision problems in representation theory\n52\n11.0.1 Littlewood-Richardson problem . . . . . . . . . . . . .\n53\n11.0.2 Kronecker problem . . . . . . . . . . . . . . . . . . . .\n53\n11.0.3 The plethysm problem . . . . . . . . . . . . . . . . . .\n53\n12 The P-barrier in representation theory\n54\n12.1 Crossing the P-barrier . . . . . . . . . . . . . . . . . . . . . .\n54\n13 Reduction\n56\n13.1 Towards easy discovery . . . . . . . . . . . . . . . . . . . . . .\n57\n13.2 From easy algorithm for discovery to easy proof of existence .\n59\n14 Standard quantum group\n61\n15 Nonstandard quantum groups\n65\n15.1 Quantization\n. . . . . . . . . . . . . . . . . . . . . . . . . . .\n66\n15.2 PH0 for couples and triples\n. . . . . . . . . . . . . . . . . . .\n70\n15.3 PH0 for class varieties . . . . . . . . . . . . . . . . . . . . . .\n70\n15.4 PH1 and SH . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n70\n16 Ultimate mystery: nonstandard Riemann hypotheses?\n71\n17 Obstructions vs. expanders\n72\n18 On relativization and P/poly-naturalization barriers\n74\n18.1 From usefulness to superpolynomial lower bounds . . . . . . .\n77\n18.2 On violation of the largeness constraint\n. . . . . . . . . . . .\n79\n18.3 On violation of the constructivity constraint . . . . . . . . . .\n80\n19 P-verifiable and P-constructible proof techniques and their\nexplicit construction complexity\n80\n19.1 P-verifiable proof technique . . . . . . . . . . . . . . . . . . .\n80\n19.2 P-barrier for verification . . . . . . . . . . . . . . . . . . . . .\n83\n3\n\n19.3 P-constructible proof technique . . . . . . . . . . . . . . . . .\n85\n19.4 General setting . . . . . . . . . . . . . . . . . . . . . . . . . .\n86\n19.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . .\n88\n19.5 Explicit construction complexity\n. . . . . . . . . . . . . . . .\n90\n19.6 Is there a simpler proof technique? . . . . . . . . . . . . . . .\n94\n1\nIntroduction\nGeometric complexity theory (GCT) is a plausible approach to the P vs\nNP [Co, Ka, Le] and related problems in complexity theory via algebraic\ngeometry and representation theory. The goal of this paper is to give a high-\nlevel overview of its basic plan and the underlying principle called the flip,\nwithout assuming any background in algebraic geometry or representation\ntheory. A detailed exposition for mathematicians will appear in [GCTflip2].\nA brief proposal and announcement appeared earlier in cf.[GCTconf]. The\nflip has been partially implemented in a series of papers [GCT1]-[GCT11].\nThis article, followed by [GCTintro], should provide an introduction to the\noverall structure of GCT for computer scientists who wish to get a high-level\npicture before going any further. We assume a few elementary notions of\nalgebraic geometry and representation theory in this introduction. They are\ndescribed in full detail in Section 2, which can be referred to if necessary. For\nthe readers looking for a quick overview, the article [GCTabs], which gives a\nnontechnical synopsis of this paper, followed by just this introduction, which\nhas been written to read as a short paper, should suffice.\nIn this article, the underlying field of computation is taken be C. In\n[GCT11], the problems that arise in the context of the flip over an alge-\nbraically closed field of positive characteristic, or a finite field are discussed.\nThe usual P ̸= NP conjecture is over a finite field of which the one over C\nis in a sense the crux and, being also a formal implication [GCT1], has to\nbe proved first anyway.\nThe flip, in essence, “reduces” the negative hypotheses (lower bound\nproblems) in complexity theory, such as the P ̸=?NP conjecture over C, to\npositive hypotheses in complexity theory (upper bound problems): specifi-\ncally, to showing that a series of decision problems in representation theory\nand algebraic geometry belong to the complexity class P. The “reduction”\nhere is only “in essence”.\nIt is not a formal Turing machine reduction.\nIf it were, it would be relativizable. It is described briefly in Section 1.4\n4\n\nbelow, and in detail in Section 13 later. This reduction basically consti-\ntutes a flip from hard, nonexistence to easy existence.\nIn [GCT6], these\ncomplexity-theoretic positive hypotheses are further reduced to mathemat-\nical positivity hypotheses, supported by the theoretical and experimental\nevidence therein. The mathematical positivity hypotheses roughly say that\ncertain nonnegative structural functions in algebraic geometry and repre-\nsentation theory have positive formulae–i.e., formulae without alternating\nsigns–akin to the usual formula for the permanent (in contrast, the usual\nformula for the determinant has alternating signs). It turns out that the\nvalidity of these mathematical positivity hypotheses is intimately linked to\nthe Riemann hypothesis over finite fields–proved in [Dl2] as a culmination\nof extensive effort in mathematics–and the related works in algebraic ge-\nometry and the theory of quantum groups [BBD, KL2, Kas3, Lu1, Lu2].\nIn [GCT6], a plan is suggested for proving them via the theory of quan-\ntum groups. Generalizations of the standard quantum group [Dri, Ji, RTF]\nneeded for this purpose, which we call nonstandard quantum groups, are\nconstructed in [GCT4, GCT7], with further conjectural extensions pointed\nout in [GCT10]. All papers of GCT together suggest that if the Riemann\nhypothesis over finite fields and the related works in the theory of standard\nquantum groups mentioned above can be systematically extended to the\nsetting of the nonstandard quantum groups that arise in GCT, then this\nmay lead to the proof of the P ̸= NP conjecture over C. This basic plan\nof GCT is summarized in Figure 1. Question marks indicate the main open\nproblems.\nThe proof in characteristic zero may eventually extend to finite fields, as\nin the usual form of the conjecture, along the lines suggested in [GCT11].\nThus the ultimate goal of the GCT flip is to deduce the ultimate nega-\ntive hypothesis of mathematics, the P ̸= NP conjecture, in essence, from\nthe ultimate positive hypotheses in mathematics, (nonstandard) Riemann\nHypotheses, thereby giving the ultimate flip shown in Figure 2.\nIn the rest of this introduction, we elaborate Figure 1 further.\nAcknowledgement\nThe author is deeply grateful to Madhav Nori, who taught him algebraic\ngeometry, Milind Sohoni, who collaborated in GCT 1-4, and Manju the\nsource of energy behind this work. The author is also grateful to A. Razborov\nfor pointing out the need for a high-level account. This article is essentially\nan elaboration of the answers to his questions. A part of this work was done\n5\n\nComplexity theoretic negative hypotheses (lower bound problems)\n|\n|\nThe flip\n|\n↓\nComplexity theoretic positive hypotheses (upper bound problems)\n|\n|\nGCT6|\n|\n↓\nMathematical positivity hypotheses |\n|\n?\n|\n↓\n(?): Nonstandard extensions of the Riemann hypothe-\nsis over finite fields, and the related works in algebraic\ngeometry and the theory of quantum groups\nFigure 1: The basic plan of GCT\n6\n\n(?): Nonstandard Riemann Hypotheses (+)\n|\n|\n?\n|\n↓\nThe P ̸= NP conjecture (-)\nFigure 2: The ultimate goal of the flip\nwhile the author was visiting I.I.T. Mumbai to which the author is grateful\nfor its hospitality. It is also a pleasure to thank the graduate students who\ntook the accompanying introductory course [GCTintro] on GCT for their\nfeedback.\n1.1\nThe flip\nWe begin with the top arrow in Figure 1: the flip. It is motivated by the\nclassical flip–from the undecidable (negative) to the decidable (positive)–\nthat occurs in G ̈odel’s incompleteness theorem.\nAll known lower bound\nresults–e.g. the hierarchy theorems in complexity theory or the lower bound\nresults in the constant depth [BS] or the PRAM model without bit oper-\nations [Mu1]–depend on flips from lower bounds to upper bounds of some\nsort. But such variations of the classical flip cannot work in the context\nof the P vs. NP problem because they are either relativizable [BGS] or\nnaturalizable [RR]. In contrast, the flip here should be nonrelativizable and\nnonnaturalizable (Section 18).\nThere are actually two flips within this flip: (1) from nonexistence to\nexistence, and (2) from hard to easy.\nHere hard means: the problem of\ndeciding if a computational circuit of size m exists for a given function\nf(x) = f(x1, . . . , xn) is hard. Accordingly, the flip from hard nonexistence\nto easy existence goes in two stages.\n1.1.1\nFrom nonexistence to existence\nThe flip from nonexistence to existence is addressed in [GCT1, GCT2]. Here\nthe nonexistence (lower bound) problem is reduced to an existence problem:\n7\n\nspecifically, to the problem of proving existence of obstructions, which serve\nas “proofs” or “witnesses” for nonexistence of an efficient computational\ncircuit for the explicit hard function in the lower bound problem under\nconsideration.\nJust as existence of a forbidden Kurotowoski minor in a\ngraph serves as an obstruction, i.e., a “proof” for nonexistence of a planar\nembedding.\nAn obstruction in [GCT1, GCT2] is intuitively defined as follows. First\na specific (co)-NP-complete function E(X) = E(x1, . . . , xn), and a specific\nP-complete function H(Y ) = H(y1, . . . , yl) are constructed in [GCT1] so as\nto have special properties that we shall describe in a moment. Using H(Y ),\na projective algebraic variety XP (l) = XP (H; l), for every positive integer l,\nis associated with the complexity class P, called the class variety associated\nwith P, or the simply the P-variety. Here, a projective algebraic variety\nmeans the zero set of a system of homogeneous polynomial equations (cf.\nSection 2.2). These are generalizations of the familar curves and surfaces.\nIt will turn out that XP (l) is a G-variety for G = GLl(C), the group of\ninvertible l × l complex matrices. This means elements of G act on this\nvariety as its transformations–i.e., move its points around–just as G acts on\nCl in the usual way. Similarly, using E(X), a projective variety XNP (n, l) =\nXNP (E; n, l), for every positive integer n and l ≥n, is associated with the\ncomplexity class NP.\nIt is called the class variety associated with NP,\nor simply the NP-variety.\nIt will again be a G-variety.\nThe functions\nE(X) and H(Y ) have been specially chosen so that these class varieties are\nexceptional and their algebraic geometry can be analyzed in depth. If E(X)\ncan be computed by a circuit of size m, then it would turn out that XNP (n, l)\ncan be embedded in XP (l) as a G-subvariety for l = O(m2). Pictorially:\nXNP (n, l) ֒→XP (l).\n(1)\nWe want to show that this embedding is impossible if m = poly(n), as\nn →∞. This would show that E(X) cannot be computed by a circuit of\nm = poly(n) size, and hence, P ̸= NP over C.\nLet R(n, l) = R(E; n, l) and S(l) = S(H; l) denote the homogeneous\ncoordinate rings of XNP (E; n, l) and XP (H; l), respectively. Here by the\ncoordinate ring of a variety, we mean the ring of polynomial “functions” (of\nsome kind) on the variety as defined in Section 2.2. These are akin to the\nring of polynomial functions on Cl. Since the class varieties are G-varieties,\nthese homogeneous coordinate rings will be G-representations (Section 2.1).\nBy a G-representation we mean a vector space on which the elements of G\nact as linear transformations, just as they do on Cl. If the embedding (1)\n8\n\nexists, then it would turn out that R(n, l) is a G-subrepresentation of S(l).\nWe say that an irreducible, i.e., a minimal, nonzero representation W of G\nis an obstruction, for given n and l, if it occurs as a G-subrepresentation\nof R(n, l), but not as a G-subrepresentation of S(l). Existence of such a\nW, for given n and l, implies that R(n, l) cannot be embedded as a G-\nsubrepresentation of S(l), and hence, the embedding (1) cannot exist. Thus\nan obstruction serves as a “witness” or a “proof” that the embedding (1)\ncannot exist.\nWe now reformulate this notion of obstruction using a few basic notions\nin representation theory described in Section 2.1. It is known that (polyno-\nmial) irreducible representations of G are in one-to-one correspondence with\nthe set of sequences, also called partitions, λ : λ1 ≥λ2 · · · λk > 0 of positive\nintegers of length k ≤l. The irreducible representation of G labelled by λ is\ncalled a Weyl-module, and is denoted by Vλ(G). It is also known that each\nfinite dimensional representation V of G can be written as a direct sum of\nirreducible representations:\nV =\nM\nλ\nmλVλ(G),\nwhere mλVλ(G) denotes the direct sum of mλ copies of Vλ(G), and each\nmλ, called the multiplicity of Vλ(G) in V , is uniquely defined. Thus Vλ(G)\noccurs in V as a subrepresentation iffthe multiplicity mλ is nonzero.\nLet R(E; n, l)d and S(H; l)d denote the subspaces in R(E; n, l) and S(H; l),\nrespectively, of forms of degree d.\nLet sλ\nd(H; l) denote the multiplicity\nof Vλ(G) in S(H; l)d.\nLet sλ\nd(E; n, l) denote the multiplicity of Vλ(G) in\nR(E; n, l)d. Then Vλ(G) is an obstruction for given n and l ifffor some d\nsλ\nd(E; n, l) is nonzero but sλ\nd(H; l) is zero. Here d is uniquely determined\nby the size P\ni λi of λ. We also say that Vλ(G) is an obstruction of degree\nd, and by an abuse of language, also that the label λ is an obstruction of\ndegree d.\nThe main algebro-geometric result of [GCT2] (Theorem 6.3) indicates\nthat such obstructions should exist in the context of the P vs. NP problem,\nwhen m = poly(n), assuming that P ̸= NP, as we expect. The goal then is\nto show that obstructions indeed exist, as expected, for all n →∞, assuming\nm = poly(n). The story is similar for other related lower bound problems.\nThis addresses the easier half of the flip from nonexistence to existence.\n9\n\n1.1.2\nFrom hard to easy\nBut how should one prove that obstructions actually exist? The main hy-\npothesis governing the flip, which addresses this question, is the following\none that constitutes the harder half of the flip: from hard to easy.\nHypothesis 1.1 (PHflip1) Consider the P vs. NP problem over C. Let\nE(X) be the explicit function in [GCT1] mentioned above. Then the follow-\ning problems are “easy”; i.e., belong to P. Specifically,\n(a) Verification of an obstruction:\ngiven n, l and the partition λ,\nwhether Vλ(G) is an obstruction for given n and l can be decided in poly(n, l, ⟨λ⟩)\ntime, where ⟨λ⟩denotes the bitlength of the specification of λ.\n(b) Explicit construction of obstructions: Suppose l = nlog n (say).\nThen, for every n →∞, a label λ(n) of an obstruction Vλ(G) for n and l\ncan be constructed explicitly in poly(n, l) time, thereby proving existence of\nan obstruction for every such n and l.\nIn view of the definition of an obstruction, the statement (a) for verifi-\ncation clearly follows from:\nHypothesis 1.2 (PHflip2) The following the decision problems are easy;\ni.e., belong to P. Specifically,\n(a) Given d, n, l and a partition λ, whether sλ\nd(E; n, l) is nonzero, i.e.,\nwhether Vλ(G) occurs as a G-subrepresentation of R(n, l)d can be decided\nin poly(⟨d⟩, ⟨λ⟩, n, l) time. Here ⟨d⟩denotes the bitlength of d.\n(b) Given d, l and a partition λ, whether sλ\nd(H; l) is nonzero, i.e., whether\nVλ(G) occurs as a G-subrepresentation of S(l)d can be decided in poly(⟨d⟩, ⟨λ⟩, l)\ntime.\nThe decision problems in Hypothesis 1.2 are the crux of the matter.\nOnce easy algorithms for these decision problems are found, the goal is to\nprove existence of an obstruction for every n →∞, when l = nlog n (say),\nby constructing such an obstruction explicitly, as per Hypothesis 1.1 (b).\nWe shall discuss how this is to done in Section 1.4 below. Assuming for\nthe moment that this transformation of easy algorithms for the decision\nproblems in Hypothesis 1.2 into an easy procedure for explicit construction\nof obstructions (Hypothesis 1.1(b)) for all n →∞, when l = nlog n, works,\nwe get the “reduction” shown in the top arrow of Figure 1: from the original\nhard nonexistence (lower bound) problem to the basic upper bound problems\nin Hypothesis 1.2.\n10\n\n1.2\nThe P-barrier and its crossing\nBut, by divine justice, the task of showing that the problems in Hypothe-\nsis 1.2 are easy turned out to be extremely hard. Thus, paradoxically, the\nhardest aspect of the flip is just to prove that the basic decision problems\nthat arise in the construction of obstructions are actually easy; i.e., belong\nto P. The best algorithms for these decision problems obtained using the\ngeneral purpose algorithms in algebraic geometry and representation theory\ntake space that is double exponential in m and time that is triple expo-\nnential in m. This means even verification of an obstruction, let alone its\ndiscovery, takes time that is triple exponential in m if one were to use the\ngeneral purpose techniques.\nThe gap between this triple exponential time bound and the polynomial\ntime bound sought in Hypothesis 1.2 is so huge that, at the surface, this\nhypothesis may seem impossible. This was the main barrier, called the P-\nbarrier (Section 8), on this path towards the P vs NP problem when the\nflip was briefly announced in [GCTconf].\nThe article [GCT6] says that it can be crossed under reasonable mathe-\nmatical assumptions. We now turn to a brief description of these results.\nFor that we need a few definitions.\nWe say that a function f(k), k a nonnegative integer, is a quasi-polynomial\nif for some integer l ≥1 there exist polynomials fi(k), 1 ≤i ≤l, such that\nf(k) = fi(k) if k = i modulo l. Here l is called the period of the quasi-\npolynomial. An important example of a quasi-polynomial is the Ehrhart\nquasi-polynomial fP(k) of a polytope P. By definition, it is the number of\ninteger points in the dilated polytope kP. This is known to be a quasi-\npolynomial [St1].\nWe say that a quasi-polynomial f(k) is positive, if the coefficients of all\nfi(k) are nonnegative. We say that it is saturated if either f1(k) is identically\nzero as a polynomial, or if not, f(1) = f1(1) ̸= 0. If f(k) is positive, it is\nclearly saturated.\nNext, let us associate with the multiplicities sλ\nd(H; l) and sλ\nd(E; n, l) the\nfollowing stretching functions:\n ̃sλ\nd(H; l)(k) = skλ\nkd(H; l),\n(2)\nand\n ̃sλ\nd(E; n, l)(k) = skλ\nkd(E; n, l).\n(3)\n11\n\nThe following is the main algebro-geometric result in [GCT6].\nTheorem 1.3 (cf. Theorem 3.4.11 in [GCT6])\n(Rationality Hypothesis): Assume that the singularities of the class varieties\nXP (H; m) and XNP(E; n, l) are “nice” (rational).\nThen the stretching functions ̃sλ\nd(H; l)(k) and ̃sλ\nd(E; n, l)(k) are quasi-\npolynomials.\nWe do not need to know the exact definition of a rational singularity here,\nwhich can be found in [Ke]. It just means that the singularities are nice.\nThis depends on the exceptional nature of the class varieties (cf. Section 4)\nand is supported by the algebro-geometric results and arguments in [GCT2,\nGCT10].\nUsing Theorem 1.3, we can now formulate the conjectural mathematical\npositivity hypotheses mentioned in the third box from above in Figure 1.\nAssume the rationality hypothesis above.\nHypothesis 1.4 (PH1:) The structural constant sλ\nd(H; l) can be expressed\nas the number of integer points in a polytope P λ\nd (H; l) of poly(l, ⟨d⟩, ⟨λ⟩)\ndimension, whose Ehrhart quasi-polynomial coincides with the stretching\nquasi-polynomial ̃sλ\nd(H; l)(k) in Theorem 1.3. Furthermore, P λ\nd (H; l) can\nbe given in the form of a poly(l, ⟨d⟩, ⟨λ⟩)-time separation oracle as in [GLS].\nThere exists a polytope P λ\nd (E; n, l) for the structural constant sλ\nd(E; n, l)\nwith similar properties.\nThis, in particular, implies that sλ\nd(H; l) and sλ\nd(E; n, l) belong to #P.\nHypothesis 1.5 PH2: The quasi-polynomials ̃sλ\nd(H; l) and ̃sλ\nd(E; n, l) in\nTheorem 1.3 are positive.\nIts weaker form is:\nHypothesis 1.6 (SH:) These quasi-polynomials are saturated.\nPH1 and SH (PH2) together say that each decision problem in Hypothe-\nsis 1.2 can be transformed in polynomial time into a special kind of an integer\nprogramming problem called saturated (resp. positive) integer programming\nproblem (Section 9.3).\n12\n\nTheorem 1.7 (cf. [GCT6]) The decision problems in Hypothesis 1.2 are\nindeed in P, assuming PH1 and SH (or more strongly PH2) above.\nThis follows from a polynomial time algorithm in [GCT6] for saturated (pos-\nitive) integer programming.\nThis result reduces the positive complexity-theoretic hypotheses in Hy-\npothesis 1.2 to the mathematical positivity hypotheses PH1 and SH, as\nshown in the middle arrow in Figure 1. The algorithms in Theorem 1.7 are\nconceptually extremely simple. They just need linear programming [GLS]\nand computation of Smith normal forms [KB].\nBut their correctness depends on the positivity hypotheses PH1 and SH\n(PH2), whose validity, in turn, is intimately linked to deep phenomena in\nalgebraic geometry and the theory of quantum groups as we shall soon see.\nAn indication of such a link is already here. Since the proof of Theorem 1.3,\nwhich is necessary to even formulate these hypotheses, needs a few funda-\nmental results in algebraic geometry; namely, [Bou] (which in turn is based\non [Hi] and other results), and [Ke, Fl]. It should not then be surprising\nif the proofs the hypotheses need far more. Indeed, the quantum-group-\ntheoretic and algebro-geometric machinery is needed in GCT essentially to\nprove these hypotheses, and hence, that these extremely simple algorithms\nare actually correct.\n1.3\nWhy should PH1 and PH2 hold?\nBut first, we need to justify why these hypotheses should hold in the first\nplace. For that, let us consider the simplest analogue of the decision prob-\nlems in Hypothesis 1.2 in representation theory:\nProblem 1.8 (Littlewood-Richardson problem) Given partitions α, β and\nλ, decide if the Littlewood-Richardson coefficient cλ\nα,β (cf. Section 2.1.3) is\npositive (nonzero). This is defined to be the multiplicity of the irreducible\nrepresentation Vλ(G) in the tensor product Vα(G) ⊗Vβ(G) (which becomes\na G-representation by letting the elements of G act on its two factors simul-\ntaneously).\nThe analogous mathematical positivity hypotheses in this setting are as\nfollows.\nDefine the stretching function\n ̃cλ\nα,β(k) = ckλ\nkα,kβ,\nk ≥0,\n13\n\nwhich is obtained by stretching the Littlewood-Richardson coefficient by a\nfactor of k. It is known to be a polynomial [Der, Ki, Rs]. Then\nHypothesis 1.9 (PH1) The Littlewood-Richardson coefficient cλ\nα,β can be\nexpressed as the number of integer points in a polytope P = P λ\nα,β of dimen-\nsion polynomial in the total length of α, β and λ. Furthermore, the Ehrhart\nquasi-polynomial of P coincides with the stretching polynomial ̃cλ\nα,β(k) and\nthe membership function of P is computable in time that is polynomial in\nthe bit lengths of α, β and λ.\nThis is shown, for example, in [BZ]. There are many choices for P λ\nα,β.\nOne choice is called a hive polytope [KT1].\nHypothesis 1.10 (PH2) The coefficients of ̃cλ\nα,β(k) are nonnegative.\nThis implies:\nHypothesis 1.11 (SH) The stretching polynomial ̃cλ\nα,β(k) is saturated.\nSince ̃cλ\nα,β(k) is a polynomial, this simply means if ckλ\nkα,kβ is nonzero for\nsome k ≥1 then cλ\nα,β is also nonzero. PH2 is still open, but has a con-\nsiderable experimental evidence in its support [KTT].\nThat SH holds is\nthe saturation theorem in [KT1]. PH1 and SH in conjunction with linear\nprogramming leads [DM2, GCT3, KT2] to a polynomial time algorithm for\nthe Littlewood-Richardson problem (Problem 1.8), and a polynomial time\nalgorithm [GCT5] for a certain generalized Littlewood-Richardson problem\nassuming SH. These results were indeed a starting motivation for Theo-\nrem 1.7.\nThe Littlewood-Richardson coefficient is a special case of a far-reaching\nclass of fundamental constants in representation theory, called plethysm\nconstants, described in Section 11. The structural constants sλ\nd(H; l) and\nsλ\nd(E; n, l) can be considered to be “hyped up” versions of the plethysm con-\nstant. Considerable theoretical and experimental evidence in support of the\nanalogous positivity hypotheses PH1 and PH2 for the plethysm constants\nis given in [GCT6]; cf. Section 11. This constitutes the main evidence in\nsupport of PH1 and PH2 for sλ\nd(H; l), sλ\nd(E; n, l) and other similar algebro-\ngeometric structural constants that arise in GCT.\n14\n\n1.4\nThe reduction\nBefore we turn to the plan suggested in [GCT6] for proving PH1 and SH,\nwe explain the nature of the reduction in the top arrow of Figure 1.\nFor this, the easy algorithms in Theorem 1.7 have to be transformed into\nan easy procedure for explicit construction of obstructions as per Hypothe-\nsis 1.1 (b). This transformation cannot be carried out at present since we\ndo not have explicit descriptions of the polytopes P λ\nd (H; l) and P λ\nd (E; n, l)\nin PH1. But it is explained in Section 13 and in detail in [GCT6] why it\nshould be possible to carry out this transformation if PH1 and SH can be\nproved and explicit descriptions of the polytopes therein become available.\nThe scheme for transformation suggested there goes in two steps:\nFirst, the easy algorithms in Theorem\n1.7 have to be used to get an\neasy poly(n, l) procedure for discovering an obstruction (label) for given n\nand l, if one exists.\nSecond, this easy algorithm for discovering an obstruction, or rather its\nstructure and the underlying techniques have to be used to prove that an\nobstruction always exists for every n →∞, assuming l = nlog n, say. That\nis, to prove that this easy algorithm always says “yes” for such n and l. Just\nas the structure of the easy Hungarian method for discovering a perfect\nmatching in a bipartite graph can be used to prove Hall’s theorem that\nevery d-regular bipartite graph always has a perfect matching.\nThis transformation of an easy algorithm for discovery into an easy (i.e.\nfeasible) constructive proof–which we shall call a P-constructive proof–also\ngives, as a side product, an easy, i.e., polynomial time algorithm for explicit\nconstruction of obstructions (labels), as in Hypothesis 1.1 (b). One may\nwonder why we are going for explicit construction of obstructions, when\njust their existence would have sufficed. Because the nature of obstructions\nhere is such that the complexity deciding their existence and of constructing\nthem explicitly, if they do, should be more or less the same; cf. Section 13.2.\nJust as the complexity of deciding if a bipartite graph has a perfect matching\nis more or less the same as that of constructing one, if it exists,\nIn the context of these transformations it is crucial that the algorithms\nin Theorem 1.7 are not only easy, i.e., polynomial-time algorithms, but\nalso have a genuinely simple structure of the right kind, being just varia-\ntions of linear programming. Of course, we can not hope to use the ellip-\nsoid algorithm for linear programming–which though simple is intricate–for\na constructive proof of existence of obstructions. Rather we have to use\n15\n\nthe structure of the underlying polytopes. The analogues of the polytopes\nP λ\nd (H; l) and P λ\nd (E; n, l) in PH1 in the simplified setting of the Littlewood-\nRichardson problem (Problem 1.8) are called hive polytopes [KT1]. These\nhave extremely regular structure. The same is expected to be the case for\nthe polytopes P λ\nd (H; l) and P λ\nd (E; n, l) that actually arise here. For this and\nother reasons given in [GCT6], it is expected that, once explicit descriptions\nof the polytopes P λ\nd (H; l) and P λ\nd (E; n, l) become available, the algorithms\nin Theorem 1.7 can be transformed into simple greedy Hungarian-type al-\ngorithms which do not even need linear programming.\nThis is the main\nreason why the transformation of these easy, polynomial time algorithms\ninto an easy (feasible) proof of existence of obstructions is expected to work\nin our setting, just as it does in the case of Hall’s theorem that we mentioned\nabove.\nAssuming that this works, we would get an explicit family {λ(n)} of\nobstructions (rather their labels), as n →∞, and l = nlog n. The existence\nof such an obstruction family would imply that P ̸= NP over C.\n1.5\nTowards PH1 and SH via PH0\nNow we turn to the basic plan suggested in [GCT6] for proving PH1 and\nSH. This will explain the bottom arrow in Figure 1.\nThis plan is motivated by the proof of PH1 (Hypothesis 1.9) in the\nsimplified setting of the Littlewood-Richardson problem via the theory of\nquantum groups [Kas1, Li, Lu2]. Specifically, it is known that this PH1 is a\nconsequence, in a nontrivial way, of a deep positivity statement in the the-\nory of standard quantum groups [Dri, Ji, RTF]–whose intuitive description\nis given later in Section 14–namely: their representations and coordinate\nrings have canonical bases [Kas2, Lu1, Lu2], whose structural constants de-\ntermining their representation-theoretic and multiplicative structure are all\nnonnegative. We shall refer to the existence of a canonical basis with this\npositivity property as PH0, the zeroth positivity hypothesis (property).\nMotivated by this work, certain positivity hypotheses, again called PH0,\nare formulated in [GCT6], and it is pointed out how and why these may\nsimilarly lead to the proof of the required PH1 and also SH (Hypotheses 1.4\nand 1.6). The PH0 hypotheses in [GCT6] may be thought of as generaliza-\ntions of PH0 in the theory of standard quantum groups. PH1 and SH for\nLittlewood-Richardson coefficients (Hypotheses 1.9 and 1.11) have purely\ncombinatorial proofs [F1, KT1], and hence, PH0 is strictly speaking not re-\nquired in this context. But in the context of the PH1 that we are finally\n16\n\ninterested in (Hypothesis 1.4) the full power of PH0 seems needed for the\nplan in [GCT8, GCT10] to work.\nA natural approach to prove PH0 in [GCT6] in the context of this PH1\nis to somehow generalize the proof of PH0 in the theory of the standard\nquantum group. But the theory of standard quantum groups does not work,\nas expected, in this context. The reason is briefly as follows.\nOne can associate a complexity class with each structural constant that\narises in GCT, which we call its index class. Roughly, if a structural con-\nstant is associated with a class variety for a complexity class C, then its\nindex class is defined to C.\nFor example, the index classes of the mul-\ntiplicities sλ\nd(H; l) and sλ\nd(E; n, l) are P and NP (over C), since they are\nassociated with P- and NP-varieties, respectively. Similarly, the index class\nof the Littlewood-Richardson coefficient is the class of circuits (of restricted\nkinds) of depth two; cf. Section 9.1. The index class of the Kronecker coef-\nficient (Section 2.1.3), which is the analogue of the Littlewood-Richardson\ncoefficient in the representation theory of the symmetric group, is NC2, the\nclass of problems that can be solved by circuits of log2 n depth and polyno-\nmial size. The Littlewood-Richardson coefficient as well as the Kronecker\ncoefficient are special cases of the plethysm constants (Section 11.0.3) which\nwe mentioned earlier. The generalized plethysm constant is not associated\nwith any class variety, but it is qualitatively similar to, though much sim-\npler than sλ\nd(E; n, l). Hence, we define its index class to be NP, with the\nunderstanding that this is to be taken only in a rough sense. The index\nclasses of the structural constants here are not be confused with their usual\ncomputational complexity classes: they are all (conjecturally) in #P by\nPH1.\nThe standard quantum group is the quantum group that occurs in the\ncontext of PH1 for Littlewood-Richardson coefficients (Hypothesis 1.9). Hence,\nwe define its index class to be the same as that of Littlewood-Richardson\ncoefficients, i.e., the class of circuits of depth two. Thus the standard quan-\ntum group is the quantum group attached to constant-depth (depth-two)\ncircuits.\nGiven a big difference between the lower bound problems for constant\nand nonconstant depth circuits, it should not be a surprise if the standard\nquantum group cannot be used in the context of PH1 for the structural\nconstants that actually arise in GCT; cf. Section 16 for an intuitive mathe-\nmatical explanation for why this is so.\n17\n\n1.6\nNonstandard quantum groups\nWhat is needed then are quantum groups that can play the role of the stan-\ndard quantum group in the context of the decision problems and positivity\nhypotheses for these structural constants. The main result in this context\nis the following:\nTheorem 1.12 [GCT4] There exists a quantum group, which is qualita-\ntively similar to the standard quantum group, that can play such a role in\nthe context of the Kronecker coefficients.\n[GCT7] More generally, there exists a (possibly singular) quantum group that\ncan play such a role in the context of the generalized plethysm constants.\nA less informal statement will be given later (Theorem 15.1). A conjectural\nscheme for generalizing these quantum groups to the ones that can play such\na role in the context of sλ\nd(E; n, l), sλ\nd(H; l) and other structural constants\nin GCT is suggested in [GCT10]. We shall call the new quantum groups in\nTheorem 1.12 nonstandard, because, though they are qualitatively similar\nto the standard quantum group, they are also fundamentally different, as\nexpected.\nThus, standard corresponds to constant depth and nonstandard to non-\nconstant depth circuits.\nThe article [GCT8] gives a conjecturally correct algorithm to construct\ncanonical bases of the irreducible representations and coordinate rings of the\nnonstandard quantum groups in [GCT4, GCT7] with the required positivity\nproperties (PH0). These are natural generalizations of the canonical basis\ndue to Kashiwara and Lusztig [Kas2, Lu1, Lu2] mentioned above for the\nirreducible representations and the coordinate ring of the standard quantum\ngroup.\n[GCT8] also gives a conjecturally correct algorithm to construct\ncanonical bases with similar positivity properties (PH0) for the nonstandard\ndeformations of the symmtric group algebra that are dually paired with the\nnonstandard quantum groups–these generalize the Kazhdan-Lusztig basis\n[KL1] of the Hecke algebra. It is also shown in [GCT7, GCT8] that PH1 for\nthe plethysm constants follows from PH0 and other conjectural properties\nof these nonstandard canonical bases and quantum objects. The story for\nthe general constants sλ\nd(E; n, l) and sλ\nd(H; l) can be expected to be similar\n[GCT10].\nAt present we can neither prove correctness of the algorithms in [GCT8]\nfor constructing nonstandard canonical bases nor the required conjectural\n18\n\nproperties for the reasons that we shall describe in a moment. But a consid-\nerable evidence is given in [GCT8] in support of PH0 for the nonstandard\nquantum group in [GCT4].\nIn the standard case, PH1 follows from PH0 in a more or less rigid\nway [Dh, Kas1, Li, Lu2]. This means the polytope that occurs in PH1 for\nthe Littlewood-Richardson coefficient (Hypothesis 1.9) is more or less de-\ntermined by the canonical basis for the standard quantum group–not com-\npletely, since there are a few choices for this polytope; e.g. a hive polytope\nin [KT1], or a polytope in [BZ]. But all these choices are intimately related.\nA common feature is that they all have extremely regular structures. The\nsame can be expected for the polytopes that should arise in the nonstandard\nsetting. This regularity is crucial for the final transformation of easy algo-\nrithms for the basic decision problems in Hypothesis 1.2 into easy algorithms\nfor explicit construction of obstructions; cf. Sections 1.4 and 13.\nExistence of nonstandard quantum groups of polylogarithmic [GCT4]\nand superpolynomial [GCT7] depth complexity, the conjecturally correct\nalgorithm in [GCT8] for constructing canonical bases (PH0) of their coordi-\nnate rings and irreducible representations, and the principle that is suggested\nby the theory of standard quantum groups–namely, once a canonical basis is\nthere (PH0), everything else in the story more or less follows a rigid path–is\nthe main reason why GCT may be expected to deliver lower bounds for\ncircuits of superpolynomial depth and size eventually.\n1.7\nNonstandard Riemann hypotheses?\nBut for this plan to work, PH0 for the nonstandard quantum groups has\nto be proved. This brings us to the main open question in this story: how\ncan we prove correctness of the algorithm in [GCT8] for constructing the\ncanonical bases (PH0) of the coordinate rings of the nonstandard quantum\ngroups?\nThere are two constructions of the canonical basis in the standard set-\nting. An algebraic construction in [Kas3], where it is called global crystal\nbasis, and a topological construction in [Lu1, Lu2]. Both constructions give\nrise to the same basis [GL]. In fact, both constructions follow the same basic\nscheme. Only the proofs of correctness of this basic scheme are different. The\ntopological proof is based on the theory of perverse sheaves [BBD], which\nin turn, is based on Riemann hypothesis over finite fields [Dl2]. In essence,\nPH0 is thus ultimately deduced in the topological proof from the Riemann\nHypothesis over finite fields, which is again a deep positive statement. Be-\n19\n\ncause its usual statement is, after all, a positive statement, and it can also be\nreformulated as stipulating positivity (nonnegativity) of some mathematical\nquantities (cf. page 458 in [Ha]). The topological proof also gives, as a side\nproduct, the only known proof of nonnegativity of the structural constants\nassociated with the canonical basis in the standard setting. Though this\nnonnegativity is not needed for proving PH1 for the Littlewood-Richardson\ncoefficients, it is crucial in the nonstandard setting for the reasons given in\n[GCT8, GCT10].\nFor this reason, the topological approach seems to be the only viable\noption in the nonstandard setting, as far as we can see. Besides, the algebraic\ncomplexity of the nonstandard quantum groups is so huge–as to be expected\nin view of the huge gap between constant and nonconstant depth circuits–\nthat a purely algebraic proof of correctness of the algorithm in [GCT8] for\nconstructing canonical bases in the nonstandard setting seems difficult.\nBut the standard Riemann hypothesis over finite fields and the related\ntechniques cannot be expected to work in the the nonstandard setting for\nthe reasons given in [GCT7, GCT8]. Again this should not be surprising\ngiven the big difference between constant and nonconstant depth circuits.\nHence what seem to be needed [GCT8] to make the topological approach\nwork in the nonstandard setting are nonstandard extensions of the Riemann\nhypothesis over finite fields and the related work on perverse sheaves. By\nnonstandard, we mean the extensions that will work in the context of the\nnonstandard quantum groups.\nThe author does not have the mathematical expertize to even formulate\nsuch hypotheses, let alone prove them. But the theoretical and experimental\nevidence in [GCT4, GCT7, GCT8] (cf. Section 16) suggests that such exten-\nsions exist, and that they ought to be provable by a systematic extension of\nthe theory of standard quantum groups to the nonstandard setting. Hence\nit is reasonable to hope that the experts would be able to do so eventually,\nleading to the proof of PH0 hypotheses along the topological lines, and fi-\nnally, to the explicit construction of obstructions as outlined above, which\nwould then imply that P ̸= NP over C. The whole picture is summarized\nin Figure 3, which is an elaboration of the earlier Figure 1. The arrows with\nquestion marks are conjectural, the double arrows are unconditional. The ?\nsigns indicate the main open problems at the heart of this approach. The\nstory over C may eventually lift to the story over finite fields along lines\nsuggested in [GCT11].\n20\n\n(?): Nonstandard Riemann Hypotheses for the quantum groups in\n[GCT4, GCT7], and their conjectural extensions in [GCT10]\n|\n|\n|\n?\n↓\nPH0 (?): Existence of canonical bases [GCT6, GCT8, GCT10] |\n|\n?\n↓\nPH1,SH (PH2)\n∥\n∥\nGCT6\n∥\n⇓\nPolynomial time algorithms for the de-\ncision problems in Hypothesis 1.2\n|\n|\nThe transformation mentioned in Section 1.4; cf Section 13 and [GCT6]\n|\n?\n↓\nExplicit construction of obstructions\n∥\n∥\n∥\n⇓\nP ̸= NP over C\nFigure 3: The basic plan for implementing the flip in GCT6\n21\n\n1.8\nObstructions vs. expanders\nAn initial motivation for going for explicit construction of obstructions as\nin Figure 3 was provided by explicit construction of expanders [LPS, Ma].\nAs explained in Section 17, the obstructions in GCT are in a certain sense\ngeneralizations of the expanders from constant depth to superpolynomial\ndepth circuits. Specifically, obstructions are to superpolynomial depth cir-\ncuits what expanders are to constant depth, in fact, depth two circuits; cf.\nFigure 4. In view of this relationship, explicit construction of obstructions as\nin Figure 3 would be in the setting of superpolynomial depth circuits what\nexplicit construction of expanders is in the setting of constant depth circuits.\nAs we remarked earlier, the standard quantum group also corresponds to\ncircuits of depth two. That is, expanders and the standard quantum group\nboth correspond to the class of depth-two circuits. Hence it does not seem to\nbe a coincidence that the Riemann hypothesis over finite fields, which enters\nin the theory of the standard quantum group, also enters in the theory of\nexpanders [Lb, Sr].\n↑\ndepth\n|\nCircuits of superpolynomial depth and size: Obstructions\n↑\n|\nCircuits of depth two: expanders\nFigure 4: The relationship between obstructions and expanders\nExistence of expanders can be proved by a simple probabilistic method.\nIn contrast, existence of expanders may not be provable by a probabilistic\nmethod. Indeed, this is roughly the main content of [RR], which says that a\nnonconstructive method, such as a probabilistic method, should not work in\nthe context of the P vs. NP problem under reasonable assumptions. This\nis why in GCT we go for explicit construction of obstructions in the spirit\nof explicit construction of expanders. The P/poly-naturalizability barrier in\n[RR] should not be applicable to such explicit, constructive proof techniques.\nThis issue is addressed in more detail in Section 18.\n22\n\n1.9\nIs there a simpler proof technique?\nFinally, one may ask if the P ̸= NP conjecture may be proved by a sub-\nstantially simpler proof technique.\nThis seems unlikely for the following\nreasons.\nThe results in complexity theory such as [A2, Re] suggest that explicit\nconstructions may be more or less essential for derandomization. In conjunc-\ntion with the hardness vs. randomness principle [KI, NW], this suggests that\nexplicit constructions may also be more or less essential for (the difficult)\nlower bound problems as well.\nHence, the difficulty in any viable proof\ntechnique for the P ̸= NP conjecture may be intimately linked to the diffi-\nculty (complexity) of the explicit construction of obstructions, i.e., “proofs\nof hardness” as per that technique. This may be so regardless of whether the\ntechnique actually constructs such obstructions explicitly or not. Because,\nas per the existence-vs-construction principle [KUW], the difficulty of decid-\ning existence may be more or less the same as that of construction in natural\nproblems. These and other considerations naturally lead to a notion of ex-\nplicit construction complexity of an easy-to-verify proof technique towards\nthe P ̸= NP conjecture, where easy-to-verify formally means P-verifiable;\ncf. Section 19.\nThe explicit construction (depth) complexity of expanders is O(1), in\nfact, two, since they can be constructed by (nonuniform) depth-two algebraic\ncircuits (over a ring of integers modulo k for some k) [LPS, Ma]. Whereas,\nas per Hypothesis 1.1, the explicit construction (depth) complexity of the\nobstructions in GCT over C is poly(m), m = nlog n (say) being the circuit\nsize parameter in the lower bound problem; cf. Figure 4. The arguments in\nSection 19 suggest that this may be essentially the best explicit construction\ncomplexity that one can expect in any P-verifiable proof technique towards\nthe P ̸= NP conjecture. In other words, the massive Ω(m) gap between the\nexplicit construction complexity of obstructions and the O(1) explicit con-\nstruction complexity of expanders, as shown in Figure 4, may be inevitable\nin any P-verifiable proof technique towards the P ̸= NP conjecture. If so,\nGCT may be among the “easiest” P-verifiable approaches to this conjecture\nas per the explicit construction complexity measure defined here, and hence,\nit may be unrealistic to expect a technique that is substantially simpler or\neasier.\nIn the rest of this article, we elaborate the plan in Figure 3 further\nand give a high-level description of the results in the GCT papers. Logical\ndependence among the GCT papers is shown in Figure 5.\n23\n\n1.10\nOrganization of the paper\nIn Section 2 we recall a few basic facts in algebraic geometry and represen-\ntation theory which are easy to state and should be easy to believe. The\nreaders not familar with these fields should be able to take these on faith.\nIn Section 3 we describe a special class of algebraic varieties, called group-\ntheoretic varieties. All class varieties in GCT are group-theoretic varieties.\nThey are described in Section 4. Obstructions are defined in Section 5. Why\nthey should exist is described in Section 6. The flip is described in Sec-\ntion 7. The main barrier in the implementation of the flip, the P-barrier,\nis described in Section 8. The main result of GCT that crosses this barrier,\nassuming the mathematical positivity hypotheses PH1 and SH (PH2), is\ndescribed in Section 9. Why PH1 and PH2 should hold is described in Sec-\ntion 10. Simpler analogues in representation theory of the decision problems\nin Hypothesis 1.2 are described in Section 11. The P-barrier in this context,\nits crossing subject to analogous PH1 and SH (PH2), along with theoretical\nresults supporting these positivity hypotheses are described in Section 12.\nThe nature of the reduction in the top arrow of Figure 1 is described in\nSection 13. The basic plan in [GCT6] to prove PH1 and SH via the the-\nory of quantum groups is described next. The standard quantum group is\nintuitively described in Section 14. The nonstandard quantum groups are\nintuitively described in Section 15. Why nonstandard Riemann hypotheses\nshould exist and their role in the theory of nonstandard quantum groups is\nbriefly described in Section 16. The relationship between obstructions and\nexpanders is described in Section 17. Why GCT should cross the relativiza-\ntion and the P/poly-naturalizability barriers is described in Section 18. Why\nGCT may be among the easiest P-verifiable approaches to the P vs. NP\nproblem as per the explicit-construction-complexity measure is described in\nSection 19.\n2\nBasics in algebraic geometry and representation\ntheory\nIn this section we describe the basic facts in algebraic geometry and repre-\nsentation theory which are needed in this article and which should be easy\nto believe for the readers not familiar with these fields. Their proofs can be\nfound in [FH, Mm1].\n24\n\nThis article\n(GCTflip1)\n|\n↓\nGCTintro\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 5: Logical dependence among the GCT papers\n25\n\n2.1\nRepresentation theory\nLet G be a group. We say that a vector space V is a representation of G, or\na G-module, if there is a homomorphism\nρ : G →GL(V ),\n(4)\nwhere GL(V ) is the general linear group of invertible transformations of\nV . We denote ρ(g)(v) by g · v–the result of the action of g on v. A G-\nsubrepresentation W ⊆V is a subspace that is invariant under G; i.e.,\ng · w ∈W for every w ∈W. If G is clear from the context, we just call\nit subrepresentation. We say that V is irreducible if it does not contain a\nproper nontrivial subrepresentation. A G-homomorphism from a G-module\nU to a G-module V is map ψ : U →V such that ψ(g · u) = g · (ψ(u)) for all\nu ∈U.\nWe say that G is reductive if every finite dimensional representation V\nof G is completely reducible. This means it can be expressed as a direct sum\nof irreducible representations in the form\nV =\nM\nλ\nmλVλ(G)\n(5)\nwhere λ ranges over all indices (labels) of irreducible representations of G,\nVλ(G) denotes the irreducible representation of G with label λ, and mλVλ(G)\ndenotes a direct sum of mλ copies of Vλ(G). Here mλ is called the multiplicity\nof Vλ(G) in V . It is a basic fact of representation theory that for reductive\ngroups, the decomposition (5) is unique; i.e., mλ’s are uniquely defined. If\nmλ > 0, we say that Vλ(G) occurs in V .\nAn example of a nonreductive group is a solvable group that is not\nabelian. In this case a subrepresentation W ⊆V need not have a comple-\nment W ⊥such that V = W ⊕W ⊥.\nEvery finite group is reductive.\nThus Sn, the symmetric group on n\nletters, is reductive. A prime example of a continuous reductive group is\nthe general linear group GLn(C) = GL(Cn), the group of nonsingular n × n\nmatrices, and its subgroup the special linear group SLn(C) = SL(Cn) of\nmatrices with determinant one.\nAny product of reductive groups is also\nreductive. These are the only kinds of reductive groups that we need to\nknow in this article. So whenever we say reductive, the reader may wish to\nassume that the group is a general or special linear group or a symmetric\ngroup or a product thereof.\n26\n\nWe say that the representation (4) of G = GLn(C) or SLn(C) is polyno-\nmial if for every g ∈G, every entry in the matrix form of ρ(g) is a polynomial\nin the entries of g.\nComplete reducibility as in eq.(5) means every finite dimensional rep-\nresentation of a reductive group is composed of irreducible representations.\nThese can be thought of as the building blocks in the representation theory\nof reductive groups, and it is important to know what these building blocks\nare.\n2.1.1\nIrreducible representations of GLn(C)\nFor GLn(C) this was done by Weyl in his classic book [W]. The polynomial\nirreducible representations of GLn(C) are in one-to-one correspondence with\nthe tuples λ = (λ1, . . . , λk) of integers, where k ≤n and λ1 ≥λ2 · · · ≥λk >\n0. Here λ is called a partition of length k and size d = P\ni λi. Its bitlength\n⟨λ⟩is defined to be the total bitlength of all λi’s.\nThus the polynomial irreducible representations of GLn(C) are labelled\nby partitions λ of length at most n, but any size. The irreducible representa-\ntion corresponding to a partition λ = (λ1, λ2, . . .) is denoted by Vλ(GLn(C)),\nand is called a Weyl module of GLn(C). When GLn(C) is clear from the\ncontext, we shall denote it by simply Vλ.\nEach partition λ corresponds to a Young diagram, which consists of\nk rows of boxes, with λi boxes in the i-th row. For example, the Young\ndiagram corresponding to (4, 2, 1) is shown below:\nWhen thinking of a partition, it is helpful to think of the corresponding\nYoung diagram. Thus each Weyl module is labelled by a Young diagram of\nheight at most n. This is a useful combinatorial tool for studying the Weyl\nmodules.\nA Weyl module Vλ is explicitly constructed as follows. This construction\nof Deyruts as well as Weyl’s original construction are given in [FH]. Let Z be\nan n × n variable matrix. Let C[Z] be the ring of polynomials in the entries\nof Z. It is a representation of GLn(C). Action of a matrix σ ∈GLn(C) on\na polynomial f ∈C[Z] is given by\n(σ · f)(Z) = f(Zσ).\n(6)\n27\n\nBy a numbering (filling), we mean filling of the boxes of a Young diagram\nby numbers in [n]; for example:\n1 2 4 3\n2 3\n1\nWe call such a numbering a (semistandard) tableau if the numbers are strictly\nincreasing in each column and weakly increasing in all rows; e.g.\n1 2 3 3\n2 3\n4\nThe partition corresponding to the Young diagram of a numbering is\ncalled the shape of the numbering.\nWith every numbering T, we associate a polynomial eT ∈C[Z], which is\na product of minors for each column of T. The l × l minor ec for a column c\nof length l is formed by the first l rows of Z and the columns indexed by the\nentries cj, 1 ≤j ≤l, of c. Thus eT = Q\nc ec, where c ranges over all columns\nin T. The Weyl module Vλ is the subrepresentation of C[Z] spanned by eT ,\nwhere T ranges over all numberings of shape λ over [n]. Its one possible\nbasis is given by {eT }, where T ranges over semistandard tableau of shape\nλ over [n].\nLet B ⊆GLn(C) be the subgroup of upper triangular matrices. It is\ncalled the Borel subgroup of GLn(C). An element vλ ∈Vλ is called a highest\nweight vector if it is an eigenvector for the action of each b ∈B. It is easy to\nshow that Vλ has a unique highest weight vector, upto a constant multiple:\nit is eT0, where T0 is the canonical tableau whose i-th row contains only i’s,\nfor each i; e.g.\n1 1 1 1\n2 2\n3\nLet P ⊆GLn(C) be the subgroup of upper block triangular matrices,\nwhere the sizes of the blocks are fixed. For example:\n28\n\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n0\n0\n∗\n∗\n∗\n∗\n0\n0\n∗\n∗\n∗\n∗\n0\n0\n0\n0\n∗\n∗\n0\n0\n0\n0\n∗\n∗\n \n \nSuch subgroups are called parabolic. Let Pλ be the (projective) stabilizer\nof the highest weight vector vλ = eT0; i.e., the set of all σ ∈GLn(C) such\nthat σ ·vλ = c(σ)vσ, for some complex number c(σ). Then it is easy to show\nthat Pλ is parabolic, where the sizes of the blocks are completely determined\nby λ.\nThe irreducible representation of GLn(C) corresponding to the Young\ndiagram that consists of just one column of length n is the determinant\nrepresentation: g →det(g). When restricted to the subgroup SLn(C) ⊆\nGLn(C) this becomes trivial. More generally, Vλ(G) and Vλ′(G) give the\nsame representation of SLn(C) if λ′ is obtained from λ by removing columns\nof length n. Hence, irreducible polynomial representations of SLn(C) are\nin one to one correspondence with partitions of length less than n, and are\nobtained from the ones of GLn(C) by restriction.\n2.1.2\nIrreducible representations of the symmetric group\nIrreducible representations of Sn, called Specht modules, are in one-to-one\ncorrespondence with the Young diagrams of size n, as opposed to those of\nlength ≤n for GLn(C). We denote the Specht module corresponding to a\npartition λ by Sλ. It is explicitly constructed as follows.\nLet C[X] = C[x1, · · · , xn] be the ring polynomials in n variables. It is a\nrepresentation of Sn: given σ ∈Sn and f ∈C[X],\n(σ · f)(x1, · · · , xn) = f(xσ(1), · · · , xσ(n)).\nGiven a numbering T of λ with distinct numbers in [n], let fT be the poly-\nnomial formed by taking a product of discriminants for all columns of T.\nThe discriminant for a column with entries ci, 1 ≤i ≤l, is Q\ni<i′(xci −xci′).\nThen Sλ is simply the subrepresentation of C[X] spanned by fT, where T\nranges over all numberings of λ with distinct entries in [n]. Its basis is given\nby {fT }, where T ranges over standard tableau of shape λ with entries in\n[n]. Here a standard tableau means the rows as well as the columns are\n29\n\nstrictly increasing; e.g.\n1 2 3 6\n4 5\n7\n2.1.3\nTensor products\nIf V and W are representations of a group G, then their tensor product\nV ⊗W is also a representation: for σ ∈G, v ∈V, w ∈W,\nσ(v ⊗w) = (σ · v) ⊗(σ · w).\nGiven two irreducible representations of a reductive group G, a fun-\ndamental problem in representation theory is to find an explicit complete\ndecomposition of their tensor product in terms of irreducible representations\nof G. The following instances of this problem are of central importance in\nGCT.\nLittlewood-Richardson coefficients\nFirst we consider this problem when G = GLn(C). Given Weyl modules Vα\nand Vβ, let\nVα ⊗Vβ =\nM\nγ\ncγ\nα,βVγ,\n(7)\nbe the complete decomposition of their tensor product into irreducible Weyl\nmodules of G. Here the multiplicities cγ\nα,β are called Littlewood-Richardson\ncoefficients. The Littlewood-Richardson rule gives the sought explicit for-\nmula for these multiplicities. It is as follows.\nAlign the left top corners of the Young diagrams for α and γ. If the\nYoung diagram for α is not contained in the one for γ, then cγ\nα,β is zero.\nOtherwise, form a skew shape γ \\ α by removing the boxes in γ belonging\nto α. A skew tableau with content β and shape γ \\ α is a filling of this skew\ndiagram with β1 ones, β2 twos and so on, such that all columns are strictly\nincreasing and all rows are weakly increasing. For example, the following is\na skew tableau of skew shape (4, 3, 3, 2) \\ (2, 2, 1):\n1 1\n2\n2 3\n1 3\n(8)\n30\n\nWe say that a skew tableau is a Littlewood-Richardson tableau if, when\nits entries are read from right to left, top to bottom, the number of i’s read\nupto any point is at most the number of (i −1)’s read up to that point,\nfor any i. For example, the skew tableau above is a Littlewood-Richardson\nskew tableau. Let Cγ\nα,β be the set of Littlewood-Richardson tableau of shape\nγ \\ α with content β. The Littlewood-Richardson coefficient cγ\nα,β is simply\nthe cardinality of Cγ\nα,β: i.e.,\ncγ\nα,β = |Cγ\nα,β| =\nX\nT∈Cγ\nα,β\n1.\n(9)\nSuch a formula is called positive, because it is like the formula for the perma-\nnent which involves only positive signs. In contrast, there are many formulae\nfor such multiplicities, based on the theory of characters of group represen-\ntations [FH], which involve alternating signs, like the usual formula for the\ndeterminant. Positivity here is a deep issue; cf. [St4].\nFormally, the Littlewood-Richardson rule implies that the Littlewood-\nRichardson coefficient belongs to the complexity class #P, just like the per-\nmanent. This is the real significance of the Littlewood-Richardson rule from\nthe complexity-theoretic perspective. Furthermore, just like the permanent,\nthe Littlewood-Richardson coefficient is #P-complete [N].\nKronecker coefficients\nNow we turn to the symmetric group.\nSince it is reductive, the tensor\nproduct of two Specht modules Sα and Sβ decomposes as: by:\nSα ⊗Sβ =\nM\nγ\nkγ\nα,βSγ,\n(10)\nwhere the multiplicities kγ\nα,β are called Kronecker coefficients.\nNo positive rule akin to the Littlewood-Richardson rule is known for\nthe Kronecker coefficients. In fact, this is a fundamental open problem in\nthe representation theory of symmetric groups, which arose almost with the\nbirth of representation theory in the work of Frobenius, Schur, Weyl and\nothers in the beginning of the twentieth century; cf. [Mc, St4] for its history\nand significance. In the language of complexity theory, the problem is:\nQuestion 2.1 Does the Kronecker coefficient belong to #P?\n31\n\nThough this is not how it was stated in representation theory. The answer\nis conjecturally yes [GCT4]. Indeed, this is the main focus of the work in\n[GCT4, GCT8]: roughly, [GCT8] says that such a rule exists assuming a\nconjecture regarding the nonstandard quantum group defined in [GCT4].\nThis is the first entry point of nonstandard quantum groups in GCT.\n2.2\nAlgebraic geometry\nLet V = Cn. Let X = (x1, . . . , xn) be the variable n-vector whose entries\nstand for the coordinates of V . An affine algebraic set Z ⊆V is the set\nof zeroes of a collection of polynomials in C[X] = C[x1, . . . , xn]. An affine\nalgebraic set is called irreducible if it cannot be expressed as the union of\ntwo proper affine algebraic subsets. An irreducible affine algebraic subset\nZ of V is called an affine variety. Its ideal I(Z) ⊆C[X] is the set of all\npolynomials that vanish on Z, and its coordinate ring C[Z] is defined to be\nC[X]/I(Z). The elements of C[Z] are polynomial functions on Z.\nLet P n−1 = P(V ) be the projective space of lines in V through the\norigin. We say that V is the affine cone of P(V ). Given a nonzero v ∈V ,\nwe also denote by v the point in P(V ) that corresponds to the line in V\npassing through v and the origin; the meaning should be clear from the\ncontext. The homogeneous coordinate ring of P(V ) is defined to be C[X].\nIts elements are homogeneous functions on V , the affine cone of P(V ). A\nprojective algebraic set Y in P(V ) is the set of zeroes of a collection of\nhomogeneous forms (polynomials). The affine cone ˆY ⊆V of Y ⊆P(V )\nis defined to be the union of the lines in V corresponding to the points in\nY . A projective algebraic set is called irreducible if it cannot be expressed\nas the union of two proper projective algebraic subsets.\nAn irreducible\nprojective algebraic subset Y of P(V ) is called a projective variety. Its ideal\nI(Y ) ⊆C[X] is the set of all homogeneous forms that vanish on Y , and its\nhomogeneous coordinate ring R(Y ) is C[X]/I(Y ). The elements of R(Y ) are\nhomogeneous functions on the affine cone ˆY of Y . The degree d-component\nR(Y )d of R(Y ) is the subspace of homogeneous forms of degree d.\nThe\nHilbert function hY (d) of Y is defined to be the dimension of R(Y )d.\nBy a (Zariski)-open subset of Y we mean the complement of an algebraic\nsubset of Y . An open subset of a projective variety is also called a quasi-\nprojective variety.\nNow suppose V is a representation of a reductive group G.\nThen G\nalso acts on P(V ), since it takes line to a line. Furthermore, C[X] is also a\n32\n\nrepresentation of G: given σ ∈G and f ∈C[X], we define\n(σ · f)(X) = f(σ−1X).\n(11)\nThe variety Y is called a G-variety if its ideal I(Y ) ⊆C[X] is a G-subrepresentation\nof G; i.e., σ·f ∈I(Y ) for all σ ∈G, f ∈I(Y ). In this case, the homogeneous\ncoordinate ring R(Y ) of Y is also a representation of G. Furthermore, given\na point p ∈Y , the point σ(p) also belongs to Y . In other words, G acts on\nthe variety Y by moving its points around.\nIf Z is a projective subvariety of Y , then it is a basic fact that there\nexists a degree preserving surjection from R(Y ) to R(Z); i.e., from R(Y )d\nto R(Z)d for every d. This surjection is obtained by simply restricting a\npolynomial function on the affine cone ˆY to the subcone ˆZ ⊆ˆY . If both\nY and Z are G-varieties, then this surjection is a G-homomorphism. By\ncomplete reducibility, it then follows that R(Z)d is a G-submodule of R(Y )d\nfor every d. Pictorially,\nR(Z)d ֒→R(Y )d,\n(12)\nfor every d.\nGiven a point v ∈P(V ), let Gv denote its G-orbit. It can be shown\nthat Gv is a quasi-projective variety.\nLet Gv = {σ | σ · v = v} be its\nstabilizer. Then Gv as a set is isomorphic to the coset set G/H, H = Gv.\nQuasiprojective varieties of the form G/H are called homogeneous spaces.\nThese have been intensively studied in algebraic geometry.\nLet ∆V [v] = Gv ⊆P(V ) denote the closure of the G-orbit of v in the\nusual complex topology 1. We call such a variety an orbit closure. It can\nbe shown that ∆V [v] is a projective G-variety. One can think of ∆V [v] as\na closure of the homogeneous space G/Gv. Such spaces are called almost-\nhomogeneous spaces [Ak]. These have also been intensively studied. Let\nRV [v] be the homogeneous coordinate ring of ∆V [v], and RV [v]d its degree\nd-component. Since G acts on ∆V [v], each RV [v]d is a finite dimensional\nrepresentation of G.\nThe simplest example of ∆V [v] arises as follows.\nLet Vλ be a Weyl\nmodule of G = GLn(C). Let vλ ∈P(Vλ) be the point corresponding to the\nhighest weight vector in Vλ; we call it the highest weight point. Then it can\nbe shown that the orbit Gvλ ∼= G/Pλ, where Pλ is the stabilizer of vλ, is\nalready closed. This is called a flag variety. It has been intensively studied\nin algebraic geometry for over a century, and its algebraic geometry is now\nmore or less completely understood; e.g. see [LLM].\n1This coincides with the closure in the Zariski-topology [Mm1].\n33\n\nThe flag varieties by their very definition are smooth. But the algebraic\ngeometry of general orbit closures can be extremely complicated, and essen-\ntially, intractable. Because, even if the orbit Gv is smooth, its closure can be\nhighly singular, and the singularities can be pathological. Indeed, the moral\nof the story that can be gained from [LV] is that the algebraic geometry of\na general orbit closure is essentially hopeless.\n3\nGroup-theoretic varieties\nFortunately, the class varieties that arise in GCT are all exceptional kinds of\norbit closures, which we call group-theoretic orbit closures or group-theoretic\nvarieties. The articles [GCT2, GCT10] together roughly say that problems\nregarding the algebraic geometry of these group-theoretic class varieties can\nbe “reduced” to problems in (quantum) group theory. This is what makes\nthem tractable, and this is how the theory of quantum groups enters in\nGCT. In this section, we shall briefly describe a group-theoretic variety in\nan abstract form.\nLet V and G be as in Section 2.2. We say that v ∈P(V ) is characterized\nby its stabilizer H = Gv ⊆G, if it is the only point in P(V ) stabilized (left\ninvariant) by H. Stabilized by H means, for every σ ∈H , σ · v = v.\nFor example, the highest weight point vλ ∈P(Vλ) (Section 2.1) is char-\nacterized by its stabilizer Pλ. This indicates that the points that are char-\nacterized by their stabilizers are very special.\nSuppose v is characterized by its stabilizer. Then v, and hence, its orbit\nclosure ∆V [v] is completely determined by the group triple:\nH = Gv ֒→G\nρ→K = GL(V ),\n(13)\nwhere ρ represents the representation map; cf. (4). This leads to the follow-\ning:\nDefinition 3.1 Assume that v is characterized by its stabilizer, and that\nthe associated group triple H ֒→G ֒→K is explicitly known. Then we say\nthat the orbit closure ∆V [v] is a group-theoretic variety. It is completely\ndetermined by the preceding group triple.\nWe call H ֒→G the primary\ncouple associated with the orbit closure ∆V [v], and G ֒→K, the secondary\ncouple. We also say that v is the characteristic point of the group triple\nH ֒→G ֒→K and the primary couple H ֒→G.\n34\n\nIf every point in V is a function on some space then we say that v is\nthe characteristic function of the triple H ֒→G ֒→K and the primary\ncouple H ֒→G. (This happens if, for example, V is the space of polynomial\nfunctions on an affine G-variety X, with the action given by (11)).\nHere explicitly means the composition factors of H as well as the con-\nnecting homomorphisms in (13) are specified explicitly; for the details re-\ngarding how, see [GCT6, GCTflip2]. All varieties that arise in GCT are\neither group-theoretic orbit-closures in the above sense, or their generaliza-\ntions, which are again essentially determined by group triples as above, and\nhence, will also be called group-theoretic varieties. The simplest example of\na group-theoretic variety is a flag variety (Section 2.2).\nIf a variety is group theoretic, then, in principle, we ought to be able to\nunderstand its algebraic geometry if we understand the structure of the as-\nsociated group triple, along with the connecting homomorphisms, in depth.\nWe shall elaborate on what in depth means later in Section 15. Briefly, it\nmeans understanding the structure of the group triple at the quantum level.\n4\nClass varieties\nNow we turn to the class varieties associated with the complexity classes\nNC, P, #P and NP in [GCT1] on which the obstructions in GCT live. All\nthe class varieties that arise in GCT will be orbit closures of the following\nspecial form.\nLet Y = [y0, · · · , yl−1] denote a variable l-vector. For n < l, let X =\n[y1, · · · , yn], and ̄X = [y0, · · · , yn] be its subvectors of size n and n + 1.\nWe also denote yi, 1 ≤i ≤n, by xi.\nLet V = Syms(Y ) be the space\nof homogeneous forms of degree s in the l variable-entries of Y . It has a\nnatural action of G = SL(Y ) = SLl(C) and ˆG = GL(Y ) = GLl(C), just as\nin (11).\nSimilarly, let W = Symr(X), r < s, be the representation of GL(X) =\nGLn(C). We have a natural embedding φ : W →V , which maps\nw ∈W →ys−rw ∈V,\n(14)\nwhere y = y0 is used as the homogenizing variable. The image φ(W) is\ncontained in ̄W = Syms( ̄X), a representation of GL( ̄X) = GLn+1(C).\nThe basic recipe for constructing class varieties is as follows. Say we\nwant to separate a complexity class C1 from a complexity class C2 ⊇C1.\n35\n\nWe pick a form g = g(Y ) = g(y0, . . . , yl−1) ∈P(V ) which is a complete\nfunction for the complexity class C1. Then the orbit closure ∆V [g; l] = ∆V [g]\nis called the class variety associated with C1, or simply the C1-variety based\non the complete function g.\nIn principle, we can let g be any complete\nfunction for the class C1. But for the algebraic geometry of ∆V [g] to be\ntractable, we have to choose g so that ∆V [g] is group-theoretic; i.e., so that\ng is characterized by its stabilizer as in Definition 3.1, or in a slightly relaxed\nsense (cf. Section 7 in [GCT1]), which is good enough for our purposes.\nSimilarly, we choose a form h = h(X) = h(x1, . . . , xn) ∈P(W) which\nis complete for the class C2. Then the orbit closure ∆W[h; n] = ∆W[h] ⊆\nP(W) is called the base class variety associated with C2, or simply the base\nC2-variety based on h. Let f = φ(h), with φ as in (14). We call the orbit\nclosure ∆V [f; n, l] = ∆V [f] ⊆P(V ) the extended class variety associated\nwith C2, or the extended C2-variety based on h. This extension is necessary\nso that the C2-variety ∆V [f] and the C1-variety ∆V [g] live in the same\nambient space P(V ). Again, h has to be chosen so that it is characterized\nby its stabilizer (almost) so that the varieties ∆W [h] and ∆V [f] are group-\ntheoretic.\nLet us suppose to the contrary that C2 ⊆C1. Then it would turn out\nthat f ∈∆V [g; l], and hence, ∆V [f; n, l] is a G-subvariety of ∆V [g; l]:\n∆V [f; n, l] ֒→∆V [g; l].\nThe goal is to show that such an embedding does not exist when l is small\nenough, say, l = nlog n, n →∞. This will show that C1 ̸= C2. Here l will be\na parameter in the lower bound problem that depends on the depth and/or\nthe size of the circuit.\nWe now demonstrate this recipe in two basic separation problems in\ncomplexity theory.\n4.1\nNC vs. P #P\nLet NC be the standard class of functions that can be computed by circuits\nof polylogarithmic depth, and #P the counting class associated with NP.\nThe determinant is complete for the class NC and the permanent for the\nclass #P [V]. The P #P ̸= NC conjecture over C [V] says that the per-\nmanent cannot be computed by a circuit over C of polylogarithmic depth.\nUsing the determinant and the permanent, we now construct class varieties\nfor NC and #P. The P #P ̸= NC conjecture over C will then be reduced\n36\n\nto showing that the extended class variety for #P is not contained in the\none for NC.\nLet Y be an m × m variable matrix, which can also be thought of as a\nvariable l-vector, l = m2, by linearly ordering its entries in any order. Let\nX be its, say, the principal bottom-right n × n submatrix, n < m, which\ncan also be thought of as a variable k-vector, k = n2. Let V = Symm(Y )\nbe the space of homogeneous forms of degree m in the variable entries of\nY , and W = Symn(X), the space of homogeneous forms of degree n in the\nvariable entries of X. We have a natural action of G = SL(Y ) = SLl(C) on\nV and the projective space P(V ): namely, σ ∈G maps a form q(Y ) ∈P(V )\nto q(σ−1Y ), where we think of Y as a variable l-vector. Similarly, we have\nan action of H = SL(X) = SLk(C) on P(W).\nUsing any entry y of Y not in X as a homogenizing variable we get an\nembedding φ : W →V , which maps any w ∈W to ym−nw ∈V .\nLet g = det(Y ) ∈P(V ) be the determinant form.\nLet ∆V [g; l] =\n∆V [g] ⊆P(V ) be its orbit closure.\nIt is shown in [GCT1] that if a form h(X) ∈P(W) can be computed by a\ncircuit of depth less than logc n, then f = φ(h) lies in ∆V [g; l] for m = 2logc n.\nConversely, if f lies in ∆V [g; l] then h[X] can be approximated infinitesimally\nclosely2 by a circuit of depth O(log2c m).\nSince, the permanent is #P-\ncomplete [V], this is not expected to happen if h = perm(X) and m =\n2O(polylog(n)). This leads to:\nConjecture 4.1 [GCT1] Let h = perm(X) ∈P(W) and f = φ(h). Then\nf ̸∈∆V [g] = ∆V [g; l], if m = 2O(polylog(n)), as n →∞. Since ∆V [g] is a\nG-variety, this is equivalent to saying that ∆V [f] ̸⊆∆V [g]. Pictorially:\n∆V [f] ̸֒→∆V [g].\nHere ∆V [g] = ∆V [g; l] is called the class variety associated with the\nclass NC, or simply the NC-variety based on the complete determinant\nfunction. We also denote it by XNC(g; l) or simply XNC(l). We call ∆W[h]\nthe base class variety and ∆V [f] the extended class variety associated with\nthe class #P, or simply the base #P-variety and the extended #P-variety,\nrespectively. We also denote them by X#P (h; n) and X#P (f; n, l), or sim-\nply, X#P(n) and X#P (n, l). The goal (Conjecture 4.1) is to show that the\n2This means, for every ǫ > 0, there exists a form ̃f ∈V , which has a circuit of depth\nO(log2c m), such that ||f − ̃f|| < ǫ, in the usual norm on V .\n37\n\nextended class variety for #P cannot be contained in the class variety for\nNC, when m = 2polylog(n): i.e.,\nX#P (n, l) ̸֒→XNC(l).\nThis will show that the permanent cannot be computed by circuits of poly-\nlogarithmic depth.\nNext we describe why these class varieties are group-theoretic. For this,\nwe need to show that the determinant and the permanent are characterized\nby their stabilizers.\nThe stabilizer of det(Y ) ∈P(V ) in G = SL(Y ) = SLm2(C) is known to\nbe a reductive subgroup Gdet which consists of linear transformations in G\nof the form (thinking of Y as an m × m matrix):\nY →AY ∗B,\n(15)\nwhere Y ∗is either Y or Y T , A, B ∈GLm(C).\nThat the determinant is\ncharacterized by its stabilizer follows from classical invariant theory [FH].\nHence the NC-variety defined here is group-theoretic. The associated group\ntriple is\nGdet ֒→G ֒→GL(V ),\n(16)\nand Gdet ֒→G the primary couple. The embedding Gdet →G almost looks\nlike the natural embedding\nGL(Cm) × GL(Cm) →GL(Cm ⊗Cm),\n(17)\ngiven by: (g, h) →g ⊗h, where g ⊗h denotes the Kronecker product. That\nis,\n(g ⊗h) · (x ⊗y) = (g · x) ⊗(h · y).\n(18)\nThe stabilizer of perm(X) ∈P(W) in SL(X) = SLn2(C) is a reduc-\ntive subgroup generated by linear transformations in SL(X) of the form\n(thinking of X as an n × n matrix):\nX →λX∗μ,\n(19)\nwhere X∗is either X or XT , λ and μ are either diagonal or permutation\nmatrices, and n ≥3. It is easy to show that the permanent is also charac-\nterized by its stabilizer. Hence the base #P-variety defined in this section\nis group theoretic; the extended #P-variety is also group-theoretic.\n38\n\n4.2\nP vs. NP problem over C\nThe class varieties associated with the classes P and NP can be constructed\nin principle using any P-complete and NP-complete functions. But again it\nis necessary to choose these functions in a special way so that the resulting\nclass varieties turn out to be group-theoretic (Section 3). Such P-complete\nand (co)-NP-complete functions, called H(Y ) = H(y1, . . . , yl) and E(X) =\nE(x1, . . . , xn) respectively, have been constructed in [GCT1]. We do not\nneed to know their definitions here.\nLet W = Symr(X) be the space of forms of degree r = deg(E(X)) in\nthe entries of X. Thus E(X) ∈P(W). Let V = Syms(Y ) be the space of\nforms of degree s = deg(H(Y )) in the entries of Y . Thus H(Y ) ∈P(V ). We\nidentify X with a suitable subset of Y , and define a map φ : P(W) →P(V )\nas in (14) by choosing a variable y in Y \\ X as a homogenizing variable.\nNow, using the recipe above, we can associate with E(X), for every n\nand l ≥n, a group-theoretic variety (orbit closure) ∆V [f; n, l] = ∆V [f] ⊆\nP(V ), where f = φ(h) and h = E(X). It is a G-variety, for G = SLl(C).\nIt will be called the (extended) class variety for NP or simply the NP-\nvariety based on the form E(X), and will be denoted by XNP (E; n, l) or\nsimply XNP (n, l). Similarly, we can associate with H(Y ) a group-theoretic\nG-variety ∆V [g; l] = ∆V [g] ⊆P(V ), where g = H(Y ).\nIt is called the\nclass variety for P or simply the P-variety based on the form H(Y ), and is\ndenoted by XP (H; l) or simply XP (l).\nRemark 4.2 The actual P-variety XP (H; l) in the P vs. NP problem is\nnot meant to be ∆V [g, l], as defined here, but rather the variety ˆ∆[H(Y )]\ndefined in Section 7 of [GCT1]. But we shall ignore that difference here.\nIt can be shown [GCT1] that if E(X) is computable by a circuit of size\nm then XNP (E; n, l) can be embedded within XP (H; l) for l = O(m2):\nXNP (n, l) = XNP (E; n, l) ֒→XP (l) = XP (H; l).\n(20)\nIn this context:\nConjecture 4.3 [GCT1] This embedding cannot exist if m = nlog n, or\nmore generally, m = 2na, for a small enough a > 0, as n →∞.\nThis will show that P ̸= NP over C. This transforms the P vs. NP problem\nover C into a problem in geometric invariant theory.\n39\n\nAgain, these class varieties are group-theoretic, in a slightly relaxed sense\nthan defined in Section 3, but which is good enough for the purposes of GCT\n[GCT1].\n5\nObstructions\nAn obstruction in the P vs. NP problem (characteristic zero) is defined to\nbe a representation that lives on the extended class variety associated with\nNP but not on the class variety associated with P. We now elaborate what\nthis means.\nLet R(n, l) = R(E; n, l) and S(l) = S(H; l) denote the homogeneous co-\nordinate rings of XNP (n, l) = XNP (E; n, l) and XP (l) = XP (H; l), respec-\ntively. We call them the class rings associated with the complexity classes\nNP and P. Let R(n, l)d and S(l)d denote their degree d-components, con-\nsisting of homogeneous polynomial functions of degree d. Since G acts on\nthe class varieties, it also acts on the class rings (see Section 2.2). That is,\neach R(n, l)d or S(l)d is a finite dimensional representation of G.\nIf the embedding (20) exists, then R(n, l)d can be embedded as a G-\nsubmodule of S(l)d, for each d; cf. (12):\nR(n, l)d ֒→S(l)d.\n(21)\nIn particular, every irreducible representation (Weyl module) Vλ = Vλ(G)\nof G that occurs within R(n, l)d as a subrepresentation also occurs within\nS(l)d as a subrepresentation.\nDefinition 5.1 We say that S = Vλ is an obstruction, for n, l and the pair\n(E, H) = (E(X), H(Y )), if it occurs in R(n, l)d but not in S(l)d, for some\nd.\nIn this case we say that Vλ is an obstruction of degree d. We also refer\nto λ as an obstruction of degree d.\nObstruction in the setting of the NC vs. P #P problem over C is defined\nsimilarly.\nThis notion of obstruction in [GCT2] is a refinement of the earlier notion in\n[GCT1].\nThe specification of an obstruction is given in the form of its label λ.\nThe existence of such an obstruction for given n and l is a “proof” that the\nembedding in (21), and hence, the one in (20) cannot exist.\n40\n\nIn this context:\nConjecture 5.2 [GCT2, GCT10] An obstruction for n, l and the pair (E, H)\nexists if m = nlog n, or more generally, m = 2na, for a small enough a > 0,\nas n →∞; recall that l = O(m2). Furthermore, there exists such an ob-\nstruction of a small degree d(n, m) = 2mb, b > 0 a large enough constant.\nSimilar conjecture can be made in the context of the NC vs. P #P prob-\nlem.\nIn this case, the degree d(n, m) can be mb, b > 0 a large enough\nconstant.\nIf such an obstruction Vλ(n) exists for every n →∞, with m as above,\nthen it follows that P ̸= NP over C. We say that {Vλ(n)} or {λ(n)} is an\nobstruction family for the P vs. NP problem over C. The goal is to prove\nexistence of such a family.\n6\nWhy should obstructions exist?\nA priori, it is not at all clear why such obstructions should even exist. In\nthis section, we explain why they should.\nAn intuitive reason for existence of obstructions is as follows. The article\n[Dl1] roughly says that (algebraic) groups are completely determined by their\nrepresentations. On the other hand, the group-theoretic class varieties are\nessentially determined by the associated group triples, and hence, as per the\nphilosophy in [Dl1], the representation-theoretic information associated with\nthese group triples. Hence, a “witness” for nonexistence of the embedding\nas in (20) ought to be present in the representation-theoretic information\nassociated with the group triples, assuming that P ̸= NP–which we take on\nfaith. This is intuitively why a representation-theoretic obstruction ought\nto exist. Specifically, there should exist a representation-theoretic witness\n(obstruction) that explains why one group-theoretic class variety, with as-\nsociated group triple H1 ֒→G →K, cannot be embedded in another group\ntheoretic variety with associated group triple H2 ֒→G →K; in our problem\nG and K in both triples would be the same.\nBut why should such a representation-theoretic obstruction be specifi-\ncally of the type as defined here?\nTo see this, let us first consider a simpler example. Instead of triples,\nlet us consider couples. Let us say we are given two couples ρ1 : H1 ֒→G,\nand ρ2 : H2 ֒→G, where G = GLl(C) = GL(W), W = Cl. This means W\n41\n\nis a representation of H1 and H2. Let us assume that it is an irreducible\nrepresentation of H1 and H2, and furthermore, that both H1 and H2 are\nreductive, and that H2 is not a conjugate of H1. Now the coset sets G/H1\nand G/H2 can be given the structure of affine algebraic varieties [Mm2].\nSince H2 is not a conjugate of H1, G/H1 cannot be embedded in G/H2 (and\nvice versa). The goal is to find a representation theoretic obstruction for the\nnonexistence of such an embedding. We say that Vλ(G) is an obstruction for\nthis pair of couples (ρ1, ρ2) if it occurs as a G-submodule in the coordinate\nring of G/H1 but not in the coordinate ring of G/H2. This is equivalent\nto saying that Vλ(G) contains an H1-invariant, when considered as an H1-\nmodule via ρ1, but not an H2-invariant, when considered as an H2-module\nvia ρ2; this is a consequence of the Peter-Weyl theorem [Sp]. This then is an\nobstruction very similar to the one in Definition 5.1. Its existence implies\nthat G/H1 cannot be embedded in G/H2. The work [LP] implies that such\nas an obstruction always exists when H1 and H2 are as above.\nConjecture 5.2 is a natural generalization of this well characterized sit-\nuation. It says that there exists a similar obstruction for the embedding\namong the group-theoretic varieties under consideration. This, as expected,\nis a much harder issue. The existence of such an obstruction depends cru-\ncially on the following conjecture concerning the algebraic geometry of the\nclass varieties under consideration.\nConjecture 6.1 (a) (cf. [GCT2]) The algebraic geometry of the class va-\nriety for NC is completely determined by the representation theory of the\nassociated group triple. Specifically, let Π be the set of G-submodules of C[V ]\nwhose duals do not contain a Gdet-invariant; i.e., the trivial Gdet-module;\ncf. (16). Let X(Π) ⊆P(V ) be the zero set of the forms in the G-modules\nin Π. Then XNC = X(Π).\n(b) (cf. [GCT10]) Analogous, but more complex, statements hold for the\nclass varieties associated with the complexity classes P, NP and #P.\nFor precise statements see [GCT2, GCT10].\nRemark 6.2 (Erratum) In [GCT2] it is conjectured that XNC = X(Π)\nas a scheme [Ha]. This stronger conjecture may not hold as it is. Rather,\nits variant, as would be described in [GCT10], is expected to hold.\nConcrete support for this conjecture is provided by the following two\nresults.\nThe first result is the second fundamental theorem of invariant\n42\n\ntheory. It says that the analogue of Conjecture 6.1 holds for flag varieties\nand their generalizations [LLM]. Thus Conjecture 6.1 may be thought of\nas a natural generalization of the second fundamental theorem of invariant\ntheory to the group-theoretic class varieties under consideration. The second\nresult, specific to the setting under consideration, is the following.\nTheorem 6.3 (Theorem 2.11 in [GCT2])\nA weaker form of Conjecture 6.1 holds for the NC-variety. Specifically,\nthere is a dense open neighbourhood U ⊆P(V ) of the orbit Gg of the deter-\nminant g = det(Y ) such that XNC ∩U = X(Π) ∩U, assuming a reasonable\ntechnical condition.\nThe article [GCT10] gives justifications for and a plan to prove Conjec-\nture 6.1. It is shown in [GCT2] that obstructions as in Definition 5.1 indeed\nexist in the context of NC vs. P #P problem, for all n →∞, assuming\n1. Conjecture 6.1 (a), and\n2. that the permanent cannot be approximated infinitesimally closely by\ncircuits of polylogarithmic depth.\nThe argument for existence of obstructions in the context of the P vs. NP\nproblem based Conjecture 6.1 (b) is similar [GCT10].\nThe first statement here crucially depends on the group-theoretic nature\nof the class variety for NC. If in place of the determinant we substitute other\nfunction, this need not hold. The second statement is a slightly strengthened\nform of the statement that we are finally trying to prove: namely, that the\npermanent cannot be computed by circuits of small depth. This circular\nreasoning tells us why obstructions should exist. But it gives no help in\nshowing that they exist unconditionally.\nWe turn to this task in the next section. A remark before we do so.\nThe existence of obstructions here crucially depends on the exceptional na-\nture of H(Y ). But we have made no use so far of the exceptional nature\nof E(X). In fact, obstructions of such kind should exist for any hard (co-\nNP-complete) function h(X) in place of E(X). But the approach for con-\nstructing obstructions described in the next section crucially depends on the\nexceptional nature of E(X)–i.e., on the group-theoretic nature of the class\nvariety XNP (E; n, l) for NP based on E(X).\n43\n\n7\nThe flip\nNow we come to the real problem: how to prove the existence of obstructions\nfor the specific E(X) under consideration. One may wish to try a probabilis-\ntic strategy for proving existence of obstructions: just choose a label λ(n) of\nhigh enough degree randomly, and show that Vλ(n) is an obstruction with a\ngood probability. But this technique would not work in the context of the P\nvs. NP problem because it is P/poly-naturalizable [RR]. Hence we shall go\nfor explicit construction of obstructions in the spirit of explicit construction\nof expanders [LPS, Ma, RVW]. The P/poly-naturalizability barrier in [RR]\nwould not apply to an approach based on explicit constructions (Section18).\nThis approach is based on the following hypothesis governing the flip:\nHypothesis 7.1 (PHflip1)\nThe following problems belong to P. Specifically:\n(a) (Verification): There exists a poly(l, n, ⟨d⟩, ⟨λ⟩)-time algorithm for de-\nciding, given l, n, d and λ, if Vλ is an obstruction of degree d for n, l and\nthe pair (E, H) (Definition 5.1). Here ⟨d⟩and ⟨λ⟩denote the bitlengths of\nd and λ, respectively.\n(b) (Explicit construction of obstructions): Suppose l = nlog n, or 2na, for\na small enough constant a > 0. Then, for every n →∞, a label λ(n) of\nan obstruction for n and l can be constructed explicitly in poly(n, l) time,\nthereby proving existence of an obstruction for every such n and l.\n(c) (Discovery of obstructions in general): There exists a poly(l, n)-time\nalgorithm for deciding if there exists an obstruction for n, l and the pair\n(E, H), and for constructing the label of one, if it exists.\nSimilar hypothesis holds for the NC vs. P #P problem.\nIn view of the definition of obstruction (Definition 5.1), The statement\n(a) for verification follows from the following:\nHypothesis 7.2 (PHflip2) (a) There exists a poly(l, n, ⟨d⟩, ⟨λ⟩)-time al-\ngorithm for deciding, given l, n, d and λ, if Vλ(G) occurs in R(n, l)d.\n(b) There exists a poly(l, ⟨d⟩, ⟨λ⟩)-time algorithm for deciding, given l, d and\nλ, if Vλ(G) occurs in S(l)d.\nSimilar hypothesis holds for the NC vs. P #P problem.\nAs mentioned in Section 1.4, once Hypothesis 7.2 is proved, the poly-\nnomial time algorithms for the decision problems therein have to be trans-\n44\n\nformed into a polynomial time algorithm for explicit construction of obstruc-\ntions as in Hypothesis 7.1 (b), thereby proving Conjecture 5.2, and hence\nthe lower bound under consideration. This issue will be addressed in detail\nin Section 13 later.\nThe whole discussion in this section is summarized in Figure 6.\nFind easy, polynomial time algorithms for the\ndecision problems in Hypothesis 7.2\n|\n|\n|\n↓\nTransform these easy algorithms into an easy algorithm for ex-\nplicit construction of obstructions as in Hypothesis 7.1 (b)\n|\n|\n|\n↓\nP ̸= NP over C\nFigure 6: The flip\n8\nWhy should the flip work?: the P-barrier\nBut why should there exist easy algorithms as in Hypotheses 7.1 and 7.2?\nThis turns out to be, paradoxically, the hardest aspect of the flip: just to\nprove easiness. In this section, we elaborate its nature further.\nClearly, the function E(X) has to be extremely special for Hypotheses 7.1\nand 7.2 to hold. If, instead of E(X), we consider a general co-NP-complete\nfunction h(X) then, obstructions can still be expected to exist (cf.\nSec-\ntion 6), but Hypotheses 7.1 and 7.2 would fail severely, as we now explain.\nSo fix a general integral function h(X) = h(x1, . . . , xn), which is co-NP-\ncomplete, when considered over F2 by reduction modulo 2. Let XNP (h; n, l) ⊆\nP(V ) be the class variety associated with it by following the recipe in Sec-\ntion 4.2 with h(X) in place of E(X). Here V = Syms(Y ) is the space of forms\nof degree s = deg(H(Y )) in l = O(m2) variable entries of Y . The dimension\n45\n\nM of the ambient projective space P(V ) here is exponential in l = O(m2),\nm being the the circuit size. Using the currently best available algorithms\nfor constructing a Gr ̈obner basis [KM], and for various problems in invariant\ntheory [St], analogues of the decision problems in Hypotheses 7.1 and 7.2\nfor h(X) can be solved in at best O(dim(C[V ]d) = O(dM) = O(d2poly(m))\nspace, where C[V ]d denotes the degree d component of C[V ], the homoge-\nneous coordinate ring of P(V ). This is so even for the decision problems in\nHypothesis 7.2 and hence for the verification problem in Hypothesis 7.1 (a).\nThis is the best that we can expect for general h(X) in view of the lower\nbound [MM] for the construction of Gr ̈obner bases. In other words, for a\ngeneral h(X) the time taken by a best procedure to even verify if Vλ(G), for\na given λ, is an obstruction would take space that is double exponential in\nm, and hence, time that is triple exponential in m.\nAs we shall argue in Section 19, for any approach towards the P ̸= NP\nconjecture to be viable, at least the problem of verifying an obstruction (i.e.,\na “proof”or “witness” of hardness as per that approach) should be easy; i.e.,\nbelong to P. Intuitively, because however hard it may be to discover a proof,\nits verification, once found, should be easy. The main P-barrier in the course\nof GCT is this huge gap between the triple exponential bound given by the\ncurrently best techniques for a general h(X) and the polynomial bound\nstipulated for verification in Hypothesis 7.1 (a) and in Hypothesis 7.2.\n9\nOn crossing the P-barrier\nWe now come to the main result of [GCT6] which crosses this P-barrier\nunder reasonable assumptions. It gives polynomial-time algorithms for the\ndecision problems in Hypothesis 1.2, and hence, for verifying an obstruction\n(Hypothesis 1.1 (a)), assuming the mathematical positivity hypotheses PH1\nand SH (Hypotheses 1.4-1.6).\n9.1\nA basic prototype with constant depth complexity\nTo motivate these positivity hypotheses, we first consider a basic prototype\nof the decision problems in Hypotheses 7.2 in a simplified setting:\nProblem 9.1 (Littlewood-Richardson problem) Given α, β and λ, decide\nif the Littlewood-Richardson coefficient cλ\nα,β (cf. Section 2.1.3) is positive\n(nonzero).\n46\n\nEquivalently, consider the diagonal homomorphism:\nρ : H = GLn(C) →G = H × H.\n(22)\nGiven an irreducible G-module Vα(H) ⊗Vβ(H), decide if an irreducible H-\nmodule Vλ(H) occur in it, when considered as an H-module via the diagonal\nhomomorphism.\nThis problem corresponds to circuits of depth two in the following sense.\nLet X be an n×n variable matrix. Let V = Sym1(X) be the space of linear\nforms in the entries of X. We have the action of G on P(V ) given by:\n((h1, h2) · f)(X) = f(h−1\n1 Xh2),\nfor any h1, h2 ∈H and f ∈P(V ).\nLet f(X) = trace(X).\nThen the\nstabilizer of f in G is precisely H, and f is characterized by its stabilizer.\nHence, f(X) = trace(X) is the characteristic function (Definition 3.1) of\nthe couple (22). It can be computed by a circuit of depth two. Hence, the\ncharacteristic class of the couple (22) can be defined to the class of circuits of\ndepth two. In this sense, the setting of the Littlewood-Richardson problem\nis roughly dual to the setting of expander graphs (Section 1.8), which too\ncorrespond to circuits of depth two.\nIn [GCT3, DM2, KT2] it is shown that this problem indeed belongs to\nP, thereby establishing the analogue of Hypothesis 7.2 in this setting. Two\nmain ingradients in this proof, in addition to linear programming, are PH1\nand SH for Littlewood-Richardson coefficients (Hypotheses 1.9 and 1.11).\nIn [GCT5], it is shown that the problem of deciding nonvanishing of a gen-\neralized Littlewood-Richardson coefficient for the classical connected reduc-\ntive groups other than GLn(C), namely the simplectic and the orthogonal\ngroups, also belongs to P, assuming the following generalized form of SH in\nthis context.\nLet ̃cλ\nα,β(k) = ckλ\nkα,kβ be the stretching function for a generalized Littlewood-\nRichardson coefficient cλ\nα,β, where α, β and λ are no longer partitions, but\nrather their generalizations [FH].\nIt is known to be a quasi-polynomial\n[BZ, DM2].\nHypothesis 9.2 (PH2): The quasi-polynomial ̃cλ\nα,β(k) is positive.\nThis was conjectured in [DM2] on the basis of considerable experimental\nevidence. Its weaker form is:\n47\n\nHypothesis 9.3 (SH): The quasi-polynomial ̃cλ\nα,β(k) is saturated.\nIn [GCT5] it is shown that the problem of deciding if a generalized\nLittlewood-Richardson coefficient is nonzero also belongs to P assuming\nPH2, or its weaker form, SH.\n9.2\nFrom constant to superpolynomial depth\nThe goal now is to lift the polynomial time algorithms and the mathematical\npositivity hypotheses PH1 and PH2 above from the simplified constant-\ndepth setting to the superpolynomial-depth setting of Hypotheses 7.1 (a)\nand 7.2. This is done in [GCT6] in two steps. We only consider the P vs.\nNP problem, considerations for the NC vs. P #P problem being similar.\nWe use the same notation as in Section 5.\nThe first step is the following mathematical result which allows formu-\nlation of the mathematical hypotheses PH1,PH2, and SH. Let sλ\nd(H; l) and\nsλ\nd(E; n, l) denote the multiplicities of the Weyl module Vλ(G) in S(H; l)d\nand R(E; n, l)d, respectively.\nLet us associate with them the following\nstretching functions:\n ̃sλ\nd(H; l)(k) = skλ\nkd(H; l),\n(23)\nand\n ̃sλ\nd(E; n, l)(k) = skλ\nkd(E; n, l).\n(24)\nThen:\nTheorem 9.4 (cf. Theorem 3.4.11 in [GCT6])\n(Rationality Hypothesis): Assume that the singularities of the class varieties\nXP (H; m) and XNP(E; n, l) are rational.\nThen the stretching functions ̃sλ\nd(H; l)(k) and ̃sλ\nd(E; n, l)(k) are quasi-\npolynomials.\nSimilar result also holds in the context of NC vs. P #P problem.\nRationality (niceness) [Ke] of singularities here is supported by the algebro-\ngeometric results and arguments in [GCT2, GCT10].\nThe second step is the following complexity-theoretic result:\n48\n\nTheorem 9.5 (cf. Theorems 3.4.11 and 3.4.13 in [GCT6]) The decision\nproblems in Hypothesis 7.2, and hence, the problem of verifying an obstruc-\ntion (Hypothesis 7.1 (a)) are indeed in P assuming the rationality hypothesis\nabove, and PH1 and PH2 (or weaker SH) in the introduction (Hypothe-\nses 1.4-1.6).\nSimilar result also holds in the context of the NC vs. P #P problem,\nassuming analogous hypotheses PH1, PH2 (or weaker SH) in this setting.\nTheorem 9.5 reduces the complexity-theoretic positive hypotheses in Hy-\npothesis 7.2 to the mathematical positivity hypotheses PH1 and SH (PH2),\nand the rationality hypothesis, unconditionally. Furthermore, [GCT6] also\ngives theoretical and experimental results in support of these positivity hy-\npotheses, and suggests a plan for proving them via the theory of quantum\ngroups. We shall discuss this plan later in Sections 15-16.\nThe whole discussion of this section is summarized in Figure 7. The top\ndouble arrow is unconditional, the bottom arrow is conjectural.\nMathematical positivity hypotheses PH1,2, and the rationality hypothesis\n∥\n∥\nGCT6\n∥\n⇓\nComplexity theoretic positivity hypotheses in Hypotheses 7.2 ∥\n∥\nTransformation in Section 13; cf. Figure 6\n∥\n?\n⇓\nP ̸= NP over C\nFigure 7: The main result of GCT6\n9.3\nSaturated and positive integer programming\nThe algorithm in Theorem 9.5 is based on a polynomial time algorithm\nin [GCT6] for a restricted form of integer programming, called saturated\n49\n\n(positive) integer programming. We briefly explain it in this section.\nLet A be an m × n integer matrix, and b an integral m-vector.\nAn\ninteger programming problem asks if the polytope P : Ax ≤b contains an\ninteger point. In general, it is NP-complete. So let us begin with the well\nknown special case of integer programming which belongs to P. This is the\nunimodular integer programming problem, wherein the constraint matrix A\nis unimodular. This means the polytope P is integral. In this case, P has\nan integer point iffP is nonempty. The latter can be checked in polynomial\ntime by standard linear programming methods.\nSaturated (positive) integer programming is a generalization of unimod-\nular integer programming, wherein a variant of linear programming still\nworks, even when P is nonintegral, provided P satisfies certain saturation\nor positivity hypothesis, which make up for the loss of unimodularity.\nIt is defined as follows. Let fP(n) be the Ehrhart quasi-polynomial of\nP [St1]. An integer programming problem is called saturated if the Ehrhart\nquasi-polynomial fP(n) is guaranteed to be saturated (cf. Section 1.2), if P\nis nonempty. It is called positive if fP (n) is guaranteed to be positive (cf.\nSection 1.2), if P is nonempty. We allow m, the number of constraints, to\nbe exponential in n. Hence, we cannot assume that A and b are explicitly\nspecified. Rather, it is assumed that the polytope P is specified in the form\nof a (polynomial-time) separation oracle as in [GLS]. Given a point x ∈Rn,\nthe separation oracle tells if x ∈P, and if not, gives a hyperplane that\nseparates x from P.\nThe following is the main complexity-theoretic result in [GCT6].\nTheorem 9.6 A saturated, and hence positive, integer programming prob-\nlem has an oracle-polynomial-time algorithm.\nFurthermore, this polynomial time algorithm is conceptually extremely\nsimple. It is essentially a variant of linear programming: it uses a general-\nization of the ellipsoid method [Kh] for linear programming in [GLS], and\na polynomial time algorithm for computing Smith normal forms in [KB].\nThus the saturated and positive integer programming paradigm, in essence,\nsays that linear programming works for integer programming provided the\nsaturation or the positivity property holds.\nTheorem 9.5 follows from Theorem 9.4 because PH1 and SH (PH2) (cf.\nHypotheses 1.4-1.6) imply that the decision problems in Hypothesis 7.2 can\nbe transformed in polynomial time into saturated (positive) integer program-\n50\n\nming problems. Thus, in essence, a variant of linear programming works for\nthe decision problems in Hypothesis 7.2, provided PH1 and SH (PH2) hold.\nBut these saturation and positivity hypotheses (PH1 and SH) are non-\ntrivial, and, as we shall see in Sections 14 to 16, their validity intimately\nseems to depend on deep phenomena in algebraic geometry and the theory\nof quantum groups. We can already see an indication of this here. For ex-\nample, even to state PH1, SH or PH2, we need to show that the stretching\nfunctions used in their statements are quasi-polynomials, as shown in The-\norem 9.4. Without it, PH1, SH and PH2 are meaningless. But the proof of\nTheorem 9.4 already depends on nontrivial machinery in algebraic geome-\ntry; e.g. the cohomology vanishing result in [Ke], and the result in [Bou],\nwhich, in turn, needs resolution of singularities in characteristic zero [Hi]\nand other cohomology vanishing results. Hence it should not be surprising\nif proving these positivity hypotheses needs far more. We shall describe the\nbasic plan in [GCT6] for proving them later (Sections 15-16).\n10\nWhy should PH1 and PH2 hold?\nBut, first, we have to explain why PH1 and PH2 should hold in the first\nplace.\nThis depends, as mentioned earlier, on the exceptional nature of\nH(Y ) and E(X). Specifically, on the fact that the associated class varieties\nXP (H; l) and XNP (E; n, l) are group-theoretic. We now elaborate on this.\nFirst, let us consider the analogue of the decision problem in Hypothe-\nsis 7.2 for the simplest group-theoretic variety, namely, a flag variety (Sec-\ntion 2.2).\nGiven a flag variety Z = Gvμ ⊆P(Vμ), where Vμ is a Weyl\nmodule of G = SLl(C), the decision problem is to decide if Vλ(G) occurs\nin R(Z)d, the degree d component of the homogeneous coordinate ring of\nZ. By the Borel-Weil theorem [FH], R(Z)d = V ∗\ndμ, the dual of Vdμ. Hence,\nVλ occurs in R(Z)d iffVλ = V ∗\ndμ. It is easy to show that this is so iffthe\nYoung diagram for λ is obtained by flipping the complement of the Young\ndiagram for dμ in the smallest rectangle containing it. This can be decided\nin poly(⟨d⟩, ⟨λ⟩, ⟨μ⟩) time. The analogues of PH1 and PH2 in this setting\nclearly hold, since the multiplicity of Vλ in R(Z)d is just 0 or 1.\nNow let us move to a general group-theoretic class variety. Let (H ֒→\nG ֒→K) be the associated group triple. Since the class variety in question\nis (essentially) determined by this triple, all questions concerning the vari-\nety should, in principle, be reducible to representation-theoretic questions\nregarding this triple; cf. [GCT10], and Sections 3 and 15.\n51\n\nIn [GCT6] and [GCT10] analogues of the decision problems in Hypoth-\nesis 7.2 for the couples H ֒→G and G ֒→K are formulated. Furthermore,\ntheoretical and experimental evidence for PH1 and PH2 for the decision\nproblems associated with these couples is provided. Since the triples are\nqualitatively similar to the couples, though much harder, this provides the\nmain evidence in support of PH1 and PH2 for the class varieties under con-\nsideration. We shall turn to this evidence in the next section.\n11\nDecision problems in representation theory\nWe now describe the decision problems associated with the couple H ֒→G,\nthe couple G ֒→K being similar. A general decision problem is as follows:\nProblem 11.1 (The subgroup restriction problem)\nLet ρ : H →G be as above, with G connected (and some mild technical\nrestrictions on ρ as described in [GCT6]). Assume that both H and G are\nreductive. Let Vπ(H) be an irreducible representation of H, and Vλ(G) an\nirreducible representation of G, where π and λ denote the classifying labels\nof these representations.\nLet mπ\nλ be the multiplicity of Vπ(H) in Vλ(G),\nconsidered as an H-module via ρ. Given specifications of the embedding ρ\nand the labels λ, π, decide nonvanishing of the multiplicity mπ\nλ.\nThe general decision problems in Hypotheses 7.2 can be thought of as harder\nvariants of this problem obtained by going from couples to triples.\nAll\ncouples that arise in GCT are either of the type in this decision problem, or\nof a hybrid type obtained by combining this type with the type considered\nearlier in connection with the flag variety, when H = Pμ is parabolic; cf.\n[GCT10] for a discussion of the hybrid types.\nProblem 11.1 is a fundamental decision problem of representation theory.\nIndeed, one of the main motivations in the classical works of representation\ntheory, e.g. [W], for classifying of all irreducible representations of reductive\ngroups was to be able to solve this problem satisfactorily. But despite all\nprogress in representation theory in the last century, this problem at its very\nheart remained open. PHflip in [GCT6] says that this fundamental decision\nproblem of representation theory has an easy polynomial time algorithm.\nHere we shall describe PHflip in only the following three special cases\nof the above decision problem, referring the reader to [GCT6] for a full\ndiscussion and results for the general decision problem.\n52\n\n11.0.1\nLittlewood-Richardson problem\nLet H = GLn(C), G = H × H, the embedding\nρ : H →H × H = G\nbeing diagonal. Then the multiplicity in Problem 11.1 is just the Littlewood-\nRichardson coefficient, because every irreducible representation of G is of\nthe form Vα ⊗Vβ, where Vα and Vβ are irreducible representations of H =\nGLn(C) for partitions α, and β, and the multiplicity of an H-module Vλ in\nVα ⊗Vβ, considered as an H-module via the diagonal map ρ, is precisely\nthe Littlewood-Richardson coefficient cλ\nα,β. We have already noted that its\nnonvanishing can be decided in polynomial time (Section 9.1).\n11.0.2\nKronecker problem\nLet H = GLn(C) × GLn(C) and\nρ : H →G = GL(Cn ⊗Cn) = GLn2(C)\nthe natural embedding given by: ρ(h1, h2) = h1 ⊗h2, for any h1, h2 ∈H.\nHere h1 ⊗h2 is the Kronecker product as defined in (18). Let kπ\nλ,μ be the\nmultiplicity of the H-module Vλ(GLn(C)) ⊗Vμ(GLn(C)) in the G-module\nVπ(G), considered as an H-module via the embedding ρ. Then it can be\nshown [FH] that the Kronecker coefficient as defined in Section 2.1.2 is a\nspecial (dual) case of this when λ, μ and π there coincide with the λ, μ and\nπ here. For this reason, we call kπ\nλ,μ a Kronecker coefficient.\nProblem 11.2 (The Kronecker problem) Given partitions λ, μ and π, de-\ncide nonvanishing of the Kronecker coefficient kπ\nλ,μ.\nThe following is an analogue of Hypothesis 7.2 in this context:\nHypothesis 11.3 [GCT6] (PHflip-kronecker) Given partitions λ, μ and π,\nnonvanishing of the Kronecker coefficient kπ\nλ,μ can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)\ntime.\n11.0.3\nThe plethysm problem\nThe Kronecker coefficient is known [Ki] to be a special case of the plethysm\ncoefficient in the following more general problem.\n53\n\nProblem 11.4 (The plethysm problem) Given partitions λ, μ and π, de-\ncide nonvanishing of the plethysm constant aπ\nλ,μ. This is the multiplicity\nof the irreducible representation Vπ(H) of H = GLn(C) in the irreducible\nrepresentation Vλ(G) of G = GL(Vμ), where Vμ = Vμ(H) is an irreducible\nrepresentation H. Here Vλ(G) is considered an H-module via the represen-\ntation map\nρ : H →G = GL(Vμ).\nThe following is an analogue of Hypothesis 7.2 in this context:\nHypothesis 11.5 [GCT6] (PHflip-plethysm) Given partitions λ, μ and π,\nnonvanishing of the plethysm constant aπ\nλ,μ can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)\ntime.\n12\nThe P-barrier in representation theory\nAt the surface, this hypothesis too seems impossible because the dimension\nof G here can be exponential in the dimension of H. This happens when the\ndimension of the representation Vμ(H) is exponential in dim(H). But the\ntotal bitlength of λ, μ and π can be polynomial in dim(H). Hypothesis 11.5\nin this case says that nonvanishing of the plethysm constant can still be\ndecided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) time.\nA priori it is not even clear that the\nplethysm constant can be evaluated in PSPACE in this case. Since the usual\ncharacter-theory-based algorithms in representation theory for its evaluation\n[FH, Mc] take space that is polynomial in the dimension of G, and hence,\nexponential in the dimension of H.\nThe main P-barrier in representation theory is this huge gap between\nthe exponential space bound for the plethysm or the general decision Prob-\nlem 11.1 given by the usual methods of representation theory and the poly-\nnomial time bound stipulated in Hypothesis 11.5 for the plethysm constant\nand the hypothesis in [GCT6] for the general decision Problem 11.1.\n12.1\nCrossing the P-barrier\nWe now describe the main results of [GCT6] which together cross this P-\nbarrier in representation theory subject to the analogous mathematical pos-\nitivity hypotheses PH1 and SH (PH2). We shall only concentrate on the\nplethysm problem, since it is the crux of the matter.\n54\n\nAssociate with a plethysm constant aπ\nλ,μ the stretching function\n ̃aπ\nλ,μ(k) = akπ\nkλ,μ.\n(25)\nNote that μ is not stretched here.\nThen the following is an (unconditional) analogue of Theorem 9.4 in this\ncontext:\nTheorem 12.1 (cf.\nTheorem 1.6.1 in [GCT6]) The stretching function\n ̃aπ\nλ,μ(k) is a quasi-polynomial function of k.\nThe following are the analogues of PH1 and PH2 in this context:\nHypothesis 12.2 (PH1)\nFor every (λ, μ, π) there exists a polytope P = P π\nλ,μ ⊆Rm with m =\npoly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) such that:\naπ\nλ,μ = φ(P),\n(26)\nwhere φ(P) is equal to the number of integer points in P, and the Ehrhart\nquasi-polynomial of P coincides with the stretching quasi-polynomial ̃aπ\nλ,μ(k)\nin Theorem 12.1. (And some additional technical constraints)\nHypothesis 12.3 (PH2)\nThe stretching quasi-polynomial ̃aπ\nλ,μ(k) is positive (cf. Section 1.2).\nPH2 implies the following saturation hypothesis:\nHypothesis 12.4 (SH)\nThe quasi-polynomial ̃aπ\nλ,μ(k) is saturated (cf. Section 1.2).\nThe following is an analogue of Theorem 9.5 in this context:\nTheorem 12.5 [GCT6] Assuming PH1 and SH (or, more strongly, PH2),\nnonvanishing of a plethysm constant aπ\nλ,μ can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)\ntime; i.e. the problem of deciding nonvanishing of a plethysm constant be-\nlongs to P, as per Hypothesis 11.5.\n55\n\nPH1 above implies that aπ\nλ,μ belongs to #P just like the Littlewood-\nRichardson coefficient. Its weaker form is:\nTheorem 12.6 The plethysm constant aπ\nλ,μ can be computed in PSPACE,\ni.e., in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) space.\nThat this holds even if the dimension of G = GL(Vμ) is exponential in\nn is crucial in the context of GCT. Because the dimension of K = GL(V )\nin the triples H ֒→G ֒→K = GL(V ) associated with the class varieties\n(Section 4) is exponential in the circuit size m. Hence, without this result,\nit is not at all clear why the structural constants sλ\nd(H, l) and sλ\nd(E; n, l) in\nHypothesis 1.4 should even belong PSPACE ⊇#P, as implied by it.\nTheorem 12.6 and Theorem 12.1 together provide good theoretical evi-\ndence for PH1 (Hypothesis 12.2). Indeed, Theorem 12.1, together with other\nevidence in [GCT6], suggests that ̃aπ\nλ,μ(k) is the Ehrhart quasi-polynomial\nof some polytope P = P π\nλ,μ. Furthermore, Theorem 12.6 says that the di-\nmension m of the ambient space Rm containing P should be polynomial in\nthe bitlengths ⟨λ⟩, ⟨μ⟩and ⟨π⟩. If not, it would not be possible to count\nthe number of integer points in P in PSPACE, since even the bitlength of\nany integer point in Rm would not be polynomial. For further theoretical\nand experimental results in support of PH1 and PH2 in this context, see\n[GCT6]. These constitute the main evidence in support of PH1 and PH2\nfor the group-theoretic class varieties (Hypotheses 1.4-1.5), because mathe-\nmatical positivity is a very abstract property, which should remain invariant\nwhen we go from couples to triples.\n13\nReduction\nNow we turn to the reduction in the top arrow in Figure 1. For this, we\nhave to describe:\n1. How to transform the easy algorithms in Theorem 9.5 into an easy\nalgorithm for discovering an obstruction as in Hypothesis 7.1 (c), and\n2. How to transform this easy algorithm for discovery into a constructive\nproof of existence of obstructions–as expected (Section 6)–for every n\nand l = nlog n, by showing how such an obstruction-label can be easily\nconstructed in this case explicitly.\n56\n\nThis would imply that P ̸= NP over C.\nThese transformations cannot be carried out at present, since we do\nnot know the polytopes P λ\nd (H; l) and P λ\nd (E; n, l) explicitly. We only know\nthat they should exist as per PH1 (Hypothesis 1.4). But once their explicit\ndescriptions become available, it should be possible to carry out the above\ntwo transformations along the lines that we now suggest.\n13.1\nTowards easy discovery\nFirst, let us describe why it should be possible to extend and transform\nthe polynomial-time algorithms in Theorem 9.5 to obtain a polynomial time\nalgorithm for discovering an obstruction (Hypothesis 7.1 (c)) once explicit\ndescriptions of the polytopes P λ\nd (H; l) and P λ\nd (E; n, l) become available.\nFor the sake of simplicity, let us assume that the quasi-polynomials in\nTheorem 9.4 are actually polynomials; i.e., their periods are one, though\nthis is not expected. In that case, it can be shown that [GCT6] PH1 and SH\nimply that there exist polytopes P(n, l) = P(E; n, l) and Q(l) = P(H; l) of\npoly(n, l) dimensions such that an obstruction for n, l and the pair (E, H)\nexists iffthe relative difference T(n, l) = P(n, l) \\ Q(l) is nonempty, and\nfurthermore, an explicit obstruction can also be constructed in polynomial\ntime once we are given a rational point in T(n, l). So it suffices to check\nif T(n, l) is nonempty, and if so, find a rational point in it. This can be\ndone in polynomial time using the convex (linear) programming algorithm in\n[GLS, Vd] if Q(l) has only poly(l) explicitly described facets. This is so even\nif P(n, l) has exponentially many facets. But if Q(l) has exponentially many\nfacets–as happens even in the context of the simpler Littlewood-Richardson\nproblem (Problem 9.1)–then an oracle-based algorithm as in [GLS] cannot\nbe used to get a polynomial time algorithm for this problem [B].\nBut this does not appear to be a serious problem.\nIndeed, a gen-\neral principle in combinatorial optimization, as illustrated in [GLS], is that\ncomplexity-theoretic properties of polytopes with exponentially many facets\nare similar to the ones with polynomially many facets if these facets have a\nwell-behaved regular structure. For example, if Q(l) and P(n, l) were perfect\nmatching polytopes for non-bipartite graphs–which can have exponentially\nmany facets–nonemptiness of T(n, l) can be easily decided in polynomial\ntime [Ve] using the polynomial time algorithm [Ed] for finding a perfect\nmatching in a nonbipartite graph. The facets of the analogues of P(n, l) and\nQ(l) in the Littlewood-Richardson problem, called Littlewood-Richardson\ncones [Z], have an explicit description with very nice algebro-geometric and\n57\n\nrepresentation-theoretic properties [Kl]. The same is expected to be the case\nin our setting.\nThis is why we expect that nonemptiness of T(n, l) and computation of\na rational point in it, if it is nonempty, can be done in polynomial time, once\nexplicit descriptions of P(n, l) and Q(l) become known. This would give a\npolynomial time algorithm for discovering an obstruction, if it exists, as per\nHypothesis 7.1 (c), assuming that the quasi-polynomials in Theorem 9.4 are\npolynomials.\nFurthermore, it is expected, for the mathematical reasons given in [GCT6],\nthat there exist genuinely simple, i.e., purely combinatorial greedy-type al-\ngorithms for the problems under consideration that do not even need linear\nprogramming.\nThat is, the story is expected to be the same as for the\nmin-cost flow problem in combinatorial optimization, for which a linear-\nprogramming-based polynomial-time algorithm was found first [Ta] to be\nfollowed by several genuinely simple and purely combinatorial polynomial\ntime algorithms; e.g. see [O]. Similarly, it is reasonable to expect that the\nalgorithms in Theorem 9.5 and the subsequent algorithm for discovery of\nobstructions can be simplified further to eventually get simple greedy al-\ngorithms for these problems akin to the Hungarian method, once explicit\ndescriptions of P(n, l) and Q(l) become known.\nSo far we are assuming that the quasi-polynomials in Theorem 9.4 are\npolynomials. This need not be so. In fact, this is not so even in the sim-\nplified setting of plethysm constants [GCT6]. When the quasi-polynomials\nin Theorem 9.4 have nontrivial periods, the obstructions can be classified\nin two types: geometric and modular [GCT6]. Geometric obstructions are\nsimilar to the ones that would arise if these quasi-polynomials were poly-\nnomials. A polynomial time algorithm for their existence and construction\nmay be designed along the lines we just described.\nLet us next describe briefly what needs to be done in the case of mod-\nular obstructions. Theorem 9.5 says that for the decision problems therein\nlinear programming in conjunction with modular techniques (computation\nof Smith normal forms [KB]) works even in the modular setting, i.e., when\nthe quasi-polynomials have nontrivial periods. Hence, once we have a poly-\nnomial time algorithm for discovering a geometric construction, it should\nbe possible to extend it to a polynomial time algorithm for discovering a\nmodular obstruction in conjunction with appropriate modular techniques;\ncf. [GCT6] for the problems that need to be addressed in this extension.\n58\n\n13.2\nFrom easy algorithm for discovery to easy proof of ex-\nistence\nAssuming that we have an easy polynomial time algorithm for discovering an\nobstruction as per Hypothesis 7.1 (c), let us now describe why it should be\npossible to prove using this algorithm, or rather the underlying structure and\ntechniques, that there always exists an obstruction, as expected (Section 6),\nfor every n →∞, assuming l = nlog n (say).\nFor the sake of simplicity, let us again assume that that the quasi-\npolynomials in Theorem 9.4 are polynomials, and that we have an easy\nHungarian-type greedy algorithm as discussed above for deciding nonempti-\nness of T(n, l), and for computing a point it it, if it is nonempty. Then we\nhave to show, using the techniques and the structure underlying this algo-\nrithm, that T(n, l) is always nonempty when l = nlog n, n →∞. Such a proof\nwould also give us a polynomial time procedure for explicit construction of\nan obstruction λ(n), for every n. Hence we shall call it a P-constructive\nproof.\nTo see how to get such a P-constructive proof, let us consider an analogy.\nLet us imagine that Q(l) is empty, so that T(n, l) = P(n, l) is a polytope,\nand that it is the perfect-matching polytope of a bipartite graph G(n, l).\nThen T(n, l) is nonempty iffG(n, l) has a perfect matching, which can be\nthought of as an obstruction in this analogy. The analogous goal then is to\nshow using the techniques and structure underlying the Hungarian method\nthat G(n, l) always has a perfect matching, as expected, when l = nlog n,\nand n →∞. In other words, we have to give a P-constructive proof for\nexistence of a perfect matching in every such G(n, l). In this analogy, the\ntechnique underneath the Hungarian method can be easily used to give a\nconstructive proof of Hall’s marriage theorem–namely, that every bipartite\ngraph H in which every subset, on any side of the graph, has at least as\nmany neighbours as the size of that subset has a perfect matching–which\nthen has to be used to show that G(n, l) always has a perfect matching\nwhenever l = nlog n, n →∞.\nNow in our setting T(n, l) is not a perfect matching polytope. But if\nit has a nice structure like the perfect matching polytope then it should be\npossible to prove structure theorems in the spirit of Hall’s marriage theorem\nfor T(n, l) using the structure of the Hungarian-type greedy algorithm for\ndeciding nonemptyness of T(n, l) and then use it to prove nonemptyness of\nT(n, l), for every n →∞, when l = nlog n.\nFor such a transformation of a polynomial time algorithm for discovery\n59\n\ninto a P-constructive proof of existence to work, it is crucial that:\n1. The polyhedral set T(n, l) has a nice, regular structure like the perfect\nmatching polytope.\nFortunately, the polytopes P(n, l) and Q(n, l)\nthat would arise in our setting should be even nicer than the per-\nfect matching polytope. For example, in the simpler setting of the\nLittlewood-Richardson problem (Problem 9.1), P(n, l) and Q(l) be-\ncome Littlewood-Richardson cones [Z], which have extremely regu-\nlar structure with remarkable representation-theoretic and algebro-\ngeometric properties [F2, Kl]. The same is expected to be the case\nfor the actual P(n, l) and Q(l).\n2. The algorithm for discovery not only works in polynomial time, but\nalso has a simple structure like the Hungarian method. The Hungarian-\ntype greedy algorithms that we expect for the problems under consid-\neration should have such structures.\nHence, it is reasonable to expect that an easy Hungarian-type algorithm\nfor deciding nonemptyness of T(n, l) can be transformed into the sought P-\nconstructive proof of obstructions. The story is expected to be similar, albeit\nmuch harder, when the quasi-polynomials in Theorem 9.4 have nontrivial\nperiods; cf [GCT6].\nThe above scheme for the transformation of an algorithm for discovery\ninto a constructive proof of existence banks on the fact that the algorithm\nto be transformed is easy, i.e., works in polynomial time, besides having\na simple structure.\nThe underlying informal principle, which cannot be\nproved, is that the mathematical complexity of an algorithmic (constructive)\nproof is intimately linked to the computational complexity of the algorithm\non which it is based; see Section 19 for a detailed treatment of this issue.\nThis is why there is no nontrivial result, comparable to Hall’s theorem, for\nHamiltonian paths.\nBecause the problem of finding such a path is NP-\ncomplete.\nThe reader may wonder why we are talking about explicit construction of\nobstructions, when, strictly speaking, we only need to know their existence.\nThis is because the nature of obstructions in our case is such that their\nexplicit construction, if they exist, can be done with only a little additional\ncost over the cost of deciding existence. To see this, let us again assume,\nfor the sake of simplicity, that the quasi-polynomials in Theorem 9.5 are\npolynomials.\nThen a technique that can decide nonemptiness of T(n, l)\nshould also be able to compute a point in it, as a proof of nonemptiness, at\n60\n\nonly a little additional cost, just as in linear programming. In other words,\nthe complexity of deciding existence of an obstruction should be more or\nless the same as that of constructing it, if it exists. This is why we mainly\ntalk of explicit construction of obstructions though, in principle, just their\nexistence would suffice.\nOur discussion so far says that PH1 and SH (PH2) are the crux of the\nmatter. If they can be proved, and explicit descriptions of the polytopes\ntherein become available, it should be possible to transform the easy algo-\nrithms in Theorem 9.5 into an easy algorithm for explicit construction of\nobstructions as per Hypothesis 7.1 (b).\n14\nStandard quantum group\nNow we proceed to the basic plan in [GCT6] for proving PH1 and SH. This\nis motivated by a story in the theory of standard quantum groups in the\ncontext of the Littlewood-Richardson problem (Problem 9.1). We describe\nthat story in this section.\nFor this we need the notion of a standard quantum group, by which we\nmean the quantum group in [Dri, Ji, RTF]. We can not formally define here\nthis object, but we can at least give an intuitive idea. Let GL(Cn) be the\ngroup of nonsingular n × n matrices. It can be thought of as the group of\nnonsingular transformations of Cn. Let xi’s denote the coordinates of Cn.\nThese commute. That is:\nxixj = xjxi.\nLet us now see what happens if the coordinates become noncommuting. This\nis precisely what happened in quantum physics. We discovered that the po-\nsition and the momentum, which for centuries we thought were commuting\nobservables, do not actually commute. Quantum groups were invented pre-\ncisely to investigate the related phenomena in theoretical physics. Let Cn\nq\ndenote the quantum space whose coordinates xi’s are noncommuting, and\nsatisfy the following relation:\nxixj = qxjxi,\ni < j\nwhere q ∈C is some fixed number. The standard quantum group GLq(Cn) is\nthe “group” of invertible linear transformations of this quantum space. This\nis not a “group” in any ordinary sense. Its precise description is given in [Dri,\n61\n\nJi, RTF]. We do not need that here. Let us just think of a quantum group\nas what a group becomes when the coordinates become noncommuting.\nLet us now explain how quantum groups enter in the story of Littlewood-\nRichardson coefficients.\nThis is because the most transparent proof of\nthe Littlewood-Richardson rule came via the theory of quantum groups\n[Kas1, Li, Lu2]. The earlier proofs, though elementary and combinatorial,\nwere highly mysterious. Moreover, the theory of quantum groups gave the\nfirst proof of the generalized Littlewood-Richardson rule [] for general (con-\nnected) reductive groups, instead of just GLn(C).\nLet us now elaborate the nature of this proof. We begin by observing\nthat the Littlewood-Richardson problem (Problem 9.1) is an instance of the\ngeneral decision Problem 11.1 associated with the diagonal group homomor-\nphism\nρ : H = GL(Cn) →H × H = GL(Cn) × GL(Cn).\n(27)\nIf we understood the structure of this homomorphism in depth, we ought\nto understand why PH1 and SH (and also PH2) hold for the Littlewood-\nRichardson coefficients.\nAs we mentioned earlier, in depth means at the\nquantum level.\nTo understand the homomorphism (27) at the quantum\nlevel, we need to quantize it. Ideally, one would want its quantization in the\nform of a homomorphism\nρq : Hq = GLq(Cn) →Hq × Hq = GLq(Cn) × GLq(Cn).\n(28)\nwhere Hq is the standard quantum group associated with H. This does not\nhold as it is; i.e., Hq is not a quantum subgroup of Hq × Hq. But this is\nessentially so. That is, it holds in a certain dual setting–this is the main\nresult in [Dri, Ji, RTF]. Thus the theory of quantum group can be regarded\nas the theory of the quantization ρq.\nOnce this theory is developed sufficiently, the Littlewood-Richardson rule\nas well as PH1 for Littlewood-Richardson coefficients (Hypothesis 1.9) turn\nout to be a consequence, in a nontrivial way, of a deep positivity result in\nthe theory of the standard quantum groups [Kas2, Kas3, Lu1, Lu2]: namely,\ntheir representations and coordinate rings have canonical bases, also called\nglobal crystal bases, whose structural constants, which determine their mul-\ntiplicative and representation theoretic structure, are all nonnegative. For\nthis reason, we say that the canonical bases are positive, and refer to exis-\ntence of a canonical basis as a positivity property (hypothesis) PH0.\nWe now give a brief intuitive description of the canonical basis. Let X\nbe an n×n variable matrix. The coordinate algebra R = O(G) of the group\n62\n\nG = GL(Cn) is defined to be the C-algebra generated by the entries xij of X\nand det(X)−1, where det(X) denotes the determinant of X. Its elements are\nregular functions on G, considered as an affine variety. There is a natural\nleft action of G on R given by f(X) →f(σ−1X), for any σ ∈G, and a\nsimilar right action.\nThese notions can now be quantized. It is possible to associate a coor-\ndinate ring Rq = O(Gq) with the standard quantum group Gq = GLq(Cn),\nwhose elements can be intuitively thought of as functions on Gq. Unlike R,\nRq is not commutative. Its precise definition can be found in [RTF]. There\nare natural left and right actions of Gq on Rq.\nA canonical basis B of Rq is a very special basis with the following\nproperties:\n(1) It is representation-theoretically well behaved. This means there is a\nfiltration\nB0 = ∅⊂B1 ⊂B2 ⊂· · · ⊂B\nwith ∪iBi = B, such that ⟨Bi⟩/⟨Bi−1⟩is an irreducible Gq-module. Here\n⟨Bi⟩denotes the span of the basis elements in Bi.\n(2) Positivity property of the multiplicative structure constants:\nGiven two elements b, b′ ∈B, let\nbb′ =\nX\nb′′∈B\nf b′′\nb,b′b′′,\nbe the expansion of the product in terms of the basis B. Then each f b′′\nb,b′ is\nan explicit polynomial in q and q−1 with nonnegative coefficients. Here f b′′\nb,b′\nare called multiplicative structure constants. What this says is that each\nmultiplicative structure constant has an explicit positive formula, akin to\nthat of the permanent. Here explicit means that each nonnegative coefficient\nof f b′′\nb,b′ has an interpretation in terms of a nonnegative topological invariant\n(akin to Betti numbers) of an algebraic variety.\n(3) Positivity property of the representation-structure constants:\nGiven any element b ∈B and a generator e of a certain algebra defined\nin [Dri, Ji], which is “dual” to Rq, let\ne · b =\nX\nb′\ngb′\ne,bb′\nbe the expansion of e · b, the result of applying e to b, in terms of the\nbasis B. Then each gb′\ne,b is also an explicit polynomial in q and q−1 with\n63\n\nnonnegative coefficients.\nThat is, each representation-structure constant\nalso has an explicit positive formula.\nThese positivity properties do not actually hold as stated–that is still a\nconjecture [Lu2]–but their slightly weaker form holds unconditionally [Lu2].\nWe shall ignore that difference here.\nIf we specialize the canonical basis at q = 1, we get a canonical basis of\nR, the coordinate ring of G, with analogous positivity property. But, as of\nnow, the only way to prove existence of such a canonical basis of R is via the\ntheory of quantum groups as above. This shows the power of this theory.\nOne can easily imagine that there ought be a connection between ex-\nistence of bases whose structural constants have explicit positive formu-\nlae (PH0) and existence of an explicit positive (polyhedral) formula for\nLittlewood-Richardson coefficients (PH1). That is indeed so, as we men-\ntioned earlier, but in a quite nontrivial way; cf. [Kas1, Li, Lu2]. We shall\nsimply take this connection on faith here. Pictorially:\nPH0 →PH1.\n(29)\nOne does not really need the full power of PH0 to deduce PH1. Just\nexistence of a local crystal basis [Kas1], which is the limit (crystalization)\nof a canonical basis as q →0, is sufficient.\nBut when we move to the\nnonstandard setting in GCT, even the full power of PH0 is needed for some\nother reasons; [GCT8, GCT10].\nThe implication (29) provides arguably the most satisfactory proof of\nPH1 for Littlewood-Richardson coefficients, which, in addition, also pro-\nvides deep additional information (existence of canonical bases) which the\ncombinatorial proofs [F1] cannot provide. Such canonical bases are central\nto the approach in [GCT6, GCT10] towards PH1 and PH2 for the group-\ntheoretic class varieties (Section 15). Hence, as far as GCT is concerned,\nquantum groups are a must.\nSH for the usual Littlewood-Richardson coefficients is the saturation\ntheorem in [KT1]. It comes from a reformulation of PH1 in terms of special\npolytopes (called Hive polytopes) and their subsequent detailed study. Thus\npictorially:\nPH1 →SH,\n(30)\nagain in a nontrivial way.\n64\n\nBut how is PH0 proved?\nThe only known proof of PH0 [Lu1, Lu2]\nis based on a deep positivity property in mathematics: the Riemann Hy-\npothesis over finite fields [Dl2], and related results [BBD]. In other words,\nnonnegativity of the structural constants associated with Hq is connected at\na profound level with the lining up of the zeros of the zeta functions of some\nalgebraic varieties on one axis. We shall denote the Riemann hypothesis\nover finite fields by PH+. Then pictorially:\nPH+ →PH0,\n(31)\nin a highly nontrivial way.\nPutting implications (29)-(31) together with the story in Section 9.1, we\narrive at Figure 8 which summarizes the story in this section.\nPH+: The Riemann hypothesis over finite fields and related results [BBD, Dl2]\n∥\n∥\n⇓\nPH0: Existence of canonical bases [Lu1, Lu2]\n∥\n∥\n⇓\nPH1 and SH [Kas1, Li, Lu2, KT1]\n∥\n∥\n∥\n⇓\nPolynomial time algorithm for deciding nonvanishing of\nLittlewood-Richardson coefficients [DM2, GCT5, KT1]\nFigure 8: A story in the theory of standard quantum groups\n15\nNonstandard quantum groups\nNow we turn to the problem of proving PH1 and SH that actually arise in\nGCT (Hypotheses 1.4-1.6, and 12.2-12.4). The basic plan in [GCT6] for this\n65\n\nis simply to lift the story in Figure 8 from height two to superpolynomial\nheight–i.e., from the circuits of height two that the Littlewood-Richardson\nproblem corresponds to to the circuits of superpolynomial height that the\ndecision problems in Hypothesis 7.2 correspond to.\nRoughly, it goes as\nfollows:\n(1) Quantization: Quantize the couples\nH ֒→G,\nG ֒→K\nand the triples\nH ֒→G ֒→K,\nassociated with the class varieties in a manner akin to the quantization (28)\nof (27) via standard quantum groups.\n(2) PH0 for couples and triples: Prove that the coordinate rings and\nrepresentations of the quantum groups that arise in this quantization have\ncanonical bases akin to the canonical bases for the standard quantum groups\nwhose structure constants, which determine their multiplicative and repre-\nsentation theoretic structure, are all nonnegative.\n(3) PH0 for class varieties: Use the canonical bases for the (quantized)\ntriples associated with the class varieties to construct analogous canonical\nbases for the coordinate rings for appropriate quantizations of the class\nvarieties with nonnegative structure constants.\n(4) PH1, SH: Deduce PH1 and SH from PH0 in the spirit of the middle\narrow in Figure 8.\nFigure 9 shows this pictorially.\nWe shall now elaborate Figure 9.\n15.1\nQuantization\nLet us begin with the first step of quantization. We shall only worry about\nthe couples. To be concrete, let\nH ֒→G = GL(Ck),\n(32)\nbe as in Problem 11.1, where H is connected, reductive subgroup of G.\nQuantization of this couple is the crux of the problem.\nAll other quan-\ntizations that are needed are hyped up versions of this, so we shall only\nconcentrate on this.\n66\n\nQuantization of couples [GCT4,\nGCT7] and triples [GCT10]\n|\n|\n|\n↓\nPH0\nfor\ncouples\nand\ntriples\n[GCT6, GCT8, GCT10]\n−−−→\nPH1 and SH for cou-\nples and triples\n|\n|\n↓\nPH0 for class varieties [GCT10]\n|\n|\n|\n↓\nPH1 and SH for class varieties\nFigure 9: The basic plan for proving PH1 and SH in [GCT6]\nThe standard theory of quantum groups can not be used for quantizing\nthis couple, as expected. Specifically, let Gq = GLq(Cn) be the standard\nquantum group associated with G. In a similar fashion, one can associate\n[Dri, Ji, RTF] a standard quantum group Hq with H. Then, Hq cannot be\nembedded as a quantum subgroup of Gq (where the notion of subgroup in\nthe quantum setting is akin to the usual notion of a subgroup). Hence the\ngoal is to associate a quantization ˆGq with G akin to the standard quantum\ngroup Gq so that the standard quantum group Hq is a quantum subgroup\nof ˆGq. In that case:\nHq ֒→ˆGq,\n(33)\ncan be considered to be a quantization of (32).\nThis quantization step is addressed in the following result for the couples\nin Problems 11.2-11.4, which are the main prototypes of the couples that\narise in GCT.\n67\n\nTheorem 15.1 (1) (cf. [GCT4]) The couple\nH = GL(Cn) × GL(Cn) →GL(Cn ⊗Cn) = G,\nassociated with the Kronecker problem (Problem 11.2) can be quantized in\nthe form:\nHq →ˆGq,\nwhere Hq is the standard quantum group associated with H and ˆGq is the\nnew nonstandard quantum group associated with G. Furthermore, ˆGq has a\nquantum unitary subgroup ˆUq in the sense of [Wo], which is a quantization\nof the unitary subgroup U = Un2(C) ⊆G = GLn2(C).\n(2) (cf. [GCT7]) More generally, the couple\nH = GLn(C) →G = GL(Vμ(H)),\n(34)\nassociated with the plethysm problem (Problem 11.4) can also be quantized\nin the form:\nHq →ˆGq,\nwhere Hq is the standard quantum group associated with H and ˆGq is the\nnew nonstandard (possibly singular) quantum group associated with G. Here\nH can even be any connected classical reductive group.\nThe nonstandard quantum group in [GCT4] is qualitatively similar to\nthe standard quantum group in [Dri, Ji, RTF] in the sense that it has a max-\nimal quantum unitary subgroup just as in the the standard case. This, in\nconjunction with work in [Wo], allows the mathematical machinery related\nto unitariness–such as harmonic analysis, existence of orthonormal bases–to\nbe transported to its theory. This is important in the context of PH0. In-\ndeed, PH0 in the theory of standard quantum groups is intimately related\nto existence of unitary quantum subgroups. Because the local crystal bases\nfor representations of the standard quantum group [Kas1], which were later\nglobalized to canonical (global crystal) bases in [Kas2], arose in the study of\nspecial orthonormal Gelfand-Tsetlin bases for representations of the stan-\ndard quantum group. This is first main reason why PH0 is expected to hold\nfor the nonstandard quantum group in [GCT4].\nThe general nonstandard quantum group [GCT7] can be singular, i.e.,\nits quantum determinant can vanish. Hence, we cannot define its quantum\nunitary subgroup in the sense of [Wo].\nFortunately, this is not matter,\nbecause analogues of the main required results in [Wo] still hold; cf. [GCT7]\n68\n\nfor a precise statement.\nHence PH0 is expected to hold for the general\nnonstandard quantum groups in [GCT7] as well.\nBut at the same time these nonstandard quantum groups are fundamen-\ntally different from the standard quantum groups. Hence the terminology\nnonstandard. For a detailed description of the differences between the stan-\ndard and nonstandard quantum groups, see [GCT4, GCT7, GCT8]. Here we\nonly give a brief description from the complexity-theoretic perspective. To-\nwards this end, we associate a complexity level with each of these quantum\ngroups. This is briefly done as follows.\nSuppose H ֒→G is a primary couple associated with a group-theoretic\nclass variety for some complexity class C (Definition 3.1). Then the com-\nplexity class of this primary couple as well as its quantization, if it exists, is\ndefined to be just C.\nAs we have already noted, the theory of the standard quantum group is\nthe theory of quantization of the couple (cf. (27))\nGLn(C) →GLn(C) × GLn(C).\nThis is a primary couple associated with orbit-closure of the trace of an\nn×n matrix (Section 9.1), which can be computed by a circuit of depth two\nusing only additions or multiplications by constants. Hence, the standard\nquantum group corresponds to the complexity class of problems that can\nbe solved by circuits of depth two using only additions or multiplications\nby constants, just like expanders (Section 17).\nThere is no lower bound\nproblem here to speak of. That is why the standard quantum group cannot\nbe used for deriving any lower bound, again like expanders.\nThe couple associated with the Kronecker problem coincides with the\nprimary couple\nGL(Cm) × GL(Cm) →GL(Cm ⊗Cm).\n(35)\nassociated with the NC-class variety; cf. (17). Not exactly. The primary\ncouple associated with the NC-variety is slightly different from this, but\nthe difference is trivial, and can be ignored.\nTheory of the nonstandard\nquantum group in Theorem 15.1 (a) is the theory of quantization of this\ncouple. Hence, the complexity class of this nonstandard quantum group can\nbe defined to be NC.\nThe couple (34) that is quantized in [GCT7] is not a primary couple\nof any class variety. But it is qualitatively similar to the primary couple\n69\n\nassociated with the NP-class-variety (Section 4.2).\nFor this reason, the\nnonstandard quantum group in [GCT7] can be roughly taken to be of su-\nperpolynomial complexity.\n15.2\nPH0 for couples and triples\nThe article [GCT8] gives a conjecturally correct algorithm to construct\ncanonical bases of the coordinate rings of the nonstandard quantum groups\nin [GCT4, GCT7]. These are natural generalizations of the canonical ba-\nsis in [Kas3, Lu2] for the coordinate ring of the standard quantum group.\nFurther theoretical and experimental evidence in support of PH0 for the\nnonstandard quantum group in [GCT4] is also given. For the problems that\nhave to be addressed in the context of the triples associated with the class\nvarieties under consideration, see [GCT10].\n15.3\nPH0 for class varieties\nSince the group-theoretic class varieties are essentially determined by the\nassociated group triples, once PH0 is proved for the triples, it should, in\npriciple, be possible to “transport” this knowledge from group theory to\nalgebraic geometry, thereby proving PH0 for the class varieties. In [GCT10]\nis basic plan for this “transport” is suggested, with a description of the\nvarious mathematical problems that need to be resolved.\nA crucial bridge between group theory and algebraic geometry for this\ntransport is provided by Conjecture 6.1, which has to be proved first. It may\nbe remarked that quantum groups were indeed brought into GCT precisely\nfor the purpose of proving this conjecture, thereby extending the proof in\n[GCT2] for its weaker form (Theorem 6.3). A basic plan for this extension\nvia nonstandard quantum groups is also suggested in [GCT10].\n15.4\nPH1 and SH\nThe journey from PH0 to PH1 in the nonstandard setting should be akin to\nthe one in the standard setting; cf. [GCT6].\nIn summary, the nonstandard quantum groups have to be used as a\nrope, as it were, to pull the proofs of the various mathematical positivity\nhypotheses from the constant depth (of the standard quantum groups) to\nsuperpolynomial depth.\n70\n\n16\nUltimate mystery: nonstandard Riemann hy-\npotheses?\nNow we come to the final chapter of this story: How to prove PH0, and\nspecifically, correctness of the algorithm in [GCT8] for constructing canon-\nical bases of the coordinate rings of the nonstandard quantum groups in\n[GCT4, GCT7] and their required conjectural properties.\nFor the standard quantum group, as we mentioned in Section 1, the\ntopological proof in [Lu1, Lu2] depends on the Riemann hypothesis over\nfinite fields [Dl2] and the related work [BBD]. The main open problem at\nthe heart of GCT is to extend this work, and use it to prove nonstandard\nPH0. But the standard Riemann hypothesis over finite fields is not expected\nto work in the nonstandard setting; cf. [GCT8]. Briefly this is because the\nrelevant quantized noncommutative algebraic varieties in the nonstandard\nsetting simply “disappear” when specialized at q = 1. Specifically, unlike in\nthe standard case, the Hilbert function of these varieties at q ̸= 1 is different\nfrom the Hilbert function of the corresponding classical varieties at q = 1.\nHence, they look very different from the classical algebraic varieties. This\nis why the Riemann hypothesis over finite fields may not be used as in the\nstandard case for proving PH0.\nThus we seem to need nonstandard extensions of the Riemann hypothesis\nover finite fields in the quantized noncommutative setting to prove the PH0’s\nunder consideration. We cannot even formulate such extensions. But we\nbelieve such nonstandard extensions exist. We now briefly explain why.\nFor this, we need to indicate the nature of the experimental evidence\n[GCT8] in support of PH0 for the most basic nonstandard quantum group\nfor the Kronecker problem in [GCT4]. Specifically, around a thousand struc-\ntural constants associated with a canonical basis for a certain dual of this\nquantum group were computed, each structural constant being a polynomial\nin q of degree more than ten. All the coefficients of these structural polyno-\nmials turned out be nonnegative. In the standard case, the cause for such\nnonnegativity was the Riemann hypothesis over finite fields. There ought\nto be a similar theoretical cause for nonnegativity in the nonstandard set-\nting. For, without a cause, the probability of over ten thousand coefficients\nbeing nonnegative would be absurdly small–naively 1/210000. This estimate,\nbeing naive, should not be taken literally. But it does suggest that the ex-\nperimental evidence for positivity should only be a shadow of the ultimate\ncause–nonstandard analogues of the Riemann hypotheses over finite fields.\n71\n\nThis leads us to believe that nonstandard extensions of the Riemann\nhypothesis over finite fields for the various nonstandard quantum groups\nthat arise in GCT exist, and now, having seen the shadow, we have to search\nfor the ultimate cause whose shadow it is. If this search succeeds, then we\ncan expect to pull the proofs of PH0 from the standard to the nonstandard\nsetting, using the rope provided by the nonstandard quantum groups, and\nthe power provided by nonstandard Riemann hypotheses, thereby leading\nto the proof of P ̸= NP conjecture in characteristic zero; cf. Figure 3.\nEventually, this whole story in characteristic zero, along with the non-\nstandard Riemann hypotheses and the accompanying positivity hypotheses,\nmay be lifted, as suggested in [GCT11], to algebraically closed fields of pos-\nitive characteristic, and finally, finite fields, thereby proving the P ̸= NP\nconjecture in its usual form. This would then constitute the ultimate flip in\nFigure 2.\n17\nObstructions vs. expanders\nWe now explain the relationship between explicit construction of obstruc-\ntions and explicit construction of expanders as shown in Figure 4.\nAs per the hardness-vs-randomness principle [A2, IW, KI, NW], deran-\ndomization is intimately linked to lower bound problems.\nIn particular,\nrestricted kinds of lower bounds follow from existence of efficient pseudo-\nrandom generators. At present, we do not have pseudo-random generators\nbased on expander-like structures that can yield a lower bound result for\nconstant depth circuits. But for the sake of discussion, let us imagine that\nthe expander in, say, [LPS, Ma] can be generalized further to obtain a hy-\npothetical structure, which we shall call a strong expander, using which we\ncan obtain an efficient pseudo-random generator, whose existence, in turn,\nimplies separation of the class NC1 from AC0. Here NC1 is the class of\nproblems that can be solved by circuits of logarithmic depth, and AC0 the\nclass of problems that can be solved by circuits of constant depth. Fur-\nthermore, let us also assume that the problem of constructing such a strong\nexpander belongs to (nonuniform, algebraic) AC0, as it does for the ex-\npander in [LPS, Ma]. Now the existence of such a family {En} of strong\nexpanders would imply that an explicit function in NC1, depending on the\npseudo-random generator, cannnot be computed by a circuit of constant\ndepth. Hence such strong expanders can be regarded as obstructions, i.e.\nproofs of hardness, for computation of an explicit NC1-function by constant-\n72\n\ndepth circuits. In this sense GCT obstructions are to superpolynomial-depth\ncircuits are what strong expanders are to constant-depth circuits. This is\npictorially depicted in Figure 10.\n↑\ndepth\n|\nSuperpolynomial depth circuit: Obstruction in GCT\n↑\n|\nConstant depth circuit: Strong expander\n↑\n|\nDepth two circuit: expander\nFigure 10: The relationship between obstructions and expanders\nThe expander in [LPS, Ma] can actually be constructed by a nonuniform\nalgebraic circuit of depth two (a basic ring operation is taken as unit cost).\nHence, it can be expected to serve as an obstruction for computation by\na circuit of depth at most two–really, just one.\nBecause the depth of a\ncircuit for computing an explicit structure whose existence separates NC\nfrom (nonuniform) ACk, the class of circuits of depth k, should be at least\nk–really higher than k. So an expander, as against the hypothetical strong\nexpander, actually belongs to depth-two circuits. But there is no nontrivial\nlower bound problem for circuits of depth one. This is why the expanders\nthat we have at present cannot be used in lower bound problems.\nNow let us compare explicit construction of expanders with the suggested\nmethod for explicit construction of obstructions in Figure 3.\nFirst, let us observe that, though the explicit construction of expanders\n[LPS, Ma] is “extremely easy” (nonuniform AC0), its correctness is based\non a nontrivial mathematical positivity hypothesis:\nPHspectral: The spectral gap of an expander is bounded below by a pos-\nitive constant.\nThe mathematical positivity hypotheses PH1 and PH2 (Hypothesis 1.4-\n1.6) can be regarded as nonspectral analogues of PHspectral in the setting\nof superpolynomial depth circuits.\n73\n\nSecond, the proof of PHspectral in [LPS] for expanders depends on the\nRiemann hypothesis over finite fields (for curves) [Dl2]. It should not be a\nsurprise then that what is needed to prove the positivity hypotheses PH1, SH\n(PH2) mentioned above is, in essence, an extension of the Riemann hypoth-\nesis over finite fields and the results surrounding it. But given the big gap\nbetween constant depth and superpolynomial depth circuits it would have\nbeen a great surprise if the existing standard Riemann Hypothesis over fi-\nnite field were to suffice. Instead, what seems to be needed are nonstandard\nextensions of the Riemann hypothesis over finite fields, and the related re-\nsults; cf. Section 16. In the case of expanders, the Riemann hypothesis over\nfinite fields is not indispensible, since there are alternative constructions of\nexpanders with proofs of correctness based on linear algebra [RVW]. But,\nagain given a big gap between constant depth and superpolynomial depth, it\nshould not be surprising if nonstandard extensions of the Riemann hypoth-\nesis turn out to be indispensible in the context of the P vs. NP problem.\n18\nOn relativization and P/poly-naturalization bar-\nriers\nIn this section we point out why the flip should be nonrelativizable and\nnon-P/poly-naturalizable.\nWe already mentioned one reason for why the flip should be nonrela-\ntivizable: namely, the “reduction” from hard nonexistence to easy existence\nis not a formal Turing machine reduction. There is also another reason.\nFor this, let us examine why the proof of IP = PSPACE result [Sh] does\nnot seem relativizable. Mainly because it is based on the construction of an\nexplicit low-degree polynomial. This seems already enough to make it non-\nrelativizable, though the proof technique is not fully explicit. (Because it\nmakes use of estimates on the number of roots of a low degree polynomial.\nAny technique based on counting or estimates is, by definition, not fully\nexplicit). In contrast, the flip is to be implemented using explicit algebro-\ngeometric and representation-theoretic constructions. This is why it should\nbe nonrelativizable.\nNow we turn to the P/poly-naturalizability barrier [RR].\nIntuitively,\nthis too should be crossed simply because everything is to be done ex-\nplicitly and constructively.\nRecall that explicit construction of obstruc-\ntions is for superpolynomial depth circuits what explicit construction of\nexpanders is for depth-two circuits (Section 17).\nThe usual probabilistic\n74\n\n(nonconstructive) proof for existence of expanders may be considered to\nbe P/poly-naturalizable–as the probabilistic proofs [BS] of lower bounds\nfor constant depth circuits–whereas the proof via explicit construction in\n[LPS, Ma, RVW] may be considered non-P/poly-naturalizable. This is only\nan analogy. Strictly speaking, there is no notion of P/poly-naturalization\nfor constant depth circuits. Rather, this barrier lies between the circuits of\nconstant depth to which the expanders correspond and the circuits of super-\npolynomial depth to which the obstructions correspond. But this analogy\nshould intuitively explain why the flip should cross this barrier.\nNow we turn to a more formal argument. We begin by recalling the no-\ntion of a P/poly-naturalizable proof [RR]. We use the formal term P/poly-\nnaturalizable proof instead of the informal term natural proof, because oth-\nerwise GCT, and hence, the algebro-geometric and quantum-group-theoretic\ntechniques that enter into it would have to be called unnatural. That may\nseem paradoxical, especially since quantum groups arose in the study of\nnatural phenomena in theoretical physics.\nLet Fn be the set of n-variable boolean functions.\nBy a property of\nboolean functions, we mean a family of subsets Cn ⊆Fn for every n. It is\ncalled useful if the circuit size of any function h(X) = h(x1, . . . , xn) ∈Cn\nis super-polynomial. It is called P/poly-natural if it contains a subset C∗\nn\nsatisfying the following two constraints:\nConstructivity: Whether a given h(X) belongs to C∗\nn can be decided in\ntime polynomial in the size N = 2n of the truth table of h(X).\nLargeness:\n|C∗\nn|/|Fn| ≥1/N k,\n(36)\nfor some fixed k.\nA proof technique based on a useful P/poly-natural property is called\nP/poly-naturalizable. The article [RR] says that the P ̸= NP conjecture\nwould not have a P/poly-naturalizable proof under reasonable assumptions.\nNext, we translate this notion to the setting wherein the base field K of\ncomputation is algebraically closed, as in this article. We assume that K = C\nor K = ̄Fp, the algebraic closure of a finite field Fp. Let Fn be the set of n-\nvariable polynomials of degree d(n) for some fixed function d(n) = 2poly(n).\nIf K = C, we assume that each polynomial in Fn is an integral polynomial\nwhose coefficients have poly(n) bitlength. If K = ̄Fp, we assume that all\ncoefficients belong to Fp, and that the bitlength ⟨p⟩= poly(n).\nLet N\ndenote the total number of coefficients of h(X). The total bitlength of the\n75\n\nspecification all coefficients of h(X) is N, ignoring a poly(n) factor. Hence\nwe let it play the role of the truth-table-size in what follows. This leads\nto the following straightforward generalization of the notion of a P/poly-\nnaturalizable proof over C or ̄Fp.\nBy a property, we now mean a subset Cn ⊆Fn, for each n.\nIt is\ncalled useful if the circuit size over K of any function h(X) ∈Cn is super-\npolynomial. It is called P/poly-natural if it contains a subset C∗\nn satisfying\nthe following two constraints:\nConstructivity: Whether a given h(X) belongs to C∗\nn can be decided in\npoly(N) time, where each operation over K is considered to be of unit cost.\nLargeness:\n|C∗\nn|/|Fn| ≥1/N k,\n(37)\nfor some fixed k.\nA proof technique based on a useful P/poly-natural property is called\nP/poly-naturalizable. The results in [RR] are proved only over a finite field.\nBut the constructivity and largeness constraints over algebraically closed\nfields here are natural extensions of the ones over finite field. Hence, we\nshall assume in what follows that they are meaningful even over algebraically\nclosed fields. It would be interesting to know if the techniques in [RR] can\nbe lifted in some form to such fields.\nIn the context of the flip, we next formulate a property which is conjec-\nturally useful and which should violate both the largeness and the construc-\ntivity constraints. This should be enough to cross the P/poly-naturalizability\nbarrier.\nLet us follow the notation as in Section 4. Let K = C. Let h(X) ∈P(W)\nbe an integral homogeneous form in Fn that belongs to co-NP (i.e., the\nproblem of deciding if it is nonzero for given xi’s belongs co-NP).\nLet UP (useful property) be the conjunction of the following two prop-\nerties.\nUP1: The form h = h(X) is co-NP-complete.\nUP2: (Characterization by stabilizers)\nThe form h, as a point in P(W), is characterized by its stabilizer Gh ⊆\nGL(W), not exactly as in Definition 3.1, but in a relaxed manner as de-\nscribed in Section 7 in [GCT1]. So also the form f = φ(h) as a point in\nP(V ). This means the associated class varieties ∆W[h; n] = ∆W [h], and\n∆V [f; n, l] = ∆V [f], as defined in Section 4.2 with h(X) playing the role of\nE(X), are group-theoretic. Let (H1 ֒→G1 ֒→K1) and (H2 ֒→G2 ֒→K1)\n76\n\nbe the group-triples associated with the varieties ∆W[h; n] and ∆V [f; n, l],\nrespectively. We assume that H1 is reductive, that its simple composition\nfactors are explicitly known, and that it is built from these composition fac-\ntors by simple operations: to keep the matters simple, we only allow direct or\nwreath products, which suffice in GCT. We also assume that all the simple\ncomposition factors are either classical connected groups, tori or alternating\ngroups, as in GCT, though, again, this is strictly not necessary. We also\nassume that all homomorphisms in these triples are explicit as defined in\n[GCT6]–this is necessary.\nHere UP1 is stipulated only so that obstructions to efficient computation\nof h(X) should exist (Section 6). Otherwise, it is never explicitly used in\nGCT. The approach may work for other hard, though not co-NP-complete\nfunctions. UP2 is the main property that GCT needs for proving existence\nof obstructions. Hence we shall only concentrate on it in what follows.\nIt is shown in [GCT1] that E(X) has property UP2; whereas UP1 over C\nis shown in [Gu]. The permanent has an analogous property, where co-NP-\ncompleteness is replaced by #P-completeness. The function H(Y ) also has\nan analogous property with P-completeness replacing co-NP-completeness.\nBut in the case of H(Y ) the class variety is not the usual orbit closure\n∆[H(Y )], but rather ˆ∆[H(Y )] as defined in [GCT1]; cf. Remark 4.2.\nIn these definitions, we can also let the base field K be a finite field\nFp, or its algebraically closure ̄Fp, since characterization by stabilizers is a\nwell-defined notion over any field.\n18.1\nFrom usefulness to superpolynomial lower bounds\nThough GCT strives to prove superpolynomial lower bounds for the partic-\nular functions E(X) and perm(X), its main techniques should, in principle,\nextend to any h satisfying UP. We now briefly indicate how. This should\njustify the name UP.\nDefine an obstruction for the pair (h(x), H(Y )) as in Definition 5.1,\nwith h(X) playing the role of E(X).\nSuch obstructions should exist for\nevery n →∞, l = nlog n, as long as h(X) is co-NP-complete (cf. Section 6).\nAssociate with the class variety ∆V [f; n, l] a stretching function ̃sλ\nd(h; n, l)(k)\nas in (24) with h(X) playing the role of E(X).\nThe results in [GCT6] now imply the following analogue of Theorem 9.4\nfor h(X):\n77\n\nTheorem 18.1 [GCT6] Assuming that the singularities of the class variety\n∆V [f; n, l] and ∆W[h; l] are rational, the stretching function ̃sλ\nd(h; n, l)(k)\nassociated with the class variety ∆V [f; n, l] is a quasi-polynomial\nIt is may be conjectured that the singularities will be rational, as needed\nhere, as long as h satisfies UP2.\nUsing this theorem, we can formulate PH1, PH2, and SH for h(X) just\nas for E(X) (cf. Hypotheses 1.4-1.6).\nRemark 18.2 The statements of PH1 and PH2 given in this paper are as-\nsuming that all simple composition factors of the reductive groups under con-\nsideration are either classical connected groups or tori or alternating groups.\nIn the presence of composition factors of other types, some variations are\nnecessary [GCT6].\nThe following is an analogue of Theorem 9.5 in this context.\nTheorem 18.3 [GCT6] Assuming the rationality hypothesis (cf.\nTheo-\nrem 18.1), PH1 and SH, analogues of the decision problems in Hypothe-\nsis 7.2 for h(X) belong to P.\nIn particular, the problem of verifying an\nobstruction for the pair (h(X), H(Y )) belongs to P.\nThese results suggest, just as for E(X), the following strategy for proving\na superpolynomial lower bound for h(X):\n(1): Let Hi ֒→Gi ֒→Ki be the triples that occur in the definition of UP2.\nQuantize the couples Hi ֒→Gi and Gi ֒→Ki. That is, prove analogues of\nTheorem 15.1 for these. Also quantize the triples along the scheme suggested\nin [GCT10].\n(2): Prove existence of canonical bases (PH0) for the coordinate rings and\nrepresentations of the quantum groups that arise in this quantization along\nthe lines of the basic scheme in [GCT8]. For formal statements of PH0 see\n[GCT6, GCT10].\n(3): Use these canonical bases to prove existence of canonical bases (PH0)\nfor the coordinate rings of the class varieties ∆V [f; n, l] and ∆W [h, l] along\nthe lines suggested in [GCT10].\n(4): Use PH0 to deduce PH1 and SH, as suggested in [GCT6]. The polytope\nin PH1 should be more or less determined once PH0 holds, just as in the\nstandard case; cf. Section 1.6.\n78\n\n(5): Theorem 18.3, in conjunction with PH1 and SH for h(X), then im-\nplies polynomial time algorithm for the analogue of the decision problem in\nHypothesis 7.2 (a) for h(X) in place of E(X).\n(6): Carry out the steps (1)-(5) for the P-complete function H(Y ) as well.\nIt will imply a polynomial time algorithm for the decision problem in Hy-\npothesis 7.2 (b) for H(Y ). This step is the same as for E(X).\n(7): Transform the easy, polynomial time algorithms in steps (5) and (6),\nalong the lines suggested in Section 13 and [GCT6], into a P-constructive\nproof of existence of an obstruction λ(n) for every n →∞, assuming that l =\nnlog n. As pointed out in [GCT6], this transformation may need additional\npositivity hypotheses in the spirit of PH1 and SH. But these can be expected\nto hold, assuming h(X) satisfies UP2. The polytope in PH1 for h(X) in the\nstep (4) can also be expected to have a regular well-behaved structure, as\nneeded for this step, assuming h(X) satisfies UP2.\n(8): Existence of an obstruction family {λ(n)} would imply a superpoly-\nnomial size circuit lower bound for h(X), and hence, that P ̸= NP over\nC.\nFor the problems that need to be addressed over a finite field, or an\nalgebraically closed field of positive characteristic, see [GCT11].\n18.2\nOn violation of the largeness constraint\nNow let us see why UP2 should imply violation of the largeness constraint.\nWe cannot prove this formally over C. But this can be proved formally over\na finite field Fp or an algebraically closed field ̄Fp of positive characteristic\n[GCT10]. In fact, it turns out that violation of the largeness constraint is\nfar more severe than what is formally required. Namely, when K is a finite\nfield, it can be shown that\n|Cn|/|Fn| ≥1/2Ω(N),\nwhere Cn is the set of h(X) which satisfy UP2. This may be compared with\n(36).\nThe proof of violation of the largeness constraint over ̄Fp does not carry\nover to C for technical reasons. Specifically, the key ingradient in this proof is\nthe Riemann hypothesis over finite fields, or rather its extension as proved in\n[Dl2]. To transport this to the case when h(x) is integral would presumably\nrequire an analogous statement in arithmetic algebraic geometry.\n79\n\nIt may be remarked that, in contrast, the proof of violation of the large-\nness constraint over a finite field is elementary. Thus the difficulty of proving\nthe violation of the largeness constraint over K seems inversely related to\nthe difficulty of proving the P ̸= NP conjecture over K. When K = Fp,\nthe conjecture is hardest to prove, and hence, the proof of violation is easy.\nWhen K = C∗, the conjecture should be easier than over ̄Fp or Fp. Accord-\ningly, proving violation of the largeness constraint formally turns out to be\nthe hardest.\n18.3\nOn violation of the constructivity constraint\nNext let us see why UP2 should also imply violation of the constructivity\nconstraint. We cannot hope to show this formally, since this is a lower bound\nstatement in itself. But rather we can give good evidence. First of all, to\ncompute the stabilizer of h(X), we have to solve a system of polynomial\nequations. Determining feasibility of a general system of polynomial equa-\ntions in k variables is NP-complete and is conjectured to take ⟨p⟩Ω(k) time,\nwhen K is the finite field Fp. Analogous conjecture may be made for the\nspecific system of polynomial equations that arises in the computation of\nthe stabilizer. Assuming this, it follows [GCT10] that deciding if h(X) has\na nontrivial stabilizer would take time that is superpolynomial in N–this is\nthe truth-table size when K = Fp.\n19\nP-verifiable and P-constructible proof techniques\nand their explicit construction complexity\nIn this section we suggest why GCT may be among the “easiest” “easy-to-\nverify” approaches to the P ̸= NP conjecture as per a certain measure of\nproof-complexity, called the explicit construction complexity. For this, we\nhave to introduce the notion of an easy-to-verify (i.e. P-verifiable) proof\ntechnique and then define its explicit construction complexity (class).\n19.1\nP-verifiable proof technique\nSuppose we are given a proof technique (approach) towards to the P vs. NP\nproblem that seeks to prove a superpolynomial lower bound for a specific\nhard function h(X) under consideration. We assume that the approach seeks\nto prove, explicitly or implicitly, existence of a specific cause for the hardness\n80\n\nof h(X), which we shall refer to as an obstruction. Thus an obstruction is,\nroughly, a “cause”, a “witness” or a “proof” of hardness.\nBut what do we mean by a proof?\nThe final proof of the P ̸= NP\nconjecture, if true, would constitute the ultimate obstruction to efficient\ncomputation of every (co)-NP-complete h(X). The size of this proof would\nbe just O(1), and so also the cost its verification. By obstruction, we do\nnot mean this final proof of hardness, but rather an intermediate proof\nof hardness whose existence the approach strives to demonstrate for every\nn →∞, when the circuit size m = nlog n, say.\nThe nature of such an obstruction will depend on the proof technique.\nWe cannot define it formally. Hence we will only give an intuitive idea with\nan example. Suppose there is an efficient pseudo-random generator whose\nexistence implies a restricted type of lower bound result in the spirit of [NW].\nThen the explicit computational circuit for this pseudo-random generator,\ni.e., for all its output bits together would be an obstruction in this context.\nBecause existence of this pseudo-random generator serves as a witness for\nhardness. If the pseudo-random generator is based on an explicit structure\nin the spirit of an expander, then this structure too can be considered to\nbe an obstruction.\nMore generally, the hardness-vs-randomness principle\n[KI, NW] suggests that proof techniques for difficult lower bounds may need\nmore or less explicit constructions of some structures.\nThese structures,\nwhich serve as witnesses for hardness, can then be taken as obstructions.\nIn the rest of this section, we confine ourselves only to those techniques\ntowards the P ̸= NP conjecture which contain, explicitly or implicitly, the\nnotion of an obstruction in this spirit–a witness for hardness–which admits\na well-defined description that can be assigned bit length. The arguments\nhenceforth are subject to this assumption.\nNext we try to formalize the notion of a “viable” proof technique to-\nwards the P vs. NP problem. For this, let us begin with a technique that\nshould certainly not be considered viable–the trivial brute-force proof tech-\nnique. This is defined as follows. Assume that the base field K is finite.\nFix any co-NP-complete function h(X) = h(x1, . . . , xn). Then this proof\ntechnique strives to prove, for every n, existence of the trivial proof of hard-\nness (obstruction), which consists of just the enumeration of all circuits of\nsize m = nlog n, with a specific value of X for each circuit on which the\nfunction evaluated by the circuit differs from h(X). The size of this trivial\nobstruction is exponential in m, and the time taken to verify it is also expo-\nnential in m. Any viable proof technique for the P vs. NP problem ought\n81\n\nto be at least better than this trivial proof technique in some well defined\nsense. One obvious sense in which it could be better is that there exists an\nobstruction whose size is not exponential in m, but rather polynomial in m,\nand the time taken to verify an obstruction is not exponential in m, but\nrather polynomial in m.\nThis leads to:\nDefinition 19.1 We say that a proof technique for the P ̸= NP conjecture\nis P-verifiable if\n1. There is a well-defined notion of obstruction, either implicit or explicit\nin the technique,\n2. There exists a short obstruction to computation of the specific function\nh(X) = h(x1, . . . , xn) under consideration by a circuit of size m =\nnlog n (say), for every n →∞, though the technique may only strive\nto prove existence of any obstruction, not necessarily short. By short,\nwe mean the obstruction has a label (combinatorial specification) of bit\nlength poly(m).\n3. The problem of verifying an obstruction is easy; i.e., belongs to P.\nSpecifically, takes time that is polynomial in n, m and the bit length of\nthe obstruction.\nThe meaning of easy here is the most obvious and natural definition in\nthe context of the P vs.\nNP problem.\nThus, intuitively a P-verifiable\nproof technique is an easy-to-verify proof technique. That is, the problem\nof discovering a proof of hardness (obstruction) in the technique belongs to\nNP. The definition above makes sense over any base field K of computation,\nwith obvious modifications in the spirit of the ones in Section 18.\nThe following naive arguments suggest that for a technique towards the\nP ̸= NP conjecture to be viable it out to be P-verifiable.\nFirst, the usual experience in mathematics suggests that however hard\nthe discovery of a proof may be its verification, once found, should be easy,\nand furthermore, the proofs that are found are usually reasonably short. In\nthe definition of P-verifiability, short and easy are given the most obvious\nand natural interpretations in the context of the P vs. NP problem: de-\nscription of polynomial size (short), and can be done in polynomial time\n(easy).\n82\n\nSecond, given a technique, it seems necessary to justify why it is better\nthan the trivial brute-force technique. A P-verifiable proof technique is bet-\nter than it as per the most obvious complexity measures: (1) space (short),\nand (2) time (cost of verification).\nThird, the article [RR] roughly says that a nonspecific approach that\nis applicable to a large fraction of hard functions should not work in the\ncontext of the P vs. NP problem. Thus approaches based on probabilistic\nmethods or estimates of various kinds–such as Bezout-type estimates in\nalgebraic geometry, or estimates for discrepancies and deviations in analysis\nor number theory–should not work.\nProof of hardness as per any such\napproach–namely, the value of the measure or the estimate which is the\ncause of hardness–should be hard to verify. Since to verify the value, we\nmay have to compute it and see that it really tallies with what is given,\nand such computations should typically take time that is exponential in\nthe bitlength of the value. Thus a P/poly-naturalizable proof should also\nbe non-P-verifiable, and hence, the definition of P-verifiability here seems\nconsistent with the arguments in [RR].\nAdmittedly, these are only naive arguments. One can ask if there exists\na viable proof technique for the P ̸= NP conjecture that is better than the\ntrivial brute-force technique as per some measure of complexity other than\nthe obvious ones–space and time. But since we cannot think of any such\nnonobvious complexity measures which are also natural in the context of\nthe P vs. NP problem, we shall confine ourselves to only P-verifiable proof\ntechniques in what follows.\nBy Hypothesis 7.1 (a), which is supported by the results that we de-\nscribed in this article, GCT is a P-verifiable proof technique over C; the\nstory over a finite field should be similar [GCT11].\n19.2\nP-barrier for verification\nEvery P-verifiable technique for the P ̸= NP-conjecture has to cross the P-\nbarrier for verification; i.e., surmount the difficulty of showing that verifica-\ntion of an obstruction is easy. The magnitude and difficulty of the P-barrier\nshould be of the same order regardless of which P-verifiable approach to the\nP ̸= NP conjecture is taken.\nThis is easy to see when the base field K is finite. Then the length of\nany obstruction in the trivial brute-force technique mentioned in the begin-\nning of this section is exponential. The main task here is to come with a\n83\n\nproof technique that admits short obstructions which can be verified easily.\nThe magnitude of this P-barrier–the difference between the exponential and\nthe polynomial–is the same regardless of which approach to the P ̸= NP\nconjecture is taken.\nNext, let us assume that K = C, as in this paper. Let n be the number\nof input parameters. Let m(n) = nlogn (say) be the circuit-size parameter,\nh(n) = nlogn the height-parameter, and d(n) ≤2h(n) the degree parameter\nin the lower bound problem under consideration. For a given n, the set of\nfunctions over C computable by circuits of size at most m = m(n), height at\nmost h = h(n) and degree at most d = d(n) is an algebraically constructible\n[Mm1] subset S of the space V of all forms in m variables of degree d(n). A\nconstructible subset means it is in the boolean algebra generated by closed\nalgebraic subsets of V ; this is a generalization of an affine variety.\nThe goal in the lower bound problem under consideration is to show\nthat h(X) does not belong to S, when m = nlog n. Let ̄S be the closure of\nS. It is an affine variety. If h(X) is co-(NP)-complete, it is reasonable to\nassume that it does not belong to ̄S as well; i.e., roughly speaking, it cannot\nbe approximated infinitesimally closely by a circuit of size m(n) and height\nh(n). So it would suffice to show this.\nAn obvious obstruction here would be a polynomial in the ideal of ̄S\nwhich does not vanish on h(X). To decide if a given polynomial belongs to\nthe ideal of ̄S, an obvious method is to compute a good basis of this ideal,\nsuch as Gr ̈obner basis, and then use it for this decision. But the problem\nof Gr ̈obner basis computation is EXPSPACE complete [MM]. This means\ncomputation of the Gr ̈obner basis of ̄S can take space that is exponential in\nthe dimension of the ambient space V , which in turn is exponential in m.\nIn other words, space that is double exponential in m, and hence, time that\nis triple exponential in m. Given that the S in our problem is really bad,\nthis is the best that we can expect from any general purpose technique for\nverifying an obstruction that reasons about S directly in this fashion.\nFor the technique to be P-verifiable, the huge gap between this triple\nexponential bound for a general purpose direct technique and the polynomial\nbound in Definition 19.1 has to be bridged. The magnitude and order of this\ngap–the P-barrier–is exactly the same that we encountered in Section 8.\nRemark 19.2 The triple exponential size of this gap when K = C as\nagainst the exponential size over K = Fp does not mean that the P ̸= NP\nconjecture is easier when K = Fp. In fact, it is the other way around. Since\nthe (nonuniform) P ̸= NP conjecture in characteristic zero (over Z) is a\n84\n\nweaker implication of the conjecture over finite field (the usual case) [GCT1].\nHence, the exponential gap over Fp would be much harder to bridge than the\ntriple exponential gap over C. See [GCT6] for the problems that need to be\naddressed over finite fields.\nThe class variety XP (l) for P (Section 4.2) is constructed in [GCT1]\nprecisely to cope up with the triple exponential gap over C. It is a nice\nalgebraic variety that contains S, or rather its projectivization. So instead\nof trying to show that a given h(X) is not in S, one strives to show that\nit is not in XP (l). Since the algebraic geometry of XP (l) is exceptional,\nthis problem becomes easier–especially when h(X) is also exceptional, like\nE(X).\nThe quantum-group and algebro-geometric machinery is needed in GCT\njust to cross the P-barrier for verification (over C).\nThis suggests that\nmathematics required for any P-verifiable approach towards the P ̸= NP\nconjecture may not be substantially simpler, or easier.\n19.3\nP-constructible proof technique\nIn fact, it may be much harder unless it is also P-constructible in the fol-\nlowing sense.\nDefinition 19.3 We say that a P-verifiable proof technique for the P ̸=\nNP conjecture is P-constructible if the discovery of an obstruction in this\ntechnique is also easy. That is, there exists an algorithm, which, given n\nand m, can decide whether there exists an obstruction in poly(m) time, and\nif so, also construct a short obstruction in poly(m) time.\nThus the problem of discovering an obstruction in a P-constructible\nproof technique. belongs to P. But the proof technique itself need not give\na polynomial time algorithm for discovering an obstruction explicitly. That\nis, this may only be implicit in the proof, or it may be left to posterity. We\ncall the technique P-constructive if it gives such an algorithm more or less\nexplicitly:\nDefinition 19.4 A P-constructible proof technique is called P-constructive,\nif it also yields a procedure to construct an obstruction explicitly in poly(n, m)\ntime, if one exists.\n85\n\nThe relationship between P-constructible (constructive) and P-verifiable\nproof strategies is akin to the relationship between P and NP. The P ̸= NP\nconjecture says that the discovery of a proof is, in general, harder than its\nverification. Hence, just as P denotes the class of easy problems within NP,\nthe P-constructible and P-constructive proof strategies are in a sense the\n“easy” ones among the P-verifiable proof strategies, wherein discovery is\nalso easy like verification.\nBy Hypothesis 7.1 (c), as supported by the positivity hypotheses, results\ndescribed in this paper, GCT is P-constructive over C; the story over a finite\nis expected to be similar [GCT11].\nThat there should exist such a P-constructive proof technique for the\nP ̸= NP conjecture may, however, seem paradoxical at the surface. Be-\ncause a P-constructive (constructible) proof technique seems to go against\nthe very philosophical essence of the P ̸= NP conjecture that discovery is\nharder than verification. This is akin to the paradox in the proof of G ̈odel’s\nincompleteness theorem: that the statement which says there exist unprov-\nable true statements is itself easy to prove. Similarly, Hypothesis 7.1 (c) says\nthat the statement which says discovery is harder than verification should\nitself be easy to discover.\n19.4\nGeneral setting\nSo far we have described P-verifiable and P-constructible proof techniques\nonly in the context of the P vs. NP problem. But these notions can be\ndefined in a much more general context, as we now briefly indicate:\nDefinition 19.5 A technique for proving a mathematical property Q(X),\nwhere X ranges over a class C of mathematical objects under consideration,\nis P-verifiable if:\n1. The technique proves, explicitly or implicitly, existence of a “proof-\ncertificate” c(X), for every X ∈C, which serves as a “witness” that\nthe property Q(X) holds.\n2. There exists a short proof certificate for every X ∈C.\nBy short,\nwe mean its size is poly(⟨X⟩), where ⟨X⟩denotes the specification-\ncomplexity of X.\n3. Verification of a proof-certificate c(X) is easy; i.e., can be done in\npoly(⟨X⟩, ⟨c(X)⟩) time, where ⟨c(X)⟩denotes the bitlength of c(X).\n86\n\nAgain, we cannot formally define what a proof-certificate means. In what\nfollows, we only consider proof techniques wherein the notion of a proof-\ncertificate is well defined. The specification complexity ⟨X⟩here depends\non the problem under consideration, as we shall see in the examples below.\nDefinition 19.6 A P-verifiable proof technique is called P-constructible if\nthere exists an algorithm which, given X ∈C, can construct a proof certifi-\ncate c(X) in poly(⟨X⟩) time.\nBut the proof technique itself need not give such an algorithm explicitly.\nDefinition 19.7 A P-constructible proof technique is called P-constructive\nif, in addition, it yields an algorithm that can construct a proof-certificate\nc(X) in poly(⟨X⟩) time.\nWe can now state an informal working hypothesis:\nHypothesis 19.8 (The P-hypothesis) (informal)\n(a) Feasible P-verifiable proof techniques–that is the P-verifiable techniques\nthat can actually be used to prove the properties Q(X) in practice–are usu-\nally P-constructible, though proving P-constructibility may turn out to be\nnontrivial, and may only be done a posteriori.\n(b) Conversely, if a P-verifiable technique is P-constructible then under rea-\nsonable conditions it may also be feasible, i.e., can be used to actually prove\nQ(X).\n(c) A major part of the effort in a P-constructible proof technique usually\ngoes towards development of a polynomial time algorithm for constructing a\nproof-certificate, though this may be done only implicitly, and may become\nclear only a posteriori. That is, a P-constructible proof can usually be ex-\ntended to a P-constructive proof with a “reasonable additional effort”, albeit\na posteriori.\n(d) Mathematical complexity of a P-constructive proof technique is inti-\nmately linked to the computational complexity of the algorithm for explicit\nconstruction of a proof certificate underlying the technique.\nAs we have already remarked, the relationship between P-constructible\nproof techniques and P-verifiable proof techniques is akin to the relationship\nbetween P and NP.\nThe class P is usually regarded as the subclass of\n87\n\nfeasible problems in NP. Hence, the P-hypothesis just says that P usually\nmeans feasible in practice.\nThe reasonable conditions in (b) means: there is a polynomial time al-\ngorithm for constructing a proof-certificate, which has, furthermore, a rea-\nsonably simple structure, and which is efficient in practice.\nThat is, the\ndefinition of P as standing for feasible is not misused.\nDefinition 19.9 A mathematical theorem, which says a property Q(X) holds\nfor every X in a class C of mathematical objects under consideration, is\ncalled P-verifiable if it has a P-verifiable proof.\nA P-constructible or a P-constructive theorem is defined similarly.\n19.4.1\nExamples\nWe now give a few examples to illustrate these notions.\nP vs NP problem\nIn this context, C is the class of tuples (n, m(n)), m(n) = nc for any con-\nstant c > 0, over all n (large enough). The property Q(X), X = (n, m(n)),\njust says that the explicit function h(X) under consideration, such as E(X)\nin [GCT1], cannot be computed by a circuit of size m(n) for every n large\nenough. Here ⟨X⟩= n+m; i.e, we assume that n and m are given in unary.\nThen the notions of P-verifiability, and P-constructibility here coincide with\nthe ones in Definitions 19.1 and 19.3. When m = nc, an obstruction would\nalways exist, assuming that h(X) is co-(NP)-complete, P ̸= NP and the\ntechnique is correct. That is why Definition 19.5 would coincide with Def-\ninition 19.1, even if in the former there is no mention of deciding if an\nobstruction exists or not. As per Hypothesis 7.1, GCT is P-constructive\nover C, and hence the P ̸= NP conjecture over C is also P-constructive; the\nsame can be hypothesized over a finite field [GCT11].\nHall’s theorem\nIn this context, C is the class of d-regular bipartite graphs. The property\nQ(X) is that every d-regular bipartite graph X ∈C has a perfect matching.\nThe bit length ⟨X⟩is the bitlength of the specification of X. The proof\ncertificate c(X) is a perfect matching in X. The problems of verifying and\nconstructing a perfect matching belong to P, the former trivially. Hence,\n88\n\nHall’s theorem is P-constructive. Hall’s original proof is P-constructible,\nthough not P-constructive, since it does not explicitly give a polynomial\ntime algorithm for constructing a perfect matching.\nBut it does contain\nmajor ingradients for such a polynomial time algorithm, which came only\nmuch later. This is consistent with the P-hypothesis.\nFour colour theorem\nIn this context, C is the collection of planar graphs. The property Q(X) is\nthat any planar graph X is four colourable. The bitlength ⟨X⟩is the bit\nlength of the specification of X. The proof certificate c(X) is a four colouring\nof X. The problems of verifying and constructing a proof certificate belong\nto P, the former trivially. Hence, any proof of the four colour theorem is\nP-constructible, and the four colour theorem is P-constructive. The actual\nproof in [AH] is also (more or less) P-constructive since it implicitly yields\nto a polynomial (quartic) time algorithm for four colouring, Indeed, major\npart of the effort in the proof implicitly goes towards development of such\nan algorithm. This is consistent with the P-hypothesis.\nA simpler P-constructive proof was subsequently given in [RSST], which\ngives a better quadratic algorithm for the same problem. This too is con-\nsistent with the P-hypothesis (d).\nForbidden minor theorem\nFix a genus g. The forbidden minor theorem [RS] says that a graph which\ndoes not contain a forbidden minor from a finite list of minors depending\non g can be embedded on a genus g surface. Here C is the class of graphs\nthat do not contain a forbidden minor, Q(X) the property above, and ⟨X⟩\nthe bitlength of the specification of X. The proof certificate c(X) is just a\ndescription that tells how to embed X on a genus g surface.\nThe forbidden minor theorem is P-constructive. Any proof technique\nfor proving the forbidden minor theorem is P-constructible: it was known\n[FMR] even before [RS] that c(X) can be constructed in polynomial O(⟨X⟩O(g))\ntime. The proof of the forbidden minor theorem in [RS] gave an O(f(g)⟨X⟩2)\nalgorithm, where f(g) depends only on g. Indeed, a major part of the effort\nin [RS] implicitly goes towards finding a polynomial time algorithm whose\nrunning time is of the form O(f(g)⟨X⟩O(1)); i.e., wherein the exponent of\n⟨X⟩does not depend on g. This is again consistent with the P-hypothesis\n(d).\n89\n\nThe Poincare conjecture\nHere we can let C be the set of simplicial decompositions of compact three\ndimensional combinatorial manifolds that are simply connected. The prop-\nerty Q(X) says that X is a (combinatorial) sphere. The bitlength ⟨X⟩is\nthe bitlength of specifying X. The article [Sc] says that the sphere recogni-\ntion problem is in NP. That is, there is a proof-certificate c(X), verifiable\nin polynomial time, which certifies that X is a sphere. It is interesting to\nknow here if the problem of constructing a proof certificate c(X), for a given\nX ∈C, belongs to P. It is plausible that the proof technique in [Pe] can\nbe extended/transformed (in the combinatorial setting) to get a polynomial\ntime algorithm which constructs a proof-certificate in this spirit, though not\nexactly the one in [Sc]. If that happens, it would mean that the Poincare\nconjecture is P-constructible (P-constructive), and that the major effort in\n[Pe] implicitly went towards getting a polynomial time algorithm for this\nproblem. This would provide support for the P-hypothesis (c).\nThus a major part of the effort in the P-verifiable proofs above indeed\nseems to go towards developing a polynomial time or a better polynomial\ntime algorithm for constructing a proof-certificate, as per the P-hypothesis\n(c), though this goal may not be stated explicitly in the proofs. In the flip,\nP-constructivity as a goal is explicitly spelled out right in the beginning,\ngiven the complexity-theoretic significance of the P vs. NP problem. But\njust as in the examples above, it may not be necessary to prove PHflip\n(Hypothesis 7.1) fully to prove P ̸= NP over C. That is, it may suffice to\ndevelop only a part of all ingradients needed to put the required problems\nin P, and the remaining part can be left to posterity. In this context, the\nbasic minimum that seems to be needed is PH1 (more or less).\n19.5\nExplicit construction complexity\nWe will now try to formalize the intuition behind the P-hypothesis (d).\nTowards that end we wish to associate a measure of proof-complexity with\na P-verifiable proof technique. This is quite different, for example, from\nKolmogrov proof-complexity.\nDefinition 19.10 Explicit construction complexity of a P-constructive tech-\nnique is the computational complexity of the algorithm underlying that tech-\nnique for explicit construction of a proof-certificate.\n90\n\nBy computational complexity, we mean the usual measures such as depth\nand size of the corresponding computational circuit. If a P-verifiable tech-\nnique is not explicitly P-constructive but naturally leads to an algorithm\nfor construction of obstructions, with additional effort, we agree to take\nthe computational complexity of this algorithm to be explicit construction\ncomplexity of the technique, albeit only a posteriori.\nDefinition 19.11 (a) Verification (complexity) class of a P-verifiable proof\ntechnique is the abstract computational complexity class of the problem of\nverifying a proof-certificate (as per that technique).\n(b) Explicit construction (complexity) class of a P-verifiable proof technique\nis the computational complexity class of the problem of explicit construction\nof a proof-certificate as per that technique.\nA computational complexity class here means an abstract computation\ncomplexity class such as P, NC, NCk, AC, Dtime(N) etc. The verifica-\ntion and explicit construction classes of a P-verifiable technique are well\ndefined regardless of whether the technique shows how to construct a proof-\ncertificate explicitly or not. But what these classes are may become clear\nonly a posteriori, possibly after extending the proof technique to a get an\nefficient algorithm for construction of a proof certificate therein.\nThe complexity measures and classes above are meaningful only for P-\nverifiable proof techniques.\nThey would not make any sense for noncon-\nstructive or estimate-based techniques in analysis, number theory and so\nforth, unless it is possible to define a specification complexity ⟨X⟩and a\nproof-certificate that is polynomial time verifiable with this definition of\n⟨X⟩naturally.\nThis following gives a notion of theorem complexity for P-verifiable the-\norems.\nDefinition 19.12 Explicit construction complexity (class) of a P-verifiable\ntheorem is the minimum explicit construction complexity (class) over all P-\nverifiable proofs of the theorem. Verification complexity (class) is defined\nsimilarly.\nThe explicit construction complexity seems to be a good measure of\ncomplexity for P-verifiable proof techniques and theorems. We shall discuss\nthe examples above a bit more in this context.\n91\n\nHalls’ theorem\nVerification class here is AC (constant depth circuits), since a perfect match-\ning can be verified in constant depth. A perfect matching in a bipartite\ngraph can be computed, if one exists, in O(m log n) time. This problem also\nbelongs to RNC [KUW, MVV]. Hence, the sequential explicit construction\nclass of Hall’s theorem is Dtime(m log n). The parallel explicit construction\nclass is RNC; possibly even NC.\nFour colour theorem\nVerification class here is AC. Explicit construction complexity of the proof\nin [AH] is O(n4), whereas that of the proof in [RSST] is O(n2) [RSST].\nThus a proof technique with lower explicit construction complexity has in-\ndeed lower proof-complexity. The sequential explicit construction class of\nthe four colour theorem is thus Dtime(n2), or lower. The parallel explicit\nconstruction class is possibly NC, in view of the parallel algorithms for four\ncolouring in special cases [He].\nForbidden minor theorem\nVerification class is AC. Explicit construction complexity of the proof in\n[RS] is O(f(g)n2), where g is an explicit function of the genus g.\nThe\nsequential explicit construction class of the forbidden minor theorem is thus\nDtime(O(n2)); it may be Dtime(n). The parallel explicit construction class\nmay be NC, since planarity testing is in NC [JS].\nPoincare’s conjecture\nVerification class of the Poincare conjecture is P [Sc], assuming that the\nproof technique in [Pe] is P-verifiable. It may be smaller. NC?. Explicit\nconstruction class may be P, plausibly smaller. NC?\nTrivial example\nWe now give a trivial example to illustrate why P-verifiability is essential\nfor the complexity measures here to make sense.\nTake a trivial mathe-\nmatical theorem: that an integer n has at most log n factors. An obvious\nproof-certificate, for a given n, is the number of its factors, which shows\n92\n\nthat it is less than log n. But verification of this proof requires factoring\nand hence is hard. Thus if n is specified in binary, this theorem should\nnot be P-verifiable. That is why explicit construction complexity of this\nproof-certificate says nothing of the actual (trivial) proof-complexity of the\ntheorem. Similarly, explicit construction complexity is not meaningful for\nestimate-centred proof techniques in mathematics. The article [RR] roughly\nsays that such techniques are not expected to work in the context of the\nP vs. NP problem since they tend to be applicable to a large fraction of\nfunctions.\nIn the context of the P vs. NP problem, Definitions 19.10 and 19.11\nbecome:\nDefinition 19.13 Explicit construction complexity of a P-constructive tech-\nnique for the P ̸= NP conjecture is the computational complexity of the algo-\nrithm underlying that technique for explicit construction of a proof-certificate\n(as per that technique).\nDefinition 19.14 (a) Verification complexity class of a P-verifiable proof\ntechnique for the P ̸= NP conjecture is the computational complexity class\nof the problem of verifying an obstruction as per that technique.\n(b) Its explicit construction complexity class is the computational complexity\nclass of the problem of explicit construction of an obstruction.\nAgain these classes are well-defined regardless of whether the technique\nshows how to construct an obstruction explicitly or not, once the notion of\nan obstruction in the proof technique is well-defined.\nOne may also define existential complexity class of a P-verifiable proof\ntechnique (for the P vs. NP problem): this is the computational complexity\nclass of the problem of deciding if there exists an obstruction for a given n\nand circuit size m.\nThe existence-vs-construction principle [KUW] says that computational\ncomplexity of a construction problem is comparable to that of the associated\nexistence problem under natural conditions.\nThis means, under natural\nconditions, existential and explicit-construction complexity classes should\ncoincide. Hence, we shall not worry about existential complexity anymore.\nIt is illuminating to compare the verification complexity of the P vs. NP\nproblem with the other problems we considered. The verification complexity\nclass of Halls’ theorem, four colour theorem, or forbidden minor theorem is\n93\n\nAC. For Poincare’s conjecture, P-verifiability is quite nontrivial [Sc]. But\nfortunately the proof is not very complex.\nIn contrast, P-verifiability is already a formidable issue in the context of\nthe P vs. NP problem.\n19.6\nIs there a simpler proof technique?\nNow we ask if there is a P-verifiable proof technique towards the P ̸= NP\nconjecture that is substantially “easier” than GCT. By easier we mean, with\nlower verification and explicit construction complexity (classes). Since GCT\nis P-verifiable and also P-constructive over C as per Hypothesis 7.1, P ̸=\nNP conjecture is conjecturally P-verifiable and also P-constructive over C.\nThe same can be conjectured over Fp or ̄Fp as well [GCT10]. Assuming this,\nit is meaningful to talk of its verification and explicit construction classes.\nSo we can ask:\nQuestion 19.15 What are the (smallest) verification and explicit construc-\ntion complexity classes of the P ̸= NP conjecture?\nThe best and the most natural answer that one can expect here is P.\nIt would really be unsettling if the answer were, say, NC. Specifically, the\nproblems of verification and explicit construction of obstructions in any P-\nverifiable approach to the P ̸= NP conjecture should be at least as hard as\nP-complete problems. This is supported by the presence of linear program-\nming, which is P-complete, in the algorithms for the basic decision problems\nin Theorem 9.5.\nIf so, GCT may be among the “easiest” P-verifiable approaches to the\nP ̸= NP conjecture over C. The story over Fp may be similar; cf. [GCT11].\nReferences\n[A1]\nM. Agrawal, N. Kayal, N. Saxena, Primes is in P, Annals of Math-\nematics, 160 (2): 781-793, 2004.\n[A2]\nM. Agrawal, Proving lower bounds via pseudo-random generators,\nProceedings of FSTTCS 2005, 92-105, 2005.\n[Ak]\nD. Akhiezer, Homogeneous complex manifolds, Encyclopaedia of\nmathematical sciences, volume 10, Springer-Verlag. 1986.\n94\n\n[AH]\nK. Appel and W. Haken, Every planar map is four colorable,\nA.M.S. Contemporary Math. 98 (1989). MR 91m:05079.\n[B]\nL. Babai, private communication.\n[BGS]\nT. Baker, J. Gill, R. Soloway, Relativization of the P =?NP ques-\ntion, SIAM J. 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Kirwan: Geometric invariant theory.\nSpringer-Verlag, 1994.\n[N]\nH. Narayanan, On the complexity of computing Kostka numbers\nand Littlewood-Richardson coefficients, J. of Algebraic combina-\ntorics, vol. 24, issue 3, Nov. 2006.\n[NW]\nN. Nisan, A. Wigderson, Hardness vs. randomness, J. Comput.\nSys. Sci., 49 (2): 149-167, 1994.\n[O]\nJ. Orlin, A faster strongly polynomial minimum cost flow algo-\nrithm, proceedings of the twentieth Annual Symposium on Theory\nof Computing, 1988.\n[Pe]\nG. Perelman, The entropy formula for the Ricci flow and its geo-\nmetric applications, arXiv:math.DG/0211159.\n[Rs]\nE. Rassart, A polynomiality property for Littlewood-Richardson\ncoefficients, arXiv:math.CO/0308101, 16 Aug. 2003.\n[RR]\nA. Razborov, S. Rudich, Natural proofs, J. Comput. System Sci.,\n55 (1997), pp. 24-35.\n[Re]\nO. Reingold, Undirected s-t-connectivity in logspace, in proceed-\nings of Annual ACM symposium on the Theory of Computing,\n376-385, 2005.\n[RVW]\nO. Reingold, S. Vadhan, A. Wigderson, Entropy waves, the zig-zag\ngraph product and new constant-degree, Ann. of Math (2). vol 155\n(2002), no. 1, 157-187.\n101\n\n[RTF]\nN. Reshetikhin, L. Takhtajan, L. Faddeev, Quantization of Lie\ngroups and Lie algebras, Leningrad Math. J., 1 (1990), 193-225.\n[RSST]\nN. Robertson, D. Sanders, P. Seymour, R. Thomas, A new proof of\nthe four colour theorem, Electronic research announcement of the\namerican mathematical society, vol. 2, number 1, August, 1996.\n[RS]\nN. Robertson, P. Seymour, Graph minors. I. Excluding a forest,\nJournal of Combinatorial theory, Series B 35 (1): 39-61.\n[Sr]\nP. Sarnak, Some applications of modular forms, Cambridge U.\nPress (1990).\n[Sc]\nS. Schleimer, Sphere recognition lies in NP, arXiv:math/0407047v1,\nJul, 2004.\n[Sc]\nA. Schrijver, Combinatorial optimization, Vol. A-C, Springer, 2004.\n[Sp]\nT. Springer, Linear algebraic groups, in Algebraic Geometry IV,\nEncyclopaedia of Mathematical Sciences, Springer-Verlag, 1989.\n[St1]\nR. Stanley, Enumerative combinatorics, vol. 1, Wadsworth and\nBrooks/Cole, Advanced Books and Software, 1986.\n[St4]\nR. Stanley, Positivity problems and conjectures in algebraic com-\nbinatorics, manuscript, to appear in Mathematics: Frontiers and\nPerpsectives, 1999.\n[Ta]\nE. Tardos, A strongly polynomial algorithm to solve combinatorial\nlinear programs, Operations Research 34 (1986), 250-256.\n[Sh]\nA. Shamir, IP=PSPACE, Journal of the ACM, vol. 39, issue 4\n,October 1992.\n[St]\nB. Sturmfels, Algorithms in invariant theory, Springer-Verlag,\n1993.\n[Vd]\nP. Vaidya, A new algorithm for minimizing convex functions over\nconvex sets, Mathematical Programming 73 (1996) 291-341.\n[V]\nL. Valiant, The complexity of computing the permanent, Theoret-\nical Computer Science 8, pp 189-201, 1979.\n[Ve]\nS. Vempala, Private communication.\n102\n\n[W]\nH. Weyl, Classical groups, Princeton University Press, 1939.\n[Wo]\nS. Woronowicz: Compact matrix pseudogroups, Commun. Math.\nPhys. 111 (1987), 613-665.\n[Z]\nA.\nZelevinsky,\nLittlewood-Richardson\nsemigroups,\narXiv:math.CO/9704228 v1 30 Apr 1997.\n103","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0709.0748v1 [cs.CC] 5 Sep 2007\nOn P vs. NP, Geometric Complexity Theory, and\nThe Flip I: a high-level view\nDedicated to Sri Ramakrishna\nKetan D. Mulmuley ∗\nThe University of Chicago\nhttp://ramakrishnadas.cs.uchicago.edu\nTechnical Report TR-2007-13, Computer Science Department,\nThe University of Chicago\nSeptember, 2007\nNovember 7, 2018\nAbstract\nGeometric complexity theory (GCT) is an approach to the P vs.\nNP and related problems through algebraic geometry and representa-\ntion theory. This article gives a high-level exposition of the basic plan\nof GCT based on the principle, called the flip, without assuming any\nbackground in algebraic geometry or representation theory.\nContents\n1\nIntroduction\n4\n1.1\nThe flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n7\n1.1.1\nFrom nonexistence to existence . . . . . . . . . . . . .\n7\n1.1.2\nFrom hard to easy . . . . . . . . . . . . . . . . . . . .\n10\n1.2\nThe P-barrier and its crossing . . . . . . . . . . . . . . . . . .\n11\n∗Part of this work was done while the author was visiting I.I.T. Mumbai\n1"},{"paragraph_id":"p2","order":2,"text":"1.3\nWhy should PH1 and PH2 hold? . . . . . . . . . . . . . . . .\n13\n1.4\nThe reduction . . . . . . . . . . . . . . . . . . . . . . . . . . .\n15\n1.5\nTowards PH1 and SH via PH0\n. . . . . . . . . . . . . . . . .\n16\n1.6\nNonstandard quantum groups . . . . . . . . . . . . . . . . . .\n18\n1.7\nNonstandard Riemann hypotheses? . . . . . . . . . . . . . . .\n19\n1.8\nObstructions vs. expanders\n. . . . . . . . . . . . . . . . . . .\n22\n1.9\nIs there a simpler proof technique? . . . . . . . . . . . . . . .\n23\n1.10 Organization of the paper . . . . . . . . . . . . . . . . . . . .\n24\n2\nBasics in algebraic geometry and representation theory\n24\n2.1\nRepresentation theory . . . . . . . . . . . . . . . . . . . . . .\n26\n2.1.1\nIrreducible representations of GLn(C) . . . . . . . . .\n27\n2.1.2\nIrreducible representations of the symmetric group . .\n29\n2.1.3\nTensor products\n. . . . . . . . . . . . . . . . . . . . .\n30\n2.2\nAlgebraic geometry . . . . . . . . . . . . . . . . . . . . . . . .\n32\n3\nGroup-theoretic varieties\n34\n4\nClass varieties\n35\n4.1\nNC vs. P #P\n. . . . . . . . . . . . . . . . . . . . . . . . . . .\n36\n4.2\nP vs. NP problem over C . . . . . . . . . . . . . . . . . . . .\n39\n5\nObstructions\n40\n6\nWhy should obstructions exist?\n41\n7\nThe flip\n44\n8\nWhy should the flip work?: the P-barrier\n45\n9\nOn crossing the P-barrier\n46\n9.1\nA basic prototype with constant depth complexity\n. . . . . .\n46\n9.2\nFrom constant to superpolynomial depth . . . . . . . . . . . .\n48\n9.3\nSaturated and positive integer programming . . . . . . . . . .\n49\n2"},{"paragraph_id":"p3","order":3,"text":"10 Why should PH1 and PH2 hold?\n51\n11 Decision problems in representation theory\n52\n11.0.1 Littlewood-Richardson problem . . . . . . . . . . . . .\n53\n11.0.2 Kronecker problem . . . . . . . . . . . . . . . . . . . .\n53\n11.0.3 The plethysm problem . . . . . . . . . . . . . . . . . .\n53\n12 The P-barrier in representation theory\n54\n12.1 Crossing the P-barrier . . . . . . . . . . . . . . . . . . . . . .\n54\n13 Reduction\n56\n13.1 Towards easy discovery . . . . . . . . . . . . . . . . . . . . . .\n57\n13.2 From easy algorithm for discovery to easy proof of existence .\n59\n14 Standard quantum group\n61\n15 Nonstandard quantum groups\n65\n15.1 Quantization\n. . . . . . . . . . . . . . . . . . . . . . . . . . .\n66\n15.2 PH0 for couples and triples\n. . . . . . . . . . . . . . . . . . .\n70\n15.3 PH0 for class varieties . . . . . . . . . . . . . . . . . . . . . .\n70\n15.4 PH1 and SH . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n70\n16 Ultimate mystery: nonstandard Riemann hypotheses?\n71\n17 Obstructions vs. expanders\n72\n18 On relativization and P/poly-naturalization barriers\n74\n18.1 From usefulness to superpolynomial lower bounds . . . . . . .\n77\n18.2 On violation of the largeness constraint\n. . . . . . . . . . . .\n79\n18.3 On violation of the constructivity constraint . . . . . . . . . .\n80\n19 P-verifiable and P-constructible proof techniques and their\nexplicit construction complexity\n80\n19.1 P-verifiable proof technique . . . . . . . . . . . . . . . . . . .\n80\n19.2 P-barrier for verification . . . . . . . . . . . . . . . . . . . . .\n83\n3"},{"paragraph_id":"p4","order":4,"text":"19.3 P-constructible proof technique . . . . . . . . . . . . . . . . .\n85\n19.4 General setting . . . . . . . . . . . . . . . . . . . . . . . . . .\n86\n19.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . .\n88\n19.5 Explicit construction complexity\n. . . . . . . . . . . . . . . .\n90\n19.6 Is there a simpler proof technique? . . . . . . . . . . . . . . .\n94\n1\nIntroduction\nGeometric complexity theory (GCT) is a plausible approach to the P vs\nNP [Co, Ka, Le] and related problems in complexity theory via algebraic\ngeometry and representation theory. The goal of this paper is to give a high-\nlevel overview of its basic plan and the underlying principle called the flip,\nwithout assuming any background in algebraic geometry or representation\ntheory. A detailed exposition for mathematicians will appear in [GCTflip2].\nA brief proposal and announcement appeared earlier in cf.[GCTconf]. The\nflip has been partially implemented in a series of papers [GCT1]-[GCT11].\nThis article, followed by [GCTintro], should provide an introduction to the\noverall structure of GCT for computer scientists who wish to get a high-level\npicture before going any further. We assume a few elementary notions of\nalgebraic geometry and representation theory in this introduction. They are\ndescribed in full detail in Section 2, which can be referred to if necessary. For\nthe readers looking for a quick overview, the article [GCTabs], which gives a\nnontechnical synopsis of this paper, followed by just this introduction, which\nhas been written to read as a short paper, should suffice.\nIn this article, the underlying field of computation is taken be C. In\n[GCT11], the problems that arise in the context of the flip over an alge-\nbraically closed field of positive characteristic, or a finite field are discussed.\nThe usual P ̸= NP conjecture is over a finite field of which the one over C\nis in a sense the crux and, being also a formal implication [GCT1], has to\nbe proved first anyway.\nThe flip, in essence, “reduces” the negative hypotheses (lower bound\nproblems) in complexity theory, such as the P ̸=?NP conjecture over C, to\npositive hypotheses in complexity theory (upper bound problems): specifi-\ncally, to showing that a series of decision problems in representation theory\nand algebraic geometry belong to the complexity class P. The “reduction”\nhere is only “in essence”.\nIt is not a formal Turing machine reduction.\nIf it were, it would be relativizable. It is described briefly in Section 1.4\n4"},{"paragraph_id":"p5","order":5,"text":"below, and in detail in Section 13 later. This reduction basically consti-\ntutes a flip from hard, nonexistence to easy existence.\nIn [GCT6], these\ncomplexity-theoretic positive hypotheses are further reduced to mathemat-\nical positivity hypotheses, supported by the theoretical and experimental\nevidence therein. The mathematical positivity hypotheses roughly say that\ncertain nonnegative structural functions in algebraic geometry and repre-\nsentation theory have positive formulae–i.e., formulae without alternating\nsigns–akin to the usual formula for the permanent (in contrast, the usual\nformula for the determinant has alternating signs). It turns out that the\nvalidity of these mathematical positivity hypotheses is intimately linked to\nthe Riemann hypothesis over finite fields–proved in [Dl2] as a culmination\nof extensive effort in mathematics–and the related works in algebraic ge-\nometry and the theory of quantum groups [BBD, KL2, Kas3, Lu1, Lu2].\nIn [GCT6], a plan is suggested for proving them via the theory of quan-\ntum groups. Generalizations of the standard quantum group [Dri, Ji, RTF]\nneeded for this purpose, which we call nonstandard quantum groups, are\nconstructed in [GCT4, GCT7], with further conjectural extensions pointed\nout in [GCT10]. All papers of GCT together suggest that if the Riemann\nhypothesis over finite fields and the related works in the theory of standard\nquantum groups mentioned above can be systematically extended to the\nsetting of the nonstandard quantum groups that arise in GCT, then this\nmay lead to the proof of the P ̸= NP conjecture over C. This basic plan\nof GCT is summarized in Figure 1. Question marks indicate the main open\nproblems.\nThe proof in characteristic zero may eventually extend to finite fields, as\nin the usual form of the conjecture, along the lines suggested in [GCT11].\nThus the ultimate goal of the GCT flip is to deduce the ultimate nega-\ntive hypothesis of mathematics, the P ̸= NP conjecture, in essence, from\nthe ultimate positive hypotheses in mathematics, (nonstandard) Riemann\nHypotheses, thereby giving the ultimate flip shown in Figure 2.\nIn the rest of this introduction, we elaborate Figure 1 further.\nAcknowledgement\nThe author is deeply grateful to Madhav Nori, who taught him algebraic\ngeometry, Milind Sohoni, who collaborated in GCT 1-4, and Manju the\nsource of energy behind this work. The author is also grateful to A. Razborov\nfor pointing out the need for a high-level account. This article is essentially\nan elaboration of the answers to his questions. A part of this work was done\n5"},{"paragraph_id":"p6","order":6,"text":"Complexity theoretic negative hypotheses (lower bound problems)\n|\n|\nThe flip\n|\n↓\nComplexity theoretic positive hypotheses (upper bound problems)\n|\n|\nGCT6|\n|\n↓\nMathematical positivity hypotheses |\n|\n?\n|\n↓\n(?): Nonstandard extensions of the Riemann hypothe-\nsis over finite fields, and the related works in algebraic\ngeometry and the theory of quantum groups\nFigure 1: The basic plan of GCT\n6"},{"paragraph_id":"p7","order":7,"text":"(?): Nonstandard Riemann Hypotheses (+)\n|\n|\n?\n|\n↓\nThe P ̸= NP conjecture (-)\nFigure 2: The ultimate goal of the flip\nwhile the author was visiting I.I.T. Mumbai to which the author is grateful\nfor its hospitality. It is also a pleasure to thank the graduate students who\ntook the accompanying introductory course [GCTintro] on GCT for their\nfeedback.\n1.1\nThe flip\nWe begin with the top arrow in Figure 1: the flip. It is motivated by the\nclassical flip–from the undecidable (negative) to the decidable (positive)–\nthat occurs in G ̈odel’s incompleteness theorem.\nAll known lower bound\nresults–e.g. the hierarchy theorems in complexity theory or the lower bound\nresults in the constant depth [BS] or the PRAM model without bit oper-\nations [Mu1]–depend on flips from lower bounds to upper bounds of some\nsort. But such variations of the classical flip cannot work in the context\nof the P vs. NP problem because they are either relativizable [BGS] or\nnaturalizable [RR]. In contrast, the flip here should be nonrelativizable and\nnonnaturalizable (Section 18).\nThere are actually two flips within this flip: (1) from nonexistence to\nexistence, and (2) from hard to easy.\nHere hard means: the problem of\ndeciding if a computational circuit of size m exists for a given function\nf(x) = f(x1, . . . , xn) is hard. Accordingly, the flip from hard nonexistence\nto easy existence goes in two stages.\n1.1.1\nFrom nonexistence to existence\nThe flip from nonexistence to existence is addressed in [GCT1, GCT2]. Here\nthe nonexistence (lower bound) problem is reduced to an existence problem:\n7"},{"paragraph_id":"p8","order":8,"text":"specifically, to the problem of proving existence of obstructions, which serve\nas “proofs” or “witnesses” for nonexistence of an efficient computational\ncircuit for the explicit hard function in the lower bound problem under\nconsideration.\nJust as existence of a forbidden Kurotowoski minor in a\ngraph serves as an obstruction, i.e., a “proof” for nonexistence of a planar\nembedding.\nAn obstruction in [GCT1, GCT2] is intuitively defined as follows. First\na specific (co)-NP-complete function E(X) = E(x1, . . . , xn), and a specific\nP-complete function H(Y ) = H(y1, . . . , yl) are constructed in [GCT1] so as\nto have special properties that we shall describe in a moment. Using H(Y ),\na projective algebraic variety XP (l) = XP (H; l), for every positive integer l,\nis associated with the complexity class P, called the class variety associated\nwith P, or the simply the P-variety. Here, a projective algebraic variety\nmeans the zero set of a system of homogeneous polynomial equations (cf.\nSection 2.2). These are generalizations of the familar curves and surfaces.\nIt will turn out that XP (l) is a G-variety for G = GLl(C), the group of\ninvertible l × l complex matrices. This means elements of G act on this\nvariety as its transformations–i.e., move its points around–just as G acts on\nCl in the usual way. Similarly, using E(X), a projective variety XNP (n, l) =\nXNP (E; n, l), for every positive integer n and l ≥n, is associated with the\ncomplexity class NP.\nIt is called the class variety associated with NP,\nor simply the NP-variety.\nIt will again be a G-variety.\nThe functions\nE(X) and H(Y ) have been specially chosen so that these class varieties are\nexceptional and their algebraic geometry can be analyzed in depth. If E(X)\ncan be computed by a circuit of size m, then it would turn out that XNP (n, l)\ncan be embedded in XP (l) as a G-subvariety for l = O(m2). Pictorially:\nXNP (n, l) ֒→XP (l).\n(1)\nWe want to show that this embedding is impossible if m = poly(n), as\nn →∞. This would show that E(X) cannot be computed by a circuit of\nm = poly(n) size, and hence, P ̸= NP over C.\nLet R(n, l) = R(E; n, l) and S(l) = S(H; l) denote the homogeneous\ncoordinate rings of XNP (E; n, l) and XP (H; l), respectively. Here by the\ncoordinate ring of a variety, we mean the ring of polynomial “functions” (of\nsome kind) on the variety as defined in Section 2.2. These are akin to the\nring of polynomial functions on Cl. Since the class varieties are G-varieties,\nthese homogeneous coordinate rings will be G-representations (Section 2.1).\nBy a G-representation we mean a vector space on which the elements of G\nact as linear transformations, just as they do on Cl. If the embedding (1)\n8"},{"paragraph_id":"p9","order":9,"text":"exists, then it would turn out that R(n, l) is a G-subrepresentation of S(l).\nWe say that an irreducible, i.e., a minimal, nonzero representation W of G\nis an obstruction, for given n and l, if it occurs as a G-subrepresentation\nof R(n, l), but not as a G-subrepresentation of S(l). Existence of such a\nW, for given n and l, implies that R(n, l) cannot be embedded as a G-\nsubrepresentation of S(l), and hence, the embedding (1) cannot exist. Thus\nan obstruction serves as a “witness” or a “proof” that the embedding (1)\ncannot exist.\nWe now reformulate this notion of obstruction using a few basic notions\nin representation theory described in Section 2.1. It is known that (polyno-\nmial) irreducible representations of G are in one-to-one correspondence with\nthe set of sequences, also called partitions, λ : λ1 ≥λ2 · · · λk > 0 of positive\nintegers of length k ≤l. The irreducible representation of G labelled by λ is\ncalled a Weyl-module, and is denoted by Vλ(G). It is also known that each\nfinite dimensional representation V of G can be written as a direct sum of\nirreducible representations:\nV =\nM\nλ\nmλVλ(G),\nwhere mλVλ(G) denotes the direct sum of mλ copies of Vλ(G), and each\nmλ, called the multiplicity of Vλ(G) in V , is uniquely defined. Thus Vλ(G)\noccurs in V as a subrepresentation iffthe multiplicity mλ is nonzero.\nLet R(E; n, l)d and S(H; l)d denote the subspaces in R(E; n, l) and S(H; l),\nrespectively, of forms of degree d.\nLet sλ\nd(H; l) denote the multiplicity\nof Vλ(G) in S(H; l)d.\nLet sλ\nd(E; n, l) denote the multiplicity of Vλ(G) in\nR(E; n, l)d. Then Vλ(G) is an obstruction for given n and l ifffor some d\nsλ\nd(E; n, l) is nonzero but sλ\nd(H; l) is zero. Here d is uniquely determined\nby the size P\ni λi of λ. We also say that Vλ(G) is an obstruction of degree\nd, and by an abuse of language, also that the label λ is an obstruction of\ndegree d.\nThe main algebro-geometric result of [GCT2] (Theorem 6.3) indicates\nthat such obstructions should exist in the context of the P vs. NP problem,\nwhen m = poly(n), assuming that P ̸= NP, as we expect. The goal then is\nto show that obstructions indeed exist, as expected, for all n →∞, assuming\nm = poly(n). The story is similar for other related lower bound problems.\nThis addresses the easier half of the flip from nonexistence to existence.\n9"},{"paragraph_id":"p10","order":10,"text":"1.1.2\nFrom hard to easy\nBut how should one prove that obstructions actually exist? The main hy-\npothesis governing the flip, which addresses this question, is the following\none that constitutes the harder half of the flip: from hard to easy.\nHypothesis 1.1 (PHflip1) Consider the P vs. NP problem over C. Let\nE(X) be the explicit function in [GCT1] mentioned above. Then the follow-\ning problems are “easy”; i.e., belong to P. Specifically,\n(a) Verification of an obstruction:\ngiven n, l and the partition λ,\nwhether Vλ(G) is an obstruction for given n and l can be decided in poly(n, l, ⟨λ⟩)\ntime, where ⟨λ⟩denotes the bitlength of the specification of λ.\n(b) Explicit construction of obstructions: Suppose l = nlog n (say).\nThen, for every n →∞, a label λ(n) of an obstruction Vλ(G) for n and l\ncan be constructed explicitly in poly(n, l) time, thereby proving existence of\nan obstruction for every such n and l.\nIn view of the definition of an obstruction, the statement (a) for verifi-\ncation clearly follows from:\nHypothesis 1.2 (PHflip2) The following the decision problems are easy;\ni.e., belong to P. Specifically,\n(a) Given d, n, l and a partition λ, whether sλ\nd(E; n, l) is nonzero, i.e.,\nwhether Vλ(G) occurs as a G-subrepresentation of R(n, l)d can be decided\nin poly(⟨d⟩, ⟨λ⟩, n, l) time. Here ⟨d⟩denotes the bitlength of d.\n(b) Given d, l and a partition λ, whether sλ\nd(H; l) is nonzero, i.e., whether\nVλ(G) occurs as a G-subrepresentation of S(l)d can be decided in poly(⟨d⟩, ⟨λ⟩, l)\ntime.\nThe decision problems in Hypothesis 1.2 are the crux of the matter.\nOnce easy algorithms for these decision problems are found, the goal is to\nprove existence of an obstruction for every n →∞, when l = nlog n (say),\nby constructing such an obstruction explicitly, as per Hypothesis 1.1 (b).\nWe shall discuss how this is to done in Section 1.4 below. Assuming for\nthe moment that this transformation of easy algorithms for the decision\nproblems in Hypothesis 1.2 into an easy procedure for explicit construction\nof obstructions (Hypothesis 1.1(b)) for all n →∞, when l = nlog n, works,\nwe get the “reduction” shown in the top arrow of Figure 1: from the original\nhard nonexistence (lower bound) problem to the basic upper bound problems\nin Hypothesis 1.2.\n10"},{"paragraph_id":"p11","order":11,"text":"1.2\nThe P-barrier and its crossing\nBut, by divine justice, the task of showing that the problems in Hypothe-\nsis 1.2 are easy turned out to be extremely hard. Thus, paradoxically, the\nhardest aspect of the flip is just to prove that the basic decision problems\nthat arise in the construction of obstructions are actually easy; i.e., belong\nto P. The best algorithms for these decision problems obtained using the\ngeneral purpose algorithms in algebraic geometry and representation theory\ntake space that is double exponential in m and time that is triple expo-\nnential in m. This means even verification of an obstruction, let alone its\ndiscovery, takes time that is triple exponential in m if one were to use the\ngeneral purpose techniques.\nThe gap between this triple exponential time bound and the polynomial\ntime bound sought in Hypothesis 1.2 is so huge that, at the surface, this\nhypothesis may seem impossible. This was the main barrier, called the P-\nbarrier (Section 8), on this path towards the P vs NP problem when the\nflip was briefly announced in [GCTconf].\nThe article [GCT6] says that it can be crossed under reasonable mathe-\nmatical assumptions. We now turn to a brief description of these results.\nFor that we need a few definitions.\nWe say that a function f(k), k a nonnegative integer, is a quasi-polynomial\nif for some integer l ≥1 there exist polynomials fi(k), 1 ≤i ≤l, such that\nf(k) = fi(k) if k = i modulo l. Here l is called the period of the quasi-\npolynomial. An important example of a quasi-polynomial is the Ehrhart\nquasi-polynomial fP(k) of a polytope P. By definition, it is the number of\ninteger points in the dilated polytope kP. This is known to be a quasi-\npolynomial [St1].\nWe say that a quasi-polynomial f(k) is positive, if the coefficients of all\nfi(k) are nonnegative. We say that it is saturated if either f1(k) is identically\nzero as a polynomial, or if not, f(1) = f1(1) ̸= 0. If f(k) is positive, it is\nclearly saturated.\nNext, let us associate with the multiplicities sλ\nd(H; l) and sλ\nd(E; n, l) the\nfollowing stretching functions:\n ̃sλ\nd(H; l)(k) = skλ\nkd(H; l),\n(2)\nand\n ̃sλ\nd(E; n, l)(k) = skλ\nkd(E; n, l).\n(3)\n11"},{"paragraph_id":"p12","order":12,"text":"The following is the main algebro-geometric result in [GCT6].\nTheorem 1.3 (cf. Theorem 3.4.11 in [GCT6])\n(Rationality Hypothesis): Assume that the singularities of the class varieties\nXP (H; m) and XNP(E; n, l) are “nice” (rational).\nThen the stretching functions ̃sλ\nd(H; l)(k) and ̃sλ\nd(E; n, l)(k) are quasi-\npolynomials.\nWe do not need to know the exact definition of a rational singularity here,\nwhich can be found in [Ke]. It just means that the singularities are nice.\nThis depends on the exceptional nature of the class varieties (cf. Section 4)\nand is supported by the algebro-geometric results and arguments in [GCT2,\nGCT10].\nUsing Theorem 1.3, we can now formulate the conjectural mathematical\npositivity hypotheses mentioned in the third box from above in Figure 1.\nAssume the rationality hypothesis above.\nHypothesis 1.4 (PH1:) The structural constant sλ\nd(H; l) can be expressed\nas the number of integer points in a polytope P λ\nd (H; l) of poly(l, ⟨d⟩, ⟨λ⟩)\ndimension, whose Ehrhart quasi-polynomial coincides with the stretching\nquasi-polynomial ̃sλ\nd(H; l)(k) in Theorem 1.3. Furthermore, P λ\nd (H; l) can\nbe given in the form of a poly(l, ⟨d⟩, ⟨λ⟩)-time separation oracle as in [GLS].\nThere exists a polytope P λ\nd (E; n, l) for the structural constant sλ\nd(E; n, l)\nwith similar properties.\nThis, in particular, implies that sλ\nd(H; l) and sλ\nd(E; n, l) belong to #P.\nHypothesis 1.5 PH2: The quasi-polynomials ̃sλ\nd(H; l) and ̃sλ\nd(E; n, l) in\nTheorem 1.3 are positive.\nIts weaker form is:\nHypothesis 1.6 (SH:) These quasi-polynomials are saturated.\nPH1 and SH (PH2) together say that each decision problem in Hypothe-\nsis 1.2 can be transformed in polynomial time into a special kind of an integer\nprogramming problem called saturated (resp. positive) integer programming\nproblem (Section 9.3).\n12"},{"paragraph_id":"p13","order":13,"text":"Theorem 1.7 (cf. [GCT6]) The decision problems in Hypothesis 1.2 are\nindeed in P, assuming PH1 and SH (or more strongly PH2) above.\nThis follows from a polynomial time algorithm in [GCT6] for saturated (pos-\nitive) integer programming.\nThis result reduces the positive complexity-theoretic hypotheses in Hy-\npothesis 1.2 to the mathematical positivity hypotheses PH1 and SH, as\nshown in the middle arrow in Figure 1. The algorithms in Theorem 1.7 are\nconceptually extremely simple. They just need linear programming [GLS]\nand computation of Smith normal forms [KB].\nBut their correctness depends on the positivity hypotheses PH1 and SH\n(PH2), whose validity, in turn, is intimately linked to deep phenomena in\nalgebraic geometry and the theory of quantum groups as we shall soon see.\nAn indication of such a link is already here. Since the proof of Theorem 1.3,\nwhich is necessary to even formulate these hypotheses, needs a few funda-\nmental results in algebraic geometry; namely, [Bou] (which in turn is based\non [Hi] and other results), and [Ke, Fl]. It should not then be surprising\nif the proofs the hypotheses need far more. Indeed, the quantum-group-\ntheoretic and algebro-geometric machinery is needed in GCT essentially to\nprove these hypotheses, and hence, that these extremely simple algorithms\nare actually correct.\n1.3\nWhy should PH1 and PH2 hold?\nBut first, we need to justify why these hypotheses should hold in the first\nplace. For that, let us consider the simplest analogue of the decision prob-\nlems in Hypothesis 1.2 in representation theory:\nProblem 1.8 (Littlewood-Richardson problem) Given partitions α, β and\nλ, decide if the Littlewood-Richardson coefficient cλ\nα,β (cf. Section 2.1.3) is\npositive (nonzero). This is defined to be the multiplicity of the irreducible\nrepresentation Vλ(G) in the tensor product Vα(G) ⊗Vβ(G) (which becomes\na G-representation by letting the elements of G act on its two factors simul-\ntaneously).\nThe analogous mathematical positivity hypotheses in this setting are as\nfollows.\nDefine the stretching function\n ̃cλ\nα,β(k) = ckλ\nkα,kβ,\nk ≥0,\n13"},{"paragraph_id":"p14","order":14,"text":"which is obtained by stretching the Littlewood-Richardson coefficient by a\nfactor of k. It is known to be a polynomial [Der, Ki, Rs]. Then\nHypothesis 1.9 (PH1) The Littlewood-Richardson coefficient cλ\nα,β can be\nexpressed as the number of integer points in a polytope P = P λ\nα,β of dimen-\nsion polynomial in the total length of α, β and λ. Furthermore, the Ehrhart\nquasi-polynomial of P coincides with the stretching polynomial ̃cλ\nα,β(k) and\nthe membership function of P is computable in time that is polynomial in\nthe bit lengths of α, β and λ.\nThis is shown, for example, in [BZ]. There are many choices for P λ\nα,β.\nOne choice is called a hive polytope [KT1].\nHypothesis 1.10 (PH2) The coefficients of ̃cλ\nα,β(k) are nonnegative.\nThis implies:\nHypothesis 1.11 (SH) The stretching polynomial ̃cλ\nα,β(k) is saturated.\nSince ̃cλ\nα,β(k) is a polynomial, this simply means if ckλ\nkα,kβ is nonzero for\nsome k ≥1 then cλ\nα,β is also nonzero. PH2 is still open, but has a con-\nsiderable experimental evidence in its support [KTT].\nThat SH holds is\nthe saturation theorem in [KT1]. PH1 and SH in conjunction with linear\nprogramming leads [DM2, GCT3, KT2] to a polynomial time algorithm for\nthe Littlewood-Richardson problem (Problem 1.8), and a polynomial time\nalgorithm [GCT5] for a certain generalized Littlewood-Richardson problem\nassuming SH. These results were indeed a starting motivation for Theo-\nrem 1.7.\nThe Littlewood-Richardson coefficient is a special case of a far-reaching\nclass of fundamental constants in representation theory, called plethysm\nconstants, described in Section 11. The structural constants sλ\nd(H; l) and\nsλ\nd(E; n, l) can be considered to be “hyped up” versions of the plethysm con-\nstant. Considerable theoretical and experimental evidence in support of the\nanalogous positivity hypotheses PH1 and PH2 for the plethysm constants\nis given in [GCT6]; cf. Section 11. This constitutes the main evidence in\nsupport of PH1 and PH2 for sλ\nd(H; l), sλ\nd(E; n, l) and other similar algebro-\ngeometric structural constants that arise in GCT.\n14"},{"paragraph_id":"p15","order":15,"text":"1.4\nThe reduction\nBefore we turn to the plan suggested in [GCT6] for proving PH1 and SH,\nwe explain the nature of the reduction in the top arrow of Figure 1.\nFor this, the easy algorithms in Theorem 1.7 have to be transformed into\nan easy procedure for explicit construction of obstructions as per Hypothe-\nsis 1.1 (b). This transformation cannot be carried out at present since we\ndo not have explicit descriptions of the polytopes P λ\nd (H; l) and P λ\nd (E; n, l)\nin PH1. But it is explained in Section 13 and in detail in [GCT6] why it\nshould be possible to carry out this transformation if PH1 and SH can be\nproved and explicit descriptions of the polytopes therein become available.\nThe scheme for transformation suggested there goes in two steps:\nFirst, the easy algorithms in Theorem\n1.7 have to be used to get an\neasy poly(n, l) procedure for discovering an obstruction (label) for given n\nand l, if one exists.\nSecond, this easy algorithm for discovering an obstruction, or rather its\nstructure and the underlying techniques have to be used to prove that an\nobstruction always exists for every n →∞, assuming l = nlog n, say. That\nis, to prove that this easy algorithm always says “yes” for such n and l. Just\nas the structure of the easy Hungarian method for discovering a perfect\nmatching in a bipartite graph can be used to prove Hall’s theorem that\nevery d-regular bipartite graph always has a perfect matching.\nThis transformation of an easy algorithm for discovery into an easy (i.e.\nfeasible) constructive proof–which we shall call a P-constructive proof–also\ngives, as a side product, an easy, i.e., polynomial time algorithm for explicit\nconstruction of obstructions (labels), as in Hypothesis 1.1 (b). One may\nwonder why we are going for explicit construction of obstructions, when\njust their existence would have sufficed. Because the nature of obstructions\nhere is such that the complexity deciding their existence and of constructing\nthem explicitly, if they do, should be more or less the same; cf. Section 13.2.\nJust as the complexity of deciding if a bipartite graph has a perfect matching\nis more or less the same as that of constructing one, if it exists,\nIn the context of these transformations it is crucial that the algorithms\nin Theorem 1.7 are not only easy, i.e., polynomial-time algorithms, but\nalso have a genuinely simple structure of the right kind, being just varia-\ntions of linear programming. Of course, we can not hope to use the ellip-\nsoid algorithm for linear programming–which though simple is intricate–for\na constructive proof of existence of obstructions. Rather we have to use\n15"},{"paragraph_id":"p16","order":16,"text":"the structure of the underlying polytopes. The analogues of the polytopes\nP λ\nd (H; l) and P λ\nd (E; n, l) in PH1 in the simplified setting of the Littlewood-\nRichardson problem (Problem 1.8) are called hive polytopes [KT1]. These\nhave extremely regular structure. The same is expected to be the case for\nthe polytopes P λ\nd (H; l) and P λ\nd (E; n, l) that actually arise here. For this and\nother reasons given in [GCT6], it is expected that, once explicit descriptions\nof the polytopes P λ\nd (H; l) and P λ\nd (E; n, l) become available, the algorithms\nin Theorem 1.7 can be transformed into simple greedy Hungarian-type al-\ngorithms which do not even need linear programming.\nThis is the main\nreason why the transformation of these easy, polynomial time algorithms\ninto an easy (feasible) proof of existence of obstructions is expected to work\nin our setting, just as it does in the case of Hall’s theorem that we mentioned\nabove.\nAssuming that this works, we would get an explicit family {λ(n)} of\nobstructions (rather their labels), as n →∞, and l = nlog n. The existence\nof such an obstruction family would imply that P ̸= NP over C.\n1.5\nTowards PH1 and SH via PH0\nNow we turn to the basic plan suggested in [GCT6] for proving PH1 and\nSH. This will explain the bottom arrow in Figure 1.\nThis plan is motivated by the proof of PH1 (Hypothesis 1.9) in the\nsimplified setting of the Littlewood-Richardson problem via the theory of\nquantum groups [Kas1, Li, Lu2]. Specifically, it is known that this PH1 is a\nconsequence, in a nontrivial way, of a deep positivity statement in the the-\nory of standard quantum groups [Dri, Ji, RTF]–whose intuitive description\nis given later in Section 14–namely: their representations and coordinate\nrings have canonical bases [Kas2, Lu1, Lu2], whose structural constants de-\ntermining their representation-theoretic and multiplicative structure are all\nnonnegative. We shall refer to the existence of a canonical basis with this\npositivity property as PH0, the zeroth positivity hypothesis (property).\nMotivated by this work, certain positivity hypotheses, again called PH0,\nare formulated in [GCT6], and it is pointed out how and why these may\nsimilarly lead to the proof of the required PH1 and also SH (Hypotheses 1.4\nand 1.6). The PH0 hypotheses in [GCT6] may be thought of as generaliza-\ntions of PH0 in the theory of standard quantum groups. PH1 and SH for\nLittlewood-Richardson coefficients (Hypotheses 1.9 and 1.11) have purely\ncombinatorial proofs [F1, KT1], and hence, PH0 is strictly speaking not re-\nquired in this context. But in the context of the PH1 that we are finally\n16"},{"paragraph_id":"p17","order":17,"text":"interested in (Hypothesis 1.4) the full power of PH0 seems needed for the\nplan in [GCT8, GCT10] to work.\nA natural approach to prove PH0 in [GCT6] in the context of this PH1\nis to somehow generalize the proof of PH0 in the theory of the standard\nquantum group. But the theory of standard quantum groups does not work,\nas expected, in this context. The reason is briefly as follows.\nOne can associate a complexity class with each structural constant that\narises in GCT, which we call its index class. Roughly, if a structural con-\nstant is associated with a class variety for a complexity class C, then its\nindex class is defined to C.\nFor example, the index classes of the mul-\ntiplicities sλ\nd(H; l) and sλ\nd(E; n, l) are P and NP (over C), since they are\nassociated with P- and NP-varieties, respectively. Similarly, the index class\nof the Littlewood-Richardson coefficient is the class of circuits (of restricted\nkinds) of depth two; cf. Section 9.1. The index class of the Kronecker coef-\nficient (Section 2.1.3), which is the analogue of the Littlewood-Richardson\ncoefficient in the representation theory of the symmetric group, is NC2, the\nclass of problems that can be solved by circuits of log2 n depth and polyno-\nmial size. The Littlewood-Richardson coefficient as well as the Kronecker\ncoefficient are special cases of the plethysm constants (Section 11.0.3) which\nwe mentioned earlier. The generalized plethysm constant is not associated\nwith any class variety, but it is qualitatively similar to, though much sim-\npler than sλ\nd(E; n, l). Hence, we define its index class to be NP, with the\nunderstanding that this is to be taken only in a rough sense. The index\nclasses of the structural constants here are not be confused with their usual\ncomputational complexity classes: they are all (conjecturally) in #P by\nPH1.\nThe standard quantum group is the quantum group that occurs in the\ncontext of PH1 for Littlewood-Richardson coefficients (Hypothesis 1.9). Hence,\nwe define its index class to be the same as that of Littlewood-Richardson\ncoefficients, i.e., the class of circuits of depth two. Thus the standard quan-\ntum group is the quantum group attached to constant-depth (depth-two)\ncircuits.\nGiven a big difference between the lower bound problems for constant\nand nonconstant depth circuits, it should not be a surprise if the standard\nquantum group cannot be used in the context of PH1 for the structural\nconstants that actually arise in GCT; cf. Section 16 for an intuitive mathe-\nmatical explanation for why this is so.\n17"},{"paragraph_id":"p18","order":18,"text":"1.6\nNonstandard quantum groups\nWhat is needed then are quantum groups that can play the role of the stan-\ndard quantum group in the context of the decision problems and positivity\nhypotheses for these structural constants. The main result in this context\nis the following:\nTheorem 1.12 [GCT4] There exists a quantum group, which is qualita-\ntively similar to the standard quantum group, that can play such a role in\nthe context of the Kronecker coefficients.\n[GCT7] More generally, there exists a (possibly singular) quantum group that\ncan play such a role in the context of the generalized plethysm constants.\nA less informal statement will be given later (Theorem 15.1). A conjectural\nscheme for generalizing these quantum groups to the ones that can play such\na role in the context of sλ\nd(E; n, l), sλ\nd(H; l) and other structural constants\nin GCT is suggested in [GCT10]. We shall call the new quantum groups in\nTheorem 1.12 nonstandard, because, though they are qualitatively similar\nto the standard quantum group, they are also fundamentally different, as\nexpected.\nThus, standard corresponds to constant depth and nonstandard to non-\nconstant depth circuits.\nThe article [GCT8] gives a conjecturally correct algorithm to construct\ncanonical bases of the irreducible representations and coordinate rings of the\nnonstandard quantum groups in [GCT4, GCT7] with the required positivity\nproperties (PH0). These are natural generalizations of the canonical basis\ndue to Kashiwara and Lusztig [Kas2, Lu1, Lu2] mentioned above for the\nirreducible representations and the coordinate ring of the standard quantum\ngroup.\n[GCT8] also gives a conjecturally correct algorithm to construct\ncanonical bases with similar positivity properties (PH0) for the nonstandard\ndeformations of the symmtric group algebra that are dually paired with the\nnonstandard quantum groups–these generalize the Kazhdan-Lusztig basis\n[KL1] of the Hecke algebra. It is also shown in [GCT7, GCT8] that PH1 for\nthe plethysm constants follows from PH0 and other conjectural properties\nof these nonstandard canonical bases and quantum objects. The story for\nthe general constants sλ\nd(E; n, l) and sλ\nd(H; l) can be expected to be similar\n[GCT10].\nAt present we can neither prove correctness of the algorithms in [GCT8]\nfor constructing nonstandard canonical bases nor the required conjectural\n18"},{"paragraph_id":"p19","order":19,"text":"properties for the reasons that we shall describe in a moment. But a consid-\nerable evidence is given in [GCT8] in support of PH0 for the nonstandard\nquantum group in [GCT4].\nIn the standard case, PH1 follows from PH0 in a more or less rigid\nway [Dh, Kas1, Li, Lu2]. This means the polytope that occurs in PH1 for\nthe Littlewood-Richardson coefficient (Hypothesis 1.9) is more or less de-\ntermined by the canonical basis for the standard quantum group–not com-\npletely, since there are a few choices for this polytope; e.g. a hive polytope\nin [KT1], or a polytope in [BZ]. But all these choices are intimately related.\nA common feature is that they all have extremely regular structures. The\nsame can be expected for the polytopes that should arise in the nonstandard\nsetting. This regularity is crucial for the final transformation of easy algo-\nrithms for the basic decision problems in Hypothesis 1.2 into easy algorithms\nfor explicit construction of obstructions; cf. Sections 1.4 and 13.\nExistence of nonstandard quantum groups of polylogarithmic [GCT4]\nand superpolynomial [GCT7] depth complexity, the conjecturally correct\nalgorithm in [GCT8] for constructing canonical bases (PH0) of their coordi-\nnate rings and irreducible representations, and the principle that is suggested\nby the theory of standard quantum groups–namely, once a canonical basis is\nthere (PH0), everything else in the story more or less follows a rigid path–is\nthe main reason why GCT may be expected to deliver lower bounds for\ncircuits of superpolynomial depth and size eventually.\n1.7\nNonstandard Riemann hypotheses?\nBut for this plan to work, PH0 for the nonstandard quantum groups has\nto be proved. This brings us to the main open question in this story: how\ncan we prove correctness of the algorithm in [GCT8] for constructing the\ncanonical bases (PH0) of the coordinate rings of the nonstandard quantum\ngroups?\nThere are two constructions of the canonical basis in the standard set-\nting. An algebraic construction in [Kas3], where it is called global crystal\nbasis, and a topological construction in [Lu1, Lu2]. Both constructions give\nrise to the same basis [GL]. In fact, both constructions follow the same basic\nscheme. Only the proofs of correctness of this basic scheme are different. The\ntopological proof is based on the theory of perverse sheaves [BBD], which\nin turn, is based on Riemann hypothesis over finite fields [Dl2]. In essence,\nPH0 is thus ultimately deduced in the topological proof from the Riemann\nHypothesis over finite fields, which is again a deep positive statement. Be-\n19"},{"paragraph_id":"p20","order":20,"text":"cause its usual statement is, after all, a positive statement, and it can also be\nreformulated as stipulating positivity (nonnegativity) of some mathematical\nquantities (cf. page 458 in [Ha]). The topological proof also gives, as a side\nproduct, the only known proof of nonnegativity of the structural constants\nassociated with the canonical basis in the standard setting. Though this\nnonnegativity is not needed for proving PH1 for the Littlewood-Richardson\ncoefficients, it is crucial in the nonstandard setting for the reasons given in\n[GCT8, GCT10].\nFor this reason, the topological approach seems to be the only viable\noption in the nonstandard setting, as far as we can see. Besides, the algebraic\ncomplexity of the nonstandard quantum groups is so huge–as to be expected\nin view of the huge gap between constant and nonconstant depth circuits–\nthat a purely algebraic proof of correctness of the algorithm in [GCT8] for\nconstructing canonical bases in the nonstandard setting seems difficult.\nBut the standard Riemann hypothesis over finite fields and the related\ntechniques cannot be expected to work in the the nonstandard setting for\nthe reasons given in [GCT7, GCT8]. Again this should not be surprising\ngiven the big difference between constant and nonconstant depth circuits.\nHence what seem to be needed [GCT8] to make the topological approach\nwork in the nonstandard setting are nonstandard extensions of the Riemann\nhypothesis over finite fields and the related work on perverse sheaves. By\nnonstandard, we mean the extensions that will work in the context of the\nnonstandard quantum groups.\nThe author does not have the mathematical expertize to even formulate\nsuch hypotheses, let alone prove them. But the theoretical and experimental\nevidence in [GCT4, GCT7, GCT8] (cf. Section 16) suggests that such exten-\nsions exist, and that they ought to be provable by a systematic extension of\nthe theory of standard quantum groups to the nonstandard setting. Hence\nit is reasonable to hope that the experts would be able to do so eventually,\nleading to the proof of PH0 hypotheses along the topological lines, and fi-\nnally, to the explicit construction of obstructions as outlined above, which\nwould then imply that P ̸= NP over C. The whole picture is summarized\nin Figure 3, which is an elaboration of the earlier Figure 1. The arrows with\nquestion marks are conjectural, the double arrows are unconditional. The ?\nsigns indicate the main open problems at the heart of this approach. The\nstory over C may eventually lift to the story over finite fields along lines\nsuggested in [GCT11].\n20"},{"paragraph_id":"p21","order":21,"text":"(?): Nonstandard Riemann Hypotheses for the quantum groups in\n[GCT4, GCT7], and their conjectural extensions in [GCT10]\n|\n|\n|\n?\n↓\nPH0 (?): Existence of canonical bases [GCT6, GCT8, GCT10] |\n|\n?\n↓\nPH1,SH (PH2)\n∥\n∥\nGCT6\n∥\n⇓\nPolynomial time algorithms for the de-\ncision problems in Hypothesis 1.2\n|\n|\nThe transformation mentioned in Section 1.4; cf Section 13 and [GCT6]\n|\n?\n↓\nExplicit construction of obstructions\n∥\n∥\n∥\n⇓\nP ̸= NP over C\nFigure 3: The basic plan for implementing the flip in GCT6\n21"},{"paragraph_id":"p22","order":22,"text":"1.8\nObstructions vs. expanders\nAn initial motivation for going for explicit construction of obstructions as\nin Figure 3 was provided by explicit construction of expanders [LPS, Ma].\nAs explained in Section 17, the obstructions in GCT are in a certain sense\ngeneralizations of the expanders from constant depth to superpolynomial\ndepth circuits. Specifically, obstructions are to superpolynomial depth cir-\ncuits what expanders are to constant depth, in fact, depth two circuits; cf.\nFigure 4. In view of this relationship, explicit construction of obstructions as\nin Figure 3 would be in the setting of superpolynomial depth circuits what\nexplicit construction of expanders is in the setting of constant depth circuits.\nAs we remarked earlier, the standard quantum group also corresponds to\ncircuits of depth two. That is, expanders and the standard quantum group\nboth correspond to the class of depth-two circuits. Hence it does not seem to\nbe a coincidence that the Riemann hypothesis over finite fields, which enters\nin the theory of the standard quantum group, also enters in the theory of\nexpanders [Lb, Sr].\n↑\ndepth\n|\nCircuits of superpolynomial depth and size: Obstructions\n↑\n|\nCircuits of depth two: expanders\nFigure 4: The relationship between obstructions and expanders\nExistence of expanders can be proved by a simple probabilistic method.\nIn contrast, existence of expanders may not be provable by a probabilistic\nmethod. Indeed, this is roughly the main content of [RR], which says that a\nnonconstructive method, such as a probabilistic method, should not work in\nthe context of the P vs. NP problem under reasonable assumptions. This\nis why in GCT we go for explicit construction of obstructions in the spirit\nof explicit construction of expanders. The P/poly-naturalizability barrier in\n[RR] should not be applicable to such explicit, constructive proof techniques.\nThis issue is addressed in more detail in Section 18.\n22"},{"paragraph_id":"p23","order":23,"text":"1.9\nIs there a simpler proof technique?\nFinally, one may ask if the P ̸= NP conjecture may be proved by a sub-\nstantially simpler proof technique.\nThis seems unlikely for the following\nreasons.\nThe results in complexity theory such as [A2, Re] suggest that explicit\nconstructions may be more or less essential for derandomization. In conjunc-\ntion with the hardness vs. randomness principle [KI, NW], this suggests that\nexplicit constructions may also be more or less essential for (the difficult)\nlower bound problems as well.\nHence, the difficulty in any viable proof\ntechnique for the P ̸= NP conjecture may be intimately linked to the diffi-\nculty (complexity) of the explicit construction of obstructions, i.e., “proofs\nof hardness” as per that technique. This may be so regardless of whether the\ntechnique actually constructs such obstructions explicitly or not. Because,\nas per the existence-vs-construction principle [KUW], the difficulty of decid-\ning existence may be more or less the same as that of construction in natural\nproblems. These and other considerations naturally lead to a notion of ex-\nplicit construction complexity of an easy-to-verify proof technique towards\nthe P ̸= NP conjecture, where easy-to-verify formally means P-verifiable;\ncf. Section 19.\nThe explicit construction (depth) complexity of expanders is O(1), in\nfact, two, since they can be constructed by (nonuniform) depth-two algebraic\ncircuits (over a ring of integers modulo k for some k) [LPS, Ma]. Whereas,\nas per Hypothesis 1.1, the explicit construction (depth) complexity of the\nobstructions in GCT over C is poly(m), m = nlog n (say) being the circuit\nsize parameter in the lower bound problem; cf. Figure 4. The arguments in\nSection 19 suggest that this may be essentially the best explicit construction\ncomplexity that one can expect in any P-verifiable proof technique towards\nthe P ̸= NP conjecture. In other words, the massive Ω(m) gap between the\nexplicit construction complexity of obstructions and the O(1) explicit con-\nstruction complexity of expanders, as shown in Figure 4, may be inevitable\nin any P-verifiable proof technique towards the P ̸= NP conjecture. If so,\nGCT may be among the “easiest” P-verifiable approaches to this conjecture\nas per the explicit construction complexity measure defined here, and hence,\nit may be unrealistic to expect a technique that is substantially simpler or\neasier.\nIn the rest of this article, we elaborate the plan in Figure 3 further\nand give a high-level description of the results in the GCT papers. Logical\ndependence among the GCT papers is shown in Figure 5.\n23"},{"paragraph_id":"p24","order":24,"text":"1.10\nOrganization of the paper\nIn Section 2 we recall a few basic facts in algebraic geometry and represen-\ntation theory which are easy to state and should be easy to believe. The\nreaders not familar with these fields should be able to take these on faith.\nIn Section 3 we describe a special class of algebraic varieties, called group-\ntheoretic varieties. All class varieties in GCT are group-theoretic varieties.\nThey are described in Section 4. Obstructions are defined in Section 5. Why\nthey should exist is described in Section 6. The flip is described in Sec-\ntion 7. The main barrier in the implementation of the flip, the P-barrier,\nis described in Section 8. The main result of GCT that crosses this barrier,\nassuming the mathematical positivity hypotheses PH1 and SH (PH2), is\ndescribed in Section 9. Why PH1 and PH2 should hold is described in Sec-\ntion 10. Simpler analogues in representation theory of the decision problems\nin Hypothesis 1.2 are described in Section 11. The P-barrier in this context,\nits crossing subject to analogous PH1 and SH (PH2), along with theoretical\nresults supporting these positivity hypotheses are described in Section 12.\nThe nature of the reduction in the top arrow of Figure 1 is described in\nSection 13. The basic plan in [GCT6] to prove PH1 and SH via the the-\nory of quantum groups is described next. The standard quantum group is\nintuitively described in Section 14. The nonstandard quantum groups are\nintuitively described in Section 15. Why nonstandard Riemann hypotheses\nshould exist and their role in the theory of nonstandard quantum groups is\nbriefly described in Section 16. The relationship between obstructions and\nexpanders is described in Section 17. Why GCT should cross the relativiza-\ntion and the P/poly-naturalizability barriers is described in Section 18. Why\nGCT may be among the easiest P-verifiable approaches to the P vs. NP\nproblem as per the explicit-construction-complexity measure is described in\nSection 19.\n2\nBasics in algebraic geometry and representation\ntheory\nIn this section we describe the basic facts in algebraic geometry and repre-\nsentation theory which are needed in this article and which should be easy\nto believe for the readers not familiar with these fields. Their proofs can be\nfound in [FH, Mm1].\n24"},{"paragraph_id":"p25","order":25,"text":"This article\n(GCTflip1)\n|\n↓\nGCTintro\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 5: Logical dependence among the GCT papers\n25"},{"paragraph_id":"p26","order":26,"text":"2.1\nRepresentation theory\nLet G be a group. We say that a vector space V is a representation of G, or\na G-module, if there is a homomorphism\nρ : G →GL(V ),\n(4)\nwhere GL(V ) is the general linear group of invertible transformations of\nV . We denote ρ(g)(v) by g · v–the result of the action of g on v. A G-\nsubrepresentation W ⊆V is a subspace that is invariant under G; i.e.,\ng · w ∈W for every w ∈W. If G is clear from the context, we just call\nit subrepresentation. We say that V is irreducible if it does not contain a\nproper nontrivial subrepresentation. A G-homomorphism from a G-module\nU to a G-module V is map ψ : U →V such that ψ(g · u) = g · (ψ(u)) for all\nu ∈U.\nWe say that G is reductive if every finite dimensional representation V\nof G is completely reducible. This means it can be expressed as a direct sum\nof irreducible representations in the form\nV =\nM\nλ\nmλVλ(G)\n(5)\nwhere λ ranges over all indices (labels) of irreducible representations of G,\nVλ(G) denotes the irreducible representation of G with label λ, and mλVλ(G)\ndenotes a direct sum of mλ copies of Vλ(G). Here mλ is called the multiplicity\nof Vλ(G) in V . It is a basic fact of representation theory that for reductive\ngroups, the decomposition (5) is unique; i.e., mλ’s are uniquely defined. If\nmλ > 0, we say that Vλ(G) occurs in V .\nAn example of a nonreductive group is a solvable group that is not\nabelian. In this case a subrepresentation W ⊆V need not have a comple-\nment W ⊥such that V = W ⊕W ⊥.\nEvery finite group is reductive.\nThus Sn, the symmetric group on n\nletters, is reductive. A prime example of a continuous reductive group is\nthe general linear group GLn(C) = GL(Cn), the group of nonsingular n × n\nmatrices, and its subgroup the special linear group SLn(C) = SL(Cn) of\nmatrices with determinant one.\nAny product of reductive groups is also\nreductive. These are the only kinds of reductive groups that we need to\nknow in this article. So whenever we say reductive, the reader may wish to\nassume that the group is a general or special linear group or a symmetric\ngroup or a product thereof.\n26"},{"paragraph_id":"p27","order":27,"text":"We say that the representation (4) of G = GLn(C) or SLn(C) is polyno-\nmial if for every g ∈G, every entry in the matrix form of ρ(g) is a polynomial\nin the entries of g.\nComplete reducibility as in eq.(5) means every finite dimensional rep-\nresentation of a reductive group is composed of irreducible representations.\nThese can be thought of as the building blocks in the representation theory\nof reductive groups, and it is important to know what these building blocks\nare.\n2.1.1\nIrreducible representations of GLn(C)\nFor GLn(C) this was done by Weyl in his classic book [W]. The polynomial\nirreducible representations of GLn(C) are in one-to-one correspondence with\nthe tuples λ = (λ1, . . . , λk) of integers, where k ≤n and λ1 ≥λ2 · · · ≥λk >\n0. Here λ is called a partition of length k and size d = P\ni λi. Its bitlength\n⟨λ⟩is defined to be the total bitlength of all λi’s.\nThus the polynomial irreducible representations of GLn(C) are labelled\nby partitions λ of length at most n, but any size. The irreducible representa-\ntion corresponding to a partition λ = (λ1, λ2, . . .) is denoted by Vλ(GLn(C)),\nand is called a Weyl module of GLn(C). When GLn(C) is clear from the\ncontext, we shall denote it by simply Vλ.\nEach partition λ corresponds to a Young diagram, which consists of\nk rows of boxes, with λi boxes in the i-th row. For example, the Young\ndiagram corresponding to (4, 2, 1) is shown below:\nWhen thinking of a partition, it is helpful to think of the corresponding\nYoung diagram. Thus each Weyl module is labelled by a Young diagram of\nheight at most n. This is a useful combinatorial tool for studying the Weyl\nmodules.\nA Weyl module Vλ is explicitly constructed as follows. This construction\nof Deyruts as well as Weyl’s original construction are given in [FH]. Let Z be\nan n × n variable matrix. Let C[Z] be the ring of polynomials in the entries\nof Z. It is a representation of GLn(C). Action of a matrix σ ∈GLn(C) on\na polynomial f ∈C[Z] is given by\n(σ · f)(Z) = f(Zσ).\n(6)\n27"},{"paragraph_id":"p28","order":28,"text":"By a numbering (filling), we mean filling of the boxes of a Young diagram\nby numbers in [n]; for example:\n1 2 4 3\n2 3\n1\nWe call such a numbering a (semistandard) tableau if the numbers are strictly\nincreasing in each column and weakly increasing in all rows; e.g.\n1 2 3 3\n2 3\n4\nThe partition corresponding to the Young diagram of a numbering is\ncalled the shape of the numbering.\nWith every numbering T, we associate a polynomial eT ∈C[Z], which is\na product of minors for each column of T. The l × l minor ec for a column c\nof length l is formed by the first l rows of Z and the columns indexed by the\nentries cj, 1 ≤j ≤l, of c. Thus eT = Q\nc ec, where c ranges over all columns\nin T. The Weyl module Vλ is the subrepresentation of C[Z] spanned by eT ,\nwhere T ranges over all numberings of shape λ over [n]. Its one possible\nbasis is given by {eT }, where T ranges over semistandard tableau of shape\nλ over [n].\nLet B ⊆GLn(C) be the subgroup of upper triangular matrices. It is\ncalled the Borel subgroup of GLn(C). An element vλ ∈Vλ is called a highest\nweight vector if it is an eigenvector for the action of each b ∈B. It is easy to\nshow that Vλ has a unique highest weight vector, upto a constant multiple:\nit is eT0, where T0 is the canonical tableau whose i-th row contains only i’s,\nfor each i; e.g.\n1 1 1 1\n2 2\n3\nLet P ⊆GLn(C) be the subgroup of upper block triangular matrices,\nwhere the sizes of the blocks are fixed. For example:\n28"},{"paragraph_id":"p29","order":29,"text":"∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n0\n0\n∗\n∗\n∗\n∗\n0\n0\n∗\n∗\n∗\n∗\n0\n0\n0\n0\n∗\n∗\n0\n0\n0\n0\n∗\n∗"},{"paragraph_id":"p30","order":30,"text":"Such subgroups are called parabolic. Let Pλ be the (projective) stabilizer\nof the highest weight vector vλ = eT0; i.e., the set of all σ ∈GLn(C) such\nthat σ ·vλ = c(σ)vσ, for some complex number c(σ). Then it is easy to show\nthat Pλ is parabolic, where the sizes of the blocks are completely determined\nby λ.\nThe irreducible representation of GLn(C) corresponding to the Young\ndiagram that consists of just one column of length n is the determinant\nrepresentation: g →det(g). When restricted to the subgroup SLn(C) ⊆\nGLn(C) this becomes trivial. More generally, Vλ(G) and Vλ′(G) give the\nsame representation of SLn(C) if λ′ is obtained from λ by removing columns\nof length n. Hence, irreducible polynomial representations of SLn(C) are\nin one to one correspondence with partitions of length less than n, and are\nobtained from the ones of GLn(C) by restriction.\n2.1.2\nIrreducible representations of the symmetric group\nIrreducible representations of Sn, called Specht modules, are in one-to-one\ncorrespondence with the Young diagrams of size n, as opposed to those of\nlength ≤n for GLn(C). We denote the Specht module corresponding to a\npartition λ by Sλ. It is explicitly constructed as follows.\nLet C[X] = C[x1, · · · , xn] be the ring polynomials in n variables. It is a\nrepresentation of Sn: given σ ∈Sn and f ∈C[X],\n(σ · f)(x1, · · · , xn) = f(xσ(1), · · · , xσ(n)).\nGiven a numbering T of λ with distinct numbers in [n], let fT be the poly-\nnomial formed by taking a product of discriminants for all columns of T.\nThe discriminant for a column with entries ci, 1 ≤i ≤l, is Q\ni<i′(xci −xci′).\nThen Sλ is simply the subrepresentation of C[X] spanned by fT, where T\nranges over all numberings of λ with distinct entries in [n]. Its basis is given\nby {fT }, where T ranges over standard tableau of shape λ with entries in\n[n]. Here a standard tableau means the rows as well as the columns are\n29"},{"paragraph_id":"p31","order":31,"text":"strictly increasing; e.g.\n1 2 3 6\n4 5\n7\n2.1.3\nTensor products\nIf V and W are representations of a group G, then their tensor product\nV ⊗W is also a representation: for σ ∈G, v ∈V, w ∈W,\nσ(v ⊗w) = (σ · v) ⊗(σ · w).\nGiven two irreducible representations of a reductive group G, a fun-\ndamental problem in representation theory is to find an explicit complete\ndecomposition of their tensor product in terms of irreducible representations\nof G. The following instances of this problem are of central importance in\nGCT.\nLittlewood-Richardson coefficients\nFirst we consider this problem when G = GLn(C). Given Weyl modules Vα\nand Vβ, let\nVα ⊗Vβ =\nM\nγ\ncγ\nα,βVγ,\n(7)\nbe the complete decomposition of their tensor product into irreducible Weyl\nmodules of G. Here the multiplicities cγ\nα,β are called Littlewood-Richardson\ncoefficients. The Littlewood-Richardson rule gives the sought explicit for-\nmula for these multiplicities. It is as follows.\nAlign the left top corners of the Young diagrams for α and γ. If the\nYoung diagram for α is not contained in the one for γ, then cγ\nα,β is zero.\nOtherwise, form a skew shape γ \\ α by removing the boxes in γ belonging\nto α. A skew tableau with content β and shape γ \\ α is a filling of this skew\ndiagram with β1 ones, β2 twos and so on, such that all columns are strictly\nincreasing and all rows are weakly increasing. For example, the following is\na skew tableau of skew shape (4, 3, 3, 2) \\ (2, 2, 1):\n1 1\n2\n2 3\n1 3\n(8)\n30"},{"paragraph_id":"p32","order":32,"text":"We say that a skew tableau is a Littlewood-Richardson tableau if, when\nits entries are read from right to left, top to bottom, the number of i’s read\nupto any point is at most the number of (i −1)’s read up to that point,\nfor any i. For example, the skew tableau above is a Littlewood-Richardson\nskew tableau. Let Cγ\nα,β be the set of Littlewood-Richardson tableau of shape\nγ \\ α with content β. The Littlewood-Richardson coefficient cγ\nα,β is simply\nthe cardinality of Cγ\nα,β: i.e.,\ncγ\nα,β = |Cγ\nα,β| =\nX\nT∈Cγ\nα,β\n1.\n(9)\nSuch a formula is called positive, because it is like the formula for the perma-\nnent which involves only positive signs. In contrast, there are many formulae\nfor such multiplicities, based on the theory of characters of group represen-\ntations [FH], which involve alternating signs, like the usual formula for the\ndeterminant. Positivity here is a deep issue; cf. [St4].\nFormally, the Littlewood-Richardson rule implies that the Littlewood-\nRichardson coefficient belongs to the complexity class #P, just like the per-\nmanent. This is the real significance of the Littlewood-Richardson rule from\nthe complexity-theoretic perspective. Furthermore, just like the permanent,\nthe Littlewood-Richardson coefficient is #P-complete [N].\nKronecker coefficients\nNow we turn to the symmetric group.\nSince it is reductive, the tensor\nproduct of two Specht modules Sα and Sβ decomposes as: by:\nSα ⊗Sβ =\nM\nγ\nkγ\nα,βSγ,\n(10)\nwhere the multiplicities kγ\nα,β are called Kronecker coefficients.\nNo positive rule akin to the Littlewood-Richardson rule is known for\nthe Kronecker coefficients. In fact, this is a fundamental open problem in\nthe representation theory of symmetric groups, which arose almost with the\nbirth of representation theory in the work of Frobenius, Schur, Weyl and\nothers in the beginning of the twentieth century; cf. [Mc, St4] for its history\nand significance. In the language of complexity theory, the problem is:\nQuestion 2.1 Does the Kronecker coefficient belong to #P?\n31"},{"paragraph_id":"p33","order":33,"text":"Though this is not how it was stated in representation theory. The answer\nis conjecturally yes [GCT4]. Indeed, this is the main focus of the work in\n[GCT4, GCT8]: roughly, [GCT8] says that such a rule exists assuming a\nconjecture regarding the nonstandard quantum group defined in [GCT4].\nThis is the first entry point of nonstandard quantum groups in GCT.\n2.2\nAlgebraic geometry\nLet V = Cn. Let X = (x1, . . . , xn) be the variable n-vector whose entries\nstand for the coordinates of V . An affine algebraic set Z ⊆V is the set\nof zeroes of a collection of polynomials in C[X] = C[x1, . . . , xn]. An affine\nalgebraic set is called irreducible if it cannot be expressed as the union of\ntwo proper affine algebraic subsets. An irreducible affine algebraic subset\nZ of V is called an affine variety. Its ideal I(Z) ⊆C[X] is the set of all\npolynomials that vanish on Z, and its coordinate ring C[Z] is defined to be\nC[X]/I(Z). The elements of C[Z] are polynomial functions on Z.\nLet P n−1 = P(V ) be the projective space of lines in V through the\norigin. We say that V is the affine cone of P(V ). Given a nonzero v ∈V ,\nwe also denote by v the point in P(V ) that corresponds to the line in V\npassing through v and the origin; the meaning should be clear from the\ncontext. The homogeneous coordinate ring of P(V ) is defined to be C[X].\nIts elements are homogeneous functions on V , the affine cone of P(V ). A\nprojective algebraic set Y in P(V ) is the set of zeroes of a collection of\nhomogeneous forms (polynomials). The affine cone ˆY ⊆V of Y ⊆P(V )\nis defined to be the union of the lines in V corresponding to the points in\nY . A projective algebraic set is called irreducible if it cannot be expressed\nas the union of two proper projective algebraic subsets.\nAn irreducible\nprojective algebraic subset Y of P(V ) is called a projective variety. Its ideal\nI(Y ) ⊆C[X] is the set of all homogeneous forms that vanish on Y , and its\nhomogeneous coordinate ring R(Y ) is C[X]/I(Y ). The elements of R(Y ) are\nhomogeneous functions on the affine cone ˆY of Y . The degree d-component\nR(Y )d of R(Y ) is the subspace of homogeneous forms of degree d.\nThe\nHilbert function hY (d) of Y is defined to be the dimension of R(Y )d.\nBy a (Zariski)-open subset of Y we mean the complement of an algebraic\nsubset of Y . An open subset of a projective variety is also called a quasi-\nprojective variety.\nNow suppose V is a representation of a reductive group G.\nThen G\nalso acts on P(V ), since it takes line to a line. Furthermore, C[X] is also a\n32"},{"paragraph_id":"p34","order":34,"text":"representation of G: given σ ∈G and f ∈C[X], we define\n(σ · f)(X) = f(σ−1X).\n(11)\nThe variety Y is called a G-variety if its ideal I(Y ) ⊆C[X] is a G-subrepresentation\nof G; i.e., σ·f ∈I(Y ) for all σ ∈G, f ∈I(Y ). In this case, the homogeneous\ncoordinate ring R(Y ) of Y is also a representation of G. Furthermore, given\na point p ∈Y , the point σ(p) also belongs to Y . In other words, G acts on\nthe variety Y by moving its points around.\nIf Z is a projective subvariety of Y , then it is a basic fact that there\nexists a degree preserving surjection from R(Y ) to R(Z); i.e., from R(Y )d\nto R(Z)d for every d. This surjection is obtained by simply restricting a\npolynomial function on the affine cone ˆY to the subcone ˆZ ⊆ˆY . If both\nY and Z are G-varieties, then this surjection is a G-homomorphism. By\ncomplete reducibility, it then follows that R(Z)d is a G-submodule of R(Y )d\nfor every d. Pictorially,\nR(Z)d ֒→R(Y )d,\n(12)\nfor every d.\nGiven a point v ∈P(V ), let Gv denote its G-orbit. It can be shown\nthat Gv is a quasi-projective variety.\nLet Gv = {σ | σ · v = v} be its\nstabilizer. Then Gv as a set is isomorphic to the coset set G/H, H = Gv.\nQuasiprojective varieties of the form G/H are called homogeneous spaces.\nThese have been intensively studied in algebraic geometry.\nLet ∆V [v] = Gv ⊆P(V ) denote the closure of the G-orbit of v in the\nusual complex topology 1. We call such a variety an orbit closure. It can\nbe shown that ∆V [v] is a projective G-variety. One can think of ∆V [v] as\na closure of the homogeneous space G/Gv. Such spaces are called almost-\nhomogeneous spaces [Ak]. These have also been intensively studied. Let\nRV [v] be the homogeneous coordinate ring of ∆V [v], and RV [v]d its degree\nd-component. Since G acts on ∆V [v], each RV [v]d is a finite dimensional\nrepresentation of G.\nThe simplest example of ∆V [v] arises as follows.\nLet Vλ be a Weyl\nmodule of G = GLn(C). Let vλ ∈P(Vλ) be the point corresponding to the\nhighest weight vector in Vλ; we call it the highest weight point. Then it can\nbe shown that the orbit Gvλ ∼= G/Pλ, where Pλ is the stabilizer of vλ, is\nalready closed. This is called a flag variety. It has been intensively studied\nin algebraic geometry for over a century, and its algebraic geometry is now\nmore or less completely understood; e.g. see [LLM].\n1This coincides with the closure in the Zariski-topology [Mm1].\n33"},{"paragraph_id":"p35","order":35,"text":"The flag varieties by their very definition are smooth. But the algebraic\ngeometry of general orbit closures can be extremely complicated, and essen-\ntially, intractable. Because, even if the orbit Gv is smooth, its closure can be\nhighly singular, and the singularities can be pathological. Indeed, the moral\nof the story that can be gained from [LV] is that the algebraic geometry of\na general orbit closure is essentially hopeless.\n3\nGroup-theoretic varieties\nFortunately, the class varieties that arise in GCT are all exceptional kinds of\norbit closures, which we call group-theoretic orbit closures or group-theoretic\nvarieties. The articles [GCT2, GCT10] together roughly say that problems\nregarding the algebraic geometry of these group-theoretic class varieties can\nbe “reduced” to problems in (quantum) group theory. This is what makes\nthem tractable, and this is how the theory of quantum groups enters in\nGCT. In this section, we shall briefly describe a group-theoretic variety in\nan abstract form.\nLet V and G be as in Section 2.2. We say that v ∈P(V ) is characterized\nby its stabilizer H = Gv ⊆G, if it is the only point in P(V ) stabilized (left\ninvariant) by H. Stabilized by H means, for every σ ∈H , σ · v = v.\nFor example, the highest weight point vλ ∈P(Vλ) (Section 2.1) is char-\nacterized by its stabilizer Pλ. This indicates that the points that are char-\nacterized by their stabilizers are very special.\nSuppose v is characterized by its stabilizer. Then v, and hence, its orbit\nclosure ∆V [v] is completely determined by the group triple:\nH = Gv ֒→G\nρ→K = GL(V ),\n(13)\nwhere ρ represents the representation map; cf. (4). This leads to the follow-\ning:\nDefinition 3.1 Assume that v is characterized by its stabilizer, and that\nthe associated group triple H ֒→G ֒→K is explicitly known. Then we say\nthat the orbit closure ∆V [v] is a group-theoretic variety. It is completely\ndetermined by the preceding group triple.\nWe call H ֒→G the primary\ncouple associated with the orbit closure ∆V [v], and G ֒→K, the secondary\ncouple. We also say that v is the characteristic point of the group triple\nH ֒→G ֒→K and the primary couple H ֒→G.\n34"},{"paragraph_id":"p36","order":36,"text":"If every point in V is a function on some space then we say that v is\nthe characteristic function of the triple H ֒→G ֒→K and the primary\ncouple H ֒→G. (This happens if, for example, V is the space of polynomial\nfunctions on an affine G-variety X, with the action given by (11)).\nHere explicitly means the composition factors of H as well as the con-\nnecting homomorphisms in (13) are specified explicitly; for the details re-\ngarding how, see [GCT6, GCTflip2]. All varieties that arise in GCT are\neither group-theoretic orbit-closures in the above sense, or their generaliza-\ntions, which are again essentially determined by group triples as above, and\nhence, will also be called group-theoretic varieties. The simplest example of\na group-theoretic variety is a flag variety (Section 2.2).\nIf a variety is group theoretic, then, in principle, we ought to be able to\nunderstand its algebraic geometry if we understand the structure of the as-\nsociated group triple, along with the connecting homomorphisms, in depth.\nWe shall elaborate on what in depth means later in Section 15. Briefly, it\nmeans understanding the structure of the group triple at the quantum level.\n4\nClass varieties\nNow we turn to the class varieties associated with the complexity classes\nNC, P, #P and NP in [GCT1] on which the obstructions in GCT live. All\nthe class varieties that arise in GCT will be orbit closures of the following\nspecial form.\nLet Y = [y0, · · · , yl−1] denote a variable l-vector. For n < l, let X =\n[y1, · · · , yn], and ̄X = [y0, · · · , yn] be its subvectors of size n and n + 1.\nWe also denote yi, 1 ≤i ≤n, by xi.\nLet V = Syms(Y ) be the space\nof homogeneous forms of degree s in the l variable-entries of Y . It has a\nnatural action of G = SL(Y ) = SLl(C) and ˆG = GL(Y ) = GLl(C), just as\nin (11).\nSimilarly, let W = Symr(X), r < s, be the representation of GL(X) =\nGLn(C). We have a natural embedding φ : W →V , which maps\nw ∈W →ys−rw ∈V,\n(14)\nwhere y = y0 is used as the homogenizing variable. The image φ(W) is\ncontained in ̄W = Syms( ̄X), a representation of GL( ̄X) = GLn+1(C).\nThe basic recipe for constructing class varieties is as follows. Say we\nwant to separate a complexity class C1 from a complexity class C2 ⊇C1.\n35"},{"paragraph_id":"p37","order":37,"text":"We pick a form g = g(Y ) = g(y0, . . . , yl−1) ∈P(V ) which is a complete\nfunction for the complexity class C1. Then the orbit closure ∆V [g; l] = ∆V [g]\nis called the class variety associated with C1, or simply the C1-variety based\non the complete function g.\nIn principle, we can let g be any complete\nfunction for the class C1. But for the algebraic geometry of ∆V [g] to be\ntractable, we have to choose g so that ∆V [g] is group-theoretic; i.e., so that\ng is characterized by its stabilizer as in Definition 3.1, or in a slightly relaxed\nsense (cf. Section 7 in [GCT1]), which is good enough for our purposes.\nSimilarly, we choose a form h = h(X) = h(x1, . . . , xn) ∈P(W) which\nis complete for the class C2. Then the orbit closure ∆W[h; n] = ∆W[h] ⊆\nP(W) is called the base class variety associated with C2, or simply the base\nC2-variety based on h. Let f = φ(h), with φ as in (14). We call the orbit\nclosure ∆V [f; n, l] = ∆V [f] ⊆P(V ) the extended class variety associated\nwith C2, or the extended C2-variety based on h. This extension is necessary\nso that the C2-variety ∆V [f] and the C1-variety ∆V [g] live in the same\nambient space P(V ). Again, h has to be chosen so that it is characterized\nby its stabilizer (almost) so that the varieties ∆W [h] and ∆V [f] are group-\ntheoretic.\nLet us suppose to the contrary that C2 ⊆C1. Then it would turn out\nthat f ∈∆V [g; l], and hence, ∆V [f; n, l] is a G-subvariety of ∆V [g; l]:\n∆V [f; n, l] ֒→∆V [g; l].\nThe goal is to show that such an embedding does not exist when l is small\nenough, say, l = nlog n, n →∞. This will show that C1 ̸= C2. Here l will be\na parameter in the lower bound problem that depends on the depth and/or\nthe size of the circuit.\nWe now demonstrate this recipe in two basic separation problems in\ncomplexity theory.\n4.1\nNC vs. P #P\nLet NC be the standard class of functions that can be computed by circuits\nof polylogarithmic depth, and #P the counting class associated with NP.\nThe determinant is complete for the class NC and the permanent for the\nclass #P [V]. The P #P ̸= NC conjecture over C [V] says that the per-\nmanent cannot be computed by a circuit over C of polylogarithmic depth.\nUsing the determinant and the permanent, we now construct class varieties\nfor NC and #P. The P #P ̸= NC conjecture over C will then be reduced\n36"},{"paragraph_id":"p38","order":38,"text":"to showing that the extended class variety for #P is not contained in the\none for NC.\nLet Y be an m × m variable matrix, which can also be thought of as a\nvariable l-vector, l = m2, by linearly ordering its entries in any order. Let\nX be its, say, the principal bottom-right n × n submatrix, n < m, which\ncan also be thought of as a variable k-vector, k = n2. Let V = Symm(Y )\nbe the space of homogeneous forms of degree m in the variable entries of\nY , and W = Symn(X), the space of homogeneous forms of degree n in the\nvariable entries of X. We have a natural action of G = SL(Y ) = SLl(C) on\nV and the projective space P(V ): namely, σ ∈G maps a form q(Y ) ∈P(V )\nto q(σ−1Y ), where we think of Y as a variable l-vector. Similarly, we have\nan action of H = SL(X) = SLk(C) on P(W).\nUsing any entry y of Y not in X as a homogenizing variable we get an\nembedding φ : W →V , which maps any w ∈W to ym−nw ∈V .\nLet g = det(Y ) ∈P(V ) be the determinant form.\nLet ∆V [g; l] =\n∆V [g] ⊆P(V ) be its orbit closure.\nIt is shown in [GCT1] that if a form h(X) ∈P(W) can be computed by a\ncircuit of depth less than logc n, then f = φ(h) lies in ∆V [g; l] for m = 2logc n.\nConversely, if f lies in ∆V [g; l] then h[X] can be approximated infinitesimally\nclosely2 by a circuit of depth O(log2c m).\nSince, the permanent is #P-\ncomplete [V], this is not expected to happen if h = perm(X) and m =\n2O(polylog(n)). This leads to:\nConjecture 4.1 [GCT1] Let h = perm(X) ∈P(W) and f = φ(h). Then\nf ̸∈∆V [g] = ∆V [g; l], if m = 2O(polylog(n)), as n →∞. Since ∆V [g] is a\nG-variety, this is equivalent to saying that ∆V [f] ̸⊆∆V [g]. Pictorially:\n∆V [f] ̸֒→∆V [g].\nHere ∆V [g] = ∆V [g; l] is called the class variety associated with the\nclass NC, or simply the NC-variety based on the complete determinant\nfunction. We also denote it by XNC(g; l) or simply XNC(l). We call ∆W[h]\nthe base class variety and ∆V [f] the extended class variety associated with\nthe class #P, or simply the base #P-variety and the extended #P-variety,\nrespectively. We also denote them by X#P (h; n) and X#P (f; n, l), or sim-\nply, X#P(n) and X#P (n, l). The goal (Conjecture 4.1) is to show that the\n2This means, for every ǫ > 0, there exists a form ̃f ∈V , which has a circuit of depth\nO(log2c m), such that ||f − ̃f|| < ǫ, in the usual norm on V .\n37"},{"paragraph_id":"p39","order":39,"text":"extended class variety for #P cannot be contained in the class variety for\nNC, when m = 2polylog(n): i.e.,\nX#P (n, l) ̸֒→XNC(l).\nThis will show that the permanent cannot be computed by circuits of poly-\nlogarithmic depth.\nNext we describe why these class varieties are group-theoretic. For this,\nwe need to show that the determinant and the permanent are characterized\nby their stabilizers.\nThe stabilizer of det(Y ) ∈P(V ) in G = SL(Y ) = SLm2(C) is known to\nbe a reductive subgroup Gdet which consists of linear transformations in G\nof the form (thinking of Y as an m × m matrix):\nY →AY ∗B,\n(15)\nwhere Y ∗is either Y or Y T , A, B ∈GLm(C).\nThat the determinant is\ncharacterized by its stabilizer follows from classical invariant theory [FH].\nHence the NC-variety defined here is group-theoretic. The associated group\ntriple is\nGdet ֒→G ֒→GL(V ),\n(16)\nand Gdet ֒→G the primary couple. The embedding Gdet →G almost looks\nlike the natural embedding\nGL(Cm) × GL(Cm) →GL(Cm ⊗Cm),\n(17)\ngiven by: (g, h) →g ⊗h, where g ⊗h denotes the Kronecker product. That\nis,\n(g ⊗h) · (x ⊗y) = (g · x) ⊗(h · y).\n(18)\nThe stabilizer of perm(X) ∈P(W) in SL(X) = SLn2(C) is a reduc-\ntive subgroup generated by linear transformations in SL(X) of the form\n(thinking of X as an n × n matrix):\nX →λX∗μ,\n(19)\nwhere X∗is either X or XT , λ and μ are either diagonal or permutation\nmatrices, and n ≥3. It is easy to show that the permanent is also charac-\nterized by its stabilizer. Hence the base #P-variety defined in this section\nis group theoretic; the extended #P-variety is also group-theoretic.\n38"},{"paragraph_id":"p40","order":40,"text":"4.2\nP vs. NP problem over C\nThe class varieties associated with the classes P and NP can be constructed\nin principle using any P-complete and NP-complete functions. But again it\nis necessary to choose these functions in a special way so that the resulting\nclass varieties turn out to be group-theoretic (Section 3). Such P-complete\nand (co)-NP-complete functions, called H(Y ) = H(y1, . . . , yl) and E(X) =\nE(x1, . . . , xn) respectively, have been constructed in [GCT1]. We do not\nneed to know their definitions here.\nLet W = Symr(X) be the space of forms of degree r = deg(E(X)) in\nthe entries of X. Thus E(X) ∈P(W). Let V = Syms(Y ) be the space of\nforms of degree s = deg(H(Y )) in the entries of Y . Thus H(Y ) ∈P(V ). We\nidentify X with a suitable subset of Y , and define a map φ : P(W) →P(V )\nas in (14) by choosing a variable y in Y \\ X as a homogenizing variable.\nNow, using the recipe above, we can associate with E(X), for every n\nand l ≥n, a group-theoretic variety (orbit closure) ∆V [f; n, l] = ∆V [f] ⊆\nP(V ), where f = φ(h) and h = E(X). It is a G-variety, for G = SLl(C).\nIt will be called the (extended) class variety for NP or simply the NP-\nvariety based on the form E(X), and will be denoted by XNP (E; n, l) or\nsimply XNP (n, l). Similarly, we can associate with H(Y ) a group-theoretic\nG-variety ∆V [g; l] = ∆V [g] ⊆P(V ), where g = H(Y ).\nIt is called the\nclass variety for P or simply the P-variety based on the form H(Y ), and is\ndenoted by XP (H; l) or simply XP (l).\nRemark 4.2 The actual P-variety XP (H; l) in the P vs. NP problem is\nnot meant to be ∆V [g, l], as defined here, but rather the variety ˆ∆[H(Y )]\ndefined in Section 7 of [GCT1]. But we shall ignore that difference here.\nIt can be shown [GCT1] that if E(X) is computable by a circuit of size\nm then XNP (E; n, l) can be embedded within XP (H; l) for l = O(m2):\nXNP (n, l) = XNP (E; n, l) ֒→XP (l) = XP (H; l).\n(20)\nIn this context:\nConjecture 4.3 [GCT1] This embedding cannot exist if m = nlog n, or\nmore generally, m = 2na, for a small enough a > 0, as n →∞.\nThis will show that P ̸= NP over C. This transforms the P vs. NP problem\nover C into a problem in geometric invariant theory.\n39"},{"paragraph_id":"p41","order":41,"text":"Again, these class varieties are group-theoretic, in a slightly relaxed sense\nthan defined in Section 3, but which is good enough for the purposes of GCT\n[GCT1].\n5\nObstructions\nAn obstruction in the P vs. NP problem (characteristic zero) is defined to\nbe a representation that lives on the extended class variety associated with\nNP but not on the class variety associated with P. We now elaborate what\nthis means.\nLet R(n, l) = R(E; n, l) and S(l) = S(H; l) denote the homogeneous co-\nordinate rings of XNP (n, l) = XNP (E; n, l) and XP (l) = XP (H; l), respec-\ntively. We call them the class rings associated with the complexity classes\nNP and P. Let R(n, l)d and S(l)d denote their degree d-components, con-\nsisting of homogeneous polynomial functions of degree d. Since G acts on\nthe class varieties, it also acts on the class rings (see Section 2.2). That is,\neach R(n, l)d or S(l)d is a finite dimensional representation of G.\nIf the embedding (20) exists, then R(n, l)d can be embedded as a G-\nsubmodule of S(l)d, for each d; cf. (12):\nR(n, l)d ֒→S(l)d.\n(21)\nIn particular, every irreducible representation (Weyl module) Vλ = Vλ(G)\nof G that occurs within R(n, l)d as a subrepresentation also occurs within\nS(l)d as a subrepresentation.\nDefinition 5.1 We say that S = Vλ is an obstruction, for n, l and the pair\n(E, H) = (E(X), H(Y )), if it occurs in R(n, l)d but not in S(l)d, for some\nd.\nIn this case we say that Vλ is an obstruction of degree d. We also refer\nto λ as an obstruction of degree d.\nObstruction in the setting of the NC vs. P #P problem over C is defined\nsimilarly.\nThis notion of obstruction in [GCT2] is a refinement of the earlier notion in\n[GCT1].\nThe specification of an obstruction is given in the form of its label λ.\nThe existence of such an obstruction for given n and l is a “proof” that the\nembedding in (21), and hence, the one in (20) cannot exist.\n40"},{"paragraph_id":"p42","order":42,"text":"In this context:\nConjecture 5.2 [GCT2, GCT10] An obstruction for n, l and the pair (E, H)\nexists if m = nlog n, or more generally, m = 2na, for a small enough a > 0,\nas n →∞; recall that l = O(m2). Furthermore, there exists such an ob-\nstruction of a small degree d(n, m) = 2mb, b > 0 a large enough constant.\nSimilar conjecture can be made in the context of the NC vs. P #P prob-\nlem.\nIn this case, the degree d(n, m) can be mb, b > 0 a large enough\nconstant.\nIf such an obstruction Vλ(n) exists for every n →∞, with m as above,\nthen it follows that P ̸= NP over C. We say that {Vλ(n)} or {λ(n)} is an\nobstruction family for the P vs. NP problem over C. The goal is to prove\nexistence of such a family.\n6\nWhy should obstructions exist?\nA priori, it is not at all clear why such obstructions should even exist. In\nthis section, we explain why they should.\nAn intuitive reason for existence of obstructions is as follows. The article\n[Dl1] roughly says that (algebraic) groups are completely determined by their\nrepresentations. On the other hand, the group-theoretic class varieties are\nessentially determined by the associated group triples, and hence, as per the\nphilosophy in [Dl1], the representation-theoretic information associated with\nthese group triples. Hence, a “witness” for nonexistence of the embedding\nas in (20) ought to be present in the representation-theoretic information\nassociated with the group triples, assuming that P ̸= NP–which we take on\nfaith. This is intuitively why a representation-theoretic obstruction ought\nto exist. Specifically, there should exist a representation-theoretic witness\n(obstruction) that explains why one group-theoretic class variety, with as-\nsociated group triple H1 ֒→G →K, cannot be embedded in another group\ntheoretic variety with associated group triple H2 ֒→G →K; in our problem\nG and K in both triples would be the same.\nBut why should such a representation-theoretic obstruction be specifi-\ncally of the type as defined here?\nTo see this, let us first consider a simpler example. Instead of triples,\nlet us consider couples. Let us say we are given two couples ρ1 : H1 ֒→G,\nand ρ2 : H2 ֒→G, where G = GLl(C) = GL(W), W = Cl. This means W\n41"},{"paragraph_id":"p43","order":43,"text":"is a representation of H1 and H2. Let us assume that it is an irreducible\nrepresentation of H1 and H2, and furthermore, that both H1 and H2 are\nreductive, and that H2 is not a conjugate of H1. Now the coset sets G/H1\nand G/H2 can be given the structure of affine algebraic varieties [Mm2].\nSince H2 is not a conjugate of H1, G/H1 cannot be embedded in G/H2 (and\nvice versa). The goal is to find a representation theoretic obstruction for the\nnonexistence of such an embedding. We say that Vλ(G) is an obstruction for\nthis pair of couples (ρ1, ρ2) if it occurs as a G-submodule in the coordinate\nring of G/H1 but not in the coordinate ring of G/H2. This is equivalent\nto saying that Vλ(G) contains an H1-invariant, when considered as an H1-\nmodule via ρ1, but not an H2-invariant, when considered as an H2-module\nvia ρ2; this is a consequence of the Peter-Weyl theorem [Sp]. This then is an\nobstruction very similar to the one in Definition 5.1. Its existence implies\nthat G/H1 cannot be embedded in G/H2. The work [LP] implies that such\nas an obstruction always exists when H1 and H2 are as above.\nConjecture 5.2 is a natural generalization of this well characterized sit-\nuation. It says that there exists a similar obstruction for the embedding\namong the group-theoretic varieties under consideration. This, as expected,\nis a much harder issue. The existence of such an obstruction depends cru-\ncially on the following conjecture concerning the algebraic geometry of the\nclass varieties under consideration.\nConjecture 6.1 (a) (cf. [GCT2]) The algebraic geometry of the class va-\nriety for NC is completely determined by the representation theory of the\nassociated group triple. Specifically, let Π be the set of G-submodules of C[V ]\nwhose duals do not contain a Gdet-invariant; i.e., the trivial Gdet-module;\ncf. (16). Let X(Π) ⊆P(V ) be the zero set of the forms in the G-modules\nin Π. Then XNC = X(Π).\n(b) (cf. [GCT10]) Analogous, but more complex, statements hold for the\nclass varieties associated with the complexity classes P, NP and #P.\nFor precise statements see [GCT2, GCT10].\nRemark 6.2 (Erratum) In [GCT2] it is conjectured that XNC = X(Π)\nas a scheme [Ha]. This stronger conjecture may not hold as it is. Rather,\nits variant, as would be described in [GCT10], is expected to hold.\nConcrete support for this conjecture is provided by the following two\nresults.\nThe first result is the second fundamental theorem of invariant\n42"},{"paragraph_id":"p44","order":44,"text":"theory. It says that the analogue of Conjecture 6.1 holds for flag varieties\nand their generalizations [LLM]. Thus Conjecture 6.1 may be thought of\nas a natural generalization of the second fundamental theorem of invariant\ntheory to the group-theoretic class varieties under consideration. The second\nresult, specific to the setting under consideration, is the following.\nTheorem 6.3 (Theorem 2.11 in [GCT2])\nA weaker form of Conjecture 6.1 holds for the NC-variety. Specifically,\nthere is a dense open neighbourhood U ⊆P(V ) of the orbit Gg of the deter-\nminant g = det(Y ) such that XNC ∩U = X(Π) ∩U, assuming a reasonable\ntechnical condition.\nThe article [GCT10] gives justifications for and a plan to prove Conjec-\nture 6.1. It is shown in [GCT2] that obstructions as in Definition 5.1 indeed\nexist in the context of NC vs. P #P problem, for all n →∞, assuming\n1. Conjecture 6.1 (a), and\n2. that the permanent cannot be approximated infinitesimally closely by\ncircuits of polylogarithmic depth.\nThe argument for existence of obstructions in the context of the P vs. NP\nproblem based Conjecture 6.1 (b) is similar [GCT10].\nThe first statement here crucially depends on the group-theoretic nature\nof the class variety for NC. If in place of the determinant we substitute other\nfunction, this need not hold. The second statement is a slightly strengthened\nform of the statement that we are finally trying to prove: namely, that the\npermanent cannot be computed by circuits of small depth. This circular\nreasoning tells us why obstructions should exist. But it gives no help in\nshowing that they exist unconditionally.\nWe turn to this task in the next section. A remark before we do so.\nThe existence of obstructions here crucially depends on the exceptional na-\nture of H(Y ). But we have made no use so far of the exceptional nature\nof E(X). In fact, obstructions of such kind should exist for any hard (co-\nNP-complete) function h(X) in place of E(X). But the approach for con-\nstructing obstructions described in the next section crucially depends on the\nexceptional nature of E(X)–i.e., on the group-theoretic nature of the class\nvariety XNP (E; n, l) for NP based on E(X).\n43"},{"paragraph_id":"p45","order":45,"text":"7\nThe flip\nNow we come to the real problem: how to prove the existence of obstructions\nfor the specific E(X) under consideration. One may wish to try a probabilis-\ntic strategy for proving existence of obstructions: just choose a label λ(n) of\nhigh enough degree randomly, and show that Vλ(n) is an obstruction with a\ngood probability. But this technique would not work in the context of the P\nvs. NP problem because it is P/poly-naturalizable [RR]. Hence we shall go\nfor explicit construction of obstructions in the spirit of explicit construction\nof expanders [LPS, Ma, RVW]. The P/poly-naturalizability barrier in [RR]\nwould not apply to an approach based on explicit constructions (Section18).\nThis approach is based on the following hypothesis governing the flip:\nHypothesis 7.1 (PHflip1)\nThe following problems belong to P. Specifically:\n(a) (Verification): There exists a poly(l, n, ⟨d⟩, ⟨λ⟩)-time algorithm for de-\nciding, given l, n, d and λ, if Vλ is an obstruction of degree d for n, l and\nthe pair (E, H) (Definition 5.1). Here ⟨d⟩and ⟨λ⟩denote the bitlengths of\nd and λ, respectively.\n(b) (Explicit construction of obstructions): Suppose l = nlog n, or 2na, for\na small enough constant a > 0. Then, for every n →∞, a label λ(n) of\nan obstruction for n and l can be constructed explicitly in poly(n, l) time,\nthereby proving existence of an obstruction for every such n and l.\n(c) (Discovery of obstructions in general): There exists a poly(l, n)-time\nalgorithm for deciding if there exists an obstruction for n, l and the pair\n(E, H), and for constructing the label of one, if it exists.\nSimilar hypothesis holds for the NC vs. P #P problem.\nIn view of the definition of obstruction (Definition 5.1), The statement\n(a) for verification follows from the following:\nHypothesis 7.2 (PHflip2) (a) There exists a poly(l, n, ⟨d⟩, ⟨λ⟩)-time al-\ngorithm for deciding, given l, n, d and λ, if Vλ(G) occurs in R(n, l)d.\n(b) There exists a poly(l, ⟨d⟩, ⟨λ⟩)-time algorithm for deciding, given l, d and\nλ, if Vλ(G) occurs in S(l)d.\nSimilar hypothesis holds for the NC vs. P #P problem.\nAs mentioned in Section 1.4, once Hypothesis 7.2 is proved, the poly-\nnomial time algorithms for the decision problems therein have to be trans-\n44"},{"paragraph_id":"p46","order":46,"text":"formed into a polynomial time algorithm for explicit construction of obstruc-\ntions as in Hypothesis 7.1 (b), thereby proving Conjecture 5.2, and hence\nthe lower bound under consideration. This issue will be addressed in detail\nin Section 13 later.\nThe whole discussion in this section is summarized in Figure 6.\nFind easy, polynomial time algorithms for the\ndecision problems in Hypothesis 7.2\n|\n|\n|\n↓\nTransform these easy algorithms into an easy algorithm for ex-\nplicit construction of obstructions as in Hypothesis 7.1 (b)\n|\n|\n|\n↓\nP ̸= NP over C\nFigure 6: The flip\n8\nWhy should the flip work?: the P-barrier\nBut why should there exist easy algorithms as in Hypotheses 7.1 and 7.2?\nThis turns out to be, paradoxically, the hardest aspect of the flip: just to\nprove easiness. In this section, we elaborate its nature further.\nClearly, the function E(X) has to be extremely special for Hypotheses 7.1\nand 7.2 to hold. If, instead of E(X), we consider a general co-NP-complete\nfunction h(X) then, obstructions can still be expected to exist (cf.\nSec-\ntion 6), but Hypotheses 7.1 and 7.2 would fail severely, as we now explain.\nSo fix a general integral function h(X) = h(x1, . . . , xn), which is co-NP-\ncomplete, when considered over F2 by reduction modulo 2. Let XNP (h; n, l) ⊆\nP(V ) be the class variety associated with it by following the recipe in Sec-\ntion 4.2 with h(X) in place of E(X). Here V = Syms(Y ) is the space of forms\nof degree s = deg(H(Y )) in l = O(m2) variable entries of Y . The dimension\n45"},{"paragraph_id":"p47","order":47,"text":"M of the ambient projective space P(V ) here is exponential in l = O(m2),\nm being the the circuit size. Using the currently best available algorithms\nfor constructing a Gr ̈obner basis [KM], and for various problems in invariant\ntheory [St], analogues of the decision problems in Hypotheses 7.1 and 7.2\nfor h(X) can be solved in at best O(dim(C[V ]d) = O(dM) = O(d2poly(m))\nspace, where C[V ]d denotes the degree d component of C[V ], the homoge-\nneous coordinate ring of P(V ). This is so even for the decision problems in\nHypothesis 7.2 and hence for the verification problem in Hypothesis 7.1 (a).\nThis is the best that we can expect for general h(X) in view of the lower\nbound [MM] for the construction of Gr ̈obner bases. In other words, for a\ngeneral h(X) the time taken by a best procedure to even verify if Vλ(G), for\na given λ, is an obstruction would take space that is double exponential in\nm, and hence, time that is triple exponential in m.\nAs we shall argue in Section 19, for any approach towards the P ̸= NP\nconjecture to be viable, at least the problem of verifying an obstruction (i.e.,\na “proof”or “witness” of hardness as per that approach) should be easy; i.e.,\nbelong to P. Intuitively, because however hard it may be to discover a proof,\nits verification, once found, should be easy. The main P-barrier in the course\nof GCT is this huge gap between the triple exponential bound given by the\ncurrently best techniques for a general h(X) and the polynomial bound\nstipulated for verification in Hypothesis 7.1 (a) and in Hypothesis 7.2.\n9\nOn crossing the P-barrier\nWe now come to the main result of [GCT6] which crosses this P-barrier\nunder reasonable assumptions. It gives polynomial-time algorithms for the\ndecision problems in Hypothesis 1.2, and hence, for verifying an obstruction\n(Hypothesis 1.1 (a)), assuming the mathematical positivity hypotheses PH1\nand SH (Hypotheses 1.4-1.6).\n9.1\nA basic prototype with constant depth complexity\nTo motivate these positivity hypotheses, we first consider a basic prototype\nof the decision problems in Hypotheses 7.2 in a simplified setting:\nProblem 9.1 (Littlewood-Richardson problem) Given α, β and λ, decide\nif the Littlewood-Richardson coefficient cλ\nα,β (cf. Section 2.1.3) is positive\n(nonzero).\n46"},{"paragraph_id":"p48","order":48,"text":"Equivalently, consider the diagonal homomorphism:\nρ : H = GLn(C) →G = H × H.\n(22)\nGiven an irreducible G-module Vα(H) ⊗Vβ(H), decide if an irreducible H-\nmodule Vλ(H) occur in it, when considered as an H-module via the diagonal\nhomomorphism.\nThis problem corresponds to circuits of depth two in the following sense.\nLet X be an n×n variable matrix. Let V = Sym1(X) be the space of linear\nforms in the entries of X. We have the action of G on P(V ) given by:\n((h1, h2) · f)(X) = f(h−1\n1 Xh2),\nfor any h1, h2 ∈H and f ∈P(V ).\nLet f(X) = trace(X).\nThen the\nstabilizer of f in G is precisely H, and f is characterized by its stabilizer.\nHence, f(X) = trace(X) is the characteristic function (Definition 3.1) of\nthe couple (22). It can be computed by a circuit of depth two. Hence, the\ncharacteristic class of the couple (22) can be defined to the class of circuits of\ndepth two. In this sense, the setting of the Littlewood-Richardson problem\nis roughly dual to the setting of expander graphs (Section 1.8), which too\ncorrespond to circuits of depth two.\nIn [GCT3, DM2, KT2] it is shown that this problem indeed belongs to\nP, thereby establishing the analogue of Hypothesis 7.2 in this setting. Two\nmain ingradients in this proof, in addition to linear programming, are PH1\nand SH for Littlewood-Richardson coefficients (Hypotheses 1.9 and 1.11).\nIn [GCT5], it is shown that the problem of deciding nonvanishing of a gen-\neralized Littlewood-Richardson coefficient for the classical connected reduc-\ntive groups other than GLn(C), namely the simplectic and the orthogonal\ngroups, also belongs to P, assuming the following generalized form of SH in\nthis context.\nLet ̃cλ\nα,β(k) = ckλ\nkα,kβ be the stretching function for a generalized Littlewood-\nRichardson coefficient cλ\nα,β, where α, β and λ are no longer partitions, but\nrather their generalizations [FH].\nIt is known to be a quasi-polynomial\n[BZ, DM2].\nHypothesis 9.2 (PH2): The quasi-polynomial ̃cλ\nα,β(k) is positive.\nThis was conjectured in [DM2] on the basis of considerable experimental\nevidence. Its weaker form is:\n47"},{"paragraph_id":"p49","order":49,"text":"Hypothesis 9.3 (SH): The quasi-polynomial ̃cλ\nα,β(k) is saturated.\nIn [GCT5] it is shown that the problem of deciding if a generalized\nLittlewood-Richardson coefficient is nonzero also belongs to P assuming\nPH2, or its weaker form, SH.\n9.2\nFrom constant to superpolynomial depth\nThe goal now is to lift the polynomial time algorithms and the mathematical\npositivity hypotheses PH1 and PH2 above from the simplified constant-\ndepth setting to the superpolynomial-depth setting of Hypotheses 7.1 (a)\nand 7.2. This is done in [GCT6] in two steps. We only consider the P vs.\nNP problem, considerations for the NC vs. P #P problem being similar.\nWe use the same notation as in Section 5.\nThe first step is the following mathematical result which allows formu-\nlation of the mathematical hypotheses PH1,PH2, and SH. Let sλ\nd(H; l) and\nsλ\nd(E; n, l) denote the multiplicities of the Weyl module Vλ(G) in S(H; l)d\nand R(E; n, l)d, respectively.\nLet us associate with them the following\nstretching functions:\n ̃sλ\nd(H; l)(k) = skλ\nkd(H; l),\n(23)\nand\n ̃sλ\nd(E; n, l)(k) = skλ\nkd(E; n, l).\n(24)\nThen:\nTheorem 9.4 (cf. Theorem 3.4.11 in [GCT6])\n(Rationality Hypothesis): Assume that the singularities of the class varieties\nXP (H; m) and XNP(E; n, l) are rational.\nThen the stretching functions ̃sλ\nd(H; l)(k) and ̃sλ\nd(E; n, l)(k) are quasi-\npolynomials.\nSimilar result also holds in the context of NC vs. P #P problem.\nRationality (niceness) [Ke] of singularities here is supported by the algebro-\ngeometric results and arguments in [GCT2, GCT10].\nThe second step is the following complexity-theoretic result:\n48"},{"paragraph_id":"p50","order":50,"text":"Theorem 9.5 (cf. Theorems 3.4.11 and 3.4.13 in [GCT6]) The decision\nproblems in Hypothesis 7.2, and hence, the problem of verifying an obstruc-\ntion (Hypothesis 7.1 (a)) are indeed in P assuming the rationality hypothesis\nabove, and PH1 and PH2 (or weaker SH) in the introduction (Hypothe-\nses 1.4-1.6).\nSimilar result also holds in the context of the NC vs. P #P problem,\nassuming analogous hypotheses PH1, PH2 (or weaker SH) in this setting.\nTheorem 9.5 reduces the complexity-theoretic positive hypotheses in Hy-\npothesis 7.2 to the mathematical positivity hypotheses PH1 and SH (PH2),\nand the rationality hypothesis, unconditionally. Furthermore, [GCT6] also\ngives theoretical and experimental results in support of these positivity hy-\npotheses, and suggests a plan for proving them via the theory of quantum\ngroups. We shall discuss this plan later in Sections 15-16.\nThe whole discussion of this section is summarized in Figure 7. The top\ndouble arrow is unconditional, the bottom arrow is conjectural.\nMathematical positivity hypotheses PH1,2, and the rationality hypothesis\n∥\n∥\nGCT6\n∥\n⇓\nComplexity theoretic positivity hypotheses in Hypotheses 7.2 ∥\n∥\nTransformation in Section 13; cf. Figure 6\n∥\n?\n⇓\nP ̸= NP over C\nFigure 7: The main result of GCT6\n9.3\nSaturated and positive integer programming\nThe algorithm in Theorem 9.5 is based on a polynomial time algorithm\nin [GCT6] for a restricted form of integer programming, called saturated\n49"},{"paragraph_id":"p51","order":51,"text":"(positive) integer programming. We briefly explain it in this section.\nLet A be an m × n integer matrix, and b an integral m-vector.\nAn\ninteger programming problem asks if the polytope P : Ax ≤b contains an\ninteger point. In general, it is NP-complete. So let us begin with the well\nknown special case of integer programming which belongs to P. This is the\nunimodular integer programming problem, wherein the constraint matrix A\nis unimodular. This means the polytope P is integral. In this case, P has\nan integer point iffP is nonempty. The latter can be checked in polynomial\ntime by standard linear programming methods.\nSaturated (positive) integer programming is a generalization of unimod-\nular integer programming, wherein a variant of linear programming still\nworks, even when P is nonintegral, provided P satisfies certain saturation\nor positivity hypothesis, which make up for the loss of unimodularity.\nIt is defined as follows. Let fP(n) be the Ehrhart quasi-polynomial of\nP [St1]. An integer programming problem is called saturated if the Ehrhart\nquasi-polynomial fP(n) is guaranteed to be saturated (cf. Section 1.2), if P\nis nonempty. It is called positive if fP (n) is guaranteed to be positive (cf.\nSection 1.2), if P is nonempty. We allow m, the number of constraints, to\nbe exponential in n. Hence, we cannot assume that A and b are explicitly\nspecified. Rather, it is assumed that the polytope P is specified in the form\nof a (polynomial-time) separation oracle as in [GLS]. Given a point x ∈Rn,\nthe separation oracle tells if x ∈P, and if not, gives a hyperplane that\nseparates x from P.\nThe following is the main complexity-theoretic result in [GCT6].\nTheorem 9.6 A saturated, and hence positive, integer programming prob-\nlem has an oracle-polynomial-time algorithm.\nFurthermore, this polynomial time algorithm is conceptually extremely\nsimple. It is essentially a variant of linear programming: it uses a general-\nization of the ellipsoid method [Kh] for linear programming in [GLS], and\na polynomial time algorithm for computing Smith normal forms in [KB].\nThus the saturated and positive integer programming paradigm, in essence,\nsays that linear programming works for integer programming provided the\nsaturation or the positivity property holds.\nTheorem 9.5 follows from Theorem 9.4 because PH1 and SH (PH2) (cf.\nHypotheses 1.4-1.6) imply that the decision problems in Hypothesis 7.2 can\nbe transformed in polynomial time into saturated (positive) integer program-\n50"},{"paragraph_id":"p52","order":52,"text":"ming problems. Thus, in essence, a variant of linear programming works for\nthe decision problems in Hypothesis 7.2, provided PH1 and SH (PH2) hold.\nBut these saturation and positivity hypotheses (PH1 and SH) are non-\ntrivial, and, as we shall see in Sections 14 to 16, their validity intimately\nseems to depend on deep phenomena in algebraic geometry and the theory\nof quantum groups. We can already see an indication of this here. For ex-\nample, even to state PH1, SH or PH2, we need to show that the stretching\nfunctions used in their statements are quasi-polynomials, as shown in The-\norem 9.4. Without it, PH1, SH and PH2 are meaningless. But the proof of\nTheorem 9.4 already depends on nontrivial machinery in algebraic geome-\ntry; e.g. the cohomology vanishing result in [Ke], and the result in [Bou],\nwhich, in turn, needs resolution of singularities in characteristic zero [Hi]\nand other cohomology vanishing results. Hence it should not be surprising\nif proving these positivity hypotheses needs far more. We shall describe the\nbasic plan in [GCT6] for proving them later (Sections 15-16).\n10\nWhy should PH1 and PH2 hold?\nBut, first, we have to explain why PH1 and PH2 should hold in the first\nplace.\nThis depends, as mentioned earlier, on the exceptional nature of\nH(Y ) and E(X). Specifically, on the fact that the associated class varieties\nXP (H; l) and XNP (E; n, l) are group-theoretic. We now elaborate on this.\nFirst, let us consider the analogue of the decision problem in Hypothe-\nsis 7.2 for the simplest group-theoretic variety, namely, a flag variety (Sec-\ntion 2.2).\nGiven a flag variety Z = Gvμ ⊆P(Vμ), where Vμ is a Weyl\nmodule of G = SLl(C), the decision problem is to decide if Vλ(G) occurs\nin R(Z)d, the degree d component of the homogeneous coordinate ring of\nZ. By the Borel-Weil theorem [FH], R(Z)d = V ∗\ndμ, the dual of Vdμ. Hence,\nVλ occurs in R(Z)d iffVλ = V ∗\ndμ. It is easy to show that this is so iffthe\nYoung diagram for λ is obtained by flipping the complement of the Young\ndiagram for dμ in the smallest rectangle containing it. This can be decided\nin poly(⟨d⟩, ⟨λ⟩, ⟨μ⟩) time. The analogues of PH1 and PH2 in this setting\nclearly hold, since the multiplicity of Vλ in R(Z)d is just 0 or 1.\nNow let us move to a general group-theoretic class variety. Let (H ֒→\nG ֒→K) be the associated group triple. Since the class variety in question\nis (essentially) determined by this triple, all questions concerning the vari-\nety should, in principle, be reducible to representation-theoretic questions\nregarding this triple; cf. [GCT10], and Sections 3 and 15.\n51"},{"paragraph_id":"p53","order":53,"text":"In [GCT6] and [GCT10] analogues of the decision problems in Hypoth-\nesis 7.2 for the couples H ֒→G and G ֒→K are formulated. Furthermore,\ntheoretical and experimental evidence for PH1 and PH2 for the decision\nproblems associated with these couples is provided. Since the triples are\nqualitatively similar to the couples, though much harder, this provides the\nmain evidence in support of PH1 and PH2 for the class varieties under con-\nsideration. We shall turn to this evidence in the next section.\n11\nDecision problems in representation theory\nWe now describe the decision problems associated with the couple H ֒→G,\nthe couple G ֒→K being similar. A general decision problem is as follows:\nProblem 11.1 (The subgroup restriction problem)\nLet ρ : H →G be as above, with G connected (and some mild technical\nrestrictions on ρ as described in [GCT6]). Assume that both H and G are\nreductive. Let Vπ(H) be an irreducible representation of H, and Vλ(G) an\nirreducible representation of G, where π and λ denote the classifying labels\nof these representations.\nLet mπ\nλ be the multiplicity of Vπ(H) in Vλ(G),\nconsidered as an H-module via ρ. Given specifications of the embedding ρ\nand the labels λ, π, decide nonvanishing of the multiplicity mπ\nλ.\nThe general decision problems in Hypotheses 7.2 can be thought of as harder\nvariants of this problem obtained by going from couples to triples.\nAll\ncouples that arise in GCT are either of the type in this decision problem, or\nof a hybrid type obtained by combining this type with the type considered\nearlier in connection with the flag variety, when H = Pμ is parabolic; cf.\n[GCT10] for a discussion of the hybrid types.\nProblem 11.1 is a fundamental decision problem of representation theory.\nIndeed, one of the main motivations in the classical works of representation\ntheory, e.g. [W], for classifying of all irreducible representations of reductive\ngroups was to be able to solve this problem satisfactorily. But despite all\nprogress in representation theory in the last century, this problem at its very\nheart remained open. PHflip in [GCT6] says that this fundamental decision\nproblem of representation theory has an easy polynomial time algorithm.\nHere we shall describe PHflip in only the following three special cases\nof the above decision problem, referring the reader to [GCT6] for a full\ndiscussion and results for the general decision problem.\n52"},{"paragraph_id":"p54","order":54,"text":"11.0.1\nLittlewood-Richardson problem\nLet H = GLn(C), G = H × H, the embedding\nρ : H →H × H = G\nbeing diagonal. Then the multiplicity in Problem 11.1 is just the Littlewood-\nRichardson coefficient, because every irreducible representation of G is of\nthe form Vα ⊗Vβ, where Vα and Vβ are irreducible representations of H =\nGLn(C) for partitions α, and β, and the multiplicity of an H-module Vλ in\nVα ⊗Vβ, considered as an H-module via the diagonal map ρ, is precisely\nthe Littlewood-Richardson coefficient cλ\nα,β. We have already noted that its\nnonvanishing can be decided in polynomial time (Section 9.1).\n11.0.2\nKronecker problem\nLet H = GLn(C) × GLn(C) and\nρ : H →G = GL(Cn ⊗Cn) = GLn2(C)\nthe natural embedding given by: ρ(h1, h2) = h1 ⊗h2, for any h1, h2 ∈H.\nHere h1 ⊗h2 is the Kronecker product as defined in (18). Let kπ\nλ,μ be the\nmultiplicity of the H-module Vλ(GLn(C)) ⊗Vμ(GLn(C)) in the G-module\nVπ(G), considered as an H-module via the embedding ρ. Then it can be\nshown [FH] that the Kronecker coefficient as defined in Section 2.1.2 is a\nspecial (dual) case of this when λ, μ and π there coincide with the λ, μ and\nπ here. For this reason, we call kπ\nλ,μ a Kronecker coefficient.\nProblem 11.2 (The Kronecker problem) Given partitions λ, μ and π, de-\ncide nonvanishing of the Kronecker coefficient kπ\nλ,μ.\nThe following is an analogue of Hypothesis 7.2 in this context:\nHypothesis 11.3 [GCT6] (PHflip-kronecker) Given partitions λ, μ and π,\nnonvanishing of the Kronecker coefficient kπ\nλ,μ can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)\ntime.\n11.0.3\nThe plethysm problem\nThe Kronecker coefficient is known [Ki] to be a special case of the plethysm\ncoefficient in the following more general problem.\n53"},{"paragraph_id":"p55","order":55,"text":"Problem 11.4 (The plethysm problem) Given partitions λ, μ and π, de-\ncide nonvanishing of the plethysm constant aπ\nλ,μ. This is the multiplicity\nof the irreducible representation Vπ(H) of H = GLn(C) in the irreducible\nrepresentation Vλ(G) of G = GL(Vμ), where Vμ = Vμ(H) is an irreducible\nrepresentation H. Here Vλ(G) is considered an H-module via the represen-\ntation map\nρ : H →G = GL(Vμ).\nThe following is an analogue of Hypothesis 7.2 in this context:\nHypothesis 11.5 [GCT6] (PHflip-plethysm) Given partitions λ, μ and π,\nnonvanishing of the plethysm constant aπ\nλ,μ can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)\ntime.\n12\nThe P-barrier in representation theory\nAt the surface, this hypothesis too seems impossible because the dimension\nof G here can be exponential in the dimension of H. This happens when the\ndimension of the representation Vμ(H) is exponential in dim(H). But the\ntotal bitlength of λ, μ and π can be polynomial in dim(H). Hypothesis 11.5\nin this case says that nonvanishing of the plethysm constant can still be\ndecided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) time.\nA priori it is not even clear that the\nplethysm constant can be evaluated in PSPACE in this case. Since the usual\ncharacter-theory-based algorithms in representation theory for its evaluation\n[FH, Mc] take space that is polynomial in the dimension of G, and hence,\nexponential in the dimension of H.\nThe main P-barrier in representation theory is this huge gap between\nthe exponential space bound for the plethysm or the general decision Prob-\nlem 11.1 given by the usual methods of representation theory and the poly-\nnomial time bound stipulated in Hypothesis 11.5 for the plethysm constant\nand the hypothesis in [GCT6] for the general decision Problem 11.1.\n12.1\nCrossing the P-barrier\nWe now describe the main results of [GCT6] which together cross this P-\nbarrier in representation theory subject to the analogous mathematical pos-\nitivity hypotheses PH1 and SH (PH2). We shall only concentrate on the\nplethysm problem, since it is the crux of the matter.\n54"},{"paragraph_id":"p56","order":56,"text":"Associate with a plethysm constant aπ\nλ,μ the stretching function\n ̃aπ\nλ,μ(k) = akπ\nkλ,μ.\n(25)\nNote that μ is not stretched here.\nThen the following is an (unconditional) analogue of Theorem 9.4 in this\ncontext:\nTheorem 12.1 (cf.\nTheorem 1.6.1 in [GCT6]) The stretching function\n ̃aπ\nλ,μ(k) is a quasi-polynomial function of k.\nThe following are the analogues of PH1 and PH2 in this context:\nHypothesis 12.2 (PH1)\nFor every (λ, μ, π) there exists a polytope P = P π\nλ,μ ⊆Rm with m =\npoly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) such that:\naπ\nλ,μ = φ(P),\n(26)\nwhere φ(P) is equal to the number of integer points in P, and the Ehrhart\nquasi-polynomial of P coincides with the stretching quasi-polynomial ̃aπ\nλ,μ(k)\nin Theorem 12.1. (And some additional technical constraints)\nHypothesis 12.3 (PH2)\nThe stretching quasi-polynomial ̃aπ\nλ,μ(k) is positive (cf. Section 1.2).\nPH2 implies the following saturation hypothesis:\nHypothesis 12.4 (SH)\nThe quasi-polynomial ̃aπ\nλ,μ(k) is saturated (cf. Section 1.2).\nThe following is an analogue of Theorem 9.5 in this context:\nTheorem 12.5 [GCT6] Assuming PH1 and SH (or, more strongly, PH2),\nnonvanishing of a plethysm constant aπ\nλ,μ can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)\ntime; i.e. the problem of deciding nonvanishing of a plethysm constant be-\nlongs to P, as per Hypothesis 11.5.\n55"},{"paragraph_id":"p57","order":57,"text":"PH1 above implies that aπ\nλ,μ belongs to #P just like the Littlewood-\nRichardson coefficient. Its weaker form is:\nTheorem 12.6 The plethysm constant aπ\nλ,μ can be computed in PSPACE,\ni.e., in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) space.\nThat this holds even if the dimension of G = GL(Vμ) is exponential in\nn is crucial in the context of GCT. Because the dimension of K = GL(V )\nin the triples H ֒→G ֒→K = GL(V ) associated with the class varieties\n(Section 4) is exponential in the circuit size m. Hence, without this result,\nit is not at all clear why the structural constants sλ\nd(H, l) and sλ\nd(E; n, l) in\nHypothesis 1.4 should even belong PSPACE ⊇#P, as implied by it.\nTheorem 12.6 and Theorem 12.1 together provide good theoretical evi-\ndence for PH1 (Hypothesis 12.2). Indeed, Theorem 12.1, together with other\nevidence in [GCT6], suggests that ̃aπ\nλ,μ(k) is the Ehrhart quasi-polynomial\nof some polytope P = P π\nλ,μ. Furthermore, Theorem 12.6 says that the di-\nmension m of the ambient space Rm containing P should be polynomial in\nthe bitlengths ⟨λ⟩, ⟨μ⟩and ⟨π⟩. If not, it would not be possible to count\nthe number of integer points in P in PSPACE, since even the bitlength of\nany integer point in Rm would not be polynomial. For further theoretical\nand experimental results in support of PH1 and PH2 in this context, see\n[GCT6]. These constitute the main evidence in support of PH1 and PH2\nfor the group-theoretic class varieties (Hypotheses 1.4-1.5), because mathe-\nmatical positivity is a very abstract property, which should remain invariant\nwhen we go from couples to triples.\n13\nReduction\nNow we turn to the reduction in the top arrow in Figure 1. For this, we\nhave to describe:\n1. How to transform the easy algorithms in Theorem 9.5 into an easy\nalgorithm for discovering an obstruction as in Hypothesis 7.1 (c), and\n2. How to transform this easy algorithm for discovery into a constructive\nproof of existence of obstructions–as expected (Section 6)–for every n\nand l = nlog n, by showing how such an obstruction-label can be easily\nconstructed in this case explicitly.\n56"},{"paragraph_id":"p58","order":58,"text":"This would imply that P ̸= NP over C.\nThese transformations cannot be carried out at present, since we do\nnot know the polytopes P λ\nd (H; l) and P λ\nd (E; n, l) explicitly. We only know\nthat they should exist as per PH1 (Hypothesis 1.4). But once their explicit\ndescriptions become available, it should be possible to carry out the above\ntwo transformations along the lines that we now suggest.\n13.1\nTowards easy discovery\nFirst, let us describe why it should be possible to extend and transform\nthe polynomial-time algorithms in Theorem 9.5 to obtain a polynomial time\nalgorithm for discovering an obstruction (Hypothesis 7.1 (c)) once explicit\ndescriptions of the polytopes P λ\nd (H; l) and P λ\nd (E; n, l) become available.\nFor the sake of simplicity, let us assume that the quasi-polynomials in\nTheorem 9.4 are actually polynomials; i.e., their periods are one, though\nthis is not expected. In that case, it can be shown that [GCT6] PH1 and SH\nimply that there exist polytopes P(n, l) = P(E; n, l) and Q(l) = P(H; l) of\npoly(n, l) dimensions such that an obstruction for n, l and the pair (E, H)\nexists iffthe relative difference T(n, l) = P(n, l) \\ Q(l) is nonempty, and\nfurthermore, an explicit obstruction can also be constructed in polynomial\ntime once we are given a rational point in T(n, l). So it suffices to check\nif T(n, l) is nonempty, and if so, find a rational point in it. This can be\ndone in polynomial time using the convex (linear) programming algorithm in\n[GLS, Vd] if Q(l) has only poly(l) explicitly described facets. This is so even\nif P(n, l) has exponentially many facets. But if Q(l) has exponentially many\nfacets–as happens even in the context of the simpler Littlewood-Richardson\nproblem (Problem 9.1)–then an oracle-based algorithm as in [GLS] cannot\nbe used to get a polynomial time algorithm for this problem [B].\nBut this does not appear to be a serious problem.\nIndeed, a gen-\neral principle in combinatorial optimization, as illustrated in [GLS], is that\ncomplexity-theoretic properties of polytopes with exponentially many facets\nare similar to the ones with polynomially many facets if these facets have a\nwell-behaved regular structure. For example, if Q(l) and P(n, l) were perfect\nmatching polytopes for non-bipartite graphs–which can have exponentially\nmany facets–nonemptiness of T(n, l) can be easily decided in polynomial\ntime [Ve] using the polynomial time algorithm [Ed] for finding a perfect\nmatching in a nonbipartite graph. The facets of the analogues of P(n, l) and\nQ(l) in the Littlewood-Richardson problem, called Littlewood-Richardson\ncones [Z], have an explicit description with very nice algebro-geometric and\n57"},{"paragraph_id":"p59","order":59,"text":"representation-theoretic properties [Kl]. The same is expected to be the case\nin our setting.\nThis is why we expect that nonemptiness of T(n, l) and computation of\na rational point in it, if it is nonempty, can be done in polynomial time, once\nexplicit descriptions of P(n, l) and Q(l) become known. This would give a\npolynomial time algorithm for discovering an obstruction, if it exists, as per\nHypothesis 7.1 (c), assuming that the quasi-polynomials in Theorem 9.4 are\npolynomials.\nFurthermore, it is expected, for the mathematical reasons given in [GCT6],\nthat there exist genuinely simple, i.e., purely combinatorial greedy-type al-\ngorithms for the problems under consideration that do not even need linear\nprogramming.\nThat is, the story is expected to be the same as for the\nmin-cost flow problem in combinatorial optimization, for which a linear-\nprogramming-based polynomial-time algorithm was found first [Ta] to be\nfollowed by several genuinely simple and purely combinatorial polynomial\ntime algorithms; e.g. see [O]. Similarly, it is reasonable to expect that the\nalgorithms in Theorem 9.5 and the subsequent algorithm for discovery of\nobstructions can be simplified further to eventually get simple greedy al-\ngorithms for these problems akin to the Hungarian method, once explicit\ndescriptions of P(n, l) and Q(l) become known.\nSo far we are assuming that the quasi-polynomials in Theorem 9.4 are\npolynomials. This need not be so. In fact, this is not so even in the sim-\nplified setting of plethysm constants [GCT6]. When the quasi-polynomials\nin Theorem 9.4 have nontrivial periods, the obstructions can be classified\nin two types: geometric and modular [GCT6]. Geometric obstructions are\nsimilar to the ones that would arise if these quasi-polynomials were poly-\nnomials. A polynomial time algorithm for their existence and construction\nmay be designed along the lines we just described.\nLet us next describe briefly what needs to be done in the case of mod-\nular obstructions. Theorem 9.5 says that for the decision problems therein\nlinear programming in conjunction with modular techniques (computation\nof Smith normal forms [KB]) works even in the modular setting, i.e., when\nthe quasi-polynomials have nontrivial periods. Hence, once we have a poly-\nnomial time algorithm for discovering a geometric construction, it should\nbe possible to extend it to a polynomial time algorithm for discovering a\nmodular obstruction in conjunction with appropriate modular techniques;\ncf. [GCT6] for the problems that need to be addressed in this extension.\n58"},{"paragraph_id":"p60","order":60,"text":"13.2\nFrom easy algorithm for discovery to easy proof of ex-\nistence\nAssuming that we have an easy polynomial time algorithm for discovering an\nobstruction as per Hypothesis 7.1 (c), let us now describe why it should be\npossible to prove using this algorithm, or rather the underlying structure and\ntechniques, that there always exists an obstruction, as expected (Section 6),\nfor every n →∞, assuming l = nlog n (say).\nFor the sake of simplicity, let us again assume that that the quasi-\npolynomials in Theorem 9.4 are polynomials, and that we have an easy\nHungarian-type greedy algorithm as discussed above for deciding nonempti-\nness of T(n, l), and for computing a point it it, if it is nonempty. Then we\nhave to show, using the techniques and the structure underlying this algo-\nrithm, that T(n, l) is always nonempty when l = nlog n, n →∞. Such a proof\nwould also give us a polynomial time procedure for explicit construction of\nan obstruction λ(n), for every n. Hence we shall call it a P-constructive\nproof.\nTo see how to get such a P-constructive proof, let us consider an analogy.\nLet us imagine that Q(l) is empty, so that T(n, l) = P(n, l) is a polytope,\nand that it is the perfect-matching polytope of a bipartite graph G(n, l).\nThen T(n, l) is nonempty iffG(n, l) has a perfect matching, which can be\nthought of as an obstruction in this analogy. The analogous goal then is to\nshow using the techniques and structure underlying the Hungarian method\nthat G(n, l) always has a perfect matching, as expected, when l = nlog n,\nand n →∞. In other words, we have to give a P-constructive proof for\nexistence of a perfect matching in every such G(n, l). In this analogy, the\ntechnique underneath the Hungarian method can be easily used to give a\nconstructive proof of Hall’s marriage theorem–namely, that every bipartite\ngraph H in which every subset, on any side of the graph, has at least as\nmany neighbours as the size of that subset has a perfect matching–which\nthen has to be used to show that G(n, l) always has a perfect matching\nwhenever l = nlog n, n →∞.\nNow in our setting T(n, l) is not a perfect matching polytope. But if\nit has a nice structure like the perfect matching polytope then it should be\npossible to prove structure theorems in the spirit of Hall’s marriage theorem\nfor T(n, l) using the structure of the Hungarian-type greedy algorithm for\ndeciding nonemptyness of T(n, l) and then use it to prove nonemptyness of\nT(n, l), for every n →∞, when l = nlog n.\nFor such a transformation of a polynomial time algorithm for discovery\n59"},{"paragraph_id":"p61","order":61,"text":"into a P-constructive proof of existence to work, it is crucial that:\n1. The polyhedral set T(n, l) has a nice, regular structure like the perfect\nmatching polytope.\nFortunately, the polytopes P(n, l) and Q(n, l)\nthat would arise in our setting should be even nicer than the per-\nfect matching polytope. For example, in the simpler setting of the\nLittlewood-Richardson problem (Problem 9.1), P(n, l) and Q(l) be-\ncome Littlewood-Richardson cones [Z], which have extremely regu-\nlar structure with remarkable representation-theoretic and algebro-\ngeometric properties [F2, Kl]. The same is expected to be the case\nfor the actual P(n, l) and Q(l).\n2. The algorithm for discovery not only works in polynomial time, but\nalso has a simple structure like the Hungarian method. The Hungarian-\ntype greedy algorithms that we expect for the problems under consid-\neration should have such structures.\nHence, it is reasonable to expect that an easy Hungarian-type algorithm\nfor deciding nonemptyness of T(n, l) can be transformed into the sought P-\nconstructive proof of obstructions. The story is expected to be similar, albeit\nmuch harder, when the quasi-polynomials in Theorem 9.4 have nontrivial\nperiods; cf [GCT6].\nThe above scheme for the transformation of an algorithm for discovery\ninto a constructive proof of existence banks on the fact that the algorithm\nto be transformed is easy, i.e., works in polynomial time, besides having\na simple structure.\nThe underlying informal principle, which cannot be\nproved, is that the mathematical complexity of an algorithmic (constructive)\nproof is intimately linked to the computational complexity of the algorithm\non which it is based; see Section 19 for a detailed treatment of this issue.\nThis is why there is no nontrivial result, comparable to Hall’s theorem, for\nHamiltonian paths.\nBecause the problem of finding such a path is NP-\ncomplete.\nThe reader may wonder why we are talking about explicit construction of\nobstructions, when, strictly speaking, we only need to know their existence.\nThis is because the nature of obstructions in our case is such that their\nexplicit construction, if they exist, can be done with only a little additional\ncost over the cost of deciding existence. To see this, let us again assume,\nfor the sake of simplicity, that the quasi-polynomials in Theorem 9.5 are\npolynomials.\nThen a technique that can decide nonemptiness of T(n, l)\nshould also be able to compute a point in it, as a proof of nonemptiness, at\n60"},{"paragraph_id":"p62","order":62,"text":"only a little additional cost, just as in linear programming. In other words,\nthe complexity of deciding existence of an obstruction should be more or\nless the same as that of constructing it, if it exists. This is why we mainly\ntalk of explicit construction of obstructions though, in principle, just their\nexistence would suffice.\nOur discussion so far says that PH1 and SH (PH2) are the crux of the\nmatter. If they can be proved, and explicit descriptions of the polytopes\ntherein become available, it should be possible to transform the easy algo-\nrithms in Theorem 9.5 into an easy algorithm for explicit construction of\nobstructions as per Hypothesis 7.1 (b).\n14\nStandard quantum group\nNow we proceed to the basic plan in [GCT6] for proving PH1 and SH. This\nis motivated by a story in the theory of standard quantum groups in the\ncontext of the Littlewood-Richardson problem (Problem 9.1). We describe\nthat story in this section.\nFor this we need the notion of a standard quantum group, by which we\nmean the quantum group in [Dri, Ji, RTF]. We can not formally define here\nthis object, but we can at least give an intuitive idea. Let GL(Cn) be the\ngroup of nonsingular n × n matrices. It can be thought of as the group of\nnonsingular transformations of Cn. Let xi’s denote the coordinates of Cn.\nThese commute. That is:\nxixj = xjxi.\nLet us now see what happens if the coordinates become noncommuting. This\nis precisely what happened in quantum physics. We discovered that the po-\nsition and the momentum, which for centuries we thought were commuting\nobservables, do not actually commute. Quantum groups were invented pre-\ncisely to investigate the related phenomena in theoretical physics. Let Cn\nq\ndenote the quantum space whose coordinates xi’s are noncommuting, and\nsatisfy the following relation:\nxixj = qxjxi,\ni < j\nwhere q ∈C is some fixed number. The standard quantum group GLq(Cn) is\nthe “group” of invertible linear transformations of this quantum space. This\nis not a “group” in any ordinary sense. Its precise description is given in [Dri,\n61"},{"paragraph_id":"p63","order":63,"text":"Ji, RTF]. We do not need that here. Let us just think of a quantum group\nas what a group becomes when the coordinates become noncommuting.\nLet us now explain how quantum groups enter in the story of Littlewood-\nRichardson coefficients.\nThis is because the most transparent proof of\nthe Littlewood-Richardson rule came via the theory of quantum groups\n[Kas1, Li, Lu2]. The earlier proofs, though elementary and combinatorial,\nwere highly mysterious. Moreover, the theory of quantum groups gave the\nfirst proof of the generalized Littlewood-Richardson rule [] for general (con-\nnected) reductive groups, instead of just GLn(C).\nLet us now elaborate the nature of this proof. We begin by observing\nthat the Littlewood-Richardson problem (Problem 9.1) is an instance of the\ngeneral decision Problem 11.1 associated with the diagonal group homomor-\nphism\nρ : H = GL(Cn) →H × H = GL(Cn) × GL(Cn).\n(27)\nIf we understood the structure of this homomorphism in depth, we ought\nto understand why PH1 and SH (and also PH2) hold for the Littlewood-\nRichardson coefficients.\nAs we mentioned earlier, in depth means at the\nquantum level.\nTo understand the homomorphism (27) at the quantum\nlevel, we need to quantize it. Ideally, one would want its quantization in the\nform of a homomorphism\nρq : Hq = GLq(Cn) →Hq × Hq = GLq(Cn) × GLq(Cn).\n(28)\nwhere Hq is the standard quantum group associated with H. This does not\nhold as it is; i.e., Hq is not a quantum subgroup of Hq × Hq. But this is\nessentially so. That is, it holds in a certain dual setting–this is the main\nresult in [Dri, Ji, RTF]. Thus the theory of quantum group can be regarded\nas the theory of the quantization ρq.\nOnce this theory is developed sufficiently, the Littlewood-Richardson rule\nas well as PH1 for Littlewood-Richardson coefficients (Hypothesis 1.9) turn\nout to be a consequence, in a nontrivial way, of a deep positivity result in\nthe theory of the standard quantum groups [Kas2, Kas3, Lu1, Lu2]: namely,\ntheir representations and coordinate rings have canonical bases, also called\nglobal crystal bases, whose structural constants, which determine their mul-\ntiplicative and representation theoretic structure, are all nonnegative. For\nthis reason, we say that the canonical bases are positive, and refer to exis-\ntence of a canonical basis as a positivity property (hypothesis) PH0.\nWe now give a brief intuitive description of the canonical basis. Let X\nbe an n×n variable matrix. The coordinate algebra R = O(G) of the group\n62"},{"paragraph_id":"p64","order":64,"text":"G = GL(Cn) is defined to be the C-algebra generated by the entries xij of X\nand det(X)−1, where det(X) denotes the determinant of X. Its elements are\nregular functions on G, considered as an affine variety. There is a natural\nleft action of G on R given by f(X) →f(σ−1X), for any σ ∈G, and a\nsimilar right action.\nThese notions can now be quantized. It is possible to associate a coor-\ndinate ring Rq = O(Gq) with the standard quantum group Gq = GLq(Cn),\nwhose elements can be intuitively thought of as functions on Gq. Unlike R,\nRq is not commutative. Its precise definition can be found in [RTF]. There\nare natural left and right actions of Gq on Rq.\nA canonical basis B of Rq is a very special basis with the following\nproperties:\n(1) It is representation-theoretically well behaved. This means there is a\nfiltration\nB0 = ∅⊂B1 ⊂B2 ⊂· · · ⊂B\nwith ∪iBi = B, such that ⟨Bi⟩/⟨Bi−1⟩is an irreducible Gq-module. Here\n⟨Bi⟩denotes the span of the basis elements in Bi.\n(2) Positivity property of the multiplicative structure constants:\nGiven two elements b, b′ ∈B, let\nbb′ =\nX\nb′′∈B\nf b′′\nb,b′b′′,\nbe the expansion of the product in terms of the basis B. Then each f b′′\nb,b′ is\nan explicit polynomial in q and q−1 with nonnegative coefficients. Here f b′′\nb,b′\nare called multiplicative structure constants. What this says is that each\nmultiplicative structure constant has an explicit positive formula, akin to\nthat of the permanent. Here explicit means that each nonnegative coefficient\nof f b′′\nb,b′ has an interpretation in terms of a nonnegative topological invariant\n(akin to Betti numbers) of an algebraic variety.\n(3) Positivity property of the representation-structure constants:\nGiven any element b ∈B and a generator e of a certain algebra defined\nin [Dri, Ji], which is “dual” to Rq, let\ne · b =\nX\nb′\ngb′\ne,bb′\nbe the expansion of e · b, the result of applying e to b, in terms of the\nbasis B. Then each gb′\ne,b is also an explicit polynomial in q and q−1 with\n63"},{"paragraph_id":"p65","order":65,"text":"nonnegative coefficients.\nThat is, each representation-structure constant\nalso has an explicit positive formula.\nThese positivity properties do not actually hold as stated–that is still a\nconjecture [Lu2]–but their slightly weaker form holds unconditionally [Lu2].\nWe shall ignore that difference here.\nIf we specialize the canonical basis at q = 1, we get a canonical basis of\nR, the coordinate ring of G, with analogous positivity property. But, as of\nnow, the only way to prove existence of such a canonical basis of R is via the\ntheory of quantum groups as above. This shows the power of this theory.\nOne can easily imagine that there ought be a connection between ex-\nistence of bases whose structural constants have explicit positive formu-\nlae (PH0) and existence of an explicit positive (polyhedral) formula for\nLittlewood-Richardson coefficients (PH1). That is indeed so, as we men-\ntioned earlier, but in a quite nontrivial way; cf. [Kas1, Li, Lu2]. We shall\nsimply take this connection on faith here. Pictorially:\nPH0 →PH1.\n(29)\nOne does not really need the full power of PH0 to deduce PH1. Just\nexistence of a local crystal basis [Kas1], which is the limit (crystalization)\nof a canonical basis as q →0, is sufficient.\nBut when we move to the\nnonstandard setting in GCT, even the full power of PH0 is needed for some\nother reasons; [GCT8, GCT10].\nThe implication (29) provides arguably the most satisfactory proof of\nPH1 for Littlewood-Richardson coefficients, which, in addition, also pro-\nvides deep additional information (existence of canonical bases) which the\ncombinatorial proofs [F1] cannot provide. Such canonical bases are central\nto the approach in [GCT6, GCT10] towards PH1 and PH2 for the group-\ntheoretic class varieties (Section 15). Hence, as far as GCT is concerned,\nquantum groups are a must.\nSH for the usual Littlewood-Richardson coefficients is the saturation\ntheorem in [KT1]. It comes from a reformulation of PH1 in terms of special\npolytopes (called Hive polytopes) and their subsequent detailed study. Thus\npictorially:\nPH1 →SH,\n(30)\nagain in a nontrivial way.\n64"},{"paragraph_id":"p66","order":66,"text":"But how is PH0 proved?\nThe only known proof of PH0 [Lu1, Lu2]\nis based on a deep positivity property in mathematics: the Riemann Hy-\npothesis over finite fields [Dl2], and related results [BBD]. In other words,\nnonnegativity of the structural constants associated with Hq is connected at\na profound level with the lining up of the zeros of the zeta functions of some\nalgebraic varieties on one axis. We shall denote the Riemann hypothesis\nover finite fields by PH+. Then pictorially:\nPH+ →PH0,\n(31)\nin a highly nontrivial way.\nPutting implications (29)-(31) together with the story in Section 9.1, we\narrive at Figure 8 which summarizes the story in this section.\nPH+: The Riemann hypothesis over finite fields and related results [BBD, Dl2]\n∥\n∥\n⇓\nPH0: Existence of canonical bases [Lu1, Lu2]\n∥\n∥\n⇓\nPH1 and SH [Kas1, Li, Lu2, KT1]\n∥\n∥\n∥\n⇓\nPolynomial time algorithm for deciding nonvanishing of\nLittlewood-Richardson coefficients [DM2, GCT5, KT1]\nFigure 8: A story in the theory of standard quantum groups\n15\nNonstandard quantum groups\nNow we turn to the problem of proving PH1 and SH that actually arise in\nGCT (Hypotheses 1.4-1.6, and 12.2-12.4). The basic plan in [GCT6] for this\n65"},{"paragraph_id":"p67","order":67,"text":"is simply to lift the story in Figure 8 from height two to superpolynomial\nheight–i.e., from the circuits of height two that the Littlewood-Richardson\nproblem corresponds to to the circuits of superpolynomial height that the\ndecision problems in Hypothesis 7.2 correspond to.\nRoughly, it goes as\nfollows:\n(1) Quantization: Quantize the couples\nH ֒→G,\nG ֒→K\nand the triples\nH ֒→G ֒→K,\nassociated with the class varieties in a manner akin to the quantization (28)\nof (27) via standard quantum groups.\n(2) PH0 for couples and triples: Prove that the coordinate rings and\nrepresentations of the quantum groups that arise in this quantization have\ncanonical bases akin to the canonical bases for the standard quantum groups\nwhose structure constants, which determine their multiplicative and repre-\nsentation theoretic structure, are all nonnegative.\n(3) PH0 for class varieties: Use the canonical bases for the (quantized)\ntriples associated with the class varieties to construct analogous canonical\nbases for the coordinate rings for appropriate quantizations of the class\nvarieties with nonnegative structure constants.\n(4) PH1, SH: Deduce PH1 and SH from PH0 in the spirit of the middle\narrow in Figure 8.\nFigure 9 shows this pictorially.\nWe shall now elaborate Figure 9.\n15.1\nQuantization\nLet us begin with the first step of quantization. We shall only worry about\nthe couples. To be concrete, let\nH ֒→G = GL(Ck),\n(32)\nbe as in Problem 11.1, where H is connected, reductive subgroup of G.\nQuantization of this couple is the crux of the problem.\nAll other quan-\ntizations that are needed are hyped up versions of this, so we shall only\nconcentrate on this.\n66"},{"paragraph_id":"p68","order":68,"text":"Quantization of couples [GCT4,\nGCT7] and triples [GCT10]\n|\n|\n|\n↓\nPH0\nfor\ncouples\nand\ntriples\n[GCT6, GCT8, GCT10]\n−−−→\nPH1 and SH for cou-\nples and triples\n|\n|\n↓\nPH0 for class varieties [GCT10]\n|\n|\n|\n↓\nPH1 and SH for class varieties\nFigure 9: The basic plan for proving PH1 and SH in [GCT6]\nThe standard theory of quantum groups can not be used for quantizing\nthis couple, as expected. Specifically, let Gq = GLq(Cn) be the standard\nquantum group associated with G. In a similar fashion, one can associate\n[Dri, Ji, RTF] a standard quantum group Hq with H. Then, Hq cannot be\nembedded as a quantum subgroup of Gq (where the notion of subgroup in\nthe quantum setting is akin to the usual notion of a subgroup). Hence the\ngoal is to associate a quantization ˆGq with G akin to the standard quantum\ngroup Gq so that the standard quantum group Hq is a quantum subgroup\nof ˆGq. In that case:\nHq ֒→ˆGq,\n(33)\ncan be considered to be a quantization of (32).\nThis quantization step is addressed in the following result for the couples\nin Problems 11.2-11.4, which are the main prototypes of the couples that\narise in GCT.\n67"},{"paragraph_id":"p69","order":69,"text":"Theorem 15.1 (1) (cf. [GCT4]) The couple\nH = GL(Cn) × GL(Cn) →GL(Cn ⊗Cn) = G,\nassociated with the Kronecker problem (Problem 11.2) can be quantized in\nthe form:\nHq →ˆGq,\nwhere Hq is the standard quantum group associated with H and ˆGq is the\nnew nonstandard quantum group associated with G. Furthermore, ˆGq has a\nquantum unitary subgroup ˆUq in the sense of [Wo], which is a quantization\nof the unitary subgroup U = Un2(C) ⊆G = GLn2(C).\n(2) (cf. [GCT7]) More generally, the couple\nH = GLn(C) →G = GL(Vμ(H)),\n(34)\nassociated with the plethysm problem (Problem 11.4) can also be quantized\nin the form:\nHq →ˆGq,\nwhere Hq is the standard quantum group associated with H and ˆGq is the\nnew nonstandard (possibly singular) quantum group associated with G. Here\nH can even be any connected classical reductive group.\nThe nonstandard quantum group in [GCT4] is qualitatively similar to\nthe standard quantum group in [Dri, Ji, RTF] in the sense that it has a max-\nimal quantum unitary subgroup just as in the the standard case. This, in\nconjunction with work in [Wo], allows the mathematical machinery related\nto unitariness–such as harmonic analysis, existence of orthonormal bases–to\nbe transported to its theory. This is important in the context of PH0. In-\ndeed, PH0 in the theory of standard quantum groups is intimately related\nto existence of unitary quantum subgroups. Because the local crystal bases\nfor representations of the standard quantum group [Kas1], which were later\nglobalized to canonical (global crystal) bases in [Kas2], arose in the study of\nspecial orthonormal Gelfand-Tsetlin bases for representations of the stan-\ndard quantum group. This is first main reason why PH0 is expected to hold\nfor the nonstandard quantum group in [GCT4].\nThe general nonstandard quantum group [GCT7] can be singular, i.e.,\nits quantum determinant can vanish. Hence, we cannot define its quantum\nunitary subgroup in the sense of [Wo].\nFortunately, this is not matter,\nbecause analogues of the main required results in [Wo] still hold; cf. [GCT7]\n68"},{"paragraph_id":"p70","order":70,"text":"for a precise statement.\nHence PH0 is expected to hold for the general\nnonstandard quantum groups in [GCT7] as well.\nBut at the same time these nonstandard quantum groups are fundamen-\ntally different from the standard quantum groups. Hence the terminology\nnonstandard. For a detailed description of the differences between the stan-\ndard and nonstandard quantum groups, see [GCT4, GCT7, GCT8]. Here we\nonly give a brief description from the complexity-theoretic perspective. To-\nwards this end, we associate a complexity level with each of these quantum\ngroups. This is briefly done as follows.\nSuppose H ֒→G is a primary couple associated with a group-theoretic\nclass variety for some complexity class C (Definition 3.1). Then the com-\nplexity class of this primary couple as well as its quantization, if it exists, is\ndefined to be just C.\nAs we have already noted, the theory of the standard quantum group is\nthe theory of quantization of the couple (cf. (27))\nGLn(C) →GLn(C) × GLn(C).\nThis is a primary couple associated with orbit-closure of the trace of an\nn×n matrix (Section 9.1), which can be computed by a circuit of depth two\nusing only additions or multiplications by constants. Hence, the standard\nquantum group corresponds to the complexity class of problems that can\nbe solved by circuits of depth two using only additions or multiplications\nby constants, just like expanders (Section 17).\nThere is no lower bound\nproblem here to speak of. That is why the standard quantum group cannot\nbe used for deriving any lower bound, again like expanders.\nThe couple associated with the Kronecker problem coincides with the\nprimary couple\nGL(Cm) × GL(Cm) →GL(Cm ⊗Cm).\n(35)\nassociated with the NC-class variety; cf. (17). Not exactly. The primary\ncouple associated with the NC-variety is slightly different from this, but\nthe difference is trivial, and can be ignored.\nTheory of the nonstandard\nquantum group in Theorem 15.1 (a) is the theory of quantization of this\ncouple. Hence, the complexity class of this nonstandard quantum group can\nbe defined to be NC.\nThe couple (34) that is quantized in [GCT7] is not a primary couple\nof any class variety. But it is qualitatively similar to the primary couple\n69"},{"paragraph_id":"p71","order":71,"text":"associated with the NP-class-variety (Section 4.2).\nFor this reason, the\nnonstandard quantum group in [GCT7] can be roughly taken to be of su-\nperpolynomial complexity.\n15.2\nPH0 for couples and triples\nThe article [GCT8] gives a conjecturally correct algorithm to construct\ncanonical bases of the coordinate rings of the nonstandard quantum groups\nin [GCT4, GCT7]. These are natural generalizations of the canonical ba-\nsis in [Kas3, Lu2] for the coordinate ring of the standard quantum group.\nFurther theoretical and experimental evidence in support of PH0 for the\nnonstandard quantum group in [GCT4] is also given. For the problems that\nhave to be addressed in the context of the triples associated with the class\nvarieties under consideration, see [GCT10].\n15.3\nPH0 for class varieties\nSince the group-theoretic class varieties are essentially determined by the\nassociated group triples, once PH0 is proved for the triples, it should, in\npriciple, be possible to “transport” this knowledge from group theory to\nalgebraic geometry, thereby proving PH0 for the class varieties. In [GCT10]\nis basic plan for this “transport” is suggested, with a description of the\nvarious mathematical problems that need to be resolved.\nA crucial bridge between group theory and algebraic geometry for this\ntransport is provided by Conjecture 6.1, which has to be proved first. It may\nbe remarked that quantum groups were indeed brought into GCT precisely\nfor the purpose of proving this conjecture, thereby extending the proof in\n[GCT2] for its weaker form (Theorem 6.3). A basic plan for this extension\nvia nonstandard quantum groups is also suggested in [GCT10].\n15.4\nPH1 and SH\nThe journey from PH0 to PH1 in the nonstandard setting should be akin to\nthe one in the standard setting; cf. [GCT6].\nIn summary, the nonstandard quantum groups have to be used as a\nrope, as it were, to pull the proofs of the various mathematical positivity\nhypotheses from the constant depth (of the standard quantum groups) to\nsuperpolynomial depth.\n70"},{"paragraph_id":"p72","order":72,"text":"16\nUltimate mystery: nonstandard Riemann hy-\npotheses?\nNow we come to the final chapter of this story: How to prove PH0, and\nspecifically, correctness of the algorithm in [GCT8] for constructing canon-\nical bases of the coordinate rings of the nonstandard quantum groups in\n[GCT4, GCT7] and their required conjectural properties.\nFor the standard quantum group, as we mentioned in Section 1, the\ntopological proof in [Lu1, Lu2] depends on the Riemann hypothesis over\nfinite fields [Dl2] and the related work [BBD]. The main open problem at\nthe heart of GCT is to extend this work, and use it to prove nonstandard\nPH0. But the standard Riemann hypothesis over finite fields is not expected\nto work in the nonstandard setting; cf. [GCT8]. Briefly this is because the\nrelevant quantized noncommutative algebraic varieties in the nonstandard\nsetting simply “disappear” when specialized at q = 1. Specifically, unlike in\nthe standard case, the Hilbert function of these varieties at q ̸= 1 is different\nfrom the Hilbert function of the corresponding classical varieties at q = 1.\nHence, they look very different from the classical algebraic varieties. This\nis why the Riemann hypothesis over finite fields may not be used as in the\nstandard case for proving PH0.\nThus we seem to need nonstandard extensions of the Riemann hypothesis\nover finite fields in the quantized noncommutative setting to prove the PH0’s\nunder consideration. We cannot even formulate such extensions. But we\nbelieve such nonstandard extensions exist. We now briefly explain why.\nFor this, we need to indicate the nature of the experimental evidence\n[GCT8] in support of PH0 for the most basic nonstandard quantum group\nfor the Kronecker problem in [GCT4]. Specifically, around a thousand struc-\ntural constants associated with a canonical basis for a certain dual of this\nquantum group were computed, each structural constant being a polynomial\nin q of degree more than ten. All the coefficients of these structural polyno-\nmials turned out be nonnegative. In the standard case, the cause for such\nnonnegativity was the Riemann hypothesis over finite fields. There ought\nto be a similar theoretical cause for nonnegativity in the nonstandard set-\nting. For, without a cause, the probability of over ten thousand coefficients\nbeing nonnegative would be absurdly small–naively 1/210000. This estimate,\nbeing naive, should not be taken literally. But it does suggest that the ex-\nperimental evidence for positivity should only be a shadow of the ultimate\ncause–nonstandard analogues of the Riemann hypotheses over finite fields.\n71"},{"paragraph_id":"p73","order":73,"text":"This leads us to believe that nonstandard extensions of the Riemann\nhypothesis over finite fields for the various nonstandard quantum groups\nthat arise in GCT exist, and now, having seen the shadow, we have to search\nfor the ultimate cause whose shadow it is. If this search succeeds, then we\ncan expect to pull the proofs of PH0 from the standard to the nonstandard\nsetting, using the rope provided by the nonstandard quantum groups, and\nthe power provided by nonstandard Riemann hypotheses, thereby leading\nto the proof of P ̸= NP conjecture in characteristic zero; cf. Figure 3.\nEventually, this whole story in characteristic zero, along with the non-\nstandard Riemann hypotheses and the accompanying positivity hypotheses,\nmay be lifted, as suggested in [GCT11], to algebraically closed fields of pos-\nitive characteristic, and finally, finite fields, thereby proving the P ̸= NP\nconjecture in its usual form. This would then constitute the ultimate flip in\nFigure 2.\n17\nObstructions vs. expanders\nWe now explain the relationship between explicit construction of obstruc-\ntions and explicit construction of expanders as shown in Figure 4.\nAs per the hardness-vs-randomness principle [A2, IW, KI, NW], deran-\ndomization is intimately linked to lower bound problems.\nIn particular,\nrestricted kinds of lower bounds follow from existence of efficient pseudo-\nrandom generators. At present, we do not have pseudo-random generators\nbased on expander-like structures that can yield a lower bound result for\nconstant depth circuits. But for the sake of discussion, let us imagine that\nthe expander in, say, [LPS, Ma] can be generalized further to obtain a hy-\npothetical structure, which we shall call a strong expander, using which we\ncan obtain an efficient pseudo-random generator, whose existence, in turn,\nimplies separation of the class NC1 from AC0. Here NC1 is the class of\nproblems that can be solved by circuits of logarithmic depth, and AC0 the\nclass of problems that can be solved by circuits of constant depth. Fur-\nthermore, let us also assume that the problem of constructing such a strong\nexpander belongs to (nonuniform, algebraic) AC0, as it does for the ex-\npander in [LPS, Ma]. Now the existence of such a family {En} of strong\nexpanders would imply that an explicit function in NC1, depending on the\npseudo-random generator, cannnot be computed by a circuit of constant\ndepth. Hence such strong expanders can be regarded as obstructions, i.e.\nproofs of hardness, for computation of an explicit NC1-function by constant-\n72"},{"paragraph_id":"p74","order":74,"text":"depth circuits. In this sense GCT obstructions are to superpolynomial-depth\ncircuits are what strong expanders are to constant-depth circuits. This is\npictorially depicted in Figure 10.\n↑\ndepth\n|\nSuperpolynomial depth circuit: Obstruction in GCT\n↑\n|\nConstant depth circuit: Strong expander\n↑\n|\nDepth two circuit: expander\nFigure 10: The relationship between obstructions and expanders\nThe expander in [LPS, Ma] can actually be constructed by a nonuniform\nalgebraic circuit of depth two (a basic ring operation is taken as unit cost).\nHence, it can be expected to serve as an obstruction for computation by\na circuit of depth at most two–really, just one.\nBecause the depth of a\ncircuit for computing an explicit structure whose existence separates NC\nfrom (nonuniform) ACk, the class of circuits of depth k, should be at least\nk–really higher than k. So an expander, as against the hypothetical strong\nexpander, actually belongs to depth-two circuits. But there is no nontrivial\nlower bound problem for circuits of depth one. This is why the expanders\nthat we have at present cannot be used in lower bound problems.\nNow let us compare explicit construction of expanders with the suggested\nmethod for explicit construction of obstructions in Figure 3.\nFirst, let us observe that, though the explicit construction of expanders\n[LPS, Ma] is “extremely easy” (nonuniform AC0), its correctness is based\non a nontrivial mathematical positivity hypothesis:\nPHspectral: The spectral gap of an expander is bounded below by a pos-\nitive constant.\nThe mathematical positivity hypotheses PH1 and PH2 (Hypothesis 1.4-\n1.6) can be regarded as nonspectral analogues of PHspectral in the setting\nof superpolynomial depth circuits.\n73"},{"paragraph_id":"p75","order":75,"text":"Second, the proof of PHspectral in [LPS] for expanders depends on the\nRiemann hypothesis over finite fields (for curves) [Dl2]. It should not be a\nsurprise then that what is needed to prove the positivity hypotheses PH1, SH\n(PH2) mentioned above is, in essence, an extension of the Riemann hypoth-\nesis over finite fields and the results surrounding it. But given the big gap\nbetween constant depth and superpolynomial depth circuits it would have\nbeen a great surprise if the existing standard Riemann Hypothesis over fi-\nnite field were to suffice. Instead, what seems to be needed are nonstandard\nextensions of the Riemann hypothesis over finite fields, and the related re-\nsults; cf. Section 16. In the case of expanders, the Riemann hypothesis over\nfinite fields is not indispensible, since there are alternative constructions of\nexpanders with proofs of correctness based on linear algebra [RVW]. But,\nagain given a big gap between constant depth and superpolynomial depth, it\nshould not be surprising if nonstandard extensions of the Riemann hypoth-\nesis turn out to be indispensible in the context of the P vs. NP problem.\n18\nOn relativization and P/poly-naturalization bar-\nriers\nIn this section we point out why the flip should be nonrelativizable and\nnon-P/poly-naturalizable.\nWe already mentioned one reason for why the flip should be nonrela-\ntivizable: namely, the “reduction” from hard nonexistence to easy existence\nis not a formal Turing machine reduction. There is also another reason.\nFor this, let us examine why the proof of IP = PSPACE result [Sh] does\nnot seem relativizable. Mainly because it is based on the construction of an\nexplicit low-degree polynomial. This seems already enough to make it non-\nrelativizable, though the proof technique is not fully explicit. (Because it\nmakes use of estimates on the number of roots of a low degree polynomial.\nAny technique based on counting or estimates is, by definition, not fully\nexplicit). In contrast, the flip is to be implemented using explicit algebro-\ngeometric and representation-theoretic constructions. This is why it should\nbe nonrelativizable.\nNow we turn to the P/poly-naturalizability barrier [RR].\nIntuitively,\nthis too should be crossed simply because everything is to be done ex-\nplicitly and constructively.\nRecall that explicit construction of obstruc-\ntions is for superpolynomial depth circuits what explicit construction of\nexpanders is for depth-two circuits (Section 17).\nThe usual probabilistic\n74"},{"paragraph_id":"p76","order":76,"text":"(nonconstructive) proof for existence of expanders may be considered to\nbe P/poly-naturalizable–as the probabilistic proofs [BS] of lower bounds\nfor constant depth circuits–whereas the proof via explicit construction in\n[LPS, Ma, RVW] may be considered non-P/poly-naturalizable. This is only\nan analogy. Strictly speaking, there is no notion of P/poly-naturalization\nfor constant depth circuits. Rather, this barrier lies between the circuits of\nconstant depth to which the expanders correspond and the circuits of super-\npolynomial depth to which the obstructions correspond. But this analogy\nshould intuitively explain why the flip should cross this barrier.\nNow we turn to a more formal argument. We begin by recalling the no-\ntion of a P/poly-naturalizable proof [RR]. We use the formal term P/poly-\nnaturalizable proof instead of the informal term natural proof, because oth-\nerwise GCT, and hence, the algebro-geometric and quantum-group-theoretic\ntechniques that enter into it would have to be called unnatural. That may\nseem paradoxical, especially since quantum groups arose in the study of\nnatural phenomena in theoretical physics.\nLet Fn be the set of n-variable boolean functions.\nBy a property of\nboolean functions, we mean a family of subsets Cn ⊆Fn for every n. It is\ncalled useful if the circuit size of any function h(X) = h(x1, . . . , xn) ∈Cn\nis super-polynomial. It is called P/poly-natural if it contains a subset C∗\nn\nsatisfying the following two constraints:\nConstructivity: Whether a given h(X) belongs to C∗\nn can be decided in\ntime polynomial in the size N = 2n of the truth table of h(X).\nLargeness:\n|C∗\nn|/|Fn| ≥1/N k,\n(36)\nfor some fixed k.\nA proof technique based on a useful P/poly-natural property is called\nP/poly-naturalizable. The article [RR] says that the P ̸= NP conjecture\nwould not have a P/poly-naturalizable proof under reasonable assumptions.\nNext, we translate this notion to the setting wherein the base field K of\ncomputation is algebraically closed, as in this article. We assume that K = C\nor K = ̄Fp, the algebraic closure of a finite field Fp. Let Fn be the set of n-\nvariable polynomials of degree d(n) for some fixed function d(n) = 2poly(n).\nIf K = C, we assume that each polynomial in Fn is an integral polynomial\nwhose coefficients have poly(n) bitlength. If K = ̄Fp, we assume that all\ncoefficients belong to Fp, and that the bitlength ⟨p⟩= poly(n).\nLet N\ndenote the total number of coefficients of h(X). The total bitlength of the\n75"},{"paragraph_id":"p77","order":77,"text":"specification all coefficients of h(X) is N, ignoring a poly(n) factor. Hence\nwe let it play the role of the truth-table-size in what follows. This leads\nto the following straightforward generalization of the notion of a P/poly-\nnaturalizable proof over C or ̄Fp.\nBy a property, we now mean a subset Cn ⊆Fn, for each n.\nIt is\ncalled useful if the circuit size over K of any function h(X) ∈Cn is super-\npolynomial. It is called P/poly-natural if it contains a subset C∗\nn satisfying\nthe following two constraints:\nConstructivity: Whether a given h(X) belongs to C∗\nn can be decided in\npoly(N) time, where each operation over K is considered to be of unit cost.\nLargeness:\n|C∗\nn|/|Fn| ≥1/N k,\n(37)\nfor some fixed k.\nA proof technique based on a useful P/poly-natural property is called\nP/poly-naturalizable. The results in [RR] are proved only over a finite field.\nBut the constructivity and largeness constraints over algebraically closed\nfields here are natural extensions of the ones over finite field. Hence, we\nshall assume in what follows that they are meaningful even over algebraically\nclosed fields. It would be interesting to know if the techniques in [RR] can\nbe lifted in some form to such fields.\nIn the context of the flip, we next formulate a property which is conjec-\nturally useful and which should violate both the largeness and the construc-\ntivity constraints. This should be enough to cross the P/poly-naturalizability\nbarrier.\nLet us follow the notation as in Section 4. Let K = C. Let h(X) ∈P(W)\nbe an integral homogeneous form in Fn that belongs to co-NP (i.e., the\nproblem of deciding if it is nonzero for given xi’s belongs co-NP).\nLet UP (useful property) be the conjunction of the following two prop-\nerties.\nUP1: The form h = h(X) is co-NP-complete.\nUP2: (Characterization by stabilizers)\nThe form h, as a point in P(W), is characterized by its stabilizer Gh ⊆\nGL(W), not exactly as in Definition 3.1, but in a relaxed manner as de-\nscribed in Section 7 in [GCT1]. So also the form f = φ(h) as a point in\nP(V ). This means the associated class varieties ∆W[h; n] = ∆W [h], and\n∆V [f; n, l] = ∆V [f], as defined in Section 4.2 with h(X) playing the role of\nE(X), are group-theoretic. Let (H1 ֒→G1 ֒→K1) and (H2 ֒→G2 ֒→K1)\n76"},{"paragraph_id":"p78","order":78,"text":"be the group-triples associated with the varieties ∆W[h; n] and ∆V [f; n, l],\nrespectively. We assume that H1 is reductive, that its simple composition\nfactors are explicitly known, and that it is built from these composition fac-\ntors by simple operations: to keep the matters simple, we only allow direct or\nwreath products, which suffice in GCT. We also assume that all the simple\ncomposition factors are either classical connected groups, tori or alternating\ngroups, as in GCT, though, again, this is strictly not necessary. We also\nassume that all homomorphisms in these triples are explicit as defined in\n[GCT6]–this is necessary.\nHere UP1 is stipulated only so that obstructions to efficient computation\nof h(X) should exist (Section 6). Otherwise, it is never explicitly used in\nGCT. The approach may work for other hard, though not co-NP-complete\nfunctions. UP2 is the main property that GCT needs for proving existence\nof obstructions. Hence we shall only concentrate on it in what follows.\nIt is shown in [GCT1] that E(X) has property UP2; whereas UP1 over C\nis shown in [Gu]. The permanent has an analogous property, where co-NP-\ncompleteness is replaced by #P-completeness. The function H(Y ) also has\nan analogous property with P-completeness replacing co-NP-completeness.\nBut in the case of H(Y ) the class variety is not the usual orbit closure\n∆[H(Y )], but rather ˆ∆[H(Y )] as defined in [GCT1]; cf. Remark 4.2.\nIn these definitions, we can also let the base field K be a finite field\nFp, or its algebraically closure ̄Fp, since characterization by stabilizers is a\nwell-defined notion over any field.\n18.1\nFrom usefulness to superpolynomial lower bounds\nThough GCT strives to prove superpolynomial lower bounds for the partic-\nular functions E(X) and perm(X), its main techniques should, in principle,\nextend to any h satisfying UP. We now briefly indicate how. This should\njustify the name UP.\nDefine an obstruction for the pair (h(x), H(Y )) as in Definition 5.1,\nwith h(X) playing the role of E(X).\nSuch obstructions should exist for\nevery n →∞, l = nlog n, as long as h(X) is co-NP-complete (cf. Section 6).\nAssociate with the class variety ∆V [f; n, l] a stretching function ̃sλ\nd(h; n, l)(k)\nas in (24) with h(X) playing the role of E(X).\nThe results in [GCT6] now imply the following analogue of Theorem 9.4\nfor h(X):\n77"},{"paragraph_id":"p79","order":79,"text":"Theorem 18.1 [GCT6] Assuming that the singularities of the class variety\n∆V [f; n, l] and ∆W[h; l] are rational, the stretching function ̃sλ\nd(h; n, l)(k)\nassociated with the class variety ∆V [f; n, l] is a quasi-polynomial\nIt is may be conjectured that the singularities will be rational, as needed\nhere, as long as h satisfies UP2.\nUsing this theorem, we can formulate PH1, PH2, and SH for h(X) just\nas for E(X) (cf. Hypotheses 1.4-1.6).\nRemark 18.2 The statements of PH1 and PH2 given in this paper are as-\nsuming that all simple composition factors of the reductive groups under con-\nsideration are either classical connected groups or tori or alternating groups.\nIn the presence of composition factors of other types, some variations are\nnecessary [GCT6].\nThe following is an analogue of Theorem 9.5 in this context.\nTheorem 18.3 [GCT6] Assuming the rationality hypothesis (cf.\nTheo-\nrem 18.1), PH1 and SH, analogues of the decision problems in Hypothe-\nsis 7.2 for h(X) belong to P.\nIn particular, the problem of verifying an\nobstruction for the pair (h(X), H(Y )) belongs to P.\nThese results suggest, just as for E(X), the following strategy for proving\na superpolynomial lower bound for h(X):\n(1): Let Hi ֒→Gi ֒→Ki be the triples that occur in the definition of UP2.\nQuantize the couples Hi ֒→Gi and Gi ֒→Ki. That is, prove analogues of\nTheorem 15.1 for these. Also quantize the triples along the scheme suggested\nin [GCT10].\n(2): Prove existence of canonical bases (PH0) for the coordinate rings and\nrepresentations of the quantum groups that arise in this quantization along\nthe lines of the basic scheme in [GCT8]. For formal statements of PH0 see\n[GCT6, GCT10].\n(3): Use these canonical bases to prove existence of canonical bases (PH0)\nfor the coordinate rings of the class varieties ∆V [f; n, l] and ∆W [h, l] along\nthe lines suggested in [GCT10].\n(4): Use PH0 to deduce PH1 and SH, as suggested in [GCT6]. The polytope\nin PH1 should be more or less determined once PH0 holds, just as in the\nstandard case; cf. Section 1.6.\n78"},{"paragraph_id":"p80","order":80,"text":"(5): Theorem 18.3, in conjunction with PH1 and SH for h(X), then im-\nplies polynomial time algorithm for the analogue of the decision problem in\nHypothesis 7.2 (a) for h(X) in place of E(X).\n(6): Carry out the steps (1)-(5) for the P-complete function H(Y ) as well.\nIt will imply a polynomial time algorithm for the decision problem in Hy-\npothesis 7.2 (b) for H(Y ). This step is the same as for E(X).\n(7): Transform the easy, polynomial time algorithms in steps (5) and (6),\nalong the lines suggested in Section 13 and [GCT6], into a P-constructive\nproof of existence of an obstruction λ(n) for every n →∞, assuming that l =\nnlog n. As pointed out in [GCT6], this transformation may need additional\npositivity hypotheses in the spirit of PH1 and SH. But these can be expected\nto hold, assuming h(X) satisfies UP2. The polytope in PH1 for h(X) in the\nstep (4) can also be expected to have a regular well-behaved structure, as\nneeded for this step, assuming h(X) satisfies UP2.\n(8): Existence of an obstruction family {λ(n)} would imply a superpoly-\nnomial size circuit lower bound for h(X), and hence, that P ̸= NP over\nC.\nFor the problems that need to be addressed over a finite field, or an\nalgebraically closed field of positive characteristic, see [GCT11].\n18.2\nOn violation of the largeness constraint\nNow let us see why UP2 should imply violation of the largeness constraint.\nWe cannot prove this formally over C. But this can be proved formally over\na finite field Fp or an algebraically closed field ̄Fp of positive characteristic\n[GCT10]. In fact, it turns out that violation of the largeness constraint is\nfar more severe than what is formally required. Namely, when K is a finite\nfield, it can be shown that\n|Cn|/|Fn| ≥1/2Ω(N),\nwhere Cn is the set of h(X) which satisfy UP2. This may be compared with\n(36).\nThe proof of violation of the largeness constraint over ̄Fp does not carry\nover to C for technical reasons. Specifically, the key ingradient in this proof is\nthe Riemann hypothesis over finite fields, or rather its extension as proved in\n[Dl2]. To transport this to the case when h(x) is integral would presumably\nrequire an analogous statement in arithmetic algebraic geometry.\n79"},{"paragraph_id":"p81","order":81,"text":"It may be remarked that, in contrast, the proof of violation of the large-\nness constraint over a finite field is elementary. Thus the difficulty of proving\nthe violation of the largeness constraint over K seems inversely related to\nthe difficulty of proving the P ̸= NP conjecture over K. When K = Fp,\nthe conjecture is hardest to prove, and hence, the proof of violation is easy.\nWhen K = C∗, the conjecture should be easier than over ̄Fp or Fp. Accord-\ningly, proving violation of the largeness constraint formally turns out to be\nthe hardest.\n18.3\nOn violation of the constructivity constraint\nNext let us see why UP2 should also imply violation of the constructivity\nconstraint. We cannot hope to show this formally, since this is a lower bound\nstatement in itself. But rather we can give good evidence. First of all, to\ncompute the stabilizer of h(X), we have to solve a system of polynomial\nequations. Determining feasibility of a general system of polynomial equa-\ntions in k variables is NP-complete and is conjectured to take ⟨p⟩Ω(k) time,\nwhen K is the finite field Fp. Analogous conjecture may be made for the\nspecific system of polynomial equations that arises in the computation of\nthe stabilizer. Assuming this, it follows [GCT10] that deciding if h(X) has\na nontrivial stabilizer would take time that is superpolynomial in N–this is\nthe truth-table size when K = Fp.\n19\nP-verifiable and P-constructible proof techniques\nand their explicit construction complexity\nIn this section we suggest why GCT may be among the “easiest” “easy-to-\nverify” approaches to the P ̸= NP conjecture as per a certain measure of\nproof-complexity, called the explicit construction complexity. For this, we\nhave to introduce the notion of an easy-to-verify (i.e. P-verifiable) proof\ntechnique and then define its explicit construction complexity (class).\n19.1\nP-verifiable proof technique\nSuppose we are given a proof technique (approach) towards to the P vs. NP\nproblem that seeks to prove a superpolynomial lower bound for a specific\nhard function h(X) under consideration. We assume that the approach seeks\nto prove, explicitly or implicitly, existence of a specific cause for the hardness\n80"},{"paragraph_id":"p82","order":82,"text":"of h(X), which we shall refer to as an obstruction. Thus an obstruction is,\nroughly, a “cause”, a “witness” or a “proof” of hardness.\nBut what do we mean by a proof?\nThe final proof of the P ̸= NP\nconjecture, if true, would constitute the ultimate obstruction to efficient\ncomputation of every (co)-NP-complete h(X). The size of this proof would\nbe just O(1), and so also the cost its verification. By obstruction, we do\nnot mean this final proof of hardness, but rather an intermediate proof\nof hardness whose existence the approach strives to demonstrate for every\nn →∞, when the circuit size m = nlog n, say.\nThe nature of such an obstruction will depend on the proof technique.\nWe cannot define it formally. Hence we will only give an intuitive idea with\nan example. Suppose there is an efficient pseudo-random generator whose\nexistence implies a restricted type of lower bound result in the spirit of [NW].\nThen the explicit computational circuit for this pseudo-random generator,\ni.e., for all its output bits together would be an obstruction in this context.\nBecause existence of this pseudo-random generator serves as a witness for\nhardness. If the pseudo-random generator is based on an explicit structure\nin the spirit of an expander, then this structure too can be considered to\nbe an obstruction.\nMore generally, the hardness-vs-randomness principle\n[KI, NW] suggests that proof techniques for difficult lower bounds may need\nmore or less explicit constructions of some structures.\nThese structures,\nwhich serve as witnesses for hardness, can then be taken as obstructions.\nIn the rest of this section, we confine ourselves only to those techniques\ntowards the P ̸= NP conjecture which contain, explicitly or implicitly, the\nnotion of an obstruction in this spirit–a witness for hardness–which admits\na well-defined description that can be assigned bit length. The arguments\nhenceforth are subject to this assumption.\nNext we try to formalize the notion of a “viable” proof technique to-\nwards the P vs. NP problem. For this, let us begin with a technique that\nshould certainly not be considered viable–the trivial brute-force proof tech-\nnique. This is defined as follows. Assume that the base field K is finite.\nFix any co-NP-complete function h(X) = h(x1, . . . , xn). Then this proof\ntechnique strives to prove, for every n, existence of the trivial proof of hard-\nness (obstruction), which consists of just the enumeration of all circuits of\nsize m = nlog n, with a specific value of X for each circuit on which the\nfunction evaluated by the circuit differs from h(X). The size of this trivial\nobstruction is exponential in m, and the time taken to verify it is also expo-\nnential in m. Any viable proof technique for the P vs. NP problem ought\n81"},{"paragraph_id":"p83","order":83,"text":"to be at least better than this trivial proof technique in some well defined\nsense. One obvious sense in which it could be better is that there exists an\nobstruction whose size is not exponential in m, but rather polynomial in m,\nand the time taken to verify an obstruction is not exponential in m, but\nrather polynomial in m.\nThis leads to:\nDefinition 19.1 We say that a proof technique for the P ̸= NP conjecture\nis P-verifiable if\n1. There is a well-defined notion of obstruction, either implicit or explicit\nin the technique,\n2. There exists a short obstruction to computation of the specific function\nh(X) = h(x1, . . . , xn) under consideration by a circuit of size m =\nnlog n (say), for every n →∞, though the technique may only strive\nto prove existence of any obstruction, not necessarily short. By short,\nwe mean the obstruction has a label (combinatorial specification) of bit\nlength poly(m).\n3. The problem of verifying an obstruction is easy; i.e., belongs to P.\nSpecifically, takes time that is polynomial in n, m and the bit length of\nthe obstruction.\nThe meaning of easy here is the most obvious and natural definition in\nthe context of the P vs.\nNP problem.\nThus, intuitively a P-verifiable\nproof technique is an easy-to-verify proof technique. That is, the problem\nof discovering a proof of hardness (obstruction) in the technique belongs to\nNP. The definition above makes sense over any base field K of computation,\nwith obvious modifications in the spirit of the ones in Section 18.\nThe following naive arguments suggest that for a technique towards the\nP ̸= NP conjecture to be viable it out to be P-verifiable.\nFirst, the usual experience in mathematics suggests that however hard\nthe discovery of a proof may be its verification, once found, should be easy,\nand furthermore, the proofs that are found are usually reasonably short. In\nthe definition of P-verifiability, short and easy are given the most obvious\nand natural interpretations in the context of the P vs. NP problem: de-\nscription of polynomial size (short), and can be done in polynomial time\n(easy).\n82"},{"paragraph_id":"p84","order":84,"text":"Second, given a technique, it seems necessary to justify why it is better\nthan the trivial brute-force technique. A P-verifiable proof technique is bet-\nter than it as per the most obvious complexity measures: (1) space (short),\nand (2) time (cost of verification).\nThird, the article [RR] roughly says that a nonspecific approach that\nis applicable to a large fraction of hard functions should not work in the\ncontext of the P vs. NP problem. Thus approaches based on probabilistic\nmethods or estimates of various kinds–such as Bezout-type estimates in\nalgebraic geometry, or estimates for discrepancies and deviations in analysis\nor number theory–should not work.\nProof of hardness as per any such\napproach–namely, the value of the measure or the estimate which is the\ncause of hardness–should be hard to verify. Since to verify the value, we\nmay have to compute it and see that it really tallies with what is given,\nand such computations should typically take time that is exponential in\nthe bitlength of the value. Thus a P/poly-naturalizable proof should also\nbe non-P-verifiable, and hence, the definition of P-verifiability here seems\nconsistent with the arguments in [RR].\nAdmittedly, these are only naive arguments. One can ask if there exists\na viable proof technique for the P ̸= NP conjecture that is better than the\ntrivial brute-force technique as per some measure of complexity other than\nthe obvious ones–space and time. But since we cannot think of any such\nnonobvious complexity measures which are also natural in the context of\nthe P vs. NP problem, we shall confine ourselves to only P-verifiable proof\ntechniques in what follows.\nBy Hypothesis 7.1 (a), which is supported by the results that we de-\nscribed in this article, GCT is a P-verifiable proof technique over C; the\nstory over a finite field should be similar [GCT11].\n19.2\nP-barrier for verification\nEvery P-verifiable technique for the P ̸= NP-conjecture has to cross the P-\nbarrier for verification; i.e., surmount the difficulty of showing that verifica-\ntion of an obstruction is easy. The magnitude and difficulty of the P-barrier\nshould be of the same order regardless of which P-verifiable approach to the\nP ̸= NP conjecture is taken.\nThis is easy to see when the base field K is finite. Then the length of\nany obstruction in the trivial brute-force technique mentioned in the begin-\nning of this section is exponential. The main task here is to come with a\n83"},{"paragraph_id":"p85","order":85,"text":"proof technique that admits short obstructions which can be verified easily.\nThe magnitude of this P-barrier–the difference between the exponential and\nthe polynomial–is the same regardless of which approach to the P ̸= NP\nconjecture is taken.\nNext, let us assume that K = C, as in this paper. Let n be the number\nof input parameters. Let m(n) = nlogn (say) be the circuit-size parameter,\nh(n) = nlogn the height-parameter, and d(n) ≤2h(n) the degree parameter\nin the lower bound problem under consideration. For a given n, the set of\nfunctions over C computable by circuits of size at most m = m(n), height at\nmost h = h(n) and degree at most d = d(n) is an algebraically constructible\n[Mm1] subset S of the space V of all forms in m variables of degree d(n). A\nconstructible subset means it is in the boolean algebra generated by closed\nalgebraic subsets of V ; this is a generalization of an affine variety.\nThe goal in the lower bound problem under consideration is to show\nthat h(X) does not belong to S, when m = nlog n. Let ̄S be the closure of\nS. It is an affine variety. If h(X) is co-(NP)-complete, it is reasonable to\nassume that it does not belong to ̄S as well; i.e., roughly speaking, it cannot\nbe approximated infinitesimally closely by a circuit of size m(n) and height\nh(n). So it would suffice to show this.\nAn obvious obstruction here would be a polynomial in the ideal of ̄S\nwhich does not vanish on h(X). To decide if a given polynomial belongs to\nthe ideal of ̄S, an obvious method is to compute a good basis of this ideal,\nsuch as Gr ̈obner basis, and then use it for this decision. But the problem\nof Gr ̈obner basis computation is EXPSPACE complete [MM]. This means\ncomputation of the Gr ̈obner basis of ̄S can take space that is exponential in\nthe dimension of the ambient space V , which in turn is exponential in m.\nIn other words, space that is double exponential in m, and hence, time that\nis triple exponential in m. Given that the S in our problem is really bad,\nthis is the best that we can expect from any general purpose technique for\nverifying an obstruction that reasons about S directly in this fashion.\nFor the technique to be P-verifiable, the huge gap between this triple\nexponential bound for a general purpose direct technique and the polynomial\nbound in Definition 19.1 has to be bridged. The magnitude and order of this\ngap–the P-barrier–is exactly the same that we encountered in Section 8.\nRemark 19.2 The triple exponential size of this gap when K = C as\nagainst the exponential size over K = Fp does not mean that the P ̸= NP\nconjecture is easier when K = Fp. In fact, it is the other way around. Since\nthe (nonuniform) P ̸= NP conjecture in characteristic zero (over Z) is a\n84"},{"paragraph_id":"p86","order":86,"text":"weaker implication of the conjecture over finite field (the usual case) [GCT1].\nHence, the exponential gap over Fp would be much harder to bridge than the\ntriple exponential gap over C. See [GCT6] for the problems that need to be\naddressed over finite fields.\nThe class variety XP (l) for P (Section 4.2) is constructed in [GCT1]\nprecisely to cope up with the triple exponential gap over C. It is a nice\nalgebraic variety that contains S, or rather its projectivization. So instead\nof trying to show that a given h(X) is not in S, one strives to show that\nit is not in XP (l). Since the algebraic geometry of XP (l) is exceptional,\nthis problem becomes easier–especially when h(X) is also exceptional, like\nE(X).\nThe quantum-group and algebro-geometric machinery is needed in GCT\njust to cross the P-barrier for verification (over C).\nThis suggests that\nmathematics required for any P-verifiable approach towards the P ̸= NP\nconjecture may not be substantially simpler, or easier.\n19.3\nP-constructible proof technique\nIn fact, it may be much harder unless it is also P-constructible in the fol-\nlowing sense.\nDefinition 19.3 We say that a P-verifiable proof technique for the P ̸=\nNP conjecture is P-constructible if the discovery of an obstruction in this\ntechnique is also easy. That is, there exists an algorithm, which, given n\nand m, can decide whether there exists an obstruction in poly(m) time, and\nif so, also construct a short obstruction in poly(m) time.\nThus the problem of discovering an obstruction in a P-constructible\nproof technique. belongs to P. But the proof technique itself need not give\na polynomial time algorithm for discovering an obstruction explicitly. That\nis, this may only be implicit in the proof, or it may be left to posterity. We\ncall the technique P-constructive if it gives such an algorithm more or less\nexplicitly:\nDefinition 19.4 A P-constructible proof technique is called P-constructive,\nif it also yields a procedure to construct an obstruction explicitly in poly(n, m)\ntime, if one exists.\n85"},{"paragraph_id":"p87","order":87,"text":"The relationship between P-constructible (constructive) and P-verifiable\nproof strategies is akin to the relationship between P and NP. The P ̸= NP\nconjecture says that the discovery of a proof is, in general, harder than its\nverification. Hence, just as P denotes the class of easy problems within NP,\nthe P-constructible and P-constructive proof strategies are in a sense the\n“easy” ones among the P-verifiable proof strategies, wherein discovery is\nalso easy like verification.\nBy Hypothesis 7.1 (c), as supported by the positivity hypotheses, results\ndescribed in this paper, GCT is P-constructive over C; the story over a finite\nis expected to be similar [GCT11].\nThat there should exist such a P-constructive proof technique for the\nP ̸= NP conjecture may, however, seem paradoxical at the surface. Be-\ncause a P-constructive (constructible) proof technique seems to go against\nthe very philosophical essence of the P ̸= NP conjecture that discovery is\nharder than verification. This is akin to the paradox in the proof of G ̈odel’s\nincompleteness theorem: that the statement which says there exist unprov-\nable true statements is itself easy to prove. Similarly, Hypothesis 7.1 (c) says\nthat the statement which says discovery is harder than verification should\nitself be easy to discover.\n19.4\nGeneral setting\nSo far we have described P-verifiable and P-constructible proof techniques\nonly in the context of the P vs. NP problem. But these notions can be\ndefined in a much more general context, as we now briefly indicate:\nDefinition 19.5 A technique for proving a mathematical property Q(X),\nwhere X ranges over a class C of mathematical objects under consideration,\nis P-verifiable if:\n1. The technique proves, explicitly or implicitly, existence of a “proof-\ncertificate” c(X), for every X ∈C, which serves as a “witness” that\nthe property Q(X) holds.\n2. There exists a short proof certificate for every X ∈C.\nBy short,\nwe mean its size is poly(⟨X⟩), where ⟨X⟩denotes the specification-\ncomplexity of X.\n3. Verification of a proof-certificate c(X) is easy; i.e., can be done in\npoly(⟨X⟩, ⟨c(X)⟩) time, where ⟨c(X)⟩denotes the bitlength of c(X).\n86"},{"paragraph_id":"p88","order":88,"text":"Again, we cannot formally define what a proof-certificate means. In what\nfollows, we only consider proof techniques wherein the notion of a proof-\ncertificate is well defined. The specification complexity ⟨X⟩here depends\non the problem under consideration, as we shall see in the examples below.\nDefinition 19.6 A P-verifiable proof technique is called P-constructible if\nthere exists an algorithm which, given X ∈C, can construct a proof certifi-\ncate c(X) in poly(⟨X⟩) time.\nBut the proof technique itself need not give such an algorithm explicitly.\nDefinition 19.7 A P-constructible proof technique is called P-constructive\nif, in addition, it yields an algorithm that can construct a proof-certificate\nc(X) in poly(⟨X⟩) time.\nWe can now state an informal working hypothesis:\nHypothesis 19.8 (The P-hypothesis) (informal)\n(a) Feasible P-verifiable proof techniques–that is the P-verifiable techniques\nthat can actually be used to prove the properties Q(X) in practice–are usu-\nally P-constructible, though proving P-constructibility may turn out to be\nnontrivial, and may only be done a posteriori.\n(b) Conversely, if a P-verifiable technique is P-constructible then under rea-\nsonable conditions it may also be feasible, i.e., can be used to actually prove\nQ(X).\n(c) A major part of the effort in a P-constructible proof technique usually\ngoes towards development of a polynomial time algorithm for constructing a\nproof-certificate, though this may be done only implicitly, and may become\nclear only a posteriori. That is, a P-constructible proof can usually be ex-\ntended to a P-constructive proof with a “reasonable additional effort”, albeit\na posteriori.\n(d) Mathematical complexity of a P-constructive proof technique is inti-\nmately linked to the computational complexity of the algorithm for explicit\nconstruction of a proof certificate underlying the technique.\nAs we have already remarked, the relationship between P-constructible\nproof techniques and P-verifiable proof techniques is akin to the relationship\nbetween P and NP.\nThe class P is usually regarded as the subclass of\n87"},{"paragraph_id":"p89","order":89,"text":"feasible problems in NP. Hence, the P-hypothesis just says that P usually\nmeans feasible in practice.\nThe reasonable conditions in (b) means: there is a polynomial time al-\ngorithm for constructing a proof-certificate, which has, furthermore, a rea-\nsonably simple structure, and which is efficient in practice.\nThat is, the\ndefinition of P as standing for feasible is not misused.\nDefinition 19.9 A mathematical theorem, which says a property Q(X) holds\nfor every X in a class C of mathematical objects under consideration, is\ncalled P-verifiable if it has a P-verifiable proof.\nA P-constructible or a P-constructive theorem is defined similarly.\n19.4.1\nExamples\nWe now give a few examples to illustrate these notions.\nP vs NP problem\nIn this context, C is the class of tuples (n, m(n)), m(n) = nc for any con-\nstant c > 0, over all n (large enough). The property Q(X), X = (n, m(n)),\njust says that the explicit function h(X) under consideration, such as E(X)\nin [GCT1], cannot be computed by a circuit of size m(n) for every n large\nenough. Here ⟨X⟩= n+m; i.e, we assume that n and m are given in unary.\nThen the notions of P-verifiability, and P-constructibility here coincide with\nthe ones in Definitions 19.1 and 19.3. When m = nc, an obstruction would\nalways exist, assuming that h(X) is co-(NP)-complete, P ̸= NP and the\ntechnique is correct. That is why Definition 19.5 would coincide with Def-\ninition 19.1, even if in the former there is no mention of deciding if an\nobstruction exists or not. As per Hypothesis 7.1, GCT is P-constructive\nover C, and hence the P ̸= NP conjecture over C is also P-constructive; the\nsame can be hypothesized over a finite field [GCT11].\nHall’s theorem\nIn this context, C is the class of d-regular bipartite graphs. The property\nQ(X) is that every d-regular bipartite graph X ∈C has a perfect matching.\nThe bit length ⟨X⟩is the bitlength of the specification of X. The proof\ncertificate c(X) is a perfect matching in X. The problems of verifying and\nconstructing a perfect matching belong to P, the former trivially. Hence,\n88"},{"paragraph_id":"p90","order":90,"text":"Hall’s theorem is P-constructive. Hall’s original proof is P-constructible,\nthough not P-constructive, since it does not explicitly give a polynomial\ntime algorithm for constructing a perfect matching.\nBut it does contain\nmajor ingradients for such a polynomial time algorithm, which came only\nmuch later. This is consistent with the P-hypothesis.\nFour colour theorem\nIn this context, C is the collection of planar graphs. The property Q(X) is\nthat any planar graph X is four colourable. The bitlength ⟨X⟩is the bit\nlength of the specification of X. The proof certificate c(X) is a four colouring\nof X. The problems of verifying and constructing a proof certificate belong\nto P, the former trivially. Hence, any proof of the four colour theorem is\nP-constructible, and the four colour theorem is P-constructive. The actual\nproof in [AH] is also (more or less) P-constructive since it implicitly yields\nto a polynomial (quartic) time algorithm for four colouring, Indeed, major\npart of the effort in the proof implicitly goes towards development of such\nan algorithm. This is consistent with the P-hypothesis.\nA simpler P-constructive proof was subsequently given in [RSST], which\ngives a better quadratic algorithm for the same problem. This too is con-\nsistent with the P-hypothesis (d).\nForbidden minor theorem\nFix a genus g. The forbidden minor theorem [RS] says that a graph which\ndoes not contain a forbidden minor from a finite list of minors depending\non g can be embedded on a genus g surface. Here C is the class of graphs\nthat do not contain a forbidden minor, Q(X) the property above, and ⟨X⟩\nthe bitlength of the specification of X. The proof certificate c(X) is just a\ndescription that tells how to embed X on a genus g surface.\nThe forbidden minor theorem is P-constructive. Any proof technique\nfor proving the forbidden minor theorem is P-constructible: it was known\n[FMR] even before [RS] that c(X) can be constructed in polynomial O(⟨X⟩O(g))\ntime. The proof of the forbidden minor theorem in [RS] gave an O(f(g)⟨X⟩2)\nalgorithm, where f(g) depends only on g. Indeed, a major part of the effort\nin [RS] implicitly goes towards finding a polynomial time algorithm whose\nrunning time is of the form O(f(g)⟨X⟩O(1)); i.e., wherein the exponent of\n⟨X⟩does not depend on g. This is again consistent with the P-hypothesis\n(d).\n89"},{"paragraph_id":"p91","order":91,"text":"The Poincare conjecture\nHere we can let C be the set of simplicial decompositions of compact three\ndimensional combinatorial manifolds that are simply connected. The prop-\nerty Q(X) says that X is a (combinatorial) sphere. The bitlength ⟨X⟩is\nthe bitlength of specifying X. The article [Sc] says that the sphere recogni-\ntion problem is in NP. That is, there is a proof-certificate c(X), verifiable\nin polynomial time, which certifies that X is a sphere. It is interesting to\nknow here if the problem of constructing a proof certificate c(X), for a given\nX ∈C, belongs to P. It is plausible that the proof technique in [Pe] can\nbe extended/transformed (in the combinatorial setting) to get a polynomial\ntime algorithm which constructs a proof-certificate in this spirit, though not\nexactly the one in [Sc]. If that happens, it would mean that the Poincare\nconjecture is P-constructible (P-constructive), and that the major effort in\n[Pe] implicitly went towards getting a polynomial time algorithm for this\nproblem. This would provide support for the P-hypothesis (c).\nThus a major part of the effort in the P-verifiable proofs above indeed\nseems to go towards developing a polynomial time or a better polynomial\ntime algorithm for constructing a proof-certificate, as per the P-hypothesis\n(c), though this goal may not be stated explicitly in the proofs. In the flip,\nP-constructivity as a goal is explicitly spelled out right in the beginning,\ngiven the complexity-theoretic significance of the P vs. NP problem. But\njust as in the examples above, it may not be necessary to prove PHflip\n(Hypothesis 7.1) fully to prove P ̸= NP over C. That is, it may suffice to\ndevelop only a part of all ingradients needed to put the required problems\nin P, and the remaining part can be left to posterity. In this context, the\nbasic minimum that seems to be needed is PH1 (more or less).\n19.5\nExplicit construction complexity\nWe will now try to formalize the intuition behind the P-hypothesis (d).\nTowards that end we wish to associate a measure of proof-complexity with\na P-verifiable proof technique. This is quite different, for example, from\nKolmogrov proof-complexity.\nDefinition 19.10 Explicit construction complexity of a P-constructive tech-\nnique is the computational complexity of the algorithm underlying that tech-\nnique for explicit construction of a proof-certificate.\n90"},{"paragraph_id":"p92","order":92,"text":"By computational complexity, we mean the usual measures such as depth\nand size of the corresponding computational circuit. If a P-verifiable tech-\nnique is not explicitly P-constructive but naturally leads to an algorithm\nfor construction of obstructions, with additional effort, we agree to take\nthe computational complexity of this algorithm to be explicit construction\ncomplexity of the technique, albeit only a posteriori.\nDefinition 19.11 (a) Verification (complexity) class of a P-verifiable proof\ntechnique is the abstract computational complexity class of the problem of\nverifying a proof-certificate (as per that technique).\n(b) Explicit construction (complexity) class of a P-verifiable proof technique\nis the computational complexity class of the problem of explicit construction\nof a proof-certificate as per that technique.\nA computational complexity class here means an abstract computation\ncomplexity class such as P, NC, NCk, AC, Dtime(N) etc. The verifica-\ntion and explicit construction classes of a P-verifiable technique are well\ndefined regardless of whether the technique shows how to construct a proof-\ncertificate explicitly or not. But what these classes are may become clear\nonly a posteriori, possibly after extending the proof technique to a get an\nefficient algorithm for construction of a proof certificate therein.\nThe complexity measures and classes above are meaningful only for P-\nverifiable proof techniques.\nThey would not make any sense for noncon-\nstructive or estimate-based techniques in analysis, number theory and so\nforth, unless it is possible to define a specification complexity ⟨X⟩and a\nproof-certificate that is polynomial time verifiable with this definition of\n⟨X⟩naturally.\nThis following gives a notion of theorem complexity for P-verifiable the-\norems.\nDefinition 19.12 Explicit construction complexity (class) of a P-verifiable\ntheorem is the minimum explicit construction complexity (class) over all P-\nverifiable proofs of the theorem. Verification complexity (class) is defined\nsimilarly.\nThe explicit construction complexity seems to be a good measure of\ncomplexity for P-verifiable proof techniques and theorems. We shall discuss\nthe examples above a bit more in this context.\n91"},{"paragraph_id":"p93","order":93,"text":"Halls’ theorem\nVerification class here is AC (constant depth circuits), since a perfect match-\ning can be verified in constant depth. A perfect matching in a bipartite\ngraph can be computed, if one exists, in O(m log n) time. This problem also\nbelongs to RNC [KUW, MVV]. Hence, the sequential explicit construction\nclass of Hall’s theorem is Dtime(m log n). The parallel explicit construction\nclass is RNC; possibly even NC.\nFour colour theorem\nVerification class here is AC. Explicit construction complexity of the proof\nin [AH] is O(n4), whereas that of the proof in [RSST] is O(n2) [RSST].\nThus a proof technique with lower explicit construction complexity has in-\ndeed lower proof-complexity. The sequential explicit construction class of\nthe four colour theorem is thus Dtime(n2), or lower. The parallel explicit\nconstruction class is possibly NC, in view of the parallel algorithms for four\ncolouring in special cases [He].\nForbidden minor theorem\nVerification class is AC. Explicit construction complexity of the proof in\n[RS] is O(f(g)n2), where g is an explicit function of the genus g.\nThe\nsequential explicit construction class of the forbidden minor theorem is thus\nDtime(O(n2)); it may be Dtime(n). The parallel explicit construction class\nmay be NC, since planarity testing is in NC [JS].\nPoincare’s conjecture\nVerification class of the Poincare conjecture is P [Sc], assuming that the\nproof technique in [Pe] is P-verifiable. It may be smaller. NC?. Explicit\nconstruction class may be P, plausibly smaller. NC?\nTrivial example\nWe now give a trivial example to illustrate why P-verifiability is essential\nfor the complexity measures here to make sense.\nTake a trivial mathe-\nmatical theorem: that an integer n has at most log n factors. An obvious\nproof-certificate, for a given n, is the number of its factors, which shows\n92"},{"paragraph_id":"p94","order":94,"text":"that it is less than log n. But verification of this proof requires factoring\nand hence is hard. Thus if n is specified in binary, this theorem should\nnot be P-verifiable. That is why explicit construction complexity of this\nproof-certificate says nothing of the actual (trivial) proof-complexity of the\ntheorem. Similarly, explicit construction complexity is not meaningful for\nestimate-centred proof techniques in mathematics. The article [RR] roughly\nsays that such techniques are not expected to work in the context of the\nP vs. NP problem since they tend to be applicable to a large fraction of\nfunctions.\nIn the context of the P vs. NP problem, Definitions 19.10 and 19.11\nbecome:\nDefinition 19.13 Explicit construction complexity of a P-constructive tech-\nnique for the P ̸= NP conjecture is the computational complexity of the algo-\nrithm underlying that technique for explicit construction of a proof-certificate\n(as per that technique).\nDefinition 19.14 (a) Verification complexity class of a P-verifiable proof\ntechnique for the P ̸= NP conjecture is the computational complexity class\nof the problem of verifying an obstruction as per that technique.\n(b) Its explicit construction complexity class is the computational complexity\nclass of the problem of explicit construction of an obstruction.\nAgain these classes are well-defined regardless of whether the technique\nshows how to construct an obstruction explicitly or not, once the notion of\nan obstruction in the proof technique is well-defined.\nOne may also define existential complexity class of a P-verifiable proof\ntechnique (for the P vs. NP problem): this is the computational complexity\nclass of the problem of deciding if there exists an obstruction for a given n\nand circuit size m.\nThe existence-vs-construction principle [KUW] says that computational\ncomplexity of a construction problem is comparable to that of the associated\nexistence problem under natural conditions.\nThis means, under natural\nconditions, existential and explicit-construction complexity classes should\ncoincide. Hence, we shall not worry about existential complexity anymore.\nIt is illuminating to compare the verification complexity of the P vs. NP\nproblem with the other problems we considered. The verification complexity\nclass of Halls’ theorem, four colour theorem, or forbidden minor theorem is\n93"},{"paragraph_id":"p95","order":95,"text":"AC. For Poincare’s conjecture, P-verifiability is quite nontrivial [Sc]. But\nfortunately the proof is not very complex.\nIn contrast, P-verifiability is already a formidable issue in the context of\nthe P vs. NP problem.\n19.6\nIs there a simpler proof technique?\nNow we ask if there is a P-verifiable proof technique towards the P ̸= NP\nconjecture that is substantially “easier” than GCT. By easier we mean, with\nlower verification and explicit construction complexity (classes). Since GCT\nis P-verifiable and also P-constructive over C as per Hypothesis 7.1, P ̸=\nNP conjecture is conjecturally P-verifiable and also P-constructive over C.\nThe same can be conjectured over Fp or ̄Fp as well [GCT10]. Assuming this,\nit is meaningful to talk of its verification and explicit construction classes.\nSo we can ask:\nQuestion 19.15 What are the (smallest) verification and explicit construc-\ntion complexity classes of the P ̸= NP conjecture?\nThe best and the most natural answer that one can expect here is P.\nIt would really be unsettling if the answer were, say, NC. Specifically, the\nproblems of verification and explicit construction of obstructions in any P-\nverifiable approach to the P ̸= NP conjecture should be at least as hard as\nP-complete problems. This is supported by the presence of linear program-\nming, which is P-complete, in the algorithms for the basic decision problems\nin Theorem 9.5.\nIf so, GCT may be among the “easiest” P-verifiable approaches to the\nP ̸= NP conjecture over C. The story over Fp may be similar; cf. [GCT11].\nReferences\n[A1]\nM. Agrawal, N. Kayal, N. Saxena, Primes is in P, Annals of Math-\nematics, 160 (2): 781-793, 2004.\n[A2]\nM. Agrawal, Proving lower bounds via pseudo-random generators,\nProceedings of FSTTCS 2005, 92-105, 2005.\n[Ak]\nD. Akhiezer, Homogeneous complex manifolds, Encyclopaedia of\nmathematical sciences, volume 10, Springer-Verlag. 1986.\n94"},{"paragraph_id":"p96","order":96,"text":"[AH]\nK. Appel and W. Haken, Every planar map is four colorable,\nA.M.S. Contemporary Math. 98 (1989). MR 91m:05079.\n[B]\nL. Babai, private communication.\n[BGS]\nT. Baker, J. Gill, R. Soloway, Relativization of the P =?NP ques-\ntion, SIAM J. Comput. 4, 431-442, 1975.\n[BBD]\nA.\nBeilinson,\nJ.\nBernstein,\nP.\nDeligne,\nFaisceaux\npervers,\nAst ́erisque 100, (1982), Soc. Math. France.\n[BZ]\nA. Berenstein, A. Zelevinsky, Tensor product multiplicities and\nconvex polytopes in partition space, J. Geom. Phys. 5(3): 453-472,\n1988.\n[BS]\nR. Boppana, M. Sipser: The complexity of finite functions, Hand-\nbook of Theoretical Computer Science, vol. A, Edited by J. van\nLeeuwen, North Holland, Amsterdam, 1990, 757–804.\n[Bou]\nJ. Boutot, Singularit’es rationelles et quotients par les groupes\nr’eductifs, Invent. Math. 88, (1987), 65-68.\n[Co]\nS. Cook: The complexity of theorem-proving procedures. Proceed-\nings of the third annual ACM Symposium on Theory of Computing.\n151-158. (1971).\n[Dh]\nR. Dehy, Combinatorial results on Demazure modules, J. of Alge-\nbra 205, 505-524 (1998).\n[Dl1]\nP. Deligne, J. Milne, Tannakien categories, Lecture notes in Math\n900.\n[Dl2]\nP. Deligne, La conjecture de Weil II, Publ. Math. Inst. Haut. ́Etud.\nSci. 52, (1980) 137-252.\n[DM1]\nJ. De Loera, T. McAllister, Vertices of Gelfand-Tsetlin polytopes,\nmath.CO/0309329, Sept. 2003.\n[DM2]\nJ. De Loera, T. McAllister, On the computation of Clebsch-Gordon\ncoefficients and the dilation effect, Experiment Math. 15, (2006),\nno. 1, 7-20.\n[Der]\nH. Derkesen, J. Weyman, On the Littlewood-Richardson polyno-\nmials, J. Algebra 255 (2002), no. 2, 247-257.\n95"},{"paragraph_id":"p97","order":97,"text":"[Dri]\nV. Drinfeld, Quantum groups, Proc. Int. Congr. Math. Berkeley,\n1986, vol. 1, Amer. Math. Soc. 1988, 798-820.\n[Ed]\nJ. Edmonds, Maximum matching and a polyhedron with 0 −1\nvertices, Journal of Research of the National Bureau of Standards\nB 69, (1965), 125-130.\n[FMR]\nI. Filotti, G. Miller, J. Reif, On determining the genus of a graph\nin O(vO(g)) steps, Proc. of the eleventh ACM Symposium on the\nTheory of Computing, 1979.\n[Fl]\nH. Flenner, Rationale quasihomogene Singularit ̈aten, Arch. Math.\n36 (1981), 35-44.\n[F1]\nW. Fulton, Young tableaux, Cambridge University Press, 1997.\n[F2]\nW. Fulton, Eigenvalues of sums of Hermitian matrices (after A.\nKlyachko), S ́eminaire Bourbaki, vol. 1997/98. Ast ́erisque No. 2523\n(1998), Exp. No. 845, 5, 255-269.\n[FH]\nW. Fulton, J. Harris, Representation theory, A first course,\nSpringer, 1991.\n[GCTabs] K.\nMulmuley,\nGeometric\ncomplexity\ntheory,\nabstract,\ntechnical\nreport\nTR-2007-12,\ncomputer\nscience\ndept.,\nThe\nuniversity\nof\nChicago,\nSept.\n2007.\nAvailable\nat:\nhttp://ramakrishnadas.cs.uchicago.edu.\n[GCTconf] K. Mulmuley, M. Sohoni, Geometric complexity theory, P vs.\nNP and explicit obstructions, in “Advances in Algebra and Geom-\netry”, Edited by C. Musili, the proceedings of the International\nConference on Algebra and Geometry, Hyderabad, 2001.\n[GCTflip2] K. Mulmuley, On P vs. NP, geometric complexity theory, and\nthe flip II, under preparation.\n[GCTintro] K.\nMulmuley,\nM.\nSohoni,\nGeometric\ncomplexity\ntheory:\nintroduction,\ntechnical\nreport TR-2007-16,\ncomputer science\ndept.,\nThe university of Chicago,\nSept. 2007. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu.\n[GCT1] K. Mulmuley, M. Sohoni, Geometric complexity theory I: an ap-\nproach to the P vs. NP and related problems, SIAM J. Comput.,\nvol 31, no 2, pp 496-526, 2001.\n96"},{"paragraph_id":"p98","order":98,"text":"[GCT2] K.\nMulmuley,\nM.\nSohoni,\nGeometric\ncomplexity\ntheory\nII:\ntowards\nexplicit\nobstructions\nfor\nembeddings\namong\nclass varieties,\nto appear in SIAM J. Comput.,\ncs. ArXiv\npreprint cs. CC/0612134,\nDecember 25,\n2006. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu\n[GCT3] K. Mulmuley, M. Sohoni, Geometric complexity theory III, on de-\nciding positivity of Littlewood-Richardson coefficients, cs. ArXiv\npreprint cs. CC/0501076 v1 26 Jan 2005.\n[GCT4] K.\nMulmuley,\nM.\nSohoni,\nGeometric\ncomplexity\nthe-\nory\nIV:\nquantum\ngroup\nfor\nthe\nKronecker\nproblem,\ncs.\nArXiv preprint cs. CC/0703110,\nMarch,\n2007. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu\n[GCT5] K. Mulmuley, H. Narayanan, Geometric complexity theory V: on\ndeciding nonvanishing of a generalized Littlewood-Richardson co-\nefficient, Technical report TR-2007-05, Comp. Sci. Dept. The uni-\nversity of chicago, May, 2007.\n[GCT6] K. Mulmuley, Geometric complexity theory VI: the flip via sat-\nurated and positive integer programming in representation the-\nory and algebraic geometry, Technical report TR 2007-04, Comp.\nSci. Dept., The University of Chicago, May, 2007. Available\nat: http://ramakrishnadas.cs.uchicago.edu. Revised version to be\navailable here.\n[GCT7] K. Mulmuley, Geometric complexity theory VII: a quantum group\nfor the plethysm problem, technical report TR-2007-14, computer\nscience dept., The university of Chicago, Sept. 2007. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu.\n[GCT8] K. Mulmuley, Geometric complexity theory VIII: On canonical\nbases for the nonstandard quantum groups, technical report TR-\n2007-15, computer science dept., The university of Chicago, Sept.\n2007. 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\nproblem, under preparation.\n[GCT10] K. Mulmuley, Geometric complexity theory X: On class varieties,\nand the natural proof barrier, under preparation.\n97"},{"paragraph_id":"p99","order":99,"text":"[GCT11] K. Mulmuley, Geometric complexity theory XI: on the flip over\nfinite or algebraically closed fields of positive characteristic, under\npreparation.\n[GL]\nI. Grojnowski, G. Lusztig, A comparison of bases of quantized en-\nveloping algebras, Contemp. Math. 153 (1993), 11-19.\n[GLS]\nM. Gr ̈otschel, L. Lov ́asz, A. Schrijver, Geometric algorithms and\ncombinatorial optimzation, Springer-Verlag, 1993.\n[Gu]\nL. Gurvits, private communication.\n[Ha]\nR. Hartshorne, Algebraic geometry, Springer, 1997.\n[He]\nX. He, Efficient parallel and sequential algorithms for 4-coloring\nperfect planar graphs, Algorithmica (1990) 5: 545-559.\n[Hi]\nH. Hironaka, Resolution of singularities of an algebraic variety over\na field of characteristic zero, Annal of math. 79 (1964). I: 109-203,\nII: 205-326.\n[IW]\nR. Impagliazzo, A. Wigderson, P=BPP if E requires exponential\nsize circuits: Derandomizing the XOR lemma, in proceedings of\nAnnual ACM Symposium on the Theory of Computing, pages 220-\n229, 1997.\n[JS]\nJ. J ́aJ ́a, J. Simon., Parallel algorithms in graph theory: planarity\ntesting, SIAM J. Computing, 11 (2): 314-328, 1982.\n[Ji]\nM. 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Magyar, Standard monomial the-\nory and applications, in Representation theories and algebraic ge-\nometry, (Ed. A. Broer), Kluwer Academic Publishers, (1997), 319-\n364.\n[LP]\nM. Larsen, R. Pink, Determining representations from invariant\ndimensions, Invent. math. 102, 377-398 (1990).\n[LLL]\nA. Lenstra, H. Lenstra, Jr., L. Lov’asz, Factoring polynomials with\nrational coefficients, Mathematische Annalen 261 (1982), 515-534.\n[Le]\nA. Levin: Universal sequential search problems. Problems of infor-\nmation transmission (translated from Problemy Peredachi Infor-\nmatsii (Russian)) 9 (1973).\n[Li]\nP. Littelmann, Paths and root operators in representation theory,\nAnn. of Math. 142 (1995), 499-525.\n[Lb]\nA. Lubotzky, Discrete groups, expanding graphs, and invariant\nmeasures, Progress in mathematics, Boston, Birkh ̈auser, 1994.\n[LPS]\nA. Lubotzky, R. Phillips, P. Sarnak: Ramanujan graphs, Combi-\nnatorica 8 (1988), 261-277.\n[LV]\nD. Luna, Th. 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Mumford, Algebraic Geometry I: complex projective varieties,\nSpringer, 1995.\n[Mm2]\nD. Mumford, J. Fogarty, F. Kirwan: Geometric invariant theory.\nSpringer-Verlag, 1994.\n[N]\nH. Narayanan, On the complexity of computing Kostka numbers\nand Littlewood-Richardson coefficients, J. of Algebraic combina-\ntorics, vol. 24, issue 3, Nov. 2006.\n[NW]\nN. Nisan, A. Wigderson, Hardness vs. randomness, J. Comput.\nSys. Sci., 49 (2): 149-167, 1994.\n[O]\nJ. Orlin, A faster strongly polynomial minimum cost flow algo-\nrithm, proceedings of the twentieth Annual Symposium on Theory\nof Computing, 1988.\n[Pe]\nG. Perelman, The entropy formula for the Ricci flow and its geo-\nmetric applications, arXiv:math.DG/0211159.\n[Rs]\nE. Rassart, A polynomiality property for Littlewood-Richardson\ncoefficients, arXiv:math.CO/0308101, 16 Aug. 2003.\n[RR]\nA. Razborov, S. Rudich, Natural proofs, J. Comput. System Sci.,\n55 (1997), pp. 24-35.\n[Re]\nO. 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Valiant, The complexity of computing the permanent, Theoret-\nical Computer Science 8, pp 189-201, 1979.\n[Ve]\nS. Vempala, Private communication.\n102"},{"paragraph_id":"p104","order":104,"text":"[W]\nH. Weyl, Classical groups, Princeton University Press, 1939.\n[Wo]\nS. Woronowicz: Compact matrix pseudogroups, Commun. Math.\nPhys. 111 (1987), 613-665.\n[Z]\nA.\nZelevinsky,\nLittlewood-Richardson\nsemigroups,\narXiv:math.CO/9704228 v1 30 Apr 1997.\n103"}],"pages":[{"page":1,"text":"arXiv:0709.0748v1 [cs.CC] 5 Sep 2007\nOn P vs. NP, Geometric Complexity Theory, and\nThe Flip I: a high-level view\nDedicated to Sri Ramakrishna\nKetan D. Mulmuley ∗\nThe University of Chicago\nhttp://ramakrishnadas.cs.uchicago.edu\nTechnical Report TR-2007-13, Computer Science Department,\nThe University of Chicago\nSeptember, 2007\nNovember 7, 2018\nAbstract\nGeometric complexity theory (GCT) is an approach to the P vs.\nNP and related problems through algebraic geometry and representa-\ntion theory. This article gives a high-level exposition of the basic plan\nof GCT based on the principle, called the flip, without assuming any\nbackground in algebraic geometry or representation theory.\nContents\n1\nIntroduction\n4\n1.1\nThe flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n7\n1.1.1\nFrom nonexistence to existence . . . . . . . . . . . . .\n7\n1.1.2\nFrom hard to easy . . . . . . . . . . . . . . . . . . . .\n10\n1.2\nThe P-barrier and its crossing . . . . . . . . . . . . . . . . . .\n11\n∗Part of this work was done while the author was visiting I.I.T. Mumbai\n1"},{"page":2,"text":"1.3\nWhy should PH1 and PH2 hold? . . . . . . . . . . . . . . . .\n13\n1.4\nThe reduction . . . . . . . . . . . . . . . . . . . . . . . . . . .\n15\n1.5\nTowards PH1 and SH via PH0\n. . . . . . . . . . . . . . . . .\n16\n1.6\nNonstandard quantum groups . . . . . . . . . . . . . . . . . .\n18\n1.7\nNonstandard Riemann hypotheses? . . . . . . . . . . . . . . .\n19\n1.8\nObstructions vs. expanders\n. . . . . . . . . . . . . . . . . . .\n22\n1.9\nIs there a simpler proof technique? . . . . . . . . . . . . . . .\n23\n1.10 Organization of the paper . . . . . . . . . . . . . . . . . . . .\n24\n2\nBasics in algebraic geometry and representation theory\n24\n2.1\nRepresentation theory . . . . . . . . . . . . . . . . . . . . . .\n26\n2.1.1\nIrreducible representations of GLn(C) . . . . . . . . .\n27\n2.1.2\nIrreducible representations of the symmetric group . .\n29\n2.1.3\nTensor products\n. . . . . . . . . . . . . . . . . . . . .\n30\n2.2\nAlgebraic geometry . . . . . . . . . . . . . . . . . . . . . . . .\n32\n3\nGroup-theoretic varieties\n34\n4\nClass varieties\n35\n4.1\nNC vs. P #P\n. . . . . . . . . . . . . . . . . . . . . . . . . . .\n36\n4.2\nP vs. NP problem over C . . . . . . . . . . . . . . . . . . . .\n39\n5\nObstructions\n40\n6\nWhy should obstructions exist?\n41\n7\nThe flip\n44\n8\nWhy should the flip work?: the P-barrier\n45\n9\nOn crossing the P-barrier\n46\n9.1\nA basic prototype with constant depth complexity\n. . . . . .\n46\n9.2\nFrom constant to superpolynomial depth . . . . . . . . . . . .\n48\n9.3\nSaturated and positive integer programming . . . . . . . . . .\n49\n2"},{"page":3,"text":"10 Why should PH1 and PH2 hold?\n51\n11 Decision problems in representation theory\n52\n11.0.1 Littlewood-Richardson problem . . . . . . . . . . . . .\n53\n11.0.2 Kronecker problem . . . . . . . . . . . . . . . . . . . .\n53\n11.0.3 The plethysm problem . . . . . . . . . . . . . . . . . .\n53\n12 The P-barrier in representation theory\n54\n12.1 Crossing the P-barrier . . . . . . . . . . . . . . . . . . . . . .\n54\n13 Reduction\n56\n13.1 Towards easy discovery . . . . . . . . . . . . . . . . . . . . . .\n57\n13.2 From easy algorithm for discovery to easy proof of existence .\n59\n14 Standard quantum group\n61\n15 Nonstandard quantum groups\n65\n15.1 Quantization\n. . . . . . . . . . . . . . . . . . . . . . . . . . .\n66\n15.2 PH0 for couples and triples\n. . . . . . . . . . . . . . . . . . .\n70\n15.3 PH0 for class varieties . . . . . . . . . . . . . . . . . . . . . .\n70\n15.4 PH1 and SH . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n70\n16 Ultimate mystery: nonstandard Riemann hypotheses?\n71\n17 Obstructions vs. expanders\n72\n18 On relativization and P/poly-naturalization barriers\n74\n18.1 From usefulness to superpolynomial lower bounds . . . . . . .\n77\n18.2 On violation of the largeness constraint\n. . . . . . . . . . . .\n79\n18.3 On violation of the constructivity constraint . . . . . . . . . .\n80\n19 P-verifiable and P-constructible proof techniques and their\nexplicit construction complexity\n80\n19.1 P-verifiable proof technique . . . . . . . . . . . . . . . . . . .\n80\n19.2 P-barrier for verification . . . . . . . . . . . . . . . . . . . . .\n83\n3"},{"page":4,"text":"19.3 P-constructible proof technique . . . . . . . . . . . . . . . . .\n85\n19.4 General setting . . . . . . . . . . . . . . . . . . . . . . . . . .\n86\n19.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . .\n88\n19.5 Explicit construction complexity\n. . . . . . . . . . . . . . . .\n90\n19.6 Is there a simpler proof technique? . . . . . . . . . . . . . . .\n94\n1\nIntroduction\nGeometric complexity theory (GCT) is a plausible approach to the P vs\nNP [Co, Ka, Le] and related problems in complexity theory via algebraic\ngeometry and representation theory. The goal of this paper is to give a high-\nlevel overview of its basic plan and the underlying principle called the flip,\nwithout assuming any background in algebraic geometry or representation\ntheory. A detailed exposition for mathematicians will appear in [GCTflip2].\nA brief proposal and announcement appeared earlier in cf.[GCTconf]. The\nflip has been partially implemented in a series of papers [GCT1]-[GCT11].\nThis article, followed by [GCTintro], should provide an introduction to the\noverall structure of GCT for computer scientists who wish to get a high-level\npicture before going any further. We assume a few elementary notions of\nalgebraic geometry and representation theory in this introduction. They are\ndescribed in full detail in Section 2, which can be referred to if necessary. For\nthe readers looking for a quick overview, the article [GCTabs], which gives a\nnontechnical synopsis of this paper, followed by just this introduction, which\nhas been written to read as a short paper, should suffice.\nIn this article, the underlying field of computation is taken be C. In\n[GCT11], the problems that arise in the context of the flip over an alge-\nbraically closed field of positive characteristic, or a finite field are discussed.\nThe usual P ̸= NP conjecture is over a finite field of which the one over C\nis in a sense the crux and, being also a formal implication [GCT1], has to\nbe proved first anyway.\nThe flip, in essence, “reduces” the negative hypotheses (lower bound\nproblems) in complexity theory, such as the P ̸=?NP conjecture over C, to\npositive hypotheses in complexity theory (upper bound problems): specifi-\ncally, to showing that a series of decision problems in representation theory\nand algebraic geometry belong to the complexity class P. The “reduction”\nhere is only “in essence”.\nIt is not a formal Turing machine reduction.\nIf it were, it would be relativizable. It is described briefly in Section 1.4\n4"},{"page":5,"text":"below, and in detail in Section 13 later. This reduction basically consti-\ntutes a flip from hard, nonexistence to easy existence.\nIn [GCT6], these\ncomplexity-theoretic positive hypotheses are further reduced to mathemat-\nical positivity hypotheses, supported by the theoretical and experimental\nevidence therein. The mathematical positivity hypotheses roughly say that\ncertain nonnegative structural functions in algebraic geometry and repre-\nsentation theory have positive formulae–i.e., formulae without alternating\nsigns–akin to the usual formula for the permanent (in contrast, the usual\nformula for the determinant has alternating signs). It turns out that the\nvalidity of these mathematical positivity hypotheses is intimately linked to\nthe Riemann hypothesis over finite fields–proved in [Dl2] as a culmination\nof extensive effort in mathematics–and the related works in algebraic ge-\nometry and the theory of quantum groups [BBD, KL2, Kas3, Lu1, Lu2].\nIn [GCT6], a plan is suggested for proving them via the theory of quan-\ntum groups. Generalizations of the standard quantum group [Dri, Ji, RTF]\nneeded for this purpose, which we call nonstandard quantum groups, are\nconstructed in [GCT4, GCT7], with further conjectural extensions pointed\nout in [GCT10]. All papers of GCT together suggest that if the Riemann\nhypothesis over finite fields and the related works in the theory of standard\nquantum groups mentioned above can be systematically extended to the\nsetting of the nonstandard quantum groups that arise in GCT, then this\nmay lead to the proof of the P ̸= NP conjecture over C. This basic plan\nof GCT is summarized in Figure 1. Question marks indicate the main open\nproblems.\nThe proof in characteristic zero may eventually extend to finite fields, as\nin the usual form of the conjecture, along the lines suggested in [GCT11].\nThus the ultimate goal of the GCT flip is to deduce the ultimate nega-\ntive hypothesis of mathematics, the P ̸= NP conjecture, in essence, from\nthe ultimate positive hypotheses in mathematics, (nonstandard) Riemann\nHypotheses, thereby giving the ultimate flip shown in Figure 2.\nIn the rest of this introduction, we elaborate Figure 1 further.\nAcknowledgement\nThe author is deeply grateful to Madhav Nori, who taught him algebraic\ngeometry, Milind Sohoni, who collaborated in GCT 1-4, and Manju the\nsource of energy behind this work. The author is also grateful to A. Razborov\nfor pointing out the need for a high-level account. This article is essentially\nan elaboration of the answers to his questions. A part of this work was done\n5"},{"page":6,"text":"Complexity theoretic negative hypotheses (lower bound problems)\n|\n|\nThe flip\n|\n↓\nComplexity theoretic positive hypotheses (upper bound problems)\n|\n|\nGCT6|\n|\n↓\nMathematical positivity hypotheses |\n|\n?\n|\n↓\n(?): Nonstandard extensions of the Riemann hypothe-\nsis over finite fields, and the related works in algebraic\ngeometry and the theory of quantum groups\nFigure 1: The basic plan of GCT\n6"},{"page":7,"text":"(?): Nonstandard Riemann Hypotheses (+)\n|\n|\n?\n|\n↓\nThe P ̸= NP conjecture (-)\nFigure 2: The ultimate goal of the flip\nwhile the author was visiting I.I.T. Mumbai to which the author is grateful\nfor its hospitality. It is also a pleasure to thank the graduate students who\ntook the accompanying introductory course [GCTintro] on GCT for their\nfeedback.\n1.1\nThe flip\nWe begin with the top arrow in Figure 1: the flip. It is motivated by the\nclassical flip–from the undecidable (negative) to the decidable (positive)–\nthat occurs in G ̈odel’s incompleteness theorem.\nAll known lower bound\nresults–e.g. the hierarchy theorems in complexity theory or the lower bound\nresults in the constant depth [BS] or the PRAM model without bit oper-\nations [Mu1]–depend on flips from lower bounds to upper bounds of some\nsort. But such variations of the classical flip cannot work in the context\nof the P vs. NP problem because they are either relativizable [BGS] or\nnaturalizable [RR]. In contrast, the flip here should be nonrelativizable and\nnonnaturalizable (Section 18).\nThere are actually two flips within this flip: (1) from nonexistence to\nexistence, and (2) from hard to easy.\nHere hard means: the problem of\ndeciding if a computational circuit of size m exists for a given function\nf(x) = f(x1, . . . , xn) is hard. Accordingly, the flip from hard nonexistence\nto easy existence goes in two stages.\n1.1.1\nFrom nonexistence to existence\nThe flip from nonexistence to existence is addressed in [GCT1, GCT2]. Here\nthe nonexistence (lower bound) problem is reduced to an existence problem:\n7"},{"page":8,"text":"specifically, to the problem of proving existence of obstructions, which serve\nas “proofs” or “witnesses” for nonexistence of an efficient computational\ncircuit for the explicit hard function in the lower bound problem under\nconsideration.\nJust as existence of a forbidden Kurotowoski minor in a\ngraph serves as an obstruction, i.e., a “proof” for nonexistence of a planar\nembedding.\nAn obstruction in [GCT1, GCT2] is intuitively defined as follows. First\na specific (co)-NP-complete function E(X) = E(x1, . . . , xn), and a specific\nP-complete function H(Y ) = H(y1, . . . , yl) are constructed in [GCT1] so as\nto have special properties that we shall describe in a moment. Using H(Y ),\na projective algebraic variety XP (l) = XP (H; l), for every positive integer l,\nis associated with the complexity class P, called the class variety associated\nwith P, or the simply the P-variety. Here, a projective algebraic variety\nmeans the zero set of a system of homogeneous polynomial equations (cf.\nSection 2.2). These are generalizations of the familar curves and surfaces.\nIt will turn out that XP (l) is a G-variety for G = GLl(C), the group of\ninvertible l × l complex matrices. This means elements of G act on this\nvariety as its transformations–i.e., move its points around–just as G acts on\nCl in the usual way. Similarly, using E(X), a projective variety XNP (n, l) =\nXNP (E; n, l), for every positive integer n and l ≥n, is associated with the\ncomplexity class NP.\nIt is called the class variety associated with NP,\nor simply the NP-variety.\nIt will again be a G-variety.\nThe functions\nE(X) and H(Y ) have been specially chosen so that these class varieties are\nexceptional and their algebraic geometry can be analyzed in depth. If E(X)\ncan be computed by a circuit of size m, then it would turn out that XNP (n, l)\ncan be embedded in XP (l) as a G-subvariety for l = O(m2). Pictorially:\nXNP (n, l) ֒→XP (l).\n(1)\nWe want to show that this embedding is impossible if m = poly(n), as\nn →∞. This would show that E(X) cannot be computed by a circuit of\nm = poly(n) size, and hence, P ̸= NP over C.\nLet R(n, l) = R(E; n, l) and S(l) = S(H; l) denote the homogeneous\ncoordinate rings of XNP (E; n, l) and XP (H; l), respectively. Here by the\ncoordinate ring of a variety, we mean the ring of polynomial “functions” (of\nsome kind) on the variety as defined in Section 2.2. These are akin to the\nring of polynomial functions on Cl. Since the class varieties are G-varieties,\nthese homogeneous coordinate rings will be G-representations (Section 2.1).\nBy a G-representation we mean a vector space on which the elements of G\nact as linear transformations, just as they do on Cl. If the embedding (1)\n8"},{"page":9,"text":"exists, then it would turn out that R(n, l) is a G-subrepresentation of S(l).\nWe say that an irreducible, i.e., a minimal, nonzero representation W of G\nis an obstruction, for given n and l, if it occurs as a G-subrepresentation\nof R(n, l), but not as a G-subrepresentation of S(l). Existence of such a\nW, for given n and l, implies that R(n, l) cannot be embedded as a G-\nsubrepresentation of S(l), and hence, the embedding (1) cannot exist. Thus\nan obstruction serves as a “witness” or a “proof” that the embedding (1)\ncannot exist.\nWe now reformulate this notion of obstruction using a few basic notions\nin representation theory described in Section 2.1. It is known that (polyno-\nmial) irreducible representations of G are in one-to-one correspondence with\nthe set of sequences, also called partitions, λ : λ1 ≥λ2 · · · λk > 0 of positive\nintegers of length k ≤l. The irreducible representation of G labelled by λ is\ncalled a Weyl-module, and is denoted by Vλ(G). It is also known that each\nfinite dimensional representation V of G can be written as a direct sum of\nirreducible representations:\nV =\nM\nλ\nmλVλ(G),\nwhere mλVλ(G) denotes the direct sum of mλ copies of Vλ(G), and each\nmλ, called the multiplicity of Vλ(G) in V , is uniquely defined. Thus Vλ(G)\noccurs in V as a subrepresentation iffthe multiplicity mλ is nonzero.\nLet R(E; n, l)d and S(H; l)d denote the subspaces in R(E; n, l) and S(H; l),\nrespectively, of forms of degree d.\nLet sλ\nd(H; l) denote the multiplicity\nof Vλ(G) in S(H; l)d.\nLet sλ\nd(E; n, l) denote the multiplicity of Vλ(G) in\nR(E; n, l)d. Then Vλ(G) is an obstruction for given n and l ifffor some d\nsλ\nd(E; n, l) is nonzero but sλ\nd(H; l) is zero. Here d is uniquely determined\nby the size P\ni λi of λ. We also say that Vλ(G) is an obstruction of degree\nd, and by an abuse of language, also that the label λ is an obstruction of\ndegree d.\nThe main algebro-geometric result of [GCT2] (Theorem 6.3) indicates\nthat such obstructions should exist in the context of the P vs. NP problem,\nwhen m = poly(n), assuming that P ̸= NP, as we expect. The goal then is\nto show that obstructions indeed exist, as expected, for all n →∞, assuming\nm = poly(n). The story is similar for other related lower bound problems.\nThis addresses the easier half of the flip from nonexistence to existence.\n9"},{"page":10,"text":"1.1.2\nFrom hard to easy\nBut how should one prove that obstructions actually exist? The main hy-\npothesis governing the flip, which addresses this question, is the following\none that constitutes the harder half of the flip: from hard to easy.\nHypothesis 1.1 (PHflip1) Consider the P vs. NP problem over C. Let\nE(X) be the explicit function in [GCT1] mentioned above. Then the follow-\ning problems are “easy”; i.e., belong to P. Specifically,\n(a) Verification of an obstruction:\ngiven n, l and the partition λ,\nwhether Vλ(G) is an obstruction for given n and l can be decided in poly(n, l, ⟨λ⟩)\ntime, where ⟨λ⟩denotes the bitlength of the specification of λ.\n(b) Explicit construction of obstructions: Suppose l = nlog n (say).\nThen, for every n →∞, a label λ(n) of an obstruction Vλ(G) for n and l\ncan be constructed explicitly in poly(n, l) time, thereby proving existence of\nan obstruction for every such n and l.\nIn view of the definition of an obstruction, the statement (a) for verifi-\ncation clearly follows from:\nHypothesis 1.2 (PHflip2) The following the decision problems are easy;\ni.e., belong to P. Specifically,\n(a) Given d, n, l and a partition λ, whether sλ\nd(E; n, l) is nonzero, i.e.,\nwhether Vλ(G) occurs as a G-subrepresentation of R(n, l)d can be decided\nin poly(⟨d⟩, ⟨λ⟩, n, l) time. Here ⟨d⟩denotes the bitlength of d.\n(b) Given d, l and a partition λ, whether sλ\nd(H; l) is nonzero, i.e., whether\nVλ(G) occurs as a G-subrepresentation of S(l)d can be decided in poly(⟨d⟩, ⟨λ⟩, l)\ntime.\nThe decision problems in Hypothesis 1.2 are the crux of the matter.\nOnce easy algorithms for these decision problems are found, the goal is to\nprove existence of an obstruction for every n →∞, when l = nlog n (say),\nby constructing such an obstruction explicitly, as per Hypothesis 1.1 (b).\nWe shall discuss how this is to done in Section 1.4 below. Assuming for\nthe moment that this transformation of easy algorithms for the decision\nproblems in Hypothesis 1.2 into an easy procedure for explicit construction\nof obstructions (Hypothesis 1.1(b)) for all n →∞, when l = nlog n, works,\nwe get the “reduction” shown in the top arrow of Figure 1: from the original\nhard nonexistence (lower bound) problem to the basic upper bound problems\nin Hypothesis 1.2.\n10"},{"page":11,"text":"1.2\nThe P-barrier and its crossing\nBut, by divine justice, the task of showing that the problems in Hypothe-\nsis 1.2 are easy turned out to be extremely hard. Thus, paradoxically, the\nhardest aspect of the flip is just to prove that the basic decision problems\nthat arise in the construction of obstructions are actually easy; i.e., belong\nto P. The best algorithms for these decision problems obtained using the\ngeneral purpose algorithms in algebraic geometry and representation theory\ntake space that is double exponential in m and time that is triple expo-\nnential in m. This means even verification of an obstruction, let alone its\ndiscovery, takes time that is triple exponential in m if one were to use the\ngeneral purpose techniques.\nThe gap between this triple exponential time bound and the polynomial\ntime bound sought in Hypothesis 1.2 is so huge that, at the surface, this\nhypothesis may seem impossible. This was the main barrier, called the P-\nbarrier (Section 8), on this path towards the P vs NP problem when the\nflip was briefly announced in [GCTconf].\nThe article [GCT6] says that it can be crossed under reasonable mathe-\nmatical assumptions. We now turn to a brief description of these results.\nFor that we need a few definitions.\nWe say that a function f(k), k a nonnegative integer, is a quasi-polynomial\nif for some integer l ≥1 there exist polynomials fi(k), 1 ≤i ≤l, such that\nf(k) = fi(k) if k = i modulo l. Here l is called the period of the quasi-\npolynomial. An important example of a quasi-polynomial is the Ehrhart\nquasi-polynomial fP(k) of a polytope P. By definition, it is the number of\ninteger points in the dilated polytope kP. This is known to be a quasi-\npolynomial [St1].\nWe say that a quasi-polynomial f(k) is positive, if the coefficients of all\nfi(k) are nonnegative. We say that it is saturated if either f1(k) is identically\nzero as a polynomial, or if not, f(1) = f1(1) ̸= 0. If f(k) is positive, it is\nclearly saturated.\nNext, let us associate with the multiplicities sλ\nd(H; l) and sλ\nd(E; n, l) the\nfollowing stretching functions:\n ̃sλ\nd(H; l)(k) = skλ\nkd(H; l),\n(2)\nand\n ̃sλ\nd(E; n, l)(k) = skλ\nkd(E; n, l).\n(3)\n11"},{"page":12,"text":"The following is the main algebro-geometric result in [GCT6].\nTheorem 1.3 (cf. Theorem 3.4.11 in [GCT6])\n(Rationality Hypothesis): Assume that the singularities of the class varieties\nXP (H; m) and XNP(E; n, l) are “nice” (rational).\nThen the stretching functions ̃sλ\nd(H; l)(k) and ̃sλ\nd(E; n, l)(k) are quasi-\npolynomials.\nWe do not need to know the exact definition of a rational singularity here,\nwhich can be found in [Ke]. It just means that the singularities are nice.\nThis depends on the exceptional nature of the class varieties (cf. Section 4)\nand is supported by the algebro-geometric results and arguments in [GCT2,\nGCT10].\nUsing Theorem 1.3, we can now formulate the conjectural mathematical\npositivity hypotheses mentioned in the third box from above in Figure 1.\nAssume the rationality hypothesis above.\nHypothesis 1.4 (PH1:) The structural constant sλ\nd(H; l) can be expressed\nas the number of integer points in a polytope P λ\nd (H; l) of poly(l, ⟨d⟩, ⟨λ⟩)\ndimension, whose Ehrhart quasi-polynomial coincides with the stretching\nquasi-polynomial ̃sλ\nd(H; l)(k) in Theorem 1.3. Furthermore, P λ\nd (H; l) can\nbe given in the form of a poly(l, ⟨d⟩, ⟨λ⟩)-time separation oracle as in [GLS].\nThere exists a polytope P λ\nd (E; n, l) for the structural constant sλ\nd(E; n, l)\nwith similar properties.\nThis, in particular, implies that sλ\nd(H; l) and sλ\nd(E; n, l) belong to #P.\nHypothesis 1.5 PH2: The quasi-polynomials ̃sλ\nd(H; l) and ̃sλ\nd(E; n, l) in\nTheorem 1.3 are positive.\nIts weaker form is:\nHypothesis 1.6 (SH:) These quasi-polynomials are saturated.\nPH1 and SH (PH2) together say that each decision problem in Hypothe-\nsis 1.2 can be transformed in polynomial time into a special kind of an integer\nprogramming problem called saturated (resp. positive) integer programming\nproblem (Section 9.3).\n12"},{"page":13,"text":"Theorem 1.7 (cf. [GCT6]) The decision problems in Hypothesis 1.2 are\nindeed in P, assuming PH1 and SH (or more strongly PH2) above.\nThis follows from a polynomial time algorithm in [GCT6] for saturated (pos-\nitive) integer programming.\nThis result reduces the positive complexity-theoretic hypotheses in Hy-\npothesis 1.2 to the mathematical positivity hypotheses PH1 and SH, as\nshown in the middle arrow in Figure 1. The algorithms in Theorem 1.7 are\nconceptually extremely simple. They just need linear programming [GLS]\nand computation of Smith normal forms [KB].\nBut their correctness depends on the positivity hypotheses PH1 and SH\n(PH2), whose validity, in turn, is intimately linked to deep phenomena in\nalgebraic geometry and the theory of quantum groups as we shall soon see.\nAn indication of such a link is already here. Since the proof of Theorem 1.3,\nwhich is necessary to even formulate these hypotheses, needs a few funda-\nmental results in algebraic geometry; namely, [Bou] (which in turn is based\non [Hi] and other results), and [Ke, Fl]. It should not then be surprising\nif the proofs the hypotheses need far more. Indeed, the quantum-group-\ntheoretic and algebro-geometric machinery is needed in GCT essentially to\nprove these hypotheses, and hence, that these extremely simple algorithms\nare actually correct.\n1.3\nWhy should PH1 and PH2 hold?\nBut first, we need to justify why these hypotheses should hold in the first\nplace. For that, let us consider the simplest analogue of the decision prob-\nlems in Hypothesis 1.2 in representation theory:\nProblem 1.8 (Littlewood-Richardson problem) Given partitions α, β and\nλ, decide if the Littlewood-Richardson coefficient cλ\nα,β (cf. Section 2.1.3) is\npositive (nonzero). This is defined to be the multiplicity of the irreducible\nrepresentation Vλ(G) in the tensor product Vα(G) ⊗Vβ(G) (which becomes\na G-representation by letting the elements of G act on its two factors simul-\ntaneously).\nThe analogous mathematical positivity hypotheses in this setting are as\nfollows.\nDefine the stretching function\n ̃cλ\nα,β(k) = ckλ\nkα,kβ,\nk ≥0,\n13"},{"page":14,"text":"which is obtained by stretching the Littlewood-Richardson coefficient by a\nfactor of k. It is known to be a polynomial [Der, Ki, Rs]. Then\nHypothesis 1.9 (PH1) The Littlewood-Richardson coefficient cλ\nα,β can be\nexpressed as the number of integer points in a polytope P = P λ\nα,β of dimen-\nsion polynomial in the total length of α, β and λ. Furthermore, the Ehrhart\nquasi-polynomial of P coincides with the stretching polynomial ̃cλ\nα,β(k) and\nthe membership function of P is computable in time that is polynomial in\nthe bit lengths of α, β and λ.\nThis is shown, for example, in [BZ]. There are many choices for P λ\nα,β.\nOne choice is called a hive polytope [KT1].\nHypothesis 1.10 (PH2) The coefficients of ̃cλ\nα,β(k) are nonnegative.\nThis implies:\nHypothesis 1.11 (SH) The stretching polynomial ̃cλ\nα,β(k) is saturated.\nSince ̃cλ\nα,β(k) is a polynomial, this simply means if ckλ\nkα,kβ is nonzero for\nsome k ≥1 then cλ\nα,β is also nonzero. PH2 is still open, but has a con-\nsiderable experimental evidence in its support [KTT].\nThat SH holds is\nthe saturation theorem in [KT1]. PH1 and SH in conjunction with linear\nprogramming leads [DM2, GCT3, KT2] to a polynomial time algorithm for\nthe Littlewood-Richardson problem (Problem 1.8), and a polynomial time\nalgorithm [GCT5] for a certain generalized Littlewood-Richardson problem\nassuming SH. These results were indeed a starting motivation for Theo-\nrem 1.7.\nThe Littlewood-Richardson coefficient is a special case of a far-reaching\nclass of fundamental constants in representation theory, called plethysm\nconstants, described in Section 11. The structural constants sλ\nd(H; l) and\nsλ\nd(E; n, l) can be considered to be “hyped up” versions of the plethysm con-\nstant. Considerable theoretical and experimental evidence in support of the\nanalogous positivity hypotheses PH1 and PH2 for the plethysm constants\nis given in [GCT6]; cf. Section 11. This constitutes the main evidence in\nsupport of PH1 and PH2 for sλ\nd(H; l), sλ\nd(E; n, l) and other similar algebro-\ngeometric structural constants that arise in GCT.\n14"},{"page":15,"text":"1.4\nThe reduction\nBefore we turn to the plan suggested in [GCT6] for proving PH1 and SH,\nwe explain the nature of the reduction in the top arrow of Figure 1.\nFor this, the easy algorithms in Theorem 1.7 have to be transformed into\nan easy procedure for explicit construction of obstructions as per Hypothe-\nsis 1.1 (b). This transformation cannot be carried out at present since we\ndo not have explicit descriptions of the polytopes P λ\nd (H; l) and P λ\nd (E; n, l)\nin PH1. But it is explained in Section 13 and in detail in [GCT6] why it\nshould be possible to carry out this transformation if PH1 and SH can be\nproved and explicit descriptions of the polytopes therein become available.\nThe scheme for transformation suggested there goes in two steps:\nFirst, the easy algorithms in Theorem\n1.7 have to be used to get an\neasy poly(n, l) procedure for discovering an obstruction (label) for given n\nand l, if one exists.\nSecond, this easy algorithm for discovering an obstruction, or rather its\nstructure and the underlying techniques have to be used to prove that an\nobstruction always exists for every n →∞, assuming l = nlog n, say. That\nis, to prove that this easy algorithm always says “yes” for such n and l. Just\nas the structure of the easy Hungarian method for discovering a perfect\nmatching in a bipartite graph can be used to prove Hall’s theorem that\nevery d-regular bipartite graph always has a perfect matching.\nThis transformation of an easy algorithm for discovery into an easy (i.e.\nfeasible) constructive proof–which we shall call a P-constructive proof–also\ngives, as a side product, an easy, i.e., polynomial time algorithm for explicit\nconstruction of obstructions (labels), as in Hypothesis 1.1 (b). One may\nwonder why we are going for explicit construction of obstructions, when\njust their existence would have sufficed. Because the nature of obstructions\nhere is such that the complexity deciding their existence and of constructing\nthem explicitly, if they do, should be more or less the same; cf. Section 13.2.\nJust as the complexity of deciding if a bipartite graph has a perfect matching\nis more or less the same as that of constructing one, if it exists,\nIn the context of these transformations it is crucial that the algorithms\nin Theorem 1.7 are not only easy, i.e., polynomial-time algorithms, but\nalso have a genuinely simple structure of the right kind, being just varia-\ntions of linear programming. Of course, we can not hope to use the ellip-\nsoid algorithm for linear programming–which though simple is intricate–for\na constructive proof of existence of obstructions. Rather we have to use\n15"},{"page":16,"text":"the structure of the underlying polytopes. The analogues of the polytopes\nP λ\nd (H; l) and P λ\nd (E; n, l) in PH1 in the simplified setting of the Littlewood-\nRichardson problem (Problem 1.8) are called hive polytopes [KT1]. These\nhave extremely regular structure. The same is expected to be the case for\nthe polytopes P λ\nd (H; l) and P λ\nd (E; n, l) that actually arise here. For this and\nother reasons given in [GCT6], it is expected that, once explicit descriptions\nof the polytopes P λ\nd (H; l) and P λ\nd (E; n, l) become available, the algorithms\nin Theorem 1.7 can be transformed into simple greedy Hungarian-type al-\ngorithms which do not even need linear programming.\nThis is the main\nreason why the transformation of these easy, polynomial time algorithms\ninto an easy (feasible) proof of existence of obstructions is expected to work\nin our setting, just as it does in the case of Hall’s theorem that we mentioned\nabove.\nAssuming that this works, we would get an explicit family {λ(n)} of\nobstructions (rather their labels), as n →∞, and l = nlog n. The existence\nof such an obstruction family would imply that P ̸= NP over C.\n1.5\nTowards PH1 and SH via PH0\nNow we turn to the basic plan suggested in [GCT6] for proving PH1 and\nSH. This will explain the bottom arrow in Figure 1.\nThis plan is motivated by the proof of PH1 (Hypothesis 1.9) in the\nsimplified setting of the Littlewood-Richardson problem via the theory of\nquantum groups [Kas1, Li, Lu2]. Specifically, it is known that this PH1 is a\nconsequence, in a nontrivial way, of a deep positivity statement in the the-\nory of standard quantum groups [Dri, Ji, RTF]–whose intuitive description\nis given later in Section 14–namely: their representations and coordinate\nrings have canonical bases [Kas2, Lu1, Lu2], whose structural constants de-\ntermining their representation-theoretic and multiplicative structure are all\nnonnegative. We shall refer to the existence of a canonical basis with this\npositivity property as PH0, the zeroth positivity hypothesis (property).\nMotivated by this work, certain positivity hypotheses, again called PH0,\nare formulated in [GCT6], and it is pointed out how and why these may\nsimilarly lead to the proof of the required PH1 and also SH (Hypotheses 1.4\nand 1.6). The PH0 hypotheses in [GCT6] may be thought of as generaliza-\ntions of PH0 in the theory of standard quantum groups. PH1 and SH for\nLittlewood-Richardson coefficients (Hypotheses 1.9 and 1.11) have purely\ncombinatorial proofs [F1, KT1], and hence, PH0 is strictly speaking not re-\nquired in this context. But in the context of the PH1 that we are finally\n16"},{"page":17,"text":"interested in (Hypothesis 1.4) the full power of PH0 seems needed for the\nplan in [GCT8, GCT10] to work.\nA natural approach to prove PH0 in [GCT6] in the context of this PH1\nis to somehow generalize the proof of PH0 in the theory of the standard\nquantum group. But the theory of standard quantum groups does not work,\nas expected, in this context. The reason is briefly as follows.\nOne can associate a complexity class with each structural constant that\narises in GCT, which we call its index class. Roughly, if a structural con-\nstant is associated with a class variety for a complexity class C, then its\nindex class is defined to C.\nFor example, the index classes of the mul-\ntiplicities sλ\nd(H; l) and sλ\nd(E; n, l) are P and NP (over C), since they are\nassociated with P- and NP-varieties, respectively. Similarly, the index class\nof the Littlewood-Richardson coefficient is the class of circuits (of restricted\nkinds) of depth two; cf. Section 9.1. The index class of the Kronecker coef-\nficient (Section 2.1.3), which is the analogue of the Littlewood-Richardson\ncoefficient in the representation theory of the symmetric group, is NC2, the\nclass of problems that can be solved by circuits of log2 n depth and polyno-\nmial size. The Littlewood-Richardson coefficient as well as the Kronecker\ncoefficient are special cases of the plethysm constants (Section 11.0.3) which\nwe mentioned earlier. The generalized plethysm constant is not associated\nwith any class variety, but it is qualitatively similar to, though much sim-\npler than sλ\nd(E; n, l). Hence, we define its index class to be NP, with the\nunderstanding that this is to be taken only in a rough sense. The index\nclasses of the structural constants here are not be confused with their usual\ncomputational complexity classes: they are all (conjecturally) in #P by\nPH1.\nThe standard quantum group is the quantum group that occurs in the\ncontext of PH1 for Littlewood-Richardson coefficients (Hypothesis 1.9). Hence,\nwe define its index class to be the same as that of Littlewood-Richardson\ncoefficients, i.e., the class of circuits of depth two. Thus the standard quan-\ntum group is the quantum group attached to constant-depth (depth-two)\ncircuits.\nGiven a big difference between the lower bound problems for constant\nand nonconstant depth circuits, it should not be a surprise if the standard\nquantum group cannot be used in the context of PH1 for the structural\nconstants that actually arise in GCT; cf. Section 16 for an intuitive mathe-\nmatical explanation for why this is so.\n17"},{"page":18,"text":"1.6\nNonstandard quantum groups\nWhat is needed then are quantum groups that can play the role of the stan-\ndard quantum group in the context of the decision problems and positivity\nhypotheses for these structural constants. The main result in this context\nis the following:\nTheorem 1.12 [GCT4] There exists a quantum group, which is qualita-\ntively similar to the standard quantum group, that can play such a role in\nthe context of the Kronecker coefficients.\n[GCT7] More generally, there exists a (possibly singular) quantum group that\ncan play such a role in the context of the generalized plethysm constants.\nA less informal statement will be given later (Theorem 15.1). A conjectural\nscheme for generalizing these quantum groups to the ones that can play such\na role in the context of sλ\nd(E; n, l), sλ\nd(H; l) and other structural constants\nin GCT is suggested in [GCT10]. We shall call the new quantum groups in\nTheorem 1.12 nonstandard, because, though they are qualitatively similar\nto the standard quantum group, they are also fundamentally different, as\nexpected.\nThus, standard corresponds to constant depth and nonstandard to non-\nconstant depth circuits.\nThe article [GCT8] gives a conjecturally correct algorithm to construct\ncanonical bases of the irreducible representations and coordinate rings of the\nnonstandard quantum groups in [GCT4, GCT7] with the required positivity\nproperties (PH0). These are natural generalizations of the canonical basis\ndue to Kashiwara and Lusztig [Kas2, Lu1, Lu2] mentioned above for the\nirreducible representations and the coordinate ring of the standard quantum\ngroup.\n[GCT8] also gives a conjecturally correct algorithm to construct\ncanonical bases with similar positivity properties (PH0) for the nonstandard\ndeformations of the symmtric group algebra that are dually paired with the\nnonstandard quantum groups–these generalize the Kazhdan-Lusztig basis\n[KL1] of the Hecke algebra. It is also shown in [GCT7, GCT8] that PH1 for\nthe plethysm constants follows from PH0 and other conjectural properties\nof these nonstandard canonical bases and quantum objects. The story for\nthe general constants sλ\nd(E; n, l) and sλ\nd(H; l) can be expected to be similar\n[GCT10].\nAt present we can neither prove correctness of the algorithms in [GCT8]\nfor constructing nonstandard canonical bases nor the required conjectural\n18"},{"page":19,"text":"properties for the reasons that we shall describe in a moment. But a consid-\nerable evidence is given in [GCT8] in support of PH0 for the nonstandard\nquantum group in [GCT4].\nIn the standard case, PH1 follows from PH0 in a more or less rigid\nway [Dh, Kas1, Li, Lu2]. This means the polytope that occurs in PH1 for\nthe Littlewood-Richardson coefficient (Hypothesis 1.9) is more or less de-\ntermined by the canonical basis for the standard quantum group–not com-\npletely, since there are a few choices for this polytope; e.g. a hive polytope\nin [KT1], or a polytope in [BZ]. But all these choices are intimately related.\nA common feature is that they all have extremely regular structures. The\nsame can be expected for the polytopes that should arise in the nonstandard\nsetting. This regularity is crucial for the final transformation of easy algo-\nrithms for the basic decision problems in Hypothesis 1.2 into easy algorithms\nfor explicit construction of obstructions; cf. Sections 1.4 and 13.\nExistence of nonstandard quantum groups of polylogarithmic [GCT4]\nand superpolynomial [GCT7] depth complexity, the conjecturally correct\nalgorithm in [GCT8] for constructing canonical bases (PH0) of their coordi-\nnate rings and irreducible representations, and the principle that is suggested\nby the theory of standard quantum groups–namely, once a canonical basis is\nthere (PH0), everything else in the story more or less follows a rigid path–is\nthe main reason why GCT may be expected to deliver lower bounds for\ncircuits of superpolynomial depth and size eventually.\n1.7\nNonstandard Riemann hypotheses?\nBut for this plan to work, PH0 for the nonstandard quantum groups has\nto be proved. This brings us to the main open question in this story: how\ncan we prove correctness of the algorithm in [GCT8] for constructing the\ncanonical bases (PH0) of the coordinate rings of the nonstandard quantum\ngroups?\nThere are two constructions of the canonical basis in the standard set-\nting. An algebraic construction in [Kas3], where it is called global crystal\nbasis, and a topological construction in [Lu1, Lu2]. Both constructions give\nrise to the same basis [GL]. In fact, both constructions follow the same basic\nscheme. Only the proofs of correctness of this basic scheme are different. The\ntopological proof is based on the theory of perverse sheaves [BBD], which\nin turn, is based on Riemann hypothesis over finite fields [Dl2]. In essence,\nPH0 is thus ultimately deduced in the topological proof from the Riemann\nHypothesis over finite fields, which is again a deep positive statement. Be-\n19"},{"page":20,"text":"cause its usual statement is, after all, a positive statement, and it can also be\nreformulated as stipulating positivity (nonnegativity) of some mathematical\nquantities (cf. page 458 in [Ha]). The topological proof also gives, as a side\nproduct, the only known proof of nonnegativity of the structural constants\nassociated with the canonical basis in the standard setting. Though this\nnonnegativity is not needed for proving PH1 for the Littlewood-Richardson\ncoefficients, it is crucial in the nonstandard setting for the reasons given in\n[GCT8, GCT10].\nFor this reason, the topological approach seems to be the only viable\noption in the nonstandard setting, as far as we can see. Besides, the algebraic\ncomplexity of the nonstandard quantum groups is so huge–as to be expected\nin view of the huge gap between constant and nonconstant depth circuits–\nthat a purely algebraic proof of correctness of the algorithm in [GCT8] for\nconstructing canonical bases in the nonstandard setting seems difficult.\nBut the standard Riemann hypothesis over finite fields and the related\ntechniques cannot be expected to work in the the nonstandard setting for\nthe reasons given in [GCT7, GCT8]. Again this should not be surprising\ngiven the big difference between constant and nonconstant depth circuits.\nHence what seem to be needed [GCT8] to make the topological approach\nwork in the nonstandard setting are nonstandard extensions of the Riemann\nhypothesis over finite fields and the related work on perverse sheaves. By\nnonstandard, we mean the extensions that will work in the context of the\nnonstandard quantum groups.\nThe author does not have the mathematical expertize to even formulate\nsuch hypotheses, let alone prove them. But the theoretical and experimental\nevidence in [GCT4, GCT7, GCT8] (cf. Section 16) suggests that such exten-\nsions exist, and that they ought to be provable by a systematic extension of\nthe theory of standard quantum groups to the nonstandard setting. Hence\nit is reasonable to hope that the experts would be able to do so eventually,\nleading to the proof of PH0 hypotheses along the topological lines, and fi-\nnally, to the explicit construction of obstructions as outlined above, which\nwould then imply that P ̸= NP over C. The whole picture is summarized\nin Figure 3, which is an elaboration of the earlier Figure 1. The arrows with\nquestion marks are conjectural, the double arrows are unconditional. The ?\nsigns indicate the main open problems at the heart of this approach. The\nstory over C may eventually lift to the story over finite fields along lines\nsuggested in [GCT11].\n20"},{"page":21,"text":"(?): Nonstandard Riemann Hypotheses for the quantum groups in\n[GCT4, GCT7], and their conjectural extensions in [GCT10]\n|\n|\n|\n?\n↓\nPH0 (?): Existence of canonical bases [GCT6, GCT8, GCT10] |\n|\n?\n↓\nPH1,SH (PH2)\n∥\n∥\nGCT6\n∥\n⇓\nPolynomial time algorithms for the de-\ncision problems in Hypothesis 1.2\n|\n|\nThe transformation mentioned in Section 1.4; cf Section 13 and [GCT6]\n|\n?\n↓\nExplicit construction of obstructions\n∥\n∥\n∥\n⇓\nP ̸= NP over C\nFigure 3: The basic plan for implementing the flip in GCT6\n21"},{"page":22,"text":"1.8\nObstructions vs. expanders\nAn initial motivation for going for explicit construction of obstructions as\nin Figure 3 was provided by explicit construction of expanders [LPS, Ma].\nAs explained in Section 17, the obstructions in GCT are in a certain sense\ngeneralizations of the expanders from constant depth to superpolynomial\ndepth circuits. Specifically, obstructions are to superpolynomial depth cir-\ncuits what expanders are to constant depth, in fact, depth two circuits; cf.\nFigure 4. In view of this relationship, explicit construction of obstructions as\nin Figure 3 would be in the setting of superpolynomial depth circuits what\nexplicit construction of expanders is in the setting of constant depth circuits.\nAs we remarked earlier, the standard quantum group also corresponds to\ncircuits of depth two. That is, expanders and the standard quantum group\nboth correspond to the class of depth-two circuits. Hence it does not seem to\nbe a coincidence that the Riemann hypothesis over finite fields, which enters\nin the theory of the standard quantum group, also enters in the theory of\nexpanders [Lb, Sr].\n↑\ndepth\n|\nCircuits of superpolynomial depth and size: Obstructions\n↑\n|\nCircuits of depth two: expanders\nFigure 4: The relationship between obstructions and expanders\nExistence of expanders can be proved by a simple probabilistic method.\nIn contrast, existence of expanders may not be provable by a probabilistic\nmethod. Indeed, this is roughly the main content of [RR], which says that a\nnonconstructive method, such as a probabilistic method, should not work in\nthe context of the P vs. NP problem under reasonable assumptions. This\nis why in GCT we go for explicit construction of obstructions in the spirit\nof explicit construction of expanders. The P/poly-naturalizability barrier in\n[RR] should not be applicable to such explicit, constructive proof techniques.\nThis issue is addressed in more detail in Section 18.\n22"},{"page":23,"text":"1.9\nIs there a simpler proof technique?\nFinally, one may ask if the P ̸= NP conjecture may be proved by a sub-\nstantially simpler proof technique.\nThis seems unlikely for the following\nreasons.\nThe results in complexity theory such as [A2, Re] suggest that explicit\nconstructions may be more or less essential for derandomization. In conjunc-\ntion with the hardness vs. randomness principle [KI, NW], this suggests that\nexplicit constructions may also be more or less essential for (the difficult)\nlower bound problems as well.\nHence, the difficulty in any viable proof\ntechnique for the P ̸= NP conjecture may be intimately linked to the diffi-\nculty (complexity) of the explicit construction of obstructions, i.e., “proofs\nof hardness” as per that technique. This may be so regardless of whether the\ntechnique actually constructs such obstructions explicitly or not. Because,\nas per the existence-vs-construction principle [KUW], the difficulty of decid-\ning existence may be more or less the same as that of construction in natural\nproblems. These and other considerations naturally lead to a notion of ex-\nplicit construction complexity of an easy-to-verify proof technique towards\nthe P ̸= NP conjecture, where easy-to-verify formally means P-verifiable;\ncf. Section 19.\nThe explicit construction (depth) complexity of expanders is O(1), in\nfact, two, since they can be constructed by (nonuniform) depth-two algebraic\ncircuits (over a ring of integers modulo k for some k) [LPS, Ma]. Whereas,\nas per Hypothesis 1.1, the explicit construction (depth) complexity of the\nobstructions in GCT over C is poly(m), m = nlog n (say) being the circuit\nsize parameter in the lower bound problem; cf. Figure 4. The arguments in\nSection 19 suggest that this may be essentially the best explicit construction\ncomplexity that one can expect in any P-verifiable proof technique towards\nthe P ̸= NP conjecture. In other words, the massive Ω(m) gap between the\nexplicit construction complexity of obstructions and the O(1) explicit con-\nstruction complexity of expanders, as shown in Figure 4, may be inevitable\nin any P-verifiable proof technique towards the P ̸= NP conjecture. If so,\nGCT may be among the “easiest” P-verifiable approaches to this conjecture\nas per the explicit construction complexity measure defined here, and hence,\nit may be unrealistic to expect a technique that is substantially simpler or\neasier.\nIn the rest of this article, we elaborate the plan in Figure 3 further\nand give a high-level description of the results in the GCT papers. Logical\ndependence among the GCT papers is shown in Figure 5.\n23"},{"page":24,"text":"1.10\nOrganization of the paper\nIn Section 2 we recall a few basic facts in algebraic geometry and represen-\ntation theory which are easy to state and should be easy to believe. The\nreaders not familar with these fields should be able to take these on faith.\nIn Section 3 we describe a special class of algebraic varieties, called group-\ntheoretic varieties. All class varieties in GCT are group-theoretic varieties.\nThey are described in Section 4. Obstructions are defined in Section 5. Why\nthey should exist is described in Section 6. The flip is described in Sec-\ntion 7. The main barrier in the implementation of the flip, the P-barrier,\nis described in Section 8. The main result of GCT that crosses this barrier,\nassuming the mathematical positivity hypotheses PH1 and SH (PH2), is\ndescribed in Section 9. Why PH1 and PH2 should hold is described in Sec-\ntion 10. Simpler analogues in representation theory of the decision problems\nin Hypothesis 1.2 are described in Section 11. The P-barrier in this context,\nits crossing subject to analogous PH1 and SH (PH2), along with theoretical\nresults supporting these positivity hypotheses are described in Section 12.\nThe nature of the reduction in the top arrow of Figure 1 is described in\nSection 13. The basic plan in [GCT6] to prove PH1 and SH via the the-\nory of quantum groups is described next. The standard quantum group is\nintuitively described in Section 14. The nonstandard quantum groups are\nintuitively described in Section 15. Why nonstandard Riemann hypotheses\nshould exist and their role in the theory of nonstandard quantum groups is\nbriefly described in Section 16. The relationship between obstructions and\nexpanders is described in Section 17. Why GCT should cross the relativiza-\ntion and the P/poly-naturalizability barriers is described in Section 18. Why\nGCT may be among the easiest P-verifiable approaches to the P vs. NP\nproblem as per the explicit-construction-complexity measure is described in\nSection 19.\n2\nBasics in algebraic geometry and representation\ntheory\nIn this section we describe the basic facts in algebraic geometry and repre-\nsentation theory which are needed in this article and which should be easy\nto believe for the readers not familiar with these fields. Their proofs can be\nfound in [FH, Mm1].\n24"},{"page":25,"text":"This article\n(GCTflip1)\n|\n↓\nGCTintro\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 5: Logical dependence among the GCT papers\n25"},{"page":26,"text":"2.1\nRepresentation theory\nLet G be a group. We say that a vector space V is a representation of G, or\na G-module, if there is a homomorphism\nρ : G →GL(V ),\n(4)\nwhere GL(V ) is the general linear group of invertible transformations of\nV . We denote ρ(g)(v) by g · v–the result of the action of g on v. A G-\nsubrepresentation W ⊆V is a subspace that is invariant under G; i.e.,\ng · w ∈W for every w ∈W. If G is clear from the context, we just call\nit subrepresentation. We say that V is irreducible if it does not contain a\nproper nontrivial subrepresentation. A G-homomorphism from a G-module\nU to a G-module V is map ψ : U →V such that ψ(g · u) = g · (ψ(u)) for all\nu ∈U.\nWe say that G is reductive if every finite dimensional representation V\nof G is completely reducible. This means it can be expressed as a direct sum\nof irreducible representations in the form\nV =\nM\nλ\nmλVλ(G)\n(5)\nwhere λ ranges over all indices (labels) of irreducible representations of G,\nVλ(G) denotes the irreducible representation of G with label λ, and mλVλ(G)\ndenotes a direct sum of mλ copies of Vλ(G). Here mλ is called the multiplicity\nof Vλ(G) in V . It is a basic fact of representation theory that for reductive\ngroups, the decomposition (5) is unique; i.e., mλ’s are uniquely defined. If\nmλ > 0, we say that Vλ(G) occurs in V .\nAn example of a nonreductive group is a solvable group that is not\nabelian. In this case a subrepresentation W ⊆V need not have a comple-\nment W ⊥such that V = W ⊕W ⊥.\nEvery finite group is reductive.\nThus Sn, the symmetric group on n\nletters, is reductive. A prime example of a continuous reductive group is\nthe general linear group GLn(C) = GL(Cn), the group of nonsingular n × n\nmatrices, and its subgroup the special linear group SLn(C) = SL(Cn) of\nmatrices with determinant one.\nAny product of reductive groups is also\nreductive. These are the only kinds of reductive groups that we need to\nknow in this article. So whenever we say reductive, the reader may wish to\nassume that the group is a general or special linear group or a symmetric\ngroup or a product thereof.\n26"},{"page":27,"text":"We say that the representation (4) of G = GLn(C) or SLn(C) is polyno-\nmial if for every g ∈G, every entry in the matrix form of ρ(g) is a polynomial\nin the entries of g.\nComplete reducibility as in eq.(5) means every finite dimensional rep-\nresentation of a reductive group is composed of irreducible representations.\nThese can be thought of as the building blocks in the representation theory\nof reductive groups, and it is important to know what these building blocks\nare.\n2.1.1\nIrreducible representations of GLn(C)\nFor GLn(C) this was done by Weyl in his classic book [W]. The polynomial\nirreducible representations of GLn(C) are in one-to-one correspondence with\nthe tuples λ = (λ1, . . . , λk) of integers, where k ≤n and λ1 ≥λ2 · · · ≥λk >\n0. Here λ is called a partition of length k and size d = P\ni λi. Its bitlength\n⟨λ⟩is defined to be the total bitlength of all λi’s.\nThus the polynomial irreducible representations of GLn(C) are labelled\nby partitions λ of length at most n, but any size. The irreducible representa-\ntion corresponding to a partition λ = (λ1, λ2, . . .) is denoted by Vλ(GLn(C)),\nand is called a Weyl module of GLn(C). When GLn(C) is clear from the\ncontext, we shall denote it by simply Vλ.\nEach partition λ corresponds to a Young diagram, which consists of\nk rows of boxes, with λi boxes in the i-th row. For example, the Young\ndiagram corresponding to (4, 2, 1) is shown below:\nWhen thinking of a partition, it is helpful to think of the corresponding\nYoung diagram. Thus each Weyl module is labelled by a Young diagram of\nheight at most n. This is a useful combinatorial tool for studying the Weyl\nmodules.\nA Weyl module Vλ is explicitly constructed as follows. This construction\nof Deyruts as well as Weyl’s original construction are given in [FH]. Let Z be\nan n × n variable matrix. Let C[Z] be the ring of polynomials in the entries\nof Z. It is a representation of GLn(C). Action of a matrix σ ∈GLn(C) on\na polynomial f ∈C[Z] is given by\n(σ · f)(Z) = f(Zσ).\n(6)\n27"},{"page":28,"text":"By a numbering (filling), we mean filling of the boxes of a Young diagram\nby numbers in [n]; for example:\n1 2 4 3\n2 3\n1\nWe call such a numbering a (semistandard) tableau if the numbers are strictly\nincreasing in each column and weakly increasing in all rows; e.g.\n1 2 3 3\n2 3\n4\nThe partition corresponding to the Young diagram of a numbering is\ncalled the shape of the numbering.\nWith every numbering T, we associate a polynomial eT ∈C[Z], which is\na product of minors for each column of T. The l × l minor ec for a column c\nof length l is formed by the first l rows of Z and the columns indexed by the\nentries cj, 1 ≤j ≤l, of c. Thus eT = Q\nc ec, where c ranges over all columns\nin T. The Weyl module Vλ is the subrepresentation of C[Z] spanned by eT ,\nwhere T ranges over all numberings of shape λ over [n]. Its one possible\nbasis is given by {eT }, where T ranges over semistandard tableau of shape\nλ over [n].\nLet B ⊆GLn(C) be the subgroup of upper triangular matrices. It is\ncalled the Borel subgroup of GLn(C). An element vλ ∈Vλ is called a highest\nweight vector if it is an eigenvector for the action of each b ∈B. It is easy to\nshow that Vλ has a unique highest weight vector, upto a constant multiple:\nit is eT0, where T0 is the canonical tableau whose i-th row contains only i’s,\nfor each i; e.g.\n1 1 1 1\n2 2\n3\nLet P ⊆GLn(C) be the subgroup of upper block triangular matrices,\nwhere the sizes of the blocks are fixed. For example:\n28"},{"page":29,"text":"∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n0\n0\n∗\n∗\n∗\n∗\n0\n0\n∗\n∗\n∗\n∗\n0\n0\n0\n0\n∗\n∗\n0\n0\n0\n0\n∗\n∗\n \n \nSuch subgroups are called parabolic. Let Pλ be the (projective) stabilizer\nof the highest weight vector vλ = eT0; i.e., the set of all σ ∈GLn(C) such\nthat σ ·vλ = c(σ)vσ, for some complex number c(σ). Then it is easy to show\nthat Pλ is parabolic, where the sizes of the blocks are completely determined\nby λ.\nThe irreducible representation of GLn(C) corresponding to the Young\ndiagram that consists of just one column of length n is the determinant\nrepresentation: g →det(g). When restricted to the subgroup SLn(C) ⊆\nGLn(C) this becomes trivial. More generally, Vλ(G) and Vλ′(G) give the\nsame representation of SLn(C) if λ′ is obtained from λ by removing columns\nof length n. Hence, irreducible polynomial representations of SLn(C) are\nin one to one correspondence with partitions of length less than n, and are\nobtained from the ones of GLn(C) by restriction.\n2.1.2\nIrreducible representations of the symmetric group\nIrreducible representations of Sn, called Specht modules, are in one-to-one\ncorrespondence with the Young diagrams of size n, as opposed to those of\nlength ≤n for GLn(C). We denote the Specht module corresponding to a\npartition λ by Sλ. It is explicitly constructed as follows.\nLet C[X] = C[x1, · · · , xn] be the ring polynomials in n variables. It is a\nrepresentation of Sn: given σ ∈Sn and f ∈C[X],\n(σ · f)(x1, · · · , xn) = f(xσ(1), · · · , xσ(n)).\nGiven a numbering T of λ with distinct numbers in [n], let fT be the poly-\nnomial formed by taking a product of discriminants for all columns of T.\nThe discriminant for a column with entries ci, 1 ≤i ≤l, is Q\ni<i′(xci −xci′).\nThen Sλ is simply the subrepresentation of C[X] spanned by fT, where T\nranges over all numberings of λ with distinct entries in [n]. Its basis is given\nby {fT }, where T ranges over standard tableau of shape λ with entries in\n[n]. Here a standard tableau means the rows as well as the columns are\n29"},{"page":30,"text":"strictly increasing; e.g.\n1 2 3 6\n4 5\n7\n2.1.3\nTensor products\nIf V and W are representations of a group G, then their tensor product\nV ⊗W is also a representation: for σ ∈G, v ∈V, w ∈W,\nσ(v ⊗w) = (σ · v) ⊗(σ · w).\nGiven two irreducible representations of a reductive group G, a fun-\ndamental problem in representation theory is to find an explicit complete\ndecomposition of their tensor product in terms of irreducible representations\nof G. The following instances of this problem are of central importance in\nGCT.\nLittlewood-Richardson coefficients\nFirst we consider this problem when G = GLn(C). Given Weyl modules Vα\nand Vβ, let\nVα ⊗Vβ =\nM\nγ\ncγ\nα,βVγ,\n(7)\nbe the complete decomposition of their tensor product into irreducible Weyl\nmodules of G. Here the multiplicities cγ\nα,β are called Littlewood-Richardson\ncoefficients. The Littlewood-Richardson rule gives the sought explicit for-\nmula for these multiplicities. It is as follows.\nAlign the left top corners of the Young diagrams for α and γ. If the\nYoung diagram for α is not contained in the one for γ, then cγ\nα,β is zero.\nOtherwise, form a skew shape γ \\ α by removing the boxes in γ belonging\nto α. A skew tableau with content β and shape γ \\ α is a filling of this skew\ndiagram with β1 ones, β2 twos and so on, such that all columns are strictly\nincreasing and all rows are weakly increasing. For example, the following is\na skew tableau of skew shape (4, 3, 3, 2) \\ (2, 2, 1):\n1 1\n2\n2 3\n1 3\n(8)\n30"},{"page":31,"text":"We say that a skew tableau is a Littlewood-Richardson tableau if, when\nits entries are read from right to left, top to bottom, the number of i’s read\nupto any point is at most the number of (i −1)’s read up to that point,\nfor any i. For example, the skew tableau above is a Littlewood-Richardson\nskew tableau. Let Cγ\nα,β be the set of Littlewood-Richardson tableau of shape\nγ \\ α with content β. The Littlewood-Richardson coefficient cγ\nα,β is simply\nthe cardinality of Cγ\nα,β: i.e.,\ncγ\nα,β = |Cγ\nα,β| =\nX\nT∈Cγ\nα,β\n1.\n(9)\nSuch a formula is called positive, because it is like the formula for the perma-\nnent which involves only positive signs. In contrast, there are many formulae\nfor such multiplicities, based on the theory of characters of group represen-\ntations [FH], which involve alternating signs, like the usual formula for the\ndeterminant. Positivity here is a deep issue; cf. [St4].\nFormally, the Littlewood-Richardson rule implies that the Littlewood-\nRichardson coefficient belongs to the complexity class #P, just like the per-\nmanent. This is the real significance of the Littlewood-Richardson rule from\nthe complexity-theoretic perspective. Furthermore, just like the permanent,\nthe Littlewood-Richardson coefficient is #P-complete [N].\nKronecker coefficients\nNow we turn to the symmetric group.\nSince it is reductive, the tensor\nproduct of two Specht modules Sα and Sβ decomposes as: by:\nSα ⊗Sβ =\nM\nγ\nkγ\nα,βSγ,\n(10)\nwhere the multiplicities kγ\nα,β are called Kronecker coefficients.\nNo positive rule akin to the Littlewood-Richardson rule is known for\nthe Kronecker coefficients. In fact, this is a fundamental open problem in\nthe representation theory of symmetric groups, which arose almost with the\nbirth of representation theory in the work of Frobenius, Schur, Weyl and\nothers in the beginning of the twentieth century; cf. [Mc, St4] for its history\nand significance. In the language of complexity theory, the problem is:\nQuestion 2.1 Does the Kronecker coefficient belong to #P?\n31"},{"page":32,"text":"Though this is not how it was stated in representation theory. The answer\nis conjecturally yes [GCT4]. Indeed, this is the main focus of the work in\n[GCT4, GCT8]: roughly, [GCT8] says that such a rule exists assuming a\nconjecture regarding the nonstandard quantum group defined in [GCT4].\nThis is the first entry point of nonstandard quantum groups in GCT.\n2.2\nAlgebraic geometry\nLet V = Cn. Let X = (x1, . . . , xn) be the variable n-vector whose entries\nstand for the coordinates of V . An affine algebraic set Z ⊆V is the set\nof zeroes of a collection of polynomials in C[X] = C[x1, . . . , xn]. An affine\nalgebraic set is called irreducible if it cannot be expressed as the union of\ntwo proper affine algebraic subsets. An irreducible affine algebraic subset\nZ of V is called an affine variety. Its ideal I(Z) ⊆C[X] is the set of all\npolynomials that vanish on Z, and its coordinate ring C[Z] is defined to be\nC[X]/I(Z). The elements of C[Z] are polynomial functions on Z.\nLet P n−1 = P(V ) be the projective space of lines in V through the\norigin. We say that V is the affine cone of P(V ). Given a nonzero v ∈V ,\nwe also denote by v the point in P(V ) that corresponds to the line in V\npassing through v and the origin; the meaning should be clear from the\ncontext. The homogeneous coordinate ring of P(V ) is defined to be C[X].\nIts elements are homogeneous functions on V , the affine cone of P(V ). A\nprojective algebraic set Y in P(V ) is the set of zeroes of a collection of\nhomogeneous forms (polynomials). The affine cone ˆY ⊆V of Y ⊆P(V )\nis defined to be the union of the lines in V corresponding to the points in\nY . A projective algebraic set is called irreducible if it cannot be expressed\nas the union of two proper projective algebraic subsets.\nAn irreducible\nprojective algebraic subset Y of P(V ) is called a projective variety. Its ideal\nI(Y ) ⊆C[X] is the set of all homogeneous forms that vanish on Y , and its\nhomogeneous coordinate ring R(Y ) is C[X]/I(Y ). The elements of R(Y ) are\nhomogeneous functions on the affine cone ˆY of Y . The degree d-component\nR(Y )d of R(Y ) is the subspace of homogeneous forms of degree d.\nThe\nHilbert function hY (d) of Y is defined to be the dimension of R(Y )d.\nBy a (Zariski)-open subset of Y we mean the complement of an algebraic\nsubset of Y . An open subset of a projective variety is also called a quasi-\nprojective variety.\nNow suppose V is a representation of a reductive group G.\nThen G\nalso acts on P(V ), since it takes line to a line. Furthermore, C[X] is also a\n32"},{"page":33,"text":"representation of G: given σ ∈G and f ∈C[X], we define\n(σ · f)(X) = f(σ−1X).\n(11)\nThe variety Y is called a G-variety if its ideal I(Y ) ⊆C[X] is a G-subrepresentation\nof G; i.e., σ·f ∈I(Y ) for all σ ∈G, f ∈I(Y ). In this case, the homogeneous\ncoordinate ring R(Y ) of Y is also a representation of G. Furthermore, given\na point p ∈Y , the point σ(p) also belongs to Y . In other words, G acts on\nthe variety Y by moving its points around.\nIf Z is a projective subvariety of Y , then it is a basic fact that there\nexists a degree preserving surjection from R(Y ) to R(Z); i.e., from R(Y )d\nto R(Z)d for every d. This surjection is obtained by simply restricting a\npolynomial function on the affine cone ˆY to the subcone ˆZ ⊆ˆY . If both\nY and Z are G-varieties, then this surjection is a G-homomorphism. By\ncomplete reducibility, it then follows that R(Z)d is a G-submodule of R(Y )d\nfor every d. Pictorially,\nR(Z)d ֒→R(Y )d,\n(12)\nfor every d.\nGiven a point v ∈P(V ), let Gv denote its G-orbit. It can be shown\nthat Gv is a quasi-projective variety.\nLet Gv = {σ | σ · v = v} be its\nstabilizer. Then Gv as a set is isomorphic to the coset set G/H, H = Gv.\nQuasiprojective varieties of the form G/H are called homogeneous spaces.\nThese have been intensively studied in algebraic geometry.\nLet ∆V [v] = Gv ⊆P(V ) denote the closure of the G-orbit of v in the\nusual complex topology 1. We call such a variety an orbit closure. It can\nbe shown that ∆V [v] is a projective G-variety. One can think of ∆V [v] as\na closure of the homogeneous space G/Gv. Such spaces are called almost-\nhomogeneous spaces [Ak]. These have also been intensively studied. Let\nRV [v] be the homogeneous coordinate ring of ∆V [v], and RV [v]d its degree\nd-component. Since G acts on ∆V [v], each RV [v]d is a finite dimensional\nrepresentation of G.\nThe simplest example of ∆V [v] arises as follows.\nLet Vλ be a Weyl\nmodule of G = GLn(C). Let vλ ∈P(Vλ) be the point corresponding to the\nhighest weight vector in Vλ; we call it the highest weight point. Then it can\nbe shown that the orbit Gvλ ∼= G/Pλ, where Pλ is the stabilizer of vλ, is\nalready closed. This is called a flag variety. It has been intensively studied\nin algebraic geometry for over a century, and its algebraic geometry is now\nmore or less completely understood; e.g. see [LLM].\n1This coincides with the closure in the Zariski-topology [Mm1].\n33"},{"page":34,"text":"The flag varieties by their very definition are smooth. But the algebraic\ngeometry of general orbit closures can be extremely complicated, and essen-\ntially, intractable. Because, even if the orbit Gv is smooth, its closure can be\nhighly singular, and the singularities can be pathological. Indeed, the moral\nof the story that can be gained from [LV] is that the algebraic geometry of\na general orbit closure is essentially hopeless.\n3\nGroup-theoretic varieties\nFortunately, the class varieties that arise in GCT are all exceptional kinds of\norbit closures, which we call group-theoretic orbit closures or group-theoretic\nvarieties. The articles [GCT2, GCT10] together roughly say that problems\nregarding the algebraic geometry of these group-theoretic class varieties can\nbe “reduced” to problems in (quantum) group theory. This is what makes\nthem tractable, and this is how the theory of quantum groups enters in\nGCT. In this section, we shall briefly describe a group-theoretic variety in\nan abstract form.\nLet V and G be as in Section 2.2. We say that v ∈P(V ) is characterized\nby its stabilizer H = Gv ⊆G, if it is the only point in P(V ) stabilized (left\ninvariant) by H. Stabilized by H means, for every σ ∈H , σ · v = v.\nFor example, the highest weight point vλ ∈P(Vλ) (Section 2.1) is char-\nacterized by its stabilizer Pλ. This indicates that the points that are char-\nacterized by their stabilizers are very special.\nSuppose v is characterized by its stabilizer. Then v, and hence, its orbit\nclosure ∆V [v] is completely determined by the group triple:\nH = Gv ֒→G\nρ→K = GL(V ),\n(13)\nwhere ρ represents the representation map; cf. (4). This leads to the follow-\ning:\nDefinition 3.1 Assume that v is characterized by its stabilizer, and that\nthe associated group triple H ֒→G ֒→K is explicitly known. Then we say\nthat the orbit closure ∆V [v] is a group-theoretic variety. It is completely\ndetermined by the preceding group triple.\nWe call H ֒→G the primary\ncouple associated with the orbit closure ∆V [v], and G ֒→K, the secondary\ncouple. We also say that v is the characteristic point of the group triple\nH ֒→G ֒→K and the primary couple H ֒→G.\n34"},{"page":35,"text":"If every point in V is a function on some space then we say that v is\nthe characteristic function of the triple H ֒→G ֒→K and the primary\ncouple H ֒→G. (This happens if, for example, V is the space of polynomial\nfunctions on an affine G-variety X, with the action given by (11)).\nHere explicitly means the composition factors of H as well as the con-\nnecting homomorphisms in (13) are specified explicitly; for the details re-\ngarding how, see [GCT6, GCTflip2]. All varieties that arise in GCT are\neither group-theoretic orbit-closures in the above sense, or their generaliza-\ntions, which are again essentially determined by group triples as above, and\nhence, will also be called group-theoretic varieties. The simplest example of\na group-theoretic variety is a flag variety (Section 2.2).\nIf a variety is group theoretic, then, in principle, we ought to be able to\nunderstand its algebraic geometry if we understand the structure of the as-\nsociated group triple, along with the connecting homomorphisms, in depth.\nWe shall elaborate on what in depth means later in Section 15. Briefly, it\nmeans understanding the structure of the group triple at the quantum level.\n4\nClass varieties\nNow we turn to the class varieties associated with the complexity classes\nNC, P, #P and NP in [GCT1] on which the obstructions in GCT live. All\nthe class varieties that arise in GCT will be orbit closures of the following\nspecial form.\nLet Y = [y0, · · · , yl−1] denote a variable l-vector. For n < l, let X =\n[y1, · · · , yn], and ̄X = [y0, · · · , yn] be its subvectors of size n and n + 1.\nWe also denote yi, 1 ≤i ≤n, by xi.\nLet V = Syms(Y ) be the space\nof homogeneous forms of degree s in the l variable-entries of Y . It has a\nnatural action of G = SL(Y ) = SLl(C) and ˆG = GL(Y ) = GLl(C), just as\nin (11).\nSimilarly, let W = Symr(X), r < s, be the representation of GL(X) =\nGLn(C). We have a natural embedding φ : W →V , which maps\nw ∈W →ys−rw ∈V,\n(14)\nwhere y = y0 is used as the homogenizing variable. The image φ(W) is\ncontained in ̄W = Syms( ̄X), a representation of GL( ̄X) = GLn+1(C).\nThe basic recipe for constructing class varieties is as follows. Say we\nwant to separate a complexity class C1 from a complexity class C2 ⊇C1.\n35"},{"page":36,"text":"We pick a form g = g(Y ) = g(y0, . . . , yl−1) ∈P(V ) which is a complete\nfunction for the complexity class C1. Then the orbit closure ∆V [g; l] = ∆V [g]\nis called the class variety associated with C1, or simply the C1-variety based\non the complete function g.\nIn principle, we can let g be any complete\nfunction for the class C1. But for the algebraic geometry of ∆V [g] to be\ntractable, we have to choose g so that ∆V [g] is group-theoretic; i.e., so that\ng is characterized by its stabilizer as in Definition 3.1, or in a slightly relaxed\nsense (cf. Section 7 in [GCT1]), which is good enough for our purposes.\nSimilarly, we choose a form h = h(X) = h(x1, . . . , xn) ∈P(W) which\nis complete for the class C2. Then the orbit closure ∆W[h; n] = ∆W[h] ⊆\nP(W) is called the base class variety associated with C2, or simply the base\nC2-variety based on h. Let f = φ(h), with φ as in (14). We call the orbit\nclosure ∆V [f; n, l] = ∆V [f] ⊆P(V ) the extended class variety associated\nwith C2, or the extended C2-variety based on h. This extension is necessary\nso that the C2-variety ∆V [f] and the C1-variety ∆V [g] live in the same\nambient space P(V ). Again, h has to be chosen so that it is characterized\nby its stabilizer (almost) so that the varieties ∆W [h] and ∆V [f] are group-\ntheoretic.\nLet us suppose to the contrary that C2 ⊆C1. Then it would turn out\nthat f ∈∆V [g; l], and hence, ∆V [f; n, l] is a G-subvariety of ∆V [g; l]:\n∆V [f; n, l] ֒→∆V [g; l].\nThe goal is to show that such an embedding does not exist when l is small\nenough, say, l = nlog n, n →∞. This will show that C1 ̸= C2. Here l will be\na parameter in the lower bound problem that depends on the depth and/or\nthe size of the circuit.\nWe now demonstrate this recipe in two basic separation problems in\ncomplexity theory.\n4.1\nNC vs. P #P\nLet NC be the standard class of functions that can be computed by circuits\nof polylogarithmic depth, and #P the counting class associated with NP.\nThe determinant is complete for the class NC and the permanent for the\nclass #P [V]. The P #P ̸= NC conjecture over C [V] says that the per-\nmanent cannot be computed by a circuit over C of polylogarithmic depth.\nUsing the determinant and the permanent, we now construct class varieties\nfor NC and #P. The P #P ̸= NC conjecture over C will then be reduced\n36"},{"page":37,"text":"to showing that the extended class variety for #P is not contained in the\none for NC.\nLet Y be an m × m variable matrix, which can also be thought of as a\nvariable l-vector, l = m2, by linearly ordering its entries in any order. Let\nX be its, say, the principal bottom-right n × n submatrix, n < m, which\ncan also be thought of as a variable k-vector, k = n2. Let V = Symm(Y )\nbe the space of homogeneous forms of degree m in the variable entries of\nY , and W = Symn(X), the space of homogeneous forms of degree n in the\nvariable entries of X. We have a natural action of G = SL(Y ) = SLl(C) on\nV and the projective space P(V ): namely, σ ∈G maps a form q(Y ) ∈P(V )\nto q(σ−1Y ), where we think of Y as a variable l-vector. Similarly, we have\nan action of H = SL(X) = SLk(C) on P(W).\nUsing any entry y of Y not in X as a homogenizing variable we get an\nembedding φ : W →V , which maps any w ∈W to ym−nw ∈V .\nLet g = det(Y ) ∈P(V ) be the determinant form.\nLet ∆V [g; l] =\n∆V [g] ⊆P(V ) be its orbit closure.\nIt is shown in [GCT1] that if a form h(X) ∈P(W) can be computed by a\ncircuit of depth less than logc n, then f = φ(h) lies in ∆V [g; l] for m = 2logc n.\nConversely, if f lies in ∆V [g; l] then h[X] can be approximated infinitesimally\nclosely2 by a circuit of depth O(log2c m).\nSince, the permanent is #P-\ncomplete [V], this is not expected to happen if h = perm(X) and m =\n2O(polylog(n)). This leads to:\nConjecture 4.1 [GCT1] Let h = perm(X) ∈P(W) and f = φ(h). Then\nf ̸∈∆V [g] = ∆V [g; l], if m = 2O(polylog(n)), as n →∞. Since ∆V [g] is a\nG-variety, this is equivalent to saying that ∆V [f] ̸⊆∆V [g]. Pictorially:\n∆V [f] ̸֒→∆V [g].\nHere ∆V [g] = ∆V [g; l] is called the class variety associated with the\nclass NC, or simply the NC-variety based on the complete determinant\nfunction. We also denote it by XNC(g; l) or simply XNC(l). We call ∆W[h]\nthe base class variety and ∆V [f] the extended class variety associated with\nthe class #P, or simply the base #P-variety and the extended #P-variety,\nrespectively. We also denote them by X#P (h; n) and X#P (f; n, l), or sim-\nply, X#P(n) and X#P (n, l). The goal (Conjecture 4.1) is to show that the\n2This means, for every ǫ > 0, there exists a form ̃f ∈V , which has a circuit of depth\nO(log2c m), such that ||f − ̃f|| < ǫ, in the usual norm on V .\n37"},{"page":38,"text":"extended class variety for #P cannot be contained in the class variety for\nNC, when m = 2polylog(n): i.e.,\nX#P (n, l) ̸֒→XNC(l).\nThis will show that the permanent cannot be computed by circuits of poly-\nlogarithmic depth.\nNext we describe why these class varieties are group-theoretic. For this,\nwe need to show that the determinant and the permanent are characterized\nby their stabilizers.\nThe stabilizer of det(Y ) ∈P(V ) in G = SL(Y ) = SLm2(C) is known to\nbe a reductive subgroup Gdet which consists of linear transformations in G\nof the form (thinking of Y as an m × m matrix):\nY →AY ∗B,\n(15)\nwhere Y ∗is either Y or Y T , A, B ∈GLm(C).\nThat the determinant is\ncharacterized by its stabilizer follows from classical invariant theory [FH].\nHence the NC-variety defined here is group-theoretic. The associated group\ntriple is\nGdet ֒→G ֒→GL(V ),\n(16)\nand Gdet ֒→G the primary couple. The embedding Gdet →G almost looks\nlike the natural embedding\nGL(Cm) × GL(Cm) →GL(Cm ⊗Cm),\n(17)\ngiven by: (g, h) →g ⊗h, where g ⊗h denotes the Kronecker product. That\nis,\n(g ⊗h) · (x ⊗y) = (g · x) ⊗(h · y).\n(18)\nThe stabilizer of perm(X) ∈P(W) in SL(X) = SLn2(C) is a reduc-\ntive subgroup generated by linear transformations in SL(X) of the form\n(thinking of X as an n × n matrix):\nX →λX∗μ,\n(19)\nwhere X∗is either X or XT , λ and μ are either diagonal or permutation\nmatrices, and n ≥3. It is easy to show that the permanent is also charac-\nterized by its stabilizer. Hence the base #P-variety defined in this section\nis group theoretic; the extended #P-variety is also group-theoretic.\n38"},{"page":39,"text":"4.2\nP vs. NP problem over C\nThe class varieties associated with the classes P and NP can be constructed\nin principle using any P-complete and NP-complete functions. But again it\nis necessary to choose these functions in a special way so that the resulting\nclass varieties turn out to be group-theoretic (Section 3). Such P-complete\nand (co)-NP-complete functions, called H(Y ) = H(y1, . . . , yl) and E(X) =\nE(x1, . . . , xn) respectively, have been constructed in [GCT1]. We do not\nneed to know their definitions here.\nLet W = Symr(X) be the space of forms of degree r = deg(E(X)) in\nthe entries of X. Thus E(X) ∈P(W). Let V = Syms(Y ) be the space of\nforms of degree s = deg(H(Y )) in the entries of Y . Thus H(Y ) ∈P(V ). We\nidentify X with a suitable subset of Y , and define a map φ : P(W) →P(V )\nas in (14) by choosing a variable y in Y \\ X as a homogenizing variable.\nNow, using the recipe above, we can associate with E(X), for every n\nand l ≥n, a group-theoretic variety (orbit closure) ∆V [f; n, l] = ∆V [f] ⊆\nP(V ), where f = φ(h) and h = E(X). It is a G-variety, for G = SLl(C).\nIt will be called the (extended) class variety for NP or simply the NP-\nvariety based on the form E(X), and will be denoted by XNP (E; n, l) or\nsimply XNP (n, l). Similarly, we can associate with H(Y ) a group-theoretic\nG-variety ∆V [g; l] = ∆V [g] ⊆P(V ), where g = H(Y ).\nIt is called the\nclass variety for P or simply the P-variety based on the form H(Y ), and is\ndenoted by XP (H; l) or simply XP (l).\nRemark 4.2 The actual P-variety XP (H; l) in the P vs. NP problem is\nnot meant to be ∆V [g, l], as defined here, but rather the variety ˆ∆[H(Y )]\ndefined in Section 7 of [GCT1]. But we shall ignore that difference here.\nIt can be shown [GCT1] that if E(X) is computable by a circuit of size\nm then XNP (E; n, l) can be embedded within XP (H; l) for l = O(m2):\nXNP (n, l) = XNP (E; n, l) ֒→XP (l) = XP (H; l).\n(20)\nIn this context:\nConjecture 4.3 [GCT1] This embedding cannot exist if m = nlog n, or\nmore generally, m = 2na, for a small enough a > 0, as n →∞.\nThis will show that P ̸= NP over C. This transforms the P vs. NP problem\nover C into a problem in geometric invariant theory.\n39"},{"page":40,"text":"Again, these class varieties are group-theoretic, in a slightly relaxed sense\nthan defined in Section 3, but which is good enough for the purposes of GCT\n[GCT1].\n5\nObstructions\nAn obstruction in the P vs. NP problem (characteristic zero) is defined to\nbe a representation that lives on the extended class variety associated with\nNP but not on the class variety associated with P. We now elaborate what\nthis means.\nLet R(n, l) = R(E; n, l) and S(l) = S(H; l) denote the homogeneous co-\nordinate rings of XNP (n, l) = XNP (E; n, l) and XP (l) = XP (H; l), respec-\ntively. We call them the class rings associated with the complexity classes\nNP and P. Let R(n, l)d and S(l)d denote their degree d-components, con-\nsisting of homogeneous polynomial functions of degree d. Since G acts on\nthe class varieties, it also acts on the class rings (see Section 2.2). That is,\neach R(n, l)d or S(l)d is a finite dimensional representation of G.\nIf the embedding (20) exists, then R(n, l)d can be embedded as a G-\nsubmodule of S(l)d, for each d; cf. (12):\nR(n, l)d ֒→S(l)d.\n(21)\nIn particular, every irreducible representation (Weyl module) Vλ = Vλ(G)\nof G that occurs within R(n, l)d as a subrepresentation also occurs within\nS(l)d as a subrepresentation.\nDefinition 5.1 We say that S = Vλ is an obstruction, for n, l and the pair\n(E, H) = (E(X), H(Y )), if it occurs in R(n, l)d but not in S(l)d, for some\nd.\nIn this case we say that Vλ is an obstruction of degree d. We also refer\nto λ as an obstruction of degree d.\nObstruction in the setting of the NC vs. P #P problem over C is defined\nsimilarly.\nThis notion of obstruction in [GCT2] is a refinement of the earlier notion in\n[GCT1].\nThe specification of an obstruction is given in the form of its label λ.\nThe existence of such an obstruction for given n and l is a “proof” that the\nembedding in (21), and hence, the one in (20) cannot exist.\n40"},{"page":41,"text":"In this context:\nConjecture 5.2 [GCT2, GCT10] An obstruction for n, l and the pair (E, H)\nexists if m = nlog n, or more generally, m = 2na, for a small enough a > 0,\nas n →∞; recall that l = O(m2). Furthermore, there exists such an ob-\nstruction of a small degree d(n, m) = 2mb, b > 0 a large enough constant.\nSimilar conjecture can be made in the context of the NC vs. P #P prob-\nlem.\nIn this case, the degree d(n, m) can be mb, b > 0 a large enough\nconstant.\nIf such an obstruction Vλ(n) exists for every n →∞, with m as above,\nthen it follows that P ̸= NP over C. We say that {Vλ(n)} or {λ(n)} is an\nobstruction family for the P vs. NP problem over C. The goal is to prove\nexistence of such a family.\n6\nWhy should obstructions exist?\nA priori, it is not at all clear why such obstructions should even exist. In\nthis section, we explain why they should.\nAn intuitive reason for existence of obstructions is as follows. The article\n[Dl1] roughly says that (algebraic) groups are completely determined by their\nrepresentations. On the other hand, the group-theoretic class varieties are\nessentially determined by the associated group triples, and hence, as per the\nphilosophy in [Dl1], the representation-theoretic information associated with\nthese group triples. Hence, a “witness” for nonexistence of the embedding\nas in (20) ought to be present in the representation-theoretic information\nassociated with the group triples, assuming that P ̸= NP–which we take on\nfaith. This is intuitively why a representation-theoretic obstruction ought\nto exist. Specifically, there should exist a representation-theoretic witness\n(obstruction) that explains why one group-theoretic class variety, with as-\nsociated group triple H1 ֒→G →K, cannot be embedded in another group\ntheoretic variety with associated group triple H2 ֒→G →K; in our problem\nG and K in both triples would be the same.\nBut why should such a representation-theoretic obstruction be specifi-\ncally of the type as defined here?\nTo see this, let us first consider a simpler example. Instead of triples,\nlet us consider couples. Let us say we are given two couples ρ1 : H1 ֒→G,\nand ρ2 : H2 ֒→G, where G = GLl(C) = GL(W), W = Cl. This means W\n41"},{"page":42,"text":"is a representation of H1 and H2. Let us assume that it is an irreducible\nrepresentation of H1 and H2, and furthermore, that both H1 and H2 are\nreductive, and that H2 is not a conjugate of H1. Now the coset sets G/H1\nand G/H2 can be given the structure of affine algebraic varieties [Mm2].\nSince H2 is not a conjugate of H1, G/H1 cannot be embedded in G/H2 (and\nvice versa). The goal is to find a representation theoretic obstruction for the\nnonexistence of such an embedding. We say that Vλ(G) is an obstruction for\nthis pair of couples (ρ1, ρ2) if it occurs as a G-submodule in the coordinate\nring of G/H1 but not in the coordinate ring of G/H2. This is equivalent\nto saying that Vλ(G) contains an H1-invariant, when considered as an H1-\nmodule via ρ1, but not an H2-invariant, when considered as an H2-module\nvia ρ2; this is a consequence of the Peter-Weyl theorem [Sp]. This then is an\nobstruction very similar to the one in Definition 5.1. Its existence implies\nthat G/H1 cannot be embedded in G/H2. The work [LP] implies that such\nas an obstruction always exists when H1 and H2 are as above.\nConjecture 5.2 is a natural generalization of this well characterized sit-\nuation. It says that there exists a similar obstruction for the embedding\namong the group-theoretic varieties under consideration. This, as expected,\nis a much harder issue. The existence of such an obstruction depends cru-\ncially on the following conjecture concerning the algebraic geometry of the\nclass varieties under consideration.\nConjecture 6.1 (a) (cf. [GCT2]) The algebraic geometry of the class va-\nriety for NC is completely determined by the representation theory of the\nassociated group triple. Specifically, let Π be the set of G-submodules of C[V ]\nwhose duals do not contain a Gdet-invariant; i.e., the trivial Gdet-module;\ncf. (16). Let X(Π) ⊆P(V ) be the zero set of the forms in the G-modules\nin Π. Then XNC = X(Π).\n(b) (cf. [GCT10]) Analogous, but more complex, statements hold for the\nclass varieties associated with the complexity classes P, NP and #P.\nFor precise statements see [GCT2, GCT10].\nRemark 6.2 (Erratum) In [GCT2] it is conjectured that XNC = X(Π)\nas a scheme [Ha]. This stronger conjecture may not hold as it is. Rather,\nits variant, as would be described in [GCT10], is expected to hold.\nConcrete support for this conjecture is provided by the following two\nresults.\nThe first result is the second fundamental theorem of invariant\n42"},{"page":43,"text":"theory. It says that the analogue of Conjecture 6.1 holds for flag varieties\nand their generalizations [LLM]. Thus Conjecture 6.1 may be thought of\nas a natural generalization of the second fundamental theorem of invariant\ntheory to the group-theoretic class varieties under consideration. The second\nresult, specific to the setting under consideration, is the following.\nTheorem 6.3 (Theorem 2.11 in [GCT2])\nA weaker form of Conjecture 6.1 holds for the NC-variety. Specifically,\nthere is a dense open neighbourhood U ⊆P(V ) of the orbit Gg of the deter-\nminant g = det(Y ) such that XNC ∩U = X(Π) ∩U, assuming a reasonable\ntechnical condition.\nThe article [GCT10] gives justifications for and a plan to prove Conjec-\nture 6.1. It is shown in [GCT2] that obstructions as in Definition 5.1 indeed\nexist in the context of NC vs. P #P problem, for all n →∞, assuming\n1. Conjecture 6.1 (a), and\n2. that the permanent cannot be approximated infinitesimally closely by\ncircuits of polylogarithmic depth.\nThe argument for existence of obstructions in the context of the P vs. NP\nproblem based Conjecture 6.1 (b) is similar [GCT10].\nThe first statement here crucially depends on the group-theoretic nature\nof the class variety for NC. If in place of the determinant we substitute other\nfunction, this need not hold. The second statement is a slightly strengthened\nform of the statement that we are finally trying to prove: namely, that the\npermanent cannot be computed by circuits of small depth. This circular\nreasoning tells us why obstructions should exist. But it gives no help in\nshowing that they exist unconditionally.\nWe turn to this task in the next section. A remark before we do so.\nThe existence of obstructions here crucially depends on the exceptional na-\nture of H(Y ). But we have made no use so far of the exceptional nature\nof E(X). In fact, obstructions of such kind should exist for any hard (co-\nNP-complete) function h(X) in place of E(X). But the approach for con-\nstructing obstructions described in the next section crucially depends on the\nexceptional nature of E(X)–i.e., on the group-theoretic nature of the class\nvariety XNP (E; n, l) for NP based on E(X).\n43"},{"page":44,"text":"7\nThe flip\nNow we come to the real problem: how to prove the existence of obstructions\nfor the specific E(X) under consideration. One may wish to try a probabilis-\ntic strategy for proving existence of obstructions: just choose a label λ(n) of\nhigh enough degree randomly, and show that Vλ(n) is an obstruction with a\ngood probability. But this technique would not work in the context of the P\nvs. NP problem because it is P/poly-naturalizable [RR]. Hence we shall go\nfor explicit construction of obstructions in the spirit of explicit construction\nof expanders [LPS, Ma, RVW]. The P/poly-naturalizability barrier in [RR]\nwould not apply to an approach based on explicit constructions (Section18).\nThis approach is based on the following hypothesis governing the flip:\nHypothesis 7.1 (PHflip1)\nThe following problems belong to P. Specifically:\n(a) (Verification): There exists a poly(l, n, ⟨d⟩, ⟨λ⟩)-time algorithm for de-\nciding, given l, n, d and λ, if Vλ is an obstruction of degree d for n, l and\nthe pair (E, H) (Definition 5.1). Here ⟨d⟩and ⟨λ⟩denote the bitlengths of\nd and λ, respectively.\n(b) (Explicit construction of obstructions): Suppose l = nlog n, or 2na, for\na small enough constant a > 0. Then, for every n →∞, a label λ(n) of\nan obstruction for n and l can be constructed explicitly in poly(n, l) time,\nthereby proving existence of an obstruction for every such n and l.\n(c) (Discovery of obstructions in general): There exists a poly(l, n)-time\nalgorithm for deciding if there exists an obstruction for n, l and the pair\n(E, H), and for constructing the label of one, if it exists.\nSimilar hypothesis holds for the NC vs. P #P problem.\nIn view of the definition of obstruction (Definition 5.1), The statement\n(a) for verification follows from the following:\nHypothesis 7.2 (PHflip2) (a) There exists a poly(l, n, ⟨d⟩, ⟨λ⟩)-time al-\ngorithm for deciding, given l, n, d and λ, if Vλ(G) occurs in R(n, l)d.\n(b) There exists a poly(l, ⟨d⟩, ⟨λ⟩)-time algorithm for deciding, given l, d and\nλ, if Vλ(G) occurs in S(l)d.\nSimilar hypothesis holds for the NC vs. P #P problem.\nAs mentioned in Section 1.4, once Hypothesis 7.2 is proved, the poly-\nnomial time algorithms for the decision problems therein have to be trans-\n44"},{"page":45,"text":"formed into a polynomial time algorithm for explicit construction of obstruc-\ntions as in Hypothesis 7.1 (b), thereby proving Conjecture 5.2, and hence\nthe lower bound under consideration. This issue will be addressed in detail\nin Section 13 later.\nThe whole discussion in this section is summarized in Figure 6.\nFind easy, polynomial time algorithms for the\ndecision problems in Hypothesis 7.2\n|\n|\n|\n↓\nTransform these easy algorithms into an easy algorithm for ex-\nplicit construction of obstructions as in Hypothesis 7.1 (b)\n|\n|\n|\n↓\nP ̸= NP over C\nFigure 6: The flip\n8\nWhy should the flip work?: the P-barrier\nBut why should there exist easy algorithms as in Hypotheses 7.1 and 7.2?\nThis turns out to be, paradoxically, the hardest aspect of the flip: just to\nprove easiness. In this section, we elaborate its nature further.\nClearly, the function E(X) has to be extremely special for Hypotheses 7.1\nand 7.2 to hold. If, instead of E(X), we consider a general co-NP-complete\nfunction h(X) then, obstructions can still be expected to exist (cf.\nSec-\ntion 6), but Hypotheses 7.1 and 7.2 would fail severely, as we now explain.\nSo fix a general integral function h(X) = h(x1, . . . , xn), which is co-NP-\ncomplete, when considered over F2 by reduction modulo 2. Let XNP (h; n, l) ⊆\nP(V ) be the class variety associated with it by following the recipe in Sec-\ntion 4.2 with h(X) in place of E(X). Here V = Syms(Y ) is the space of forms\nof degree s = deg(H(Y )) in l = O(m2) variable entries of Y . The dimension\n45"},{"page":46,"text":"M of the ambient projective space P(V ) here is exponential in l = O(m2),\nm being the the circuit size. Using the currently best available algorithms\nfor constructing a Gr ̈obner basis [KM], and for various problems in invariant\ntheory [St], analogues of the decision problems in Hypotheses 7.1 and 7.2\nfor h(X) can be solved in at best O(dim(C[V ]d) = O(dM) = O(d2poly(m))\nspace, where C[V ]d denotes the degree d component of C[V ], the homoge-\nneous coordinate ring of P(V ). This is so even for the decision problems in\nHypothesis 7.2 and hence for the verification problem in Hypothesis 7.1 (a).\nThis is the best that we can expect for general h(X) in view of the lower\nbound [MM] for the construction of Gr ̈obner bases. In other words, for a\ngeneral h(X) the time taken by a best procedure to even verify if Vλ(G), for\na given λ, is an obstruction would take space that is double exponential in\nm, and hence, time that is triple exponential in m.\nAs we shall argue in Section 19, for any approach towards the P ̸= NP\nconjecture to be viable, at least the problem of verifying an obstruction (i.e.,\na “proof”or “witness” of hardness as per that approach) should be easy; i.e.,\nbelong to P. Intuitively, because however hard it may be to discover a proof,\nits verification, once found, should be easy. The main P-barrier in the course\nof GCT is this huge gap between the triple exponential bound given by the\ncurrently best techniques for a general h(X) and the polynomial bound\nstipulated for verification in Hypothesis 7.1 (a) and in Hypothesis 7.2.\n9\nOn crossing the P-barrier\nWe now come to the main result of [GCT6] which crosses this P-barrier\nunder reasonable assumptions. It gives polynomial-time algorithms for the\ndecision problems in Hypothesis 1.2, and hence, for verifying an obstruction\n(Hypothesis 1.1 (a)), assuming the mathematical positivity hypotheses PH1\nand SH (Hypotheses 1.4-1.6).\n9.1\nA basic prototype with constant depth complexity\nTo motivate these positivity hypotheses, we first consider a basic prototype\nof the decision problems in Hypotheses 7.2 in a simplified setting:\nProblem 9.1 (Littlewood-Richardson problem) Given α, β and λ, decide\nif the Littlewood-Richardson coefficient cλ\nα,β (cf. Section 2.1.3) is positive\n(nonzero).\n46"},{"page":47,"text":"Equivalently, consider the diagonal homomorphism:\nρ : H = GLn(C) →G = H × H.\n(22)\nGiven an irreducible G-module Vα(H) ⊗Vβ(H), decide if an irreducible H-\nmodule Vλ(H) occur in it, when considered as an H-module via the diagonal\nhomomorphism.\nThis problem corresponds to circuits of depth two in the following sense.\nLet X be an n×n variable matrix. Let V = Sym1(X) be the space of linear\nforms in the entries of X. We have the action of G on P(V ) given by:\n((h1, h2) · f)(X) = f(h−1\n1 Xh2),\nfor any h1, h2 ∈H and f ∈P(V ).\nLet f(X) = trace(X).\nThen the\nstabilizer of f in G is precisely H, and f is characterized by its stabilizer.\nHence, f(X) = trace(X) is the characteristic function (Definition 3.1) of\nthe couple (22). It can be computed by a circuit of depth two. Hence, the\ncharacteristic class of the couple (22) can be defined to the class of circuits of\ndepth two. In this sense, the setting of the Littlewood-Richardson problem\nis roughly dual to the setting of expander graphs (Section 1.8), which too\ncorrespond to circuits of depth two.\nIn [GCT3, DM2, KT2] it is shown that this problem indeed belongs to\nP, thereby establishing the analogue of Hypothesis 7.2 in this setting. Two\nmain ingradients in this proof, in addition to linear programming, are PH1\nand SH for Littlewood-Richardson coefficients (Hypotheses 1.9 and 1.11).\nIn [GCT5], it is shown that the problem of deciding nonvanishing of a gen-\neralized Littlewood-Richardson coefficient for the classical connected reduc-\ntive groups other than GLn(C), namely the simplectic and the orthogonal\ngroups, also belongs to P, assuming the following generalized form of SH in\nthis context.\nLet ̃cλ\nα,β(k) = ckλ\nkα,kβ be the stretching function for a generalized Littlewood-\nRichardson coefficient cλ\nα,β, where α, β and λ are no longer partitions, but\nrather their generalizations [FH].\nIt is known to be a quasi-polynomial\n[BZ, DM2].\nHypothesis 9.2 (PH2): The quasi-polynomial ̃cλ\nα,β(k) is positive.\nThis was conjectured in [DM2] on the basis of considerable experimental\nevidence. Its weaker form is:\n47"},{"page":48,"text":"Hypothesis 9.3 (SH): The quasi-polynomial ̃cλ\nα,β(k) is saturated.\nIn [GCT5] it is shown that the problem of deciding if a generalized\nLittlewood-Richardson coefficient is nonzero also belongs to P assuming\nPH2, or its weaker form, SH.\n9.2\nFrom constant to superpolynomial depth\nThe goal now is to lift the polynomial time algorithms and the mathematical\npositivity hypotheses PH1 and PH2 above from the simplified constant-\ndepth setting to the superpolynomial-depth setting of Hypotheses 7.1 (a)\nand 7.2. This is done in [GCT6] in two steps. We only consider the P vs.\nNP problem, considerations for the NC vs. P #P problem being similar.\nWe use the same notation as in Section 5.\nThe first step is the following mathematical result which allows formu-\nlation of the mathematical hypotheses PH1,PH2, and SH. Let sλ\nd(H; l) and\nsλ\nd(E; n, l) denote the multiplicities of the Weyl module Vλ(G) in S(H; l)d\nand R(E; n, l)d, respectively.\nLet us associate with them the following\nstretching functions:\n ̃sλ\nd(H; l)(k) = skλ\nkd(H; l),\n(23)\nand\n ̃sλ\nd(E; n, l)(k) = skλ\nkd(E; n, l).\n(24)\nThen:\nTheorem 9.4 (cf. Theorem 3.4.11 in [GCT6])\n(Rationality Hypothesis): Assume that the singularities of the class varieties\nXP (H; m) and XNP(E; n, l) are rational.\nThen the stretching functions ̃sλ\nd(H; l)(k) and ̃sλ\nd(E; n, l)(k) are quasi-\npolynomials.\nSimilar result also holds in the context of NC vs. P #P problem.\nRationality (niceness) [Ke] of singularities here is supported by the algebro-\ngeometric results and arguments in [GCT2, GCT10].\nThe second step is the following complexity-theoretic result:\n48"},{"page":49,"text":"Theorem 9.5 (cf. Theorems 3.4.11 and 3.4.13 in [GCT6]) The decision\nproblems in Hypothesis 7.2, and hence, the problem of verifying an obstruc-\ntion (Hypothesis 7.1 (a)) are indeed in P assuming the rationality hypothesis\nabove, and PH1 and PH2 (or weaker SH) in the introduction (Hypothe-\nses 1.4-1.6).\nSimilar result also holds in the context of the NC vs. P #P problem,\nassuming analogous hypotheses PH1, PH2 (or weaker SH) in this setting.\nTheorem 9.5 reduces the complexity-theoretic positive hypotheses in Hy-\npothesis 7.2 to the mathematical positivity hypotheses PH1 and SH (PH2),\nand the rationality hypothesis, unconditionally. Furthermore, [GCT6] also\ngives theoretical and experimental results in support of these positivity hy-\npotheses, and suggests a plan for proving them via the theory of quantum\ngroups. We shall discuss this plan later in Sections 15-16.\nThe whole discussion of this section is summarized in Figure 7. The top\ndouble arrow is unconditional, the bottom arrow is conjectural.\nMathematical positivity hypotheses PH1,2, and the rationality hypothesis\n∥\n∥\nGCT6\n∥\n⇓\nComplexity theoretic positivity hypotheses in Hypotheses 7.2 ∥\n∥\nTransformation in Section 13; cf. Figure 6\n∥\n?\n⇓\nP ̸= NP over C\nFigure 7: The main result of GCT6\n9.3\nSaturated and positive integer programming\nThe algorithm in Theorem 9.5 is based on a polynomial time algorithm\nin [GCT6] for a restricted form of integer programming, called saturated\n49"},{"page":50,"text":"(positive) integer programming. We briefly explain it in this section.\nLet A be an m × n integer matrix, and b an integral m-vector.\nAn\ninteger programming problem asks if the polytope P : Ax ≤b contains an\ninteger point. In general, it is NP-complete. So let us begin with the well\nknown special case of integer programming which belongs to P. This is the\nunimodular integer programming problem, wherein the constraint matrix A\nis unimodular. This means the polytope P is integral. In this case, P has\nan integer point iffP is nonempty. The latter can be checked in polynomial\ntime by standard linear programming methods.\nSaturated (positive) integer programming is a generalization of unimod-\nular integer programming, wherein a variant of linear programming still\nworks, even when P is nonintegral, provided P satisfies certain saturation\nor positivity hypothesis, which make up for the loss of unimodularity.\nIt is defined as follows. Let fP(n) be the Ehrhart quasi-polynomial of\nP [St1]. An integer programming problem is called saturated if the Ehrhart\nquasi-polynomial fP(n) is guaranteed to be saturated (cf. Section 1.2), if P\nis nonempty. It is called positive if fP (n) is guaranteed to be positive (cf.\nSection 1.2), if P is nonempty. We allow m, the number of constraints, to\nbe exponential in n. Hence, we cannot assume that A and b are explicitly\nspecified. Rather, it is assumed that the polytope P is specified in the form\nof a (polynomial-time) separation oracle as in [GLS]. Given a point x ∈Rn,\nthe separation oracle tells if x ∈P, and if not, gives a hyperplane that\nseparates x from P.\nThe following is the main complexity-theoretic result in [GCT6].\nTheorem 9.6 A saturated, and hence positive, integer programming prob-\nlem has an oracle-polynomial-time algorithm.\nFurthermore, this polynomial time algorithm is conceptually extremely\nsimple. It is essentially a variant of linear programming: it uses a general-\nization of the ellipsoid method [Kh] for linear programming in [GLS], and\na polynomial time algorithm for computing Smith normal forms in [KB].\nThus the saturated and positive integer programming paradigm, in essence,\nsays that linear programming works for integer programming provided the\nsaturation or the positivity property holds.\nTheorem 9.5 follows from Theorem 9.4 because PH1 and SH (PH2) (cf.\nHypotheses 1.4-1.6) imply that the decision problems in Hypothesis 7.2 can\nbe transformed in polynomial time into saturated (positive) integer program-\n50"},{"page":51,"text":"ming problems. Thus, in essence, a variant of linear programming works for\nthe decision problems in Hypothesis 7.2, provided PH1 and SH (PH2) hold.\nBut these saturation and positivity hypotheses (PH1 and SH) are non-\ntrivial, and, as we shall see in Sections 14 to 16, their validity intimately\nseems to depend on deep phenomena in algebraic geometry and the theory\nof quantum groups. We can already see an indication of this here. For ex-\nample, even to state PH1, SH or PH2, we need to show that the stretching\nfunctions used in their statements are quasi-polynomials, as shown in The-\norem 9.4. Without it, PH1, SH and PH2 are meaningless. But the proof of\nTheorem 9.4 already depends on nontrivial machinery in algebraic geome-\ntry; e.g. the cohomology vanishing result in [Ke], and the result in [Bou],\nwhich, in turn, needs resolution of singularities in characteristic zero [Hi]\nand other cohomology vanishing results. Hence it should not be surprising\nif proving these positivity hypotheses needs far more. We shall describe the\nbasic plan in [GCT6] for proving them later (Sections 15-16).\n10\nWhy should PH1 and PH2 hold?\nBut, first, we have to explain why PH1 and PH2 should hold in the first\nplace.\nThis depends, as mentioned earlier, on the exceptional nature of\nH(Y ) and E(X). Specifically, on the fact that the associated class varieties\nXP (H; l) and XNP (E; n, l) are group-theoretic. We now elaborate on this.\nFirst, let us consider the analogue of the decision problem in Hypothe-\nsis 7.2 for the simplest group-theoretic variety, namely, a flag variety (Sec-\ntion 2.2).\nGiven a flag variety Z = Gvμ ⊆P(Vμ), where Vμ is a Weyl\nmodule of G = SLl(C), the decision problem is to decide if Vλ(G) occurs\nin R(Z)d, the degree d component of the homogeneous coordinate ring of\nZ. By the Borel-Weil theorem [FH], R(Z)d = V ∗\ndμ, the dual of Vdμ. Hence,\nVλ occurs in R(Z)d iffVλ = V ∗\ndμ. It is easy to show that this is so iffthe\nYoung diagram for λ is obtained by flipping the complement of the Young\ndiagram for dμ in the smallest rectangle containing it. This can be decided\nin poly(⟨d⟩, ⟨λ⟩, ⟨μ⟩) time. The analogues of PH1 and PH2 in this setting\nclearly hold, since the multiplicity of Vλ in R(Z)d is just 0 or 1.\nNow let us move to a general group-theoretic class variety. Let (H ֒→\nG ֒→K) be the associated group triple. Since the class variety in question\nis (essentially) determined by this triple, all questions concerning the vari-\nety should, in principle, be reducible to representation-theoretic questions\nregarding this triple; cf. [GCT10], and Sections 3 and 15.\n51"},{"page":52,"text":"In [GCT6] and [GCT10] analogues of the decision problems in Hypoth-\nesis 7.2 for the couples H ֒→G and G ֒→K are formulated. Furthermore,\ntheoretical and experimental evidence for PH1 and PH2 for the decision\nproblems associated with these couples is provided. Since the triples are\nqualitatively similar to the couples, though much harder, this provides the\nmain evidence in support of PH1 and PH2 for the class varieties under con-\nsideration. We shall turn to this evidence in the next section.\n11\nDecision problems in representation theory\nWe now describe the decision problems associated with the couple H ֒→G,\nthe couple G ֒→K being similar. A general decision problem is as follows:\nProblem 11.1 (The subgroup restriction problem)\nLet ρ : H →G be as above, with G connected (and some mild technical\nrestrictions on ρ as described in [GCT6]). Assume that both H and G are\nreductive. Let Vπ(H) be an irreducible representation of H, and Vλ(G) an\nirreducible representation of G, where π and λ denote the classifying labels\nof these representations.\nLet mπ\nλ be the multiplicity of Vπ(H) in Vλ(G),\nconsidered as an H-module via ρ. Given specifications of the embedding ρ\nand the labels λ, π, decide nonvanishing of the multiplicity mπ\nλ.\nThe general decision problems in Hypotheses 7.2 can be thought of as harder\nvariants of this problem obtained by going from couples to triples.\nAll\ncouples that arise in GCT are either of the type in this decision problem, or\nof a hybrid type obtained by combining this type with the type considered\nearlier in connection with the flag variety, when H = Pμ is parabolic; cf.\n[GCT10] for a discussion of the hybrid types.\nProblem 11.1 is a fundamental decision problem of representation theory.\nIndeed, one of the main motivations in the classical works of representation\ntheory, e.g. [W], for classifying of all irreducible representations of reductive\ngroups was to be able to solve this problem satisfactorily. But despite all\nprogress in representation theory in the last century, this problem at its very\nheart remained open. PHflip in [GCT6] says that this fundamental decision\nproblem of representation theory has an easy polynomial time algorithm.\nHere we shall describe PHflip in only the following three special cases\nof the above decision problem, referring the reader to [GCT6] for a full\ndiscussion and results for the general decision problem.\n52"},{"page":53,"text":"11.0.1\nLittlewood-Richardson problem\nLet H = GLn(C), G = H × H, the embedding\nρ : H →H × H = G\nbeing diagonal. Then the multiplicity in Problem 11.1 is just the Littlewood-\nRichardson coefficient, because every irreducible representation of G is of\nthe form Vα ⊗Vβ, where Vα and Vβ are irreducible representations of H =\nGLn(C) for partitions α, and β, and the multiplicity of an H-module Vλ in\nVα ⊗Vβ, considered as an H-module via the diagonal map ρ, is precisely\nthe Littlewood-Richardson coefficient cλ\nα,β. We have already noted that its\nnonvanishing can be decided in polynomial time (Section 9.1).\n11.0.2\nKronecker problem\nLet H = GLn(C) × GLn(C) and\nρ : H →G = GL(Cn ⊗Cn) = GLn2(C)\nthe natural embedding given by: ρ(h1, h2) = h1 ⊗h2, for any h1, h2 ∈H.\nHere h1 ⊗h2 is the Kronecker product as defined in (18). Let kπ\nλ,μ be the\nmultiplicity of the H-module Vλ(GLn(C)) ⊗Vμ(GLn(C)) in the G-module\nVπ(G), considered as an H-module via the embedding ρ. Then it can be\nshown [FH] that the Kronecker coefficient as defined in Section 2.1.2 is a\nspecial (dual) case of this when λ, μ and π there coincide with the λ, μ and\nπ here. For this reason, we call kπ\nλ,μ a Kronecker coefficient.\nProblem 11.2 (The Kronecker problem) Given partitions λ, μ and π, de-\ncide nonvanishing of the Kronecker coefficient kπ\nλ,μ.\nThe following is an analogue of Hypothesis 7.2 in this context:\nHypothesis 11.3 [GCT6] (PHflip-kronecker) Given partitions λ, μ and π,\nnonvanishing of the Kronecker coefficient kπ\nλ,μ can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)\ntime.\n11.0.3\nThe plethysm problem\nThe Kronecker coefficient is known [Ki] to be a special case of the plethysm\ncoefficient in the following more general problem.\n53"},{"page":54,"text":"Problem 11.4 (The plethysm problem) Given partitions λ, μ and π, de-\ncide nonvanishing of the plethysm constant aπ\nλ,μ. This is the multiplicity\nof the irreducible representation Vπ(H) of H = GLn(C) in the irreducible\nrepresentation Vλ(G) of G = GL(Vμ), where Vμ = Vμ(H) is an irreducible\nrepresentation H. Here Vλ(G) is considered an H-module via the represen-\ntation map\nρ : H →G = GL(Vμ).\nThe following is an analogue of Hypothesis 7.2 in this context:\nHypothesis 11.5 [GCT6] (PHflip-plethysm) Given partitions λ, μ and π,\nnonvanishing of the plethysm constant aπ\nλ,μ can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)\ntime.\n12\nThe P-barrier in representation theory\nAt the surface, this hypothesis too seems impossible because the dimension\nof G here can be exponential in the dimension of H. This happens when the\ndimension of the representation Vμ(H) is exponential in dim(H). But the\ntotal bitlength of λ, μ and π can be polynomial in dim(H). Hypothesis 11.5\nin this case says that nonvanishing of the plethysm constant can still be\ndecided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) time.\nA priori it is not even clear that the\nplethysm constant can be evaluated in PSPACE in this case. Since the usual\ncharacter-theory-based algorithms in representation theory for its evaluation\n[FH, Mc] take space that is polynomial in the dimension of G, and hence,\nexponential in the dimension of H.\nThe main P-barrier in representation theory is this huge gap between\nthe exponential space bound for the plethysm or the general decision Prob-\nlem 11.1 given by the usual methods of representation theory and the poly-\nnomial time bound stipulated in Hypothesis 11.5 for the plethysm constant\nand the hypothesis in [GCT6] for the general decision Problem 11.1.\n12.1\nCrossing the P-barrier\nWe now describe the main results of [GCT6] which together cross this P-\nbarrier in representation theory subject to the analogous mathematical pos-\nitivity hypotheses PH1 and SH (PH2). We shall only concentrate on the\nplethysm problem, since it is the crux of the matter.\n54"},{"page":55,"text":"Associate with a plethysm constant aπ\nλ,μ the stretching function\n ̃aπ\nλ,μ(k) = akπ\nkλ,μ.\n(25)\nNote that μ is not stretched here.\nThen the following is an (unconditional) analogue of Theorem 9.4 in this\ncontext:\nTheorem 12.1 (cf.\nTheorem 1.6.1 in [GCT6]) The stretching function\n ̃aπ\nλ,μ(k) is a quasi-polynomial function of k.\nThe following are the analogues of PH1 and PH2 in this context:\nHypothesis 12.2 (PH1)\nFor every (λ, μ, π) there exists a polytope P = P π\nλ,μ ⊆Rm with m =\npoly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) such that:\naπ\nλ,μ = φ(P),\n(26)\nwhere φ(P) is equal to the number of integer points in P, and the Ehrhart\nquasi-polynomial of P coincides with the stretching quasi-polynomial ̃aπ\nλ,μ(k)\nin Theorem 12.1. (And some additional technical constraints)\nHypothesis 12.3 (PH2)\nThe stretching quasi-polynomial ̃aπ\nλ,μ(k) is positive (cf. Section 1.2).\nPH2 implies the following saturation hypothesis:\nHypothesis 12.4 (SH)\nThe quasi-polynomial ̃aπ\nλ,μ(k) is saturated (cf. Section 1.2).\nThe following is an analogue of Theorem 9.5 in this context:\nTheorem 12.5 [GCT6] Assuming PH1 and SH (or, more strongly, PH2),\nnonvanishing of a plethysm constant aπ\nλ,μ can be decided in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩)\ntime; i.e. the problem of deciding nonvanishing of a plethysm constant be-\nlongs to P, as per Hypothesis 11.5.\n55"},{"page":56,"text":"PH1 above implies that aπ\nλ,μ belongs to #P just like the Littlewood-\nRichardson coefficient. Its weaker form is:\nTheorem 12.6 The plethysm constant aπ\nλ,μ can be computed in PSPACE,\ni.e., in poly(⟨λ⟩, ⟨μ⟩, ⟨π⟩) space.\nThat this holds even if the dimension of G = GL(Vμ) is exponential in\nn is crucial in the context of GCT. Because the dimension of K = GL(V )\nin the triples H ֒→G ֒→K = GL(V ) associated with the class varieties\n(Section 4) is exponential in the circuit size m. Hence, without this result,\nit is not at all clear why the structural constants sλ\nd(H, l) and sλ\nd(E; n, l) in\nHypothesis 1.4 should even belong PSPACE ⊇#P, as implied by it.\nTheorem 12.6 and Theorem 12.1 together provide good theoretical evi-\ndence for PH1 (Hypothesis 12.2). Indeed, Theorem 12.1, together with other\nevidence in [GCT6], suggests that ̃aπ\nλ,μ(k) is the Ehrhart quasi-polynomial\nof some polytope P = P π\nλ,μ. Furthermore, Theorem 12.6 says that the di-\nmension m of the ambient space Rm containing P should be polynomial in\nthe bitlengths ⟨λ⟩, ⟨μ⟩and ⟨π⟩. If not, it would not be possible to count\nthe number of integer points in P in PSPACE, since even the bitlength of\nany integer point in Rm would not be polynomial. For further theoretical\nand experimental results in support of PH1 and PH2 in this context, see\n[GCT6]. These constitute the main evidence in support of PH1 and PH2\nfor the group-theoretic class varieties (Hypotheses 1.4-1.5), because mathe-\nmatical positivity is a very abstract property, which should remain invariant\nwhen we go from couples to triples.\n13\nReduction\nNow we turn to the reduction in the top arrow in Figure 1. For this, we\nhave to describe:\n1. How to transform the easy algorithms in Theorem 9.5 into an easy\nalgorithm for discovering an obstruction as in Hypothesis 7.1 (c), and\n2. How to transform this easy algorithm for discovery into a constructive\nproof of existence of obstructions–as expected (Section 6)–for every n\nand l = nlog n, by showing how such an obstruction-label can be easily\nconstructed in this case explicitly.\n56"},{"page":57,"text":"This would imply that P ̸= NP over C.\nThese transformations cannot be carried out at present, since we do\nnot know the polytopes P λ\nd (H; l) and P λ\nd (E; n, l) explicitly. We only know\nthat they should exist as per PH1 (Hypothesis 1.4). But once their explicit\ndescriptions become available, it should be possible to carry out the above\ntwo transformations along the lines that we now suggest.\n13.1\nTowards easy discovery\nFirst, let us describe why it should be possible to extend and transform\nthe polynomial-time algorithms in Theorem 9.5 to obtain a polynomial time\nalgorithm for discovering an obstruction (Hypothesis 7.1 (c)) once explicit\ndescriptions of the polytopes P λ\nd (H; l) and P λ\nd (E; n, l) become available.\nFor the sake of simplicity, let us assume that the quasi-polynomials in\nTheorem 9.4 are actually polynomials; i.e., their periods are one, though\nthis is not expected. In that case, it can be shown that [GCT6] PH1 and SH\nimply that there exist polytopes P(n, l) = P(E; n, l) and Q(l) = P(H; l) of\npoly(n, l) dimensions such that an obstruction for n, l and the pair (E, H)\nexists iffthe relative difference T(n, l) = P(n, l) \\ Q(l) is nonempty, and\nfurthermore, an explicit obstruction can also be constructed in polynomial\ntime once we are given a rational point in T(n, l). So it suffices to check\nif T(n, l) is nonempty, and if so, find a rational point in it. This can be\ndone in polynomial time using the convex (linear) programming algorithm in\n[GLS, Vd] if Q(l) has only poly(l) explicitly described facets. This is so even\nif P(n, l) has exponentially many facets. But if Q(l) has exponentially many\nfacets–as happens even in the context of the simpler Littlewood-Richardson\nproblem (Problem 9.1)–then an oracle-based algorithm as in [GLS] cannot\nbe used to get a polynomial time algorithm for this problem [B].\nBut this does not appear to be a serious problem.\nIndeed, a gen-\neral principle in combinatorial optimization, as illustrated in [GLS], is that\ncomplexity-theoretic properties of polytopes with exponentially many facets\nare similar to the ones with polynomially many facets if these facets have a\nwell-behaved regular structure. For example, if Q(l) and P(n, l) were perfect\nmatching polytopes for non-bipartite graphs–which can have exponentially\nmany facets–nonemptiness of T(n, l) can be easily decided in polynomial\ntime [Ve] using the polynomial time algorithm [Ed] for finding a perfect\nmatching in a nonbipartite graph. The facets of the analogues of P(n, l) and\nQ(l) in the Littlewood-Richardson problem, called Littlewood-Richardson\ncones [Z], have an explicit description with very nice algebro-geometric and\n57"},{"page":58,"text":"representation-theoretic properties [Kl]. The same is expected to be the case\nin our setting.\nThis is why we expect that nonemptiness of T(n, l) and computation of\na rational point in it, if it is nonempty, can be done in polynomial time, once\nexplicit descriptions of P(n, l) and Q(l) become known. This would give a\npolynomial time algorithm for discovering an obstruction, if it exists, as per\nHypothesis 7.1 (c), assuming that the quasi-polynomials in Theorem 9.4 are\npolynomials.\nFurthermore, it is expected, for the mathematical reasons given in [GCT6],\nthat there exist genuinely simple, i.e., purely combinatorial greedy-type al-\ngorithms for the problems under consideration that do not even need linear\nprogramming.\nThat is, the story is expected to be the same as for the\nmin-cost flow problem in combinatorial optimization, for which a linear-\nprogramming-based polynomial-time algorithm was found first [Ta] to be\nfollowed by several genuinely simple and purely combinatorial polynomial\ntime algorithms; e.g. see [O]. Similarly, it is reasonable to expect that the\nalgorithms in Theorem 9.5 and the subsequent algorithm for discovery of\nobstructions can be simplified further to eventually get simple greedy al-\ngorithms for these problems akin to the Hungarian method, once explicit\ndescriptions of P(n, l) and Q(l) become known.\nSo far we are assuming that the quasi-polynomials in Theorem 9.4 are\npolynomials. This need not be so. In fact, this is not so even in the sim-\nplified setting of plethysm constants [GCT6]. When the quasi-polynomials\nin Theorem 9.4 have nontrivial periods, the obstructions can be classified\nin two types: geometric and modular [GCT6]. Geometric obstructions are\nsimilar to the ones that would arise if these quasi-polynomials were poly-\nnomials. A polynomial time algorithm for their existence and construction\nmay be designed along the lines we just described.\nLet us next describe briefly what needs to be done in the case of mod-\nular obstructions. Theorem 9.5 says that for the decision problems therein\nlinear programming in conjunction with modular techniques (computation\nof Smith normal forms [KB]) works even in the modular setting, i.e., when\nthe quasi-polynomials have nontrivial periods. Hence, once we have a poly-\nnomial time algorithm for discovering a geometric construction, it should\nbe possible to extend it to a polynomial time algorithm for discovering a\nmodular obstruction in conjunction with appropriate modular techniques;\ncf. [GCT6] for the problems that need to be addressed in this extension.\n58"},{"page":59,"text":"13.2\nFrom easy algorithm for discovery to easy proof of ex-\nistence\nAssuming that we have an easy polynomial time algorithm for discovering an\nobstruction as per Hypothesis 7.1 (c), let us now describe why it should be\npossible to prove using this algorithm, or rather the underlying structure and\ntechniques, that there always exists an obstruction, as expected (Section 6),\nfor every n →∞, assuming l = nlog n (say).\nFor the sake of simplicity, let us again assume that that the quasi-\npolynomials in Theorem 9.4 are polynomials, and that we have an easy\nHungarian-type greedy algorithm as discussed above for deciding nonempti-\nness of T(n, l), and for computing a point it it, if it is nonempty. Then we\nhave to show, using the techniques and the structure underlying this algo-\nrithm, that T(n, l) is always nonempty when l = nlog n, n →∞. Such a proof\nwould also give us a polynomial time procedure for explicit construction of\nan obstruction λ(n), for every n. Hence we shall call it a P-constructive\nproof.\nTo see how to get such a P-constructive proof, let us consider an analogy.\nLet us imagine that Q(l) is empty, so that T(n, l) = P(n, l) is a polytope,\nand that it is the perfect-matching polytope of a bipartite graph G(n, l).\nThen T(n, l) is nonempty iffG(n, l) has a perfect matching, which can be\nthought of as an obstruction in this analogy. The analogous goal then is to\nshow using the techniques and structure underlying the Hungarian method\nthat G(n, l) always has a perfect matching, as expected, when l = nlog n,\nand n →∞. In other words, we have to give a P-constructive proof for\nexistence of a perfect matching in every such G(n, l). In this analogy, the\ntechnique underneath the Hungarian method can be easily used to give a\nconstructive proof of Hall’s marriage theorem–namely, that every bipartite\ngraph H in which every subset, on any side of the graph, has at least as\nmany neighbours as the size of that subset has a perfect matching–which\nthen has to be used to show that G(n, l) always has a perfect matching\nwhenever l = nlog n, n →∞.\nNow in our setting T(n, l) is not a perfect matching polytope. But if\nit has a nice structure like the perfect matching polytope then it should be\npossible to prove structure theorems in the spirit of Hall’s marriage theorem\nfor T(n, l) using the structure of the Hungarian-type greedy algorithm for\ndeciding nonemptyness of T(n, l) and then use it to prove nonemptyness of\nT(n, l), for every n →∞, when l = nlog n.\nFor such a transformation of a polynomial time algorithm for discovery\n59"},{"page":60,"text":"into a P-constructive proof of existence to work, it is crucial that:\n1. The polyhedral set T(n, l) has a nice, regular structure like the perfect\nmatching polytope.\nFortunately, the polytopes P(n, l) and Q(n, l)\nthat would arise in our setting should be even nicer than the per-\nfect matching polytope. For example, in the simpler setting of the\nLittlewood-Richardson problem (Problem 9.1), P(n, l) and Q(l) be-\ncome Littlewood-Richardson cones [Z], which have extremely regu-\nlar structure with remarkable representation-theoretic and algebro-\ngeometric properties [F2, Kl]. The same is expected to be the case\nfor the actual P(n, l) and Q(l).\n2. The algorithm for discovery not only works in polynomial time, but\nalso has a simple structure like the Hungarian method. The Hungarian-\ntype greedy algorithms that we expect for the problems under consid-\neration should have such structures.\nHence, it is reasonable to expect that an easy Hungarian-type algorithm\nfor deciding nonemptyness of T(n, l) can be transformed into the sought P-\nconstructive proof of obstructions. The story is expected to be similar, albeit\nmuch harder, when the quasi-polynomials in Theorem 9.4 have nontrivial\nperiods; cf [GCT6].\nThe above scheme for the transformation of an algorithm for discovery\ninto a constructive proof of existence banks on the fact that the algorithm\nto be transformed is easy, i.e., works in polynomial time, besides having\na simple structure.\nThe underlying informal principle, which cannot be\nproved, is that the mathematical complexity of an algorithmic (constructive)\nproof is intimately linked to the computational complexity of the algorithm\non which it is based; see Section 19 for a detailed treatment of this issue.\nThis is why there is no nontrivial result, comparable to Hall’s theorem, for\nHamiltonian paths.\nBecause the problem of finding such a path is NP-\ncomplete.\nThe reader may wonder why we are talking about explicit construction of\nobstructions, when, strictly speaking, we only need to know their existence.\nThis is because the nature of obstructions in our case is such that their\nexplicit construction, if they exist, can be done with only a little additional\ncost over the cost of deciding existence. To see this, let us again assume,\nfor the sake of simplicity, that the quasi-polynomials in Theorem 9.5 are\npolynomials.\nThen a technique that can decide nonemptiness of T(n, l)\nshould also be able to compute a point in it, as a proof of nonemptiness, at\n60"},{"page":61,"text":"only a little additional cost, just as in linear programming. In other words,\nthe complexity of deciding existence of an obstruction should be more or\nless the same as that of constructing it, if it exists. This is why we mainly\ntalk of explicit construction of obstructions though, in principle, just their\nexistence would suffice.\nOur discussion so far says that PH1 and SH (PH2) are the crux of the\nmatter. If they can be proved, and explicit descriptions of the polytopes\ntherein become available, it should be possible to transform the easy algo-\nrithms in Theorem 9.5 into an easy algorithm for explicit construction of\nobstructions as per Hypothesis 7.1 (b).\n14\nStandard quantum group\nNow we proceed to the basic plan in [GCT6] for proving PH1 and SH. This\nis motivated by a story in the theory of standard quantum groups in the\ncontext of the Littlewood-Richardson problem (Problem 9.1). We describe\nthat story in this section.\nFor this we need the notion of a standard quantum group, by which we\nmean the quantum group in [Dri, Ji, RTF]. We can not formally define here\nthis object, but we can at least give an intuitive idea. Let GL(Cn) be the\ngroup of nonsingular n × n matrices. It can be thought of as the group of\nnonsingular transformations of Cn. Let xi’s denote the coordinates of Cn.\nThese commute. That is:\nxixj = xjxi.\nLet us now see what happens if the coordinates become noncommuting. This\nis precisely what happened in quantum physics. We discovered that the po-\nsition and the momentum, which for centuries we thought were commuting\nobservables, do not actually commute. Quantum groups were invented pre-\ncisely to investigate the related phenomena in theoretical physics. Let Cn\nq\ndenote the quantum space whose coordinates xi’s are noncommuting, and\nsatisfy the following relation:\nxixj = qxjxi,\ni < j\nwhere q ∈C is some fixed number. The standard quantum group GLq(Cn) is\nthe “group” of invertible linear transformations of this quantum space. This\nis not a “group” in any ordinary sense. Its precise description is given in [Dri,\n61"},{"page":62,"text":"Ji, RTF]. We do not need that here. Let us just think of a quantum group\nas what a group becomes when the coordinates become noncommuting.\nLet us now explain how quantum groups enter in the story of Littlewood-\nRichardson coefficients.\nThis is because the most transparent proof of\nthe Littlewood-Richardson rule came via the theory of quantum groups\n[Kas1, Li, Lu2]. The earlier proofs, though elementary and combinatorial,\nwere highly mysterious. Moreover, the theory of quantum groups gave the\nfirst proof of the generalized Littlewood-Richardson rule [] for general (con-\nnected) reductive groups, instead of just GLn(C).\nLet us now elaborate the nature of this proof. We begin by observing\nthat the Littlewood-Richardson problem (Problem 9.1) is an instance of the\ngeneral decision Problem 11.1 associated with the diagonal group homomor-\nphism\nρ : H = GL(Cn) →H × H = GL(Cn) × GL(Cn).\n(27)\nIf we understood the structure of this homomorphism in depth, we ought\nto understand why PH1 and SH (and also PH2) hold for the Littlewood-\nRichardson coefficients.\nAs we mentioned earlier, in depth means at the\nquantum level.\nTo understand the homomorphism (27) at the quantum\nlevel, we need to quantize it. Ideally, one would want its quantization in the\nform of a homomorphism\nρq : Hq = GLq(Cn) →Hq × Hq = GLq(Cn) × GLq(Cn).\n(28)\nwhere Hq is the standard quantum group associated with H. This does not\nhold as it is; i.e., Hq is not a quantum subgroup of Hq × Hq. But this is\nessentially so. That is, it holds in a certain dual setting–this is the main\nresult in [Dri, Ji, RTF]. Thus the theory of quantum group can be regarded\nas the theory of the quantization ρq.\nOnce this theory is developed sufficiently, the Littlewood-Richardson rule\nas well as PH1 for Littlewood-Richardson coefficients (Hypothesis 1.9) turn\nout to be a consequence, in a nontrivial way, of a deep positivity result in\nthe theory of the standard quantum groups [Kas2, Kas3, Lu1, Lu2]: namely,\ntheir representations and coordinate rings have canonical bases, also called\nglobal crystal bases, whose structural constants, which determine their mul-\ntiplicative and representation theoretic structure, are all nonnegative. For\nthis reason, we say that the canonical bases are positive, and refer to exis-\ntence of a canonical basis as a positivity property (hypothesis) PH0.\nWe now give a brief intuitive description of the canonical basis. Let X\nbe an n×n variable matrix. The coordinate algebra R = O(G) of the group\n62"},{"page":63,"text":"G = GL(Cn) is defined to be the C-algebra generated by the entries xij of X\nand det(X)−1, where det(X) denotes the determinant of X. Its elements are\nregular functions on G, considered as an affine variety. There is a natural\nleft action of G on R given by f(X) →f(σ−1X), for any σ ∈G, and a\nsimilar right action.\nThese notions can now be quantized. It is possible to associate a coor-\ndinate ring Rq = O(Gq) with the standard quantum group Gq = GLq(Cn),\nwhose elements can be intuitively thought of as functions on Gq. Unlike R,\nRq is not commutative. Its precise definition can be found in [RTF]. There\nare natural left and right actions of Gq on Rq.\nA canonical basis B of Rq is a very special basis with the following\nproperties:\n(1) It is representation-theoretically well behaved. This means there is a\nfiltration\nB0 = ∅⊂B1 ⊂B2 ⊂· · · ⊂B\nwith ∪iBi = B, such that ⟨Bi⟩/⟨Bi−1⟩is an irreducible Gq-module. Here\n⟨Bi⟩denotes the span of the basis elements in Bi.\n(2) Positivity property of the multiplicative structure constants:\nGiven two elements b, b′ ∈B, let\nbb′ =\nX\nb′′∈B\nf b′′\nb,b′b′′,\nbe the expansion of the product in terms of the basis B. Then each f b′′\nb,b′ is\nan explicit polynomial in q and q−1 with nonnegative coefficients. Here f b′′\nb,b′\nare called multiplicative structure constants. What this says is that each\nmultiplicative structure constant has an explicit positive formula, akin to\nthat of the permanent. Here explicit means that each nonnegative coefficient\nof f b′′\nb,b′ has an interpretation in terms of a nonnegative topological invariant\n(akin to Betti numbers) of an algebraic variety.\n(3) Positivity property of the representation-structure constants:\nGiven any element b ∈B and a generator e of a certain algebra defined\nin [Dri, Ji], which is “dual” to Rq, let\ne · b =\nX\nb′\ngb′\ne,bb′\nbe the expansion of e · b, the result of applying e to b, in terms of the\nbasis B. Then each gb′\ne,b is also an explicit polynomial in q and q−1 with\n63"},{"page":64,"text":"nonnegative coefficients.\nThat is, each representation-structure constant\nalso has an explicit positive formula.\nThese positivity properties do not actually hold as stated–that is still a\nconjecture [Lu2]–but their slightly weaker form holds unconditionally [Lu2].\nWe shall ignore that difference here.\nIf we specialize the canonical basis at q = 1, we get a canonical basis of\nR, the coordinate ring of G, with analogous positivity property. But, as of\nnow, the only way to prove existence of such a canonical basis of R is via the\ntheory of quantum groups as above. This shows the power of this theory.\nOne can easily imagine that there ought be a connection between ex-\nistence of bases whose structural constants have explicit positive formu-\nlae (PH0) and existence of an explicit positive (polyhedral) formula for\nLittlewood-Richardson coefficients (PH1). That is indeed so, as we men-\ntioned earlier, but in a quite nontrivial way; cf. [Kas1, Li, Lu2]. We shall\nsimply take this connection on faith here. Pictorially:\nPH0 →PH1.\n(29)\nOne does not really need the full power of PH0 to deduce PH1. Just\nexistence of a local crystal basis [Kas1], which is the limit (crystalization)\nof a canonical basis as q →0, is sufficient.\nBut when we move to the\nnonstandard setting in GCT, even the full power of PH0 is needed for some\nother reasons; [GCT8, GCT10].\nThe implication (29) provides arguably the most satisfactory proof of\nPH1 for Littlewood-Richardson coefficients, which, in addition, also pro-\nvides deep additional information (existence of canonical bases) which the\ncombinatorial proofs [F1] cannot provide. Such canonical bases are central\nto the approach in [GCT6, GCT10] towards PH1 and PH2 for the group-\ntheoretic class varieties (Section 15). Hence, as far as GCT is concerned,\nquantum groups are a must.\nSH for the usual Littlewood-Richardson coefficients is the saturation\ntheorem in [KT1]. It comes from a reformulation of PH1 in terms of special\npolytopes (called Hive polytopes) and their subsequent detailed study. Thus\npictorially:\nPH1 →SH,\n(30)\nagain in a nontrivial way.\n64"},{"page":65,"text":"But how is PH0 proved?\nThe only known proof of PH0 [Lu1, Lu2]\nis based on a deep positivity property in mathematics: the Riemann Hy-\npothesis over finite fields [Dl2], and related results [BBD]. In other words,\nnonnegativity of the structural constants associated with Hq is connected at\na profound level with the lining up of the zeros of the zeta functions of some\nalgebraic varieties on one axis. We shall denote the Riemann hypothesis\nover finite fields by PH+. Then pictorially:\nPH+ →PH0,\n(31)\nin a highly nontrivial way.\nPutting implications (29)-(31) together with the story in Section 9.1, we\narrive at Figure 8 which summarizes the story in this section.\nPH+: The Riemann hypothesis over finite fields and related results [BBD, Dl2]\n∥\n∥\n⇓\nPH0: Existence of canonical bases [Lu1, Lu2]\n∥\n∥\n⇓\nPH1 and SH [Kas1, Li, Lu2, KT1]\n∥\n∥\n∥\n⇓\nPolynomial time algorithm for deciding nonvanishing of\nLittlewood-Richardson coefficients [DM2, GCT5, KT1]\nFigure 8: A story in the theory of standard quantum groups\n15\nNonstandard quantum groups\nNow we turn to the problem of proving PH1 and SH that actually arise in\nGCT (Hypotheses 1.4-1.6, and 12.2-12.4). The basic plan in [GCT6] for this\n65"},{"page":66,"text":"is simply to lift the story in Figure 8 from height two to superpolynomial\nheight–i.e., from the circuits of height two that the Littlewood-Richardson\nproblem corresponds to to the circuits of superpolynomial height that the\ndecision problems in Hypothesis 7.2 correspond to.\nRoughly, it goes as\nfollows:\n(1) Quantization: Quantize the couples\nH ֒→G,\nG ֒→K\nand the triples\nH ֒→G ֒→K,\nassociated with the class varieties in a manner akin to the quantization (28)\nof (27) via standard quantum groups.\n(2) PH0 for couples and triples: Prove that the coordinate rings and\nrepresentations of the quantum groups that arise in this quantization have\ncanonical bases akin to the canonical bases for the standard quantum groups\nwhose structure constants, which determine their multiplicative and repre-\nsentation theoretic structure, are all nonnegative.\n(3) PH0 for class varieties: Use the canonical bases for the (quantized)\ntriples associated with the class varieties to construct analogous canonical\nbases for the coordinate rings for appropriate quantizations of the class\nvarieties with nonnegative structure constants.\n(4) PH1, SH: Deduce PH1 and SH from PH0 in the spirit of the middle\narrow in Figure 8.\nFigure 9 shows this pictorially.\nWe shall now elaborate Figure 9.\n15.1\nQuantization\nLet us begin with the first step of quantization. We shall only worry about\nthe couples. To be concrete, let\nH ֒→G = GL(Ck),\n(32)\nbe as in Problem 11.1, where H is connected, reductive subgroup of G.\nQuantization of this couple is the crux of the problem.\nAll other quan-\ntizations that are needed are hyped up versions of this, so we shall only\nconcentrate on this.\n66"},{"page":67,"text":"Quantization of couples [GCT4,\nGCT7] and triples [GCT10]\n|\n|\n|\n↓\nPH0\nfor\ncouples\nand\ntriples\n[GCT6, GCT8, GCT10]\n−−−→\nPH1 and SH for cou-\nples and triples\n|\n|\n↓\nPH0 for class varieties [GCT10]\n|\n|\n|\n↓\nPH1 and SH for class varieties\nFigure 9: The basic plan for proving PH1 and SH in [GCT6]\nThe standard theory of quantum groups can not be used for quantizing\nthis couple, as expected. Specifically, let Gq = GLq(Cn) be the standard\nquantum group associated with G. In a similar fashion, one can associate\n[Dri, Ji, RTF] a standard quantum group Hq with H. Then, Hq cannot be\nembedded as a quantum subgroup of Gq (where the notion of subgroup in\nthe quantum setting is akin to the usual notion of a subgroup). Hence the\ngoal is to associate a quantization ˆGq with G akin to the standard quantum\ngroup Gq so that the standard quantum group Hq is a quantum subgroup\nof ˆGq. In that case:\nHq ֒→ˆGq,\n(33)\ncan be considered to be a quantization of (32).\nThis quantization step is addressed in the following result for the couples\nin Problems 11.2-11.4, which are the main prototypes of the couples that\narise in GCT.\n67"},{"page":68,"text":"Theorem 15.1 (1) (cf. [GCT4]) The couple\nH = GL(Cn) × GL(Cn) →GL(Cn ⊗Cn) = G,\nassociated with the Kronecker problem (Problem 11.2) can be quantized in\nthe form:\nHq →ˆGq,\nwhere Hq is the standard quantum group associated with H and ˆGq is the\nnew nonstandard quantum group associated with G. Furthermore, ˆGq has a\nquantum unitary subgroup ˆUq in the sense of [Wo], which is a quantization\nof the unitary subgroup U = Un2(C) ⊆G = GLn2(C).\n(2) (cf. [GCT7]) More generally, the couple\nH = GLn(C) →G = GL(Vμ(H)),\n(34)\nassociated with the plethysm problem (Problem 11.4) can also be quantized\nin the form:\nHq →ˆGq,\nwhere Hq is the standard quantum group associated with H and ˆGq is the\nnew nonstandard (possibly singular) quantum group associated with G. Here\nH can even be any connected classical reductive group.\nThe nonstandard quantum group in [GCT4] is qualitatively similar to\nthe standard quantum group in [Dri, Ji, RTF] in the sense that it has a max-\nimal quantum unitary subgroup just as in the the standard case. This, in\nconjunction with work in [Wo], allows the mathematical machinery related\nto unitariness–such as harmonic analysis, existence of orthonormal bases–to\nbe transported to its theory. This is important in the context of PH0. In-\ndeed, PH0 in the theory of standard quantum groups is intimately related\nto existence of unitary quantum subgroups. Because the local crystal bases\nfor representations of the standard quantum group [Kas1], which were later\nglobalized to canonical (global crystal) bases in [Kas2], arose in the study of\nspecial orthonormal Gelfand-Tsetlin bases for representations of the stan-\ndard quantum group. This is first main reason why PH0 is expected to hold\nfor the nonstandard quantum group in [GCT4].\nThe general nonstandard quantum group [GCT7] can be singular, i.e.,\nits quantum determinant can vanish. Hence, we cannot define its quantum\nunitary subgroup in the sense of [Wo].\nFortunately, this is not matter,\nbecause analogues of the main required results in [Wo] still hold; cf. [GCT7]\n68"},{"page":69,"text":"for a precise statement.\nHence PH0 is expected to hold for the general\nnonstandard quantum groups in [GCT7] as well.\nBut at the same time these nonstandard quantum groups are fundamen-\ntally different from the standard quantum groups. Hence the terminology\nnonstandard. For a detailed description of the differences between the stan-\ndard and nonstandard quantum groups, see [GCT4, GCT7, GCT8]. Here we\nonly give a brief description from the complexity-theoretic perspective. To-\nwards this end, we associate a complexity level with each of these quantum\ngroups. This is briefly done as follows.\nSuppose H ֒→G is a primary couple associated with a group-theoretic\nclass variety for some complexity class C (Definition 3.1). Then the com-\nplexity class of this primary couple as well as its quantization, if it exists, is\ndefined to be just C.\nAs we have already noted, the theory of the standard quantum group is\nthe theory of quantization of the couple (cf. (27))\nGLn(C) →GLn(C) × GLn(C).\nThis is a primary couple associated with orbit-closure of the trace of an\nn×n matrix (Section 9.1), which can be computed by a circuit of depth two\nusing only additions or multiplications by constants. Hence, the standard\nquantum group corresponds to the complexity class of problems that can\nbe solved by circuits of depth two using only additions or multiplications\nby constants, just like expanders (Section 17).\nThere is no lower bound\nproblem here to speak of. That is why the standard quantum group cannot\nbe used for deriving any lower bound, again like expanders.\nThe couple associated with the Kronecker problem coincides with the\nprimary couple\nGL(Cm) × GL(Cm) →GL(Cm ⊗Cm).\n(35)\nassociated with the NC-class variety; cf. (17). Not exactly. The primary\ncouple associated with the NC-variety is slightly different from this, but\nthe difference is trivial, and can be ignored.\nTheory of the nonstandard\nquantum group in Theorem 15.1 (a) is the theory of quantization of this\ncouple. Hence, the complexity class of this nonstandard quantum group can\nbe defined to be NC.\nThe couple (34) that is quantized in [GCT7] is not a primary couple\nof any class variety. But it is qualitatively similar to the primary couple\n69"},{"page":70,"text":"associated with the NP-class-variety (Section 4.2).\nFor this reason, the\nnonstandard quantum group in [GCT7] can be roughly taken to be of su-\nperpolynomial complexity.\n15.2\nPH0 for couples and triples\nThe article [GCT8] gives a conjecturally correct algorithm to construct\ncanonical bases of the coordinate rings of the nonstandard quantum groups\nin [GCT4, GCT7]. These are natural generalizations of the canonical ba-\nsis in [Kas3, Lu2] for the coordinate ring of the standard quantum group.\nFurther theoretical and experimental evidence in support of PH0 for the\nnonstandard quantum group in [GCT4] is also given. For the problems that\nhave to be addressed in the context of the triples associated with the class\nvarieties under consideration, see [GCT10].\n15.3\nPH0 for class varieties\nSince the group-theoretic class varieties are essentially determined by the\nassociated group triples, once PH0 is proved for the triples, it should, in\npriciple, be possible to “transport” this knowledge from group theory to\nalgebraic geometry, thereby proving PH0 for the class varieties. In [GCT10]\nis basic plan for this “transport” is suggested, with a description of the\nvarious mathematical problems that need to be resolved.\nA crucial bridge between group theory and algebraic geometry for this\ntransport is provided by Conjecture 6.1, which has to be proved first. It may\nbe remarked that quantum groups were indeed brought into GCT precisely\nfor the purpose of proving this conjecture, thereby extending the proof in\n[GCT2] for its weaker form (Theorem 6.3). A basic plan for this extension\nvia nonstandard quantum groups is also suggested in [GCT10].\n15.4\nPH1 and SH\nThe journey from PH0 to PH1 in the nonstandard setting should be akin to\nthe one in the standard setting; cf. [GCT6].\nIn summary, the nonstandard quantum groups have to be used as a\nrope, as it were, to pull the proofs of the various mathematical positivity\nhypotheses from the constant depth (of the standard quantum groups) to\nsuperpolynomial depth.\n70"},{"page":71,"text":"16\nUltimate mystery: nonstandard Riemann hy-\npotheses?\nNow we come to the final chapter of this story: How to prove PH0, and\nspecifically, correctness of the algorithm in [GCT8] for constructing canon-\nical bases of the coordinate rings of the nonstandard quantum groups in\n[GCT4, GCT7] and their required conjectural properties.\nFor the standard quantum group, as we mentioned in Section 1, the\ntopological proof in [Lu1, Lu2] depends on the Riemann hypothesis over\nfinite fields [Dl2] and the related work [BBD]. The main open problem at\nthe heart of GCT is to extend this work, and use it to prove nonstandard\nPH0. But the standard Riemann hypothesis over finite fields is not expected\nto work in the nonstandard setting; cf. [GCT8]. Briefly this is because the\nrelevant quantized noncommutative algebraic varieties in the nonstandard\nsetting simply “disappear” when specialized at q = 1. Specifically, unlike in\nthe standard case, the Hilbert function of these varieties at q ̸= 1 is different\nfrom the Hilbert function of the corresponding classical varieties at q = 1.\nHence, they look very different from the classical algebraic varieties. This\nis why the Riemann hypothesis over finite fields may not be used as in the\nstandard case for proving PH0.\nThus we seem to need nonstandard extensions of the Riemann hypothesis\nover finite fields in the quantized noncommutative setting to prove the PH0’s\nunder consideration. We cannot even formulate such extensions. But we\nbelieve such nonstandard extensions exist. We now briefly explain why.\nFor this, we need to indicate the nature of the experimental evidence\n[GCT8] in support of PH0 for the most basic nonstandard quantum group\nfor the Kronecker problem in [GCT4]. Specifically, around a thousand struc-\ntural constants associated with a canonical basis for a certain dual of this\nquantum group were computed, each structural constant being a polynomial\nin q of degree more than ten. All the coefficients of these structural polyno-\nmials turned out be nonnegative. In the standard case, the cause for such\nnonnegativity was the Riemann hypothesis over finite fields. There ought\nto be a similar theoretical cause for nonnegativity in the nonstandard set-\nting. For, without a cause, the probability of over ten thousand coefficients\nbeing nonnegative would be absurdly small–naively 1/210000. This estimate,\nbeing naive, should not be taken literally. But it does suggest that the ex-\nperimental evidence for positivity should only be a shadow of the ultimate\ncause–nonstandard analogues of the Riemann hypotheses over finite fields.\n71"},{"page":72,"text":"This leads us to believe that nonstandard extensions of the Riemann\nhypothesis over finite fields for the various nonstandard quantum groups\nthat arise in GCT exist, and now, having seen the shadow, we have to search\nfor the ultimate cause whose shadow it is. If this search succeeds, then we\ncan expect to pull the proofs of PH0 from the standard to the nonstandard\nsetting, using the rope provided by the nonstandard quantum groups, and\nthe power provided by nonstandard Riemann hypotheses, thereby leading\nto the proof of P ̸= NP conjecture in characteristic zero; cf. Figure 3.\nEventually, this whole story in characteristic zero, along with the non-\nstandard Riemann hypotheses and the accompanying positivity hypotheses,\nmay be lifted, as suggested in [GCT11], to algebraically closed fields of pos-\nitive characteristic, and finally, finite fields, thereby proving the P ̸= NP\nconjecture in its usual form. This would then constitute the ultimate flip in\nFigure 2.\n17\nObstructions vs. expanders\nWe now explain the relationship between explicit construction of obstruc-\ntions and explicit construction of expanders as shown in Figure 4.\nAs per the hardness-vs-randomness principle [A2, IW, KI, NW], deran-\ndomization is intimately linked to lower bound problems.\nIn particular,\nrestricted kinds of lower bounds follow from existence of efficient pseudo-\nrandom generators. At present, we do not have pseudo-random generators\nbased on expander-like structures that can yield a lower bound result for\nconstant depth circuits. But for the sake of discussion, let us imagine that\nthe expander in, say, [LPS, Ma] can be generalized further to obtain a hy-\npothetical structure, which we shall call a strong expander, using which we\ncan obtain an efficient pseudo-random generator, whose existence, in turn,\nimplies separation of the class NC1 from AC0. Here NC1 is the class of\nproblems that can be solved by circuits of logarithmic depth, and AC0 the\nclass of problems that can be solved by circuits of constant depth. Fur-\nthermore, let us also assume that the problem of constructing such a strong\nexpander belongs to (nonuniform, algebraic) AC0, as it does for the ex-\npander in [LPS, Ma]. Now the existence of such a family {En} of strong\nexpanders would imply that an explicit function in NC1, depending on the\npseudo-random generator, cannnot be computed by a circuit of constant\ndepth. Hence such strong expanders can be regarded as obstructions, i.e.\nproofs of hardness, for computation of an explicit NC1-function by constant-\n72"},{"page":73,"text":"depth circuits. In this sense GCT obstructions are to superpolynomial-depth\ncircuits are what strong expanders are to constant-depth circuits. This is\npictorially depicted in Figure 10.\n↑\ndepth\n|\nSuperpolynomial depth circuit: Obstruction in GCT\n↑\n|\nConstant depth circuit: Strong expander\n↑\n|\nDepth two circuit: expander\nFigure 10: The relationship between obstructions and expanders\nThe expander in [LPS, Ma] can actually be constructed by a nonuniform\nalgebraic circuit of depth two (a basic ring operation is taken as unit cost).\nHence, it can be expected to serve as an obstruction for computation by\na circuit of depth at most two–really, just one.\nBecause the depth of a\ncircuit for computing an explicit structure whose existence separates NC\nfrom (nonuniform) ACk, the class of circuits of depth k, should be at least\nk–really higher than k. So an expander, as against the hypothetical strong\nexpander, actually belongs to depth-two circuits. But there is no nontrivial\nlower bound problem for circuits of depth one. This is why the expanders\nthat we have at present cannot be used in lower bound problems.\nNow let us compare explicit construction of expanders with the suggested\nmethod for explicit construction of obstructions in Figure 3.\nFirst, let us observe that, though the explicit construction of expanders\n[LPS, Ma] is “extremely easy” (nonuniform AC0), its correctness is based\non a nontrivial mathematical positivity hypothesis:\nPHspectral: The spectral gap of an expander is bounded below by a pos-\nitive constant.\nThe mathematical positivity hypotheses PH1 and PH2 (Hypothesis 1.4-\n1.6) can be regarded as nonspectral analogues of PHspectral in the setting\nof superpolynomial depth circuits.\n73"},{"page":74,"text":"Second, the proof of PHspectral in [LPS] for expanders depends on the\nRiemann hypothesis over finite fields (for curves) [Dl2]. It should not be a\nsurprise then that what is needed to prove the positivity hypotheses PH1, SH\n(PH2) mentioned above is, in essence, an extension of the Riemann hypoth-\nesis over finite fields and the results surrounding it. But given the big gap\nbetween constant depth and superpolynomial depth circuits it would have\nbeen a great surprise if the existing standard Riemann Hypothesis over fi-\nnite field were to suffice. Instead, what seems to be needed are nonstandard\nextensions of the Riemann hypothesis over finite fields, and the related re-\nsults; cf. Section 16. In the case of expanders, the Riemann hypothesis over\nfinite fields is not indispensible, since there are alternative constructions of\nexpanders with proofs of correctness based on linear algebra [RVW]. But,\nagain given a big gap between constant depth and superpolynomial depth, it\nshould not be surprising if nonstandard extensions of the Riemann hypoth-\nesis turn out to be indispensible in the context of the P vs. NP problem.\n18\nOn relativization and P/poly-naturalization bar-\nriers\nIn this section we point out why the flip should be nonrelativizable and\nnon-P/poly-naturalizable.\nWe already mentioned one reason for why the flip should be nonrela-\ntivizable: namely, the “reduction” from hard nonexistence to easy existence\nis not a formal Turing machine reduction. There is also another reason.\nFor this, let us examine why the proof of IP = PSPACE result [Sh] does\nnot seem relativizable. Mainly because it is based on the construction of an\nexplicit low-degree polynomial. This seems already enough to make it non-\nrelativizable, though the proof technique is not fully explicit. (Because it\nmakes use of estimates on the number of roots of a low degree polynomial.\nAny technique based on counting or estimates is, by definition, not fully\nexplicit). In contrast, the flip is to be implemented using explicit algebro-\ngeometric and representation-theoretic constructions. This is why it should\nbe nonrelativizable.\nNow we turn to the P/poly-naturalizability barrier [RR].\nIntuitively,\nthis too should be crossed simply because everything is to be done ex-\nplicitly and constructively.\nRecall that explicit construction of obstruc-\ntions is for superpolynomial depth circuits what explicit construction of\nexpanders is for depth-two circuits (Section 17).\nThe usual probabilistic\n74"},{"page":75,"text":"(nonconstructive) proof for existence of expanders may be considered to\nbe P/poly-naturalizable–as the probabilistic proofs [BS] of lower bounds\nfor constant depth circuits–whereas the proof via explicit construction in\n[LPS, Ma, RVW] may be considered non-P/poly-naturalizable. This is only\nan analogy. Strictly speaking, there is no notion of P/poly-naturalization\nfor constant depth circuits. Rather, this barrier lies between the circuits of\nconstant depth to which the expanders correspond and the circuits of super-\npolynomial depth to which the obstructions correspond. But this analogy\nshould intuitively explain why the flip should cross this barrier.\nNow we turn to a more formal argument. We begin by recalling the no-\ntion of a P/poly-naturalizable proof [RR]. We use the formal term P/poly-\nnaturalizable proof instead of the informal term natural proof, because oth-\nerwise GCT, and hence, the algebro-geometric and quantum-group-theoretic\ntechniques that enter into it would have to be called unnatural. That may\nseem paradoxical, especially since quantum groups arose in the study of\nnatural phenomena in theoretical physics.\nLet Fn be the set of n-variable boolean functions.\nBy a property of\nboolean functions, we mean a family of subsets Cn ⊆Fn for every n. It is\ncalled useful if the circuit size of any function h(X) = h(x1, . . . , xn) ∈Cn\nis super-polynomial. It is called P/poly-natural if it contains a subset C∗\nn\nsatisfying the following two constraints:\nConstructivity: Whether a given h(X) belongs to C∗\nn can be decided in\ntime polynomial in the size N = 2n of the truth table of h(X).\nLargeness:\n|C∗\nn|/|Fn| ≥1/N k,\n(36)\nfor some fixed k.\nA proof technique based on a useful P/poly-natural property is called\nP/poly-naturalizable. The article [RR] says that the P ̸= NP conjecture\nwould not have a P/poly-naturalizable proof under reasonable assumptions.\nNext, we translate this notion to the setting wherein the base field K of\ncomputation is algebraically closed, as in this article. We assume that K = C\nor K = ̄Fp, the algebraic closure of a finite field Fp. Let Fn be the set of n-\nvariable polynomials of degree d(n) for some fixed function d(n) = 2poly(n).\nIf K = C, we assume that each polynomial in Fn is an integral polynomial\nwhose coefficients have poly(n) bitlength. If K = ̄Fp, we assume that all\ncoefficients belong to Fp, and that the bitlength ⟨p⟩= poly(n).\nLet N\ndenote the total number of coefficients of h(X). The total bitlength of the\n75"},{"page":76,"text":"specification all coefficients of h(X) is N, ignoring a poly(n) factor. Hence\nwe let it play the role of the truth-table-size in what follows. This leads\nto the following straightforward generalization of the notion of a P/poly-\nnaturalizable proof over C or ̄Fp.\nBy a property, we now mean a subset Cn ⊆Fn, for each n.\nIt is\ncalled useful if the circuit size over K of any function h(X) ∈Cn is super-\npolynomial. It is called P/poly-natural if it contains a subset C∗\nn satisfying\nthe following two constraints:\nConstructivity: Whether a given h(X) belongs to C∗\nn can be decided in\npoly(N) time, where each operation over K is considered to be of unit cost.\nLargeness:\n|C∗\nn|/|Fn| ≥1/N k,\n(37)\nfor some fixed k.\nA proof technique based on a useful P/poly-natural property is called\nP/poly-naturalizable. The results in [RR] are proved only over a finite field.\nBut the constructivity and largeness constraints over algebraically closed\nfields here are natural extensions of the ones over finite field. Hence, we\nshall assume in what follows that they are meaningful even over algebraically\nclosed fields. It would be interesting to know if the techniques in [RR] can\nbe lifted in some form to such fields.\nIn the context of the flip, we next formulate a property which is conjec-\nturally useful and which should violate both the largeness and the construc-\ntivity constraints. This should be enough to cross the P/poly-naturalizability\nbarrier.\nLet us follow the notation as in Section 4. Let K = C. Let h(X) ∈P(W)\nbe an integral homogeneous form in Fn that belongs to co-NP (i.e., the\nproblem of deciding if it is nonzero for given xi’s belongs co-NP).\nLet UP (useful property) be the conjunction of the following two prop-\nerties.\nUP1: The form h = h(X) is co-NP-complete.\nUP2: (Characterization by stabilizers)\nThe form h, as a point in P(W), is characterized by its stabilizer Gh ⊆\nGL(W), not exactly as in Definition 3.1, but in a relaxed manner as de-\nscribed in Section 7 in [GCT1]. So also the form f = φ(h) as a point in\nP(V ). This means the associated class varieties ∆W[h; n] = ∆W [h], and\n∆V [f; n, l] = ∆V [f], as defined in Section 4.2 with h(X) playing the role of\nE(X), are group-theoretic. Let (H1 ֒→G1 ֒→K1) and (H2 ֒→G2 ֒→K1)\n76"},{"page":77,"text":"be the group-triples associated with the varieties ∆W[h; n] and ∆V [f; n, l],\nrespectively. We assume that H1 is reductive, that its simple composition\nfactors are explicitly known, and that it is built from these composition fac-\ntors by simple operations: to keep the matters simple, we only allow direct or\nwreath products, which suffice in GCT. We also assume that all the simple\ncomposition factors are either classical connected groups, tori or alternating\ngroups, as in GCT, though, again, this is strictly not necessary. We also\nassume that all homomorphisms in these triples are explicit as defined in\n[GCT6]–this is necessary.\nHere UP1 is stipulated only so that obstructions to efficient computation\nof h(X) should exist (Section 6). Otherwise, it is never explicitly used in\nGCT. The approach may work for other hard, though not co-NP-complete\nfunctions. UP2 is the main property that GCT needs for proving existence\nof obstructions. Hence we shall only concentrate on it in what follows.\nIt is shown in [GCT1] that E(X) has property UP2; whereas UP1 over C\nis shown in [Gu]. The permanent has an analogous property, where co-NP-\ncompleteness is replaced by #P-completeness. The function H(Y ) also has\nan analogous property with P-completeness replacing co-NP-completeness.\nBut in the case of H(Y ) the class variety is not the usual orbit closure\n∆[H(Y )], but rather ˆ∆[H(Y )] as defined in [GCT1]; cf. Remark 4.2.\nIn these definitions, we can also let the base field K be a finite field\nFp, or its algebraically closure ̄Fp, since characterization by stabilizers is a\nwell-defined notion over any field.\n18.1\nFrom usefulness to superpolynomial lower bounds\nThough GCT strives to prove superpolynomial lower bounds for the partic-\nular functions E(X) and perm(X), its main techniques should, in principle,\nextend to any h satisfying UP. We now briefly indicate how. This should\njustify the name UP.\nDefine an obstruction for the pair (h(x), H(Y )) as in Definition 5.1,\nwith h(X) playing the role of E(X).\nSuch obstructions should exist for\nevery n →∞, l = nlog n, as long as h(X) is co-NP-complete (cf. Section 6).\nAssociate with the class variety ∆V [f; n, l] a stretching function ̃sλ\nd(h; n, l)(k)\nas in (24) with h(X) playing the role of E(X).\nThe results in [GCT6] now imply the following analogue of Theorem 9.4\nfor h(X):\n77"},{"page":78,"text":"Theorem 18.1 [GCT6] Assuming that the singularities of the class variety\n∆V [f; n, l] and ∆W[h; l] are rational, the stretching function ̃sλ\nd(h; n, l)(k)\nassociated with the class variety ∆V [f; n, l] is a quasi-polynomial\nIt is may be conjectured that the singularities will be rational, as needed\nhere, as long as h satisfies UP2.\nUsing this theorem, we can formulate PH1, PH2, and SH for h(X) just\nas for E(X) (cf. Hypotheses 1.4-1.6).\nRemark 18.2 The statements of PH1 and PH2 given in this paper are as-\nsuming that all simple composition factors of the reductive groups under con-\nsideration are either classical connected groups or tori or alternating groups.\nIn the presence of composition factors of other types, some variations are\nnecessary [GCT6].\nThe following is an analogue of Theorem 9.5 in this context.\nTheorem 18.3 [GCT6] Assuming the rationality hypothesis (cf.\nTheo-\nrem 18.1), PH1 and SH, analogues of the decision problems in Hypothe-\nsis 7.2 for h(X) belong to P.\nIn particular, the problem of verifying an\nobstruction for the pair (h(X), H(Y )) belongs to P.\nThese results suggest, just as for E(X), the following strategy for proving\na superpolynomial lower bound for h(X):\n(1): Let Hi ֒→Gi ֒→Ki be the triples that occur in the definition of UP2.\nQuantize the couples Hi ֒→Gi and Gi ֒→Ki. That is, prove analogues of\nTheorem 15.1 for these. Also quantize the triples along the scheme suggested\nin [GCT10].\n(2): Prove existence of canonical bases (PH0) for the coordinate rings and\nrepresentations of the quantum groups that arise in this quantization along\nthe lines of the basic scheme in [GCT8]. For formal statements of PH0 see\n[GCT6, GCT10].\n(3): Use these canonical bases to prove existence of canonical bases (PH0)\nfor the coordinate rings of the class varieties ∆V [f; n, l] and ∆W [h, l] along\nthe lines suggested in [GCT10].\n(4): Use PH0 to deduce PH1 and SH, as suggested in [GCT6]. The polytope\nin PH1 should be more or less determined once PH0 holds, just as in the\nstandard case; cf. Section 1.6.\n78"},{"page":79,"text":"(5): Theorem 18.3, in conjunction with PH1 and SH for h(X), then im-\nplies polynomial time algorithm for the analogue of the decision problem in\nHypothesis 7.2 (a) for h(X) in place of E(X).\n(6): Carry out the steps (1)-(5) for the P-complete function H(Y ) as well.\nIt will imply a polynomial time algorithm for the decision problem in Hy-\npothesis 7.2 (b) for H(Y ). This step is the same as for E(X).\n(7): Transform the easy, polynomial time algorithms in steps (5) and (6),\nalong the lines suggested in Section 13 and [GCT6], into a P-constructive\nproof of existence of an obstruction λ(n) for every n →∞, assuming that l =\nnlog n. As pointed out in [GCT6], this transformation may need additional\npositivity hypotheses in the spirit of PH1 and SH. But these can be expected\nto hold, assuming h(X) satisfies UP2. The polytope in PH1 for h(X) in the\nstep (4) can also be expected to have a regular well-behaved structure, as\nneeded for this step, assuming h(X) satisfies UP2.\n(8): Existence of an obstruction family {λ(n)} would imply a superpoly-\nnomial size circuit lower bound for h(X), and hence, that P ̸= NP over\nC.\nFor the problems that need to be addressed over a finite field, or an\nalgebraically closed field of positive characteristic, see [GCT11].\n18.2\nOn violation of the largeness constraint\nNow let us see why UP2 should imply violation of the largeness constraint.\nWe cannot prove this formally over C. But this can be proved formally over\na finite field Fp or an algebraically closed field ̄Fp of positive characteristic\n[GCT10]. In fact, it turns out that violation of the largeness constraint is\nfar more severe than what is formally required. Namely, when K is a finite\nfield, it can be shown that\n|Cn|/|Fn| ≥1/2Ω(N),\nwhere Cn is the set of h(X) which satisfy UP2. This may be compared with\n(36).\nThe proof of violation of the largeness constraint over ̄Fp does not carry\nover to C for technical reasons. Specifically, the key ingradient in this proof is\nthe Riemann hypothesis over finite fields, or rather its extension as proved in\n[Dl2]. To transport this to the case when h(x) is integral would presumably\nrequire an analogous statement in arithmetic algebraic geometry.\n79"},{"page":80,"text":"It may be remarked that, in contrast, the proof of violation of the large-\nness constraint over a finite field is elementary. Thus the difficulty of proving\nthe violation of the largeness constraint over K seems inversely related to\nthe difficulty of proving the P ̸= NP conjecture over K. When K = Fp,\nthe conjecture is hardest to prove, and hence, the proof of violation is easy.\nWhen K = C∗, the conjecture should be easier than over ̄Fp or Fp. Accord-\ningly, proving violation of the largeness constraint formally turns out to be\nthe hardest.\n18.3\nOn violation of the constructivity constraint\nNext let us see why UP2 should also imply violation of the constructivity\nconstraint. We cannot hope to show this formally, since this is a lower bound\nstatement in itself. But rather we can give good evidence. First of all, to\ncompute the stabilizer of h(X), we have to solve a system of polynomial\nequations. Determining feasibility of a general system of polynomial equa-\ntions in k variables is NP-complete and is conjectured to take ⟨p⟩Ω(k) time,\nwhen K is the finite field Fp. Analogous conjecture may be made for the\nspecific system of polynomial equations that arises in the computation of\nthe stabilizer. Assuming this, it follows [GCT10] that deciding if h(X) has\na nontrivial stabilizer would take time that is superpolynomial in N–this is\nthe truth-table size when K = Fp.\n19\nP-verifiable and P-constructible proof techniques\nand their explicit construction complexity\nIn this section we suggest why GCT may be among the “easiest” “easy-to-\nverify” approaches to the P ̸= NP conjecture as per a certain measure of\nproof-complexity, called the explicit construction complexity. For this, we\nhave to introduce the notion of an easy-to-verify (i.e. P-verifiable) proof\ntechnique and then define its explicit construction complexity (class).\n19.1\nP-verifiable proof technique\nSuppose we are given a proof technique (approach) towards to the P vs. NP\nproblem that seeks to prove a superpolynomial lower bound for a specific\nhard function h(X) under consideration. We assume that the approach seeks\nto prove, explicitly or implicitly, existence of a specific cause for the hardness\n80"},{"page":81,"text":"of h(X), which we shall refer to as an obstruction. Thus an obstruction is,\nroughly, a “cause”, a “witness” or a “proof” of hardness.\nBut what do we mean by a proof?\nThe final proof of the P ̸= NP\nconjecture, if true, would constitute the ultimate obstruction to efficient\ncomputation of every (co)-NP-complete h(X). The size of this proof would\nbe just O(1), and so also the cost its verification. By obstruction, we do\nnot mean this final proof of hardness, but rather an intermediate proof\nof hardness whose existence the approach strives to demonstrate for every\nn →∞, when the circuit size m = nlog n, say.\nThe nature of such an obstruction will depend on the proof technique.\nWe cannot define it formally. Hence we will only give an intuitive idea with\nan example. Suppose there is an efficient pseudo-random generator whose\nexistence implies a restricted type of lower bound result in the spirit of [NW].\nThen the explicit computational circuit for this pseudo-random generator,\ni.e., for all its output bits together would be an obstruction in this context.\nBecause existence of this pseudo-random generator serves as a witness for\nhardness. If the pseudo-random generator is based on an explicit structure\nin the spirit of an expander, then this structure too can be considered to\nbe an obstruction.\nMore generally, the hardness-vs-randomness principle\n[KI, NW] suggests that proof techniques for difficult lower bounds may need\nmore or less explicit constructions of some structures.\nThese structures,\nwhich serve as witnesses for hardness, can then be taken as obstructions.\nIn the rest of this section, we confine ourselves only to those techniques\ntowards the P ̸= NP conjecture which contain, explicitly or implicitly, the\nnotion of an obstruction in this spirit–a witness for hardness–which admits\na well-defined description that can be assigned bit length. The arguments\nhenceforth are subject to this assumption.\nNext we try to formalize the notion of a “viable” proof technique to-\nwards the P vs. NP problem. For this, let us begin with a technique that\nshould certainly not be considered viable–the trivial brute-force proof tech-\nnique. This is defined as follows. Assume that the base field K is finite.\nFix any co-NP-complete function h(X) = h(x1, . . . , xn). Then this proof\ntechnique strives to prove, for every n, existence of the trivial proof of hard-\nness (obstruction), which consists of just the enumeration of all circuits of\nsize m = nlog n, with a specific value of X for each circuit on which the\nfunction evaluated by the circuit differs from h(X). The size of this trivial\nobstruction is exponential in m, and the time taken to verify it is also expo-\nnential in m. Any viable proof technique for the P vs. NP problem ought\n81"},{"page":82,"text":"to be at least better than this trivial proof technique in some well defined\nsense. One obvious sense in which it could be better is that there exists an\nobstruction whose size is not exponential in m, but rather polynomial in m,\nand the time taken to verify an obstruction is not exponential in m, but\nrather polynomial in m.\nThis leads to:\nDefinition 19.1 We say that a proof technique for the P ̸= NP conjecture\nis P-verifiable if\n1. There is a well-defined notion of obstruction, either implicit or explicit\nin the technique,\n2. There exists a short obstruction to computation of the specific function\nh(X) = h(x1, . . . , xn) under consideration by a circuit of size m =\nnlog n (say), for every n →∞, though the technique may only strive\nto prove existence of any obstruction, not necessarily short. By short,\nwe mean the obstruction has a label (combinatorial specification) of bit\nlength poly(m).\n3. The problem of verifying an obstruction is easy; i.e., belongs to P.\nSpecifically, takes time that is polynomial in n, m and the bit length of\nthe obstruction.\nThe meaning of easy here is the most obvious and natural definition in\nthe context of the P vs.\nNP problem.\nThus, intuitively a P-verifiable\nproof technique is an easy-to-verify proof technique. That is, the problem\nof discovering a proof of hardness (obstruction) in the technique belongs to\nNP. The definition above makes sense over any base field K of computation,\nwith obvious modifications in the spirit of the ones in Section 18.\nThe following naive arguments suggest that for a technique towards the\nP ̸= NP conjecture to be viable it out to be P-verifiable.\nFirst, the usual experience in mathematics suggests that however hard\nthe discovery of a proof may be its verification, once found, should be easy,\nand furthermore, the proofs that are found are usually reasonably short. In\nthe definition of P-verifiability, short and easy are given the most obvious\nand natural interpretations in the context of the P vs. NP problem: de-\nscription of polynomial size (short), and can be done in polynomial time\n(easy).\n82"},{"page":83,"text":"Second, given a technique, it seems necessary to justify why it is better\nthan the trivial brute-force technique. A P-verifiable proof technique is bet-\nter than it as per the most obvious complexity measures: (1) space (short),\nand (2) time (cost of verification).\nThird, the article [RR] roughly says that a nonspecific approach that\nis applicable to a large fraction of hard functions should not work in the\ncontext of the P vs. NP problem. Thus approaches based on probabilistic\nmethods or estimates of various kinds–such as Bezout-type estimates in\nalgebraic geometry, or estimates for discrepancies and deviations in analysis\nor number theory–should not work.\nProof of hardness as per any such\napproach–namely, the value of the measure or the estimate which is the\ncause of hardness–should be hard to verify. Since to verify the value, we\nmay have to compute it and see that it really tallies with what is given,\nand such computations should typically take time that is exponential in\nthe bitlength of the value. Thus a P/poly-naturalizable proof should also\nbe non-P-verifiable, and hence, the definition of P-verifiability here seems\nconsistent with the arguments in [RR].\nAdmittedly, these are only naive arguments. One can ask if there exists\na viable proof technique for the P ̸= NP conjecture that is better than the\ntrivial brute-force technique as per some measure of complexity other than\nthe obvious ones–space and time. But since we cannot think of any such\nnonobvious complexity measures which are also natural in the context of\nthe P vs. NP problem, we shall confine ourselves to only P-verifiable proof\ntechniques in what follows.\nBy Hypothesis 7.1 (a), which is supported by the results that we de-\nscribed in this article, GCT is a P-verifiable proof technique over C; the\nstory over a finite field should be similar [GCT11].\n19.2\nP-barrier for verification\nEvery P-verifiable technique for the P ̸= NP-conjecture has to cross the P-\nbarrier for verification; i.e., surmount the difficulty of showing that verifica-\ntion of an obstruction is easy. The magnitude and difficulty of the P-barrier\nshould be of the same order regardless of which P-verifiable approach to the\nP ̸= NP conjecture is taken.\nThis is easy to see when the base field K is finite. Then the length of\nany obstruction in the trivial brute-force technique mentioned in the begin-\nning of this section is exponential. The main task here is to come with a\n83"},{"page":84,"text":"proof technique that admits short obstructions which can be verified easily.\nThe magnitude of this P-barrier–the difference between the exponential and\nthe polynomial–is the same regardless of which approach to the P ̸= NP\nconjecture is taken.\nNext, let us assume that K = C, as in this paper. Let n be the number\nof input parameters. Let m(n) = nlogn (say) be the circuit-size parameter,\nh(n) = nlogn the height-parameter, and d(n) ≤2h(n) the degree parameter\nin the lower bound problem under consideration. For a given n, the set of\nfunctions over C computable by circuits of size at most m = m(n), height at\nmost h = h(n) and degree at most d = d(n) is an algebraically constructible\n[Mm1] subset S of the space V of all forms in m variables of degree d(n). A\nconstructible subset means it is in the boolean algebra generated by closed\nalgebraic subsets of V ; this is a generalization of an affine variety.\nThe goal in the lower bound problem under consideration is to show\nthat h(X) does not belong to S, when m = nlog n. Let ̄S be the closure of\nS. It is an affine variety. If h(X) is co-(NP)-complete, it is reasonable to\nassume that it does not belong to ̄S as well; i.e., roughly speaking, it cannot\nbe approximated infinitesimally closely by a circuit of size m(n) and height\nh(n). So it would suffice to show this.\nAn obvious obstruction here would be a polynomial in the ideal of ̄S\nwhich does not vanish on h(X). To decide if a given polynomial belongs to\nthe ideal of ̄S, an obvious method is to compute a good basis of this ideal,\nsuch as Gr ̈obner basis, and then use it for this decision. But the problem\nof Gr ̈obner basis computation is EXPSPACE complete [MM]. This means\ncomputation of the Gr ̈obner basis of ̄S can take space that is exponential in\nthe dimension of the ambient space V , which in turn is exponential in m.\nIn other words, space that is double exponential in m, and hence, time that\nis triple exponential in m. Given that the S in our problem is really bad,\nthis is the best that we can expect from any general purpose technique for\nverifying an obstruction that reasons about S directly in this fashion.\nFor the technique to be P-verifiable, the huge gap between this triple\nexponential bound for a general purpose direct technique and the polynomial\nbound in Definition 19.1 has to be bridged. The magnitude and order of this\ngap–the P-barrier–is exactly the same that we encountered in Section 8.\nRemark 19.2 The triple exponential size of this gap when K = C as\nagainst the exponential size over K = Fp does not mean that the P ̸= NP\nconjecture is easier when K = Fp. In fact, it is the other way around. Since\nthe (nonuniform) P ̸= NP conjecture in characteristic zero (over Z) is a\n84"},{"page":85,"text":"weaker implication of the conjecture over finite field (the usual case) [GCT1].\nHence, the exponential gap over Fp would be much harder to bridge than the\ntriple exponential gap over C. See [GCT6] for the problems that need to be\naddressed over finite fields.\nThe class variety XP (l) for P (Section 4.2) is constructed in [GCT1]\nprecisely to cope up with the triple exponential gap over C. It is a nice\nalgebraic variety that contains S, or rather its projectivization. So instead\nof trying to show that a given h(X) is not in S, one strives to show that\nit is not in XP (l). Since the algebraic geometry of XP (l) is exceptional,\nthis problem becomes easier–especially when h(X) is also exceptional, like\nE(X).\nThe quantum-group and algebro-geometric machinery is needed in GCT\njust to cross the P-barrier for verification (over C).\nThis suggests that\nmathematics required for any P-verifiable approach towards the P ̸= NP\nconjecture may not be substantially simpler, or easier.\n19.3\nP-constructible proof technique\nIn fact, it may be much harder unless it is also P-constructible in the fol-\nlowing sense.\nDefinition 19.3 We say that a P-verifiable proof technique for the P ̸=\nNP conjecture is P-constructible if the discovery of an obstruction in this\ntechnique is also easy. That is, there exists an algorithm, which, given n\nand m, can decide whether there exists an obstruction in poly(m) time, and\nif so, also construct a short obstruction in poly(m) time.\nThus the problem of discovering an obstruction in a P-constructible\nproof technique. belongs to P. But the proof technique itself need not give\na polynomial time algorithm for discovering an obstruction explicitly. That\nis, this may only be implicit in the proof, or it may be left to posterity. We\ncall the technique P-constructive if it gives such an algorithm more or less\nexplicitly:\nDefinition 19.4 A P-constructible proof technique is called P-constructive,\nif it also yields a procedure to construct an obstruction explicitly in poly(n, m)\ntime, if one exists.\n85"},{"page":86,"text":"The relationship between P-constructible (constructive) and P-verifiable\nproof strategies is akin to the relationship between P and NP. The P ̸= NP\nconjecture says that the discovery of a proof is, in general, harder than its\nverification. Hence, just as P denotes the class of easy problems within NP,\nthe P-constructible and P-constructive proof strategies are in a sense the\n“easy” ones among the P-verifiable proof strategies, wherein discovery is\nalso easy like verification.\nBy Hypothesis 7.1 (c), as supported by the positivity hypotheses, results\ndescribed in this paper, GCT is P-constructive over C; the story over a finite\nis expected to be similar [GCT11].\nThat there should exist such a P-constructive proof technique for the\nP ̸= NP conjecture may, however, seem paradoxical at the surface. Be-\ncause a P-constructive (constructible) proof technique seems to go against\nthe very philosophical essence of the P ̸= NP conjecture that discovery is\nharder than verification. This is akin to the paradox in the proof of G ̈odel’s\nincompleteness theorem: that the statement which says there exist unprov-\nable true statements is itself easy to prove. Similarly, Hypothesis 7.1 (c) says\nthat the statement which says discovery is harder than verification should\nitself be easy to discover.\n19.4\nGeneral setting\nSo far we have described P-verifiable and P-constructible proof techniques\nonly in the context of the P vs. NP problem. But these notions can be\ndefined in a much more general context, as we now briefly indicate:\nDefinition 19.5 A technique for proving a mathematical property Q(X),\nwhere X ranges over a class C of mathematical objects under consideration,\nis P-verifiable if:\n1. The technique proves, explicitly or implicitly, existence of a “proof-\ncertificate” c(X), for every X ∈C, which serves as a “witness” that\nthe property Q(X) holds.\n2. There exists a short proof certificate for every X ∈C.\nBy short,\nwe mean its size is poly(⟨X⟩), where ⟨X⟩denotes the specification-\ncomplexity of X.\n3. Verification of a proof-certificate c(X) is easy; i.e., can be done in\npoly(⟨X⟩, ⟨c(X)⟩) time, where ⟨c(X)⟩denotes the bitlength of c(X).\n86"},{"page":87,"text":"Again, we cannot formally define what a proof-certificate means. In what\nfollows, we only consider proof techniques wherein the notion of a proof-\ncertificate is well defined. The specification complexity ⟨X⟩here depends\non the problem under consideration, as we shall see in the examples below.\nDefinition 19.6 A P-verifiable proof technique is called P-constructible if\nthere exists an algorithm which, given X ∈C, can construct a proof certifi-\ncate c(X) in poly(⟨X⟩) time.\nBut the proof technique itself need not give such an algorithm explicitly.\nDefinition 19.7 A P-constructible proof technique is called P-constructive\nif, in addition, it yields an algorithm that can construct a proof-certificate\nc(X) in poly(⟨X⟩) time.\nWe can now state an informal working hypothesis:\nHypothesis 19.8 (The P-hypothesis) (informal)\n(a) Feasible P-verifiable proof techniques–that is the P-verifiable techniques\nthat can actually be used to prove the properties Q(X) in practice–are usu-\nally P-constructible, though proving P-constructibility may turn out to be\nnontrivial, and may only be done a posteriori.\n(b) Conversely, if a P-verifiable technique is P-constructible then under rea-\nsonable conditions it may also be feasible, i.e., can be used to actually prove\nQ(X).\n(c) A major part of the effort in a P-constructible proof technique usually\ngoes towards development of a polynomial time algorithm for constructing a\nproof-certificate, though this may be done only implicitly, and may become\nclear only a posteriori. That is, a P-constructible proof can usually be ex-\ntended to a P-constructive proof with a “reasonable additional effort”, albeit\na posteriori.\n(d) Mathematical complexity of a P-constructive proof technique is inti-\nmately linked to the computational complexity of the algorithm for explicit\nconstruction of a proof certificate underlying the technique.\nAs we have already remarked, the relationship between P-constructible\nproof techniques and P-verifiable proof techniques is akin to the relationship\nbetween P and NP.\nThe class P is usually regarded as the subclass of\n87"},{"page":88,"text":"feasible problems in NP. Hence, the P-hypothesis just says that P usually\nmeans feasible in practice.\nThe reasonable conditions in (b) means: there is a polynomial time al-\ngorithm for constructing a proof-certificate, which has, furthermore, a rea-\nsonably simple structure, and which is efficient in practice.\nThat is, the\ndefinition of P as standing for feasible is not misused.\nDefinition 19.9 A mathematical theorem, which says a property Q(X) holds\nfor every X in a class C of mathematical objects under consideration, is\ncalled P-verifiable if it has a P-verifiable proof.\nA P-constructible or a P-constructive theorem is defined similarly.\n19.4.1\nExamples\nWe now give a few examples to illustrate these notions.\nP vs NP problem\nIn this context, C is the class of tuples (n, m(n)), m(n) = nc for any con-\nstant c > 0, over all n (large enough). The property Q(X), X = (n, m(n)),\njust says that the explicit function h(X) under consideration, such as E(X)\nin [GCT1], cannot be computed by a circuit of size m(n) for every n large\nenough. Here ⟨X⟩= n+m; i.e, we assume that n and m are given in unary.\nThen the notions of P-verifiability, and P-constructibility here coincide with\nthe ones in Definitions 19.1 and 19.3. When m = nc, an obstruction would\nalways exist, assuming that h(X) is co-(NP)-complete, P ̸= NP and the\ntechnique is correct. That is why Definition 19.5 would coincide with Def-\ninition 19.1, even if in the former there is no mention of deciding if an\nobstruction exists or not. As per Hypothesis 7.1, GCT is P-constructive\nover C, and hence the P ̸= NP conjecture over C is also P-constructive; the\nsame can be hypothesized over a finite field [GCT11].\nHall’s theorem\nIn this context, C is the class of d-regular bipartite graphs. The property\nQ(X) is that every d-regular bipartite graph X ∈C has a perfect matching.\nThe bit length ⟨X⟩is the bitlength of the specification of X. The proof\ncertificate c(X) is a perfect matching in X. The problems of verifying and\nconstructing a perfect matching belong to P, the former trivially. Hence,\n88"},{"page":89,"text":"Hall’s theorem is P-constructive. Hall’s original proof is P-constructible,\nthough not P-constructive, since it does not explicitly give a polynomial\ntime algorithm for constructing a perfect matching.\nBut it does contain\nmajor ingradients for such a polynomial time algorithm, which came only\nmuch later. This is consistent with the P-hypothesis.\nFour colour theorem\nIn this context, C is the collection of planar graphs. The property Q(X) is\nthat any planar graph X is four colourable. The bitlength ⟨X⟩is the bit\nlength of the specification of X. The proof certificate c(X) is a four colouring\nof X. The problems of verifying and constructing a proof certificate belong\nto P, the former trivially. Hence, any proof of the four colour theorem is\nP-constructible, and the four colour theorem is P-constructive. The actual\nproof in [AH] is also (more or less) P-constructive since it implicitly yields\nto a polynomial (quartic) time algorithm for four colouring, Indeed, major\npart of the effort in the proof implicitly goes towards development of such\nan algorithm. This is consistent with the P-hypothesis.\nA simpler P-constructive proof was subsequently given in [RSST], which\ngives a better quadratic algorithm for the same problem. This too is con-\nsistent with the P-hypothesis (d).\nForbidden minor theorem\nFix a genus g. The forbidden minor theorem [RS] says that a graph which\ndoes not contain a forbidden minor from a finite list of minors depending\non g can be embedded on a genus g surface. Here C is the class of graphs\nthat do not contain a forbidden minor, Q(X) the property above, and ⟨X⟩\nthe bitlength of the specification of X. The proof certificate c(X) is just a\ndescription that tells how to embed X on a genus g surface.\nThe forbidden minor theorem is P-constructive. Any proof technique\nfor proving the forbidden minor theorem is P-constructible: it was known\n[FMR] even before [RS] that c(X) can be constructed in polynomial O(⟨X⟩O(g))\ntime. The proof of the forbidden minor theorem in [RS] gave an O(f(g)⟨X⟩2)\nalgorithm, where f(g) depends only on g. Indeed, a major part of the effort\nin [RS] implicitly goes towards finding a polynomial time algorithm whose\nrunning time is of the form O(f(g)⟨X⟩O(1)); i.e., wherein the exponent of\n⟨X⟩does not depend on g. This is again consistent with the P-hypothesis\n(d).\n89"},{"page":90,"text":"The Poincare conjecture\nHere we can let C be the set of simplicial decompositions of compact three\ndimensional combinatorial manifolds that are simply connected. The prop-\nerty Q(X) says that X is a (combinatorial) sphere. The bitlength ⟨X⟩is\nthe bitlength of specifying X. The article [Sc] says that the sphere recogni-\ntion problem is in NP. That is, there is a proof-certificate c(X), verifiable\nin polynomial time, which certifies that X is a sphere. It is interesting to\nknow here if the problem of constructing a proof certificate c(X), for a given\nX ∈C, belongs to P. It is plausible that the proof technique in [Pe] can\nbe extended/transformed (in the combinatorial setting) to get a polynomial\ntime algorithm which constructs a proof-certificate in this spirit, though not\nexactly the one in [Sc]. If that happens, it would mean that the Poincare\nconjecture is P-constructible (P-constructive), and that the major effort in\n[Pe] implicitly went towards getting a polynomial time algorithm for this\nproblem. This would provide support for the P-hypothesis (c).\nThus a major part of the effort in the P-verifiable proofs above indeed\nseems to go towards developing a polynomial time or a better polynomial\ntime algorithm for constructing a proof-certificate, as per the P-hypothesis\n(c), though this goal may not be stated explicitly in the proofs. In the flip,\nP-constructivity as a goal is explicitly spelled out right in the beginning,\ngiven the complexity-theoretic significance of the P vs. NP problem. But\njust as in the examples above, it may not be necessary to prove PHflip\n(Hypothesis 7.1) fully to prove P ̸= NP over C. That is, it may suffice to\ndevelop only a part of all ingradients needed to put the required problems\nin P, and the remaining part can be left to posterity. In this context, the\nbasic minimum that seems to be needed is PH1 (more or less).\n19.5\nExplicit construction complexity\nWe will now try to formalize the intuition behind the P-hypothesis (d).\nTowards that end we wish to associate a measure of proof-complexity with\na P-verifiable proof technique. This is quite different, for example, from\nKolmogrov proof-complexity.\nDefinition 19.10 Explicit construction complexity of a P-constructive tech-\nnique is the computational complexity of the algorithm underlying that tech-\nnique for explicit construction of a proof-certificate.\n90"},{"page":91,"text":"By computational complexity, we mean the usual measures such as depth\nand size of the corresponding computational circuit. If a P-verifiable tech-\nnique is not explicitly P-constructive but naturally leads to an algorithm\nfor construction of obstructions, with additional effort, we agree to take\nthe computational complexity of this algorithm to be explicit construction\ncomplexity of the technique, albeit only a posteriori.\nDefinition 19.11 (a) Verification (complexity) class of a P-verifiable proof\ntechnique is the abstract computational complexity class of the problem of\nverifying a proof-certificate (as per that technique).\n(b) Explicit construction (complexity) class of a P-verifiable proof technique\nis the computational complexity class of the problem of explicit construction\nof a proof-certificate as per that technique.\nA computational complexity class here means an abstract computation\ncomplexity class such as P, NC, NCk, AC, Dtime(N) etc. The verifica-\ntion and explicit construction classes of a P-verifiable technique are well\ndefined regardless of whether the technique shows how to construct a proof-\ncertificate explicitly or not. But what these classes are may become clear\nonly a posteriori, possibly after extending the proof technique to a get an\nefficient algorithm for construction of a proof certificate therein.\nThe complexity measures and classes above are meaningful only for P-\nverifiable proof techniques.\nThey would not make any sense for noncon-\nstructive or estimate-based techniques in analysis, number theory and so\nforth, unless it is possible to define a specification complexity ⟨X⟩and a\nproof-certificate that is polynomial time verifiable with this definition of\n⟨X⟩naturally.\nThis following gives a notion of theorem complexity for P-verifiable the-\norems.\nDefinition 19.12 Explicit construction complexity (class) of a P-verifiable\ntheorem is the minimum explicit construction complexity (class) over all P-\nverifiable proofs of the theorem. Verification complexity (class) is defined\nsimilarly.\nThe explicit construction complexity seems to be a good measure of\ncomplexity for P-verifiable proof techniques and theorems. We shall discuss\nthe examples above a bit more in this context.\n91"},{"page":92,"text":"Halls’ theorem\nVerification class here is AC (constant depth circuits), since a perfect match-\ning can be verified in constant depth. A perfect matching in a bipartite\ngraph can be computed, if one exists, in O(m log n) time. This problem also\nbelongs to RNC [KUW, MVV]. Hence, the sequential explicit construction\nclass of Hall’s theorem is Dtime(m log n). The parallel explicit construction\nclass is RNC; possibly even NC.\nFour colour theorem\nVerification class here is AC. Explicit construction complexity of the proof\nin [AH] is O(n4), whereas that of the proof in [RSST] is O(n2) [RSST].\nThus a proof technique with lower explicit construction complexity has in-\ndeed lower proof-complexity. The sequential explicit construction class of\nthe four colour theorem is thus Dtime(n2), or lower. The parallel explicit\nconstruction class is possibly NC, in view of the parallel algorithms for four\ncolouring in special cases [He].\nForbidden minor theorem\nVerification class is AC. Explicit construction complexity of the proof in\n[RS] is O(f(g)n2), where g is an explicit function of the genus g.\nThe\nsequential explicit construction class of the forbidden minor theorem is thus\nDtime(O(n2)); it may be Dtime(n). The parallel explicit construction class\nmay be NC, since planarity testing is in NC [JS].\nPoincare’s conjecture\nVerification class of the Poincare conjecture is P [Sc], assuming that the\nproof technique in [Pe] is P-verifiable. It may be smaller. NC?. Explicit\nconstruction class may be P, plausibly smaller. NC?\nTrivial example\nWe now give a trivial example to illustrate why P-verifiability is essential\nfor the complexity measures here to make sense.\nTake a trivial mathe-\nmatical theorem: that an integer n has at most log n factors. An obvious\nproof-certificate, for a given n, is the number of its factors, which shows\n92"},{"page":93,"text":"that it is less than log n. But verification of this proof requires factoring\nand hence is hard. Thus if n is specified in binary, this theorem should\nnot be P-verifiable. That is why explicit construction complexity of this\nproof-certificate says nothing of the actual (trivial) proof-complexity of the\ntheorem. Similarly, explicit construction complexity is not meaningful for\nestimate-centred proof techniques in mathematics. The article [RR] roughly\nsays that such techniques are not expected to work in the context of the\nP vs. NP problem since they tend to be applicable to a large fraction of\nfunctions.\nIn the context of the P vs. NP problem, Definitions 19.10 and 19.11\nbecome:\nDefinition 19.13 Explicit construction complexity of a P-constructive tech-\nnique for the P ̸= NP conjecture is the computational complexity of the algo-\nrithm underlying that technique for explicit construction of a proof-certificate\n(as per that technique).\nDefinition 19.14 (a) Verification complexity class of a P-verifiable proof\ntechnique for the P ̸= NP conjecture is the computational complexity class\nof the problem of verifying an obstruction as per that technique.\n(b) Its explicit construction complexity class is the computational complexity\nclass of the problem of explicit construction of an obstruction.\nAgain these classes are well-defined regardless of whether the technique\nshows how to construct an obstruction explicitly or not, once the notion of\nan obstruction in the proof technique is well-defined.\nOne may also define existential complexity class of a P-verifiable proof\ntechnique (for the P vs. NP problem): this is the computational complexity\nclass of the problem of deciding if there exists an obstruction for a given n\nand circuit size m.\nThe existence-vs-construction principle [KUW] says that computational\ncomplexity of a construction problem is comparable to that of the associated\nexistence problem under natural conditions.\nThis means, under natural\nconditions, existential and explicit-construction complexity classes should\ncoincide. Hence, we shall not worry about existential complexity anymore.\nIt is illuminating to compare the verification complexity of the P vs. NP\nproblem with the other problems we considered. The verification complexity\nclass of Halls’ theorem, four colour theorem, or forbidden minor theorem is\n93"},{"page":94,"text":"AC. For Poincare’s conjecture, P-verifiability is quite nontrivial [Sc]. But\nfortunately the proof is not very complex.\nIn contrast, P-verifiability is already a formidable issue in the context of\nthe P vs. NP problem.\n19.6\nIs there a simpler proof technique?\nNow we ask if there is a P-verifiable proof technique towards the P ̸= NP\nconjecture that is substantially “easier” than GCT. By easier we mean, with\nlower verification and explicit construction complexity (classes). Since GCT\nis P-verifiable and also P-constructive over C as per Hypothesis 7.1, P ̸=\nNP conjecture is conjecturally P-verifiable and also P-constructive over C.\nThe same can be conjectured over Fp or ̄Fp as well [GCT10]. Assuming this,\nit is meaningful to talk of its verification and explicit construction classes.\nSo we can ask:\nQuestion 19.15 What are the (smallest) verification and explicit construc-\ntion complexity classes of the P ̸= NP conjecture?\nThe best and the most natural answer that one can expect here is P.\nIt would really be unsettling if the answer were, say, NC. Specifically, the\nproblems of verification and explicit construction of obstructions in any P-\nverifiable approach to the P ̸= NP conjecture should be at least as hard as\nP-complete problems. This is supported by the presence of linear program-\nming, which is P-complete, in the algorithms for the basic decision problems\nin Theorem 9.5.\nIf so, GCT may be among the “easiest” P-verifiable approaches to the\nP ̸= NP conjecture over C. The story over Fp may be similar; cf. [GCT11].\nReferences\n[A1]\nM. Agrawal, N. Kayal, N. Saxena, Primes is in P, Annals of Math-\nematics, 160 (2): 781-793, 2004.\n[A2]\nM. Agrawal, Proving lower bounds via pseudo-random generators,\nProceedings of FSTTCS 2005, 92-105, 2005.\n[Ak]\nD. Akhiezer, Homogeneous complex manifolds, Encyclopaedia of\nmathematical sciences, volume 10, Springer-Verlag. 1986.\n94"},{"page":95,"text":"[AH]\nK. Appel and W. Haken, Every planar map is four colorable,\nA.M.S. Contemporary Math. 98 (1989). MR 91m:05079.\n[B]\nL. Babai, private communication.\n[BGS]\nT. Baker, J. Gill, R. Soloway, Relativization of the P =?NP ques-\ntion, SIAM J. 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Fulton, Young tableaux, Cambridge University Press, 1997.\n[F2]\nW. Fulton, Eigenvalues of sums of Hermitian matrices (after A.\nKlyachko), S ́eminaire Bourbaki, vol. 1997/98. Ast ́erisque No. 2523\n(1998), Exp. No. 845, 5, 255-269.\n[FH]\nW. Fulton, J. Harris, Representation theory, A first course,\nSpringer, 1991.\n[GCTabs] K.\nMulmuley,\nGeometric\ncomplexity\ntheory,\nabstract,\ntechnical\nreport\nTR-2007-12,\ncomputer\nscience\ndept.,\nThe\nuniversity\nof\nChicago,\nSept.\n2007.\nAvailable\nat:\nhttp://ramakrishnadas.cs.uchicago.edu.\n[GCTconf] K. Mulmuley, M. Sohoni, Geometric complexity theory, P vs.\nNP and explicit obstructions, in “Advances in Algebra and Geom-\netry”, Edited by C. Musili, the proceedings of the International\nConference on Algebra and Geometry, Hyderabad, 2001.\n[GCTflip2] K. Mulmuley, On P vs. NP, geometric complexity theory, and\nthe flip II, under preparation.\n[GCTintro] K.\nMulmuley,\nM.\nSohoni,\nGeometric\ncomplexity\ntheory:\nintroduction,\ntechnical\nreport TR-2007-16,\ncomputer science\ndept.,\nThe university of Chicago,\nSept. 2007. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu.\n[GCT1] K. Mulmuley, M. Sohoni, Geometric complexity theory I: an ap-\nproach to the P vs. NP and related problems, SIAM J. Comput.,\nvol 31, no 2, pp 496-526, 2001.\n96"},{"page":97,"text":"[GCT2] K.\nMulmuley,\nM.\nSohoni,\nGeometric\ncomplexity\ntheory\nII:\ntowards\nexplicit\nobstructions\nfor\nembeddings\namong\nclass varieties,\nto appear in SIAM J. Comput.,\ncs. ArXiv\npreprint cs. CC/0612134,\nDecember 25,\n2006. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu\n[GCT3] K. Mulmuley, M. Sohoni, Geometric complexity theory III, on de-\nciding positivity of Littlewood-Richardson coefficients, cs. ArXiv\npreprint cs. CC/0501076 v1 26 Jan 2005.\n[GCT4] K.\nMulmuley,\nM.\nSohoni,\nGeometric\ncomplexity\nthe-\nory\nIV:\nquantum\ngroup\nfor\nthe\nKronecker\nproblem,\ncs.\nArXiv preprint cs. CC/0703110,\nMarch,\n2007. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu\n[GCT5] K. Mulmuley, H. Narayanan, Geometric complexity theory V: on\ndeciding nonvanishing of a generalized Littlewood-Richardson co-\nefficient, Technical report TR-2007-05, Comp. Sci. Dept. The uni-\nversity of chicago, May, 2007.\n[GCT6] K. Mulmuley, Geometric complexity theory VI: the flip via sat-\nurated and positive integer programming in representation the-\nory and algebraic geometry, Technical report TR 2007-04, Comp.\nSci. Dept., The University of Chicago, May, 2007. Available\nat: http://ramakrishnadas.cs.uchicago.edu. Revised version to be\navailable here.\n[GCT7] K. Mulmuley, Geometric complexity theory VII: a quantum group\nfor the plethysm problem, technical report TR-2007-14, computer\nscience dept., The university of Chicago, Sept. 2007. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu.\n[GCT8] K. Mulmuley, Geometric complexity theory VIII: On canonical\nbases for the nonstandard quantum groups, technical report TR-\n2007-15, computer science dept., The university of Chicago, Sept.\n2007. 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\nproblem, under preparation.\n[GCT10] K. Mulmuley, Geometric complexity theory X: On class varieties,\nand the natural proof barrier, under preparation.\n97"},{"page":98,"text":"[GCT11] K. Mulmuley, Geometric complexity theory XI: on the flip over\nfinite or algebraically closed fields of positive characteristic, under\npreparation.\n[GL]\nI. Grojnowski, G. Lusztig, A comparison of bases of quantized en-\nveloping algebras, Contemp. Math. 153 (1993), 11-19.\n[GLS]\nM. Gr ̈otschel, L. Lov ́asz, A. Schrijver, Geometric algorithms and\ncombinatorial optimzation, Springer-Verlag, 1993.\n[Gu]\nL. Gurvits, private communication.\n[Ha]\nR. Hartshorne, Algebraic geometry, Springer, 1997.\n[He]\nX. He, Efficient parallel and sequential algorithms for 4-coloring\nperfect planar graphs, Algorithmica (1990) 5: 545-559.\n[Hi]\nH. Hironaka, Resolution of singularities of an algebraic variety over\na field of characteristic zero, Annal of math. 79 (1964). I: 109-203,\nII: 205-326.\n[IW]\nR. Impagliazzo, A. Wigderson, P=BPP if E requires exponential\nsize circuits: Derandomizing the XOR lemma, in proceedings of\nAnnual ACM Symposium on the Theory of Computing, pages 220-\n229, 1997.\n[JS]\nJ. J ́aJ ́a, J. Simon., Parallel algorithms in graph theory: planarity\ntesting, SIAM J. Computing, 11 (2): 314-328, 1982.\n[Ji]\nM. 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Math.\nPhys. 111 (1987), 613-665.\n[Z]\nA.\nZelevinsky,\nLittlewood-Richardson\nsemigroups,\narXiv:math.CO/9704228 v1 30 Apr 1997.\n103"}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"The usual P ̸= NP conjecture is over a finite field of which the one over C","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"problems) in complexity theory, such as the P ̸=?NP conjecture over C, to","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"may lead to the proof of the P ̸= NP conjecture over C. This basic plan","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"tive hypothesis of mathematics, the P ̸= NP conjecture, in essence, from","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"The P ̸= NP conjecture (-)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"f(x) = f(x1, . . . , xn) is hard. Accordingly, the flip from hard nonexistence","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"a specific (co)-NP-complete function E(X) = E(x1, . . . , xn), and a specific","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"P-complete function H(Y ) = H(y1, . . . , yl) are constructed in [GCT1] so as","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"a projective algebraic variety XP (l) = XP (H; l), for every positive integer l,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"It will turn out that XP (l) is a G-variety for G = GLl(C), the group of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"Cl in the usual way. Similarly, using E(X), a projective variety XNP (n, l) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"can be embedded in XP (l) as a G-subvariety for l = O(m2). Pictorially:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"We want to show that this embedding is impossible if m = poly(n), as","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"m = poly(n) size, and hence, P ̸= NP over C.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"Let R(n, l) = R(E; n, l) and S(l) = S(H; l) denote the homogeneous","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"V =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"when m = poly(n), assuming that P ̸= NP, as we expect. The goal then is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"m = poly(n). The story is similar for other related lower bound problems.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"(b) Explicit construction of obstructions: Suppose l = nlog n (say).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"prove existence of an obstruction for every n →∞, when l = nlog n (say),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"of obstructions (Hypothesis 1.1(b)) for all n →∞, when l = nlog n, works,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"f(k) = fi(k) if k = i modulo l. Here l is called the period of the quasi-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"zero as a polynomial, or if not, f(1) = f1(1) ̸= 0. If f(k) is positive, it is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"d(H; l)(k) = skλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"d(E; n, l)(k) = skλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"α,β(k) = ckλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"expressed as the number of integer points in a polytope P = P λ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"obstruction always exists for every n →∞, assuming l = nlog n, say. That","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"obstructions (rather their labels), as n →∞, and l = nlog n. The existence","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"of such an obstruction family would imply that P ̸= NP over C.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"would then imply that P ̸= NP over C. The whole picture is summarized","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"P ̸= NP over C","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"Finally, one may ask if the P ̸= NP conjecture may be proved by a sub-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"technique for the P ̸= NP conjecture may be intimately linked to the diffi-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"the P ̸= NP conjecture, where easy-to-verify formally means P-verifiable;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"obstructions in GCT over C is poly(m), m = nlog n (say) being the circuit","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"the P ̸= NP conjecture. In other words, the massive Ω(m) gap between the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"in any P-verifiable proof technique towards the P ̸= NP conjecture. If so,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"U to a G-module V is map ψ : U →V such that ψ(g · u) = g · (ψ(u)) for all","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"V =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"ment W ⊥such that V = W ⊕W ⊥.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"the general linear group GLn(C) = GL(Cn), the group of nonsingular n × n","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"matrices, and its subgroup the special linear group SLn(C) = SL(Cn) of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"We say that the representation (4) of G = GLn(C) or SLn(C) is polyno-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"the tuples λ = (λ1, . . . , λk) of integers, where k ≤n and λ1 ≥λ2 · · · ≥λk >","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"0. Here λ is called a partition of length k and size d = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"tion corresponding to a partition λ = (λ1, λ2, . . .) is denoted by Vλ(GLn(C)),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"(σ · f)(Z) = f(Zσ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"entries cj, 1 ≤j ≤l, of c. Thus eT = Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"of the highest weight vector vλ = eT0; i.e., the set of all σ ∈GLn(C) such","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"that σ ·vλ = c(σ)vσ, for some complex number c(σ). Then it is easy to show","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"Let C[X] = C[x1, · · · , xn] be the ring polynomials in n variables. It is a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"(σ · f)(x1, · · · , xn) = f(xσ(1), · · · , xσ(n)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"σ(v ⊗w) = (σ · v) ⊗(σ · w).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"First we consider this problem when G = GLn(C). Given Weyl modules Vα","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"Vα ⊗Vβ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"α,β = |Cγ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"Sα ⊗Sβ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"Let V = Cn. Let X = (x1, . . . , xn) be the variable n-vector whose entries","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"of zeroes of a collection of polynomials in C[X] = C[x1, . . . , xn]. An affine","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"Let P n−1 = P(V ) be the projective space of lines in V through the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"(σ · f)(X) = f(σ−1X).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"Let Gv = {σ | σ · v = v} be its","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"stabilizer. Then Gv as a set is isomorphic to the coset set G/H, H = Gv.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq65","equation_number":null,"raw_text":"Let ∆V [v] = Gv ⊆P(V ) denote the closure of the G-orbit of v in the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq66","equation_number":null,"raw_text":"module of G = GLn(C). Let vλ ∈P(Vλ) be the point corresponding to the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq67","equation_number":null,"raw_text":"be shown that the orbit Gvλ ∼= G/Pλ, where Pλ is the stabilizer of vλ, is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq68","equation_number":null,"raw_text":"by its stabilizer H = Gv ⊆G, if it is the only point in P(V ) stabilized (left","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq69","equation_number":null,"raw_text":"invariant) by H. Stabilized by H means, for every σ ∈H , σ · v = v.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq70","equation_number":null,"raw_text":"H = Gv ֒→G","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq71","equation_number":null,"raw_text":"ρ→K = GL(V ),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq72","equation_number":null,"raw_text":"Let Y = [y0, · · · , yl−1] denote a variable l-vector. For n < l, let X =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq73","equation_number":null,"raw_text":"[y1, · · · , yn], and ̄X = [y0, · · · , yn] be its subvectors of size n and n + 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq74","equation_number":null,"raw_text":"Let V = Syms(Y ) be the space","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq75","equation_number":null,"raw_text":"natural action of G = SL(Y ) = SLl(C) and ˆG = GL(Y ) = GLl(C), just as","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq76","equation_number":null,"raw_text":"Similarly, let W = Symr(X), r < s, be the representation of GL(X) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq77","equation_number":null,"raw_text":"where y = y0 is used as the homogenizing variable. The image φ(W) is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq78","equation_number":null,"raw_text":"contained in ̄W = Syms( ̄X), a representation of GL( ̄X) = GLn+1(C).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq79","equation_number":null,"raw_text":"We pick a form g = g(Y ) = g(y0, . . . , yl−1) ∈P(V ) which is a complete","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq80","equation_number":null,"raw_text":"function for the complexity class C1. Then the orbit closure ∆V [g; l] = ∆V [g]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq81","equation_number":null,"raw_text":"Similarly, we choose a form h = h(X) = h(x1, . . . , xn) ∈P(W) which","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq82","equation_number":null,"raw_text":"is complete for the class C2. Then the orbit closure ∆W[h; n] = ∆W[h] ⊆","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq83","equation_number":null,"raw_text":"C2-variety based on h. Let f = φ(h), with φ as in (14). We call the orbit","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq84","equation_number":null,"raw_text":"closure ∆V [f; n, l] = ∆V [f] ⊆P(V ) the extended class variety associated","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq85","equation_number":null,"raw_text":"enough, say, l = nlog n, n →∞. This will show that C1 ̸= C2. Here l will be","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq86","equation_number":null,"raw_text":"class #P [V]. The P #P ̸= NC conjecture over C [V] says that the per-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq87","equation_number":null,"raw_text":"for NC and #P. The P #P ̸= NC conjecture over C will then be reduced","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq88","equation_number":null,"raw_text":"variable l-vector, l = m2, by linearly ordering its entries in any order. Let","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq89","equation_number":null,"raw_text":"can also be thought of as a variable k-vector, k = n2. Let V = Symm(Y )","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq90","equation_number":null,"raw_text":"Y , and W = Symn(X), the space of homogeneous forms of degree n in the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq91","equation_number":null,"raw_text":"variable entries of X. We have a natural action of G = SL(Y ) = SLl(C) on","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq92","equation_number":null,"raw_text":"an action of H = SL(X) = SLk(C) on P(W).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq93","equation_number":null,"raw_text":"Let g = det(Y ) ∈P(V ) be the determinant form.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq94","equation_number":null,"raw_text":"Let ∆V [g; l] =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq95","equation_number":null,"raw_text":"circuit of depth less than logc n, then f = φ(h) lies in ∆V [g; l] for m = 2logc n.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq96","equation_number":null,"raw_text":"complete [V], this is not expected to happen if h = perm(X) and m =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq97","equation_number":null,"raw_text":"Conjecture 4.1 [GCT1] Let h = perm(X) ∈P(W) and f = φ(h). Then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq98","equation_number":null,"raw_text":"f ̸∈∆V [g] = ∆V [g; l], if m = 2O(polylog(n)), as n →∞. Since ∆V [g] is a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq99","equation_number":null,"raw_text":"Here ∆V [g] = ∆V [g; l] is called the class variety associated with the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq100","equation_number":null,"raw_text":"NC, when m = 2polylog(n): i.e.,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq101","equation_number":null,"raw_text":"The stabilizer of det(Y ) ∈P(V ) in G = SL(Y ) = SLm2(C) is known to","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq102","equation_number":null,"raw_text":"(g ⊗h) · (x ⊗y) = (g · x) ⊗(h · y).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq103","equation_number":null,"raw_text":"The stabilizer of perm(X) ∈P(W) in SL(X) = SLn2(C) is a reduc-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq104","equation_number":null,"raw_text":"and (co)-NP-complete functions, called H(Y ) = H(y1, . . . , yl) and E(X) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq105","equation_number":null,"raw_text":"Let W = Symr(X) be the space of forms of degree r = deg(E(X)) in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq106","equation_number":null,"raw_text":"the entries of X. Thus E(X) ∈P(W). Let V = Syms(Y ) be the space of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq107","equation_number":null,"raw_text":"forms of degree s = deg(H(Y )) in the entries of Y . Thus H(Y ) ∈P(V ). We","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq108","equation_number":null,"raw_text":"and l ≥n, a group-theoretic variety (orbit closure) ∆V [f; n, l] = ∆V [f] ⊆","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq109","equation_number":null,"raw_text":"P(V ), where f = φ(h) and h = E(X). It is a G-variety, for G = SLl(C).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq110","equation_number":null,"raw_text":"G-variety ∆V [g; l] = ∆V [g] ⊆P(V ), where g = H(Y ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq111","equation_number":null,"raw_text":"m then XNP (E; n, l) can be embedded within XP (H; l) for l = O(m2):","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq112","equation_number":null,"raw_text":"XNP (n, l) = XNP (E; n, l) ֒→XP (l) = XP (H; l).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq113","equation_number":null,"raw_text":"Conjecture 4.3 [GCT1] This embedding cannot exist if m = nlog n, or","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq114","equation_number":null,"raw_text":"more generally, m = 2na, for a small enough a > 0, as n →∞.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq115","equation_number":null,"raw_text":"This will show that P ̸= NP over C. This transforms the P vs. NP problem","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq116","equation_number":null,"raw_text":"Let R(n, l) = R(E; n, l) and S(l) = S(H; l) denote the homogeneous co-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq117","equation_number":null,"raw_text":"ordinate rings of XNP (n, l) = XNP (E; n, l) and XP (l) = XP (H; l), respec-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq118","equation_number":null,"raw_text":"In particular, every irreducible representation (Weyl module) Vλ = Vλ(G)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq119","equation_number":null,"raw_text":"Definition 5.1 We say that S = Vλ is an obstruction, for n, l and the pair","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq120","equation_number":null,"raw_text":"(E, H) = (E(X), H(Y )), if it occurs in R(n, l)d but not in S(l)d, for some","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq121","equation_number":null,"raw_text":"exists if m = nlog n, or more generally, m = 2na, for a small enough a > 0,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq122","equation_number":null,"raw_text":"as n →∞; recall that l = O(m2). Furthermore, there exists such an ob-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq123","equation_number":null,"raw_text":"struction of a small degree d(n, m) = 2mb, b > 0 a large enough constant.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq124","equation_number":null,"raw_text":"then it follows that P ̸= NP over C. We say that {Vλ(n)} or {λ(n)} is an","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq125","equation_number":null,"raw_text":"associated with the group triples, assuming that P ̸= NP–which we take on","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq126","equation_number":null,"raw_text":"and ρ2 : H2 ֒→G, where G = GLl(C) = GL(W), W = Cl. This means W","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq127","equation_number":null,"raw_text":"in Π. Then XNC = X(Π).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq128","equation_number":null,"raw_text":"Remark 6.2 (Erratum) In [GCT2] it is conjectured that XNC = X(Π)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq129","equation_number":null,"raw_text":"minant g = det(Y ) such that XNC ∩U = X(Π) ∩U, assuming a reasonable","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq130","equation_number":null,"raw_text":"(b) (Explicit construction of obstructions): Suppose l = nlog n, or 2na, for","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq131","equation_number":null,"raw_text":"P ̸= NP over C","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq132","equation_number":null,"raw_text":"So fix a general integral function h(X) = h(x1, . . . , xn), which is co-NP-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq133","equation_number":null,"raw_text":"tion 4.2 with h(X) in place of E(X). Here V = Syms(Y ) is the space of forms","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq134","equation_number":null,"raw_text":"of degree s = deg(H(Y )) in l = O(m2) variable entries of Y . The dimension","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq135","equation_number":null,"raw_text":"M of the ambient projective space P(V ) here is exponential in l = O(m2),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq136","equation_number":null,"raw_text":"for h(X) can be solved in at best O(dim(C[V ]d) = O(dM) = O(d2poly(m))","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq137","equation_number":null,"raw_text":"As we shall argue in Section 19, for any approach towards the P ̸= NP","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq138","equation_number":null,"raw_text":"ρ : H = GLn(C) →G = H × H.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq139","equation_number":null,"raw_text":"Let X be an n×n variable matrix. Let V = Sym1(X) be the space of linear","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq140","equation_number":null,"raw_text":"((h1, h2) · f)(X) = f(h−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq141","equation_number":null,"raw_text":"Let f(X) = trace(X).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq142","equation_number":null,"raw_text":"Hence, f(X) = trace(X) is the characteristic function (Definition 3.1) of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq143","equation_number":null,"raw_text":"α,β(k) = ckλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq144","equation_number":null,"raw_text":"d(H; l)(k) = skλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq145","equation_number":null,"raw_text":"d(E; n, l)(k) = skλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq146","equation_number":null,"raw_text":"P ̸= NP over C","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq147","equation_number":null,"raw_text":"Given a flag variety Z = Gvμ ⊆P(Vμ), where Vμ is a Weyl","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq148","equation_number":null,"raw_text":"module of G = SLl(C), the decision problem is to decide if Vλ(G) occurs","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq149","equation_number":null,"raw_text":"Z. By the Borel-Weil theorem [FH], R(Z)d = V ∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq150","equation_number":null,"raw_text":"Vλ occurs in R(Z)d iffVλ = V ∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq151","equation_number":null,"raw_text":"earlier in connection with the flag variety, when H = Pμ is parabolic; cf.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq152","equation_number":null,"raw_text":"Let H = GLn(C), G = H × H, the embedding","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq153","equation_number":null,"raw_text":"ρ : H →H × H = G","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq154","equation_number":null,"raw_text":"the form Vα ⊗Vβ, where Vα and Vβ are irreducible representations of H =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq155","equation_number":null,"raw_text":"Let H = GLn(C) × GLn(C) and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq156","equation_number":null,"raw_text":"ρ : H →G = GL(Cn ⊗Cn) = GLn2(C)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq157","equation_number":null,"raw_text":"the natural embedding given by: ρ(h1, h2) = h1 ⊗h2, for any h1, h2 ∈H.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq158","equation_number":null,"raw_text":"of the irreducible representation Vπ(H) of H = GLn(C) in the irreducible","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq159","equation_number":null,"raw_text":"representation Vλ(G) of G = GL(Vμ), where Vμ = Vμ(H) is an irreducible","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq160","equation_number":null,"raw_text":"ρ : H →G = GL(Vμ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq161","equation_number":null,"raw_text":"λ,μ(k) = akπ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq162","equation_number":null,"raw_text":"For every (λ, μ, π) there exists a polytope P = P π","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq163","equation_number":null,"raw_text":"λ,μ ⊆Rm with m =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq164","equation_number":null,"raw_text":"λ,μ = φ(P),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq165","equation_number":null,"raw_text":"That this holds even if the dimension of G = GL(Vμ) is exponential in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq166","equation_number":null,"raw_text":"n is crucial in the context of GCT. Because the dimension of K = GL(V )","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq167","equation_number":null,"raw_text":"in the triples H ֒→G ֒→K = GL(V ) associated with the class varieties","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq168","equation_number":null,"raw_text":"of some polytope P = P π","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq169","equation_number":null,"raw_text":"and l = nlog n, by showing how such an obstruction-label can be easily","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq170","equation_number":null,"raw_text":"This would imply that P ̸= NP over C.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq171","equation_number":null,"raw_text":"imply that there exist polytopes P(n, l) = P(E; n, l) and Q(l) = P(H; l) of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq172","equation_number":null,"raw_text":"exists iffthe relative difference T(n, l) = P(n, l) \\ Q(l) is nonempty, and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq173","equation_number":null,"raw_text":"for every n →∞, assuming l = nlog n (say).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq174","equation_number":null,"raw_text":"rithm, that T(n, l) is always nonempty when l = nlog n, n →∞. Such a proof","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq175","equation_number":null,"raw_text":"Let us imagine that Q(l) is empty, so that T(n, l) = P(n, l) is a polytope,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq176","equation_number":null,"raw_text":"that G(n, l) always has a perfect matching, as expected, when l = nlog n,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq177","equation_number":null,"raw_text":"whenever l = nlog n, n →∞.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq178","equation_number":null,"raw_text":"T(n, l), for every n →∞, when l = nlog n.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq179","equation_number":null,"raw_text":"xixj = xjxi.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq180","equation_number":null,"raw_text":"xixj = qxjxi,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq181","equation_number":null,"raw_text":"ρ : H = GL(Cn) →H × H = GL(Cn) × GL(Cn).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq182","equation_number":null,"raw_text":"ρq : Hq = GLq(Cn) →Hq × Hq = GLq(Cn) × GLq(Cn).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq183","equation_number":null,"raw_text":"be an n×n variable matrix. The coordinate algebra R = O(G) of the group","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq184","equation_number":null,"raw_text":"G = GL(Cn) is defined to be the C-algebra generated by the entries xij of X","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq185","equation_number":null,"raw_text":"dinate ring Rq = O(Gq) with the standard quantum group Gq = GLq(Cn),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq186","equation_number":null,"raw_text":"B0 = ∅⊂B1 ⊂B2 ⊂· · · ⊂B","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq187","equation_number":null,"raw_text":"with ∪iBi = B, such that ⟨Bi⟩/⟨Bi−1⟩is an irreducible Gq-module. Here","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq188","equation_number":null,"raw_text":"bb′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq189","equation_number":null,"raw_text":"e · b =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq190","equation_number":null,"raw_text":"If we specialize the canonical basis at q = 1, we get a canonical basis of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq191","equation_number":null,"raw_text":"H ֒→G = GL(Ck),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq192","equation_number":null,"raw_text":"this couple, as expected. Specifically, let Gq = GLq(Cn) be the standard","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq193","equation_number":null,"raw_text":"H = GL(Cn) × GL(Cn) →GL(Cn ⊗Cn) = G,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq194","equation_number":null,"raw_text":"of the unitary subgroup U = Un2(C) ⊆G = GLn2(C).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq195","equation_number":null,"raw_text":"H = GLn(C) →G = GL(Vμ(H)),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq196","equation_number":null,"raw_text":"setting simply “disappear” when specialized at q = 1. Specifically, unlike in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq197","equation_number":null,"raw_text":"the standard case, the Hilbert function of these varieties at q ̸= 1 is different","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq198","equation_number":null,"raw_text":"from the Hilbert function of the corresponding classical varieties at q = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq199","equation_number":null,"raw_text":"to the proof of P ̸= NP conjecture in characteristic zero; cf. Figure 3.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq200","equation_number":null,"raw_text":"itive characteristic, and finally, finite fields, thereby proving the P ̸= NP","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq201","equation_number":null,"raw_text":"For this, let us examine why the proof of IP = PSPACE result [Sh] does","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq202","equation_number":null,"raw_text":"called useful if the circuit size of any function h(X) = h(x1, . . . , xn) ∈Cn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq203","equation_number":null,"raw_text":"time polynomial in the size N = 2n of the truth table of h(X).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq204","equation_number":null,"raw_text":"P/poly-naturalizable. The article [RR] says that the P ̸= NP conjecture","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq205","equation_number":null,"raw_text":"computation is algebraically closed, as in this article. We assume that K = C","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq206","equation_number":null,"raw_text":"or K = ̄Fp, the algebraic closure of a finite field Fp. Let Fn be the set of n-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq207","equation_number":null,"raw_text":"variable polynomials of degree d(n) for some fixed function d(n) = 2poly(n).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq208","equation_number":null,"raw_text":"If K = C, we assume that each polynomial in Fn is an integral polynomial","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq209","equation_number":null,"raw_text":"whose coefficients have poly(n) bitlength. If K = ̄Fp, we assume that all","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq210","equation_number":null,"raw_text":"coefficients belong to Fp, and that the bitlength ⟨p⟩= poly(n).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq211","equation_number":null,"raw_text":"Let us follow the notation as in Section 4. Let K = C. Let h(X) ∈P(W)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq212","equation_number":null,"raw_text":"UP1: The form h = h(X) is co-NP-complete.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq213","equation_number":null,"raw_text":"scribed in Section 7 in [GCT1]. So also the form f = φ(h) as a point in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq214","equation_number":null,"raw_text":"P(V ). This means the associated class varieties ∆W[h; n] = ∆W [h], and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq215","equation_number":null,"raw_text":"∆V [f; n, l] = ∆V [f], as defined in Section 4.2 with h(X) playing the role of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq216","equation_number":null,"raw_text":"every n →∞, l = nlog n, as long as h(X) is co-NP-complete (cf. Section 6).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq217","equation_number":null,"raw_text":"proof of existence of an obstruction λ(n) for every n →∞, assuming that l =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq218","equation_number":null,"raw_text":"nomial size circuit lower bound for h(X), and hence, that P ̸= NP over","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq219","equation_number":null,"raw_text":"the difficulty of proving the P ̸= NP conjecture over K. When K = Fp,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq220","equation_number":null,"raw_text":"When K = C∗, the conjecture should be easier than over ̄Fp or Fp. Accord-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq221","equation_number":null,"raw_text":"the truth-table size when K = Fp.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq222","equation_number":null,"raw_text":"verify” approaches to the P ̸= NP conjecture as per a certain measure of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq223","equation_number":null,"raw_text":"The final proof of the P ̸= NP","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq224","equation_number":null,"raw_text":"n →∞, when the circuit size m = nlog n, say.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq225","equation_number":null,"raw_text":"towards the P ̸= NP conjecture which contain, explicitly or implicitly, the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq226","equation_number":null,"raw_text":"Fix any co-NP-complete function h(X) = h(x1, . . . , xn). Then this proof","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq227","equation_number":null,"raw_text":"size m = nlog n, with a specific value of X for each circuit on which the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq228","equation_number":null,"raw_text":"Definition 19.1 We say that a proof technique for the P ̸= NP conjecture","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq229","equation_number":null,"raw_text":"h(X) = h(x1, . . . , xn) under consideration by a circuit of size m =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq230","equation_number":null,"raw_text":"P ̸= NP conjecture to be viable it out to be P-verifiable.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq231","equation_number":null,"raw_text":"a viable proof technique for the P ̸= NP conjecture that is better than the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq232","equation_number":null,"raw_text":"Every P-verifiable technique for the P ̸= NP-conjecture has to cross the P-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq233","equation_number":null,"raw_text":"P ̸= NP conjecture is taken.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq234","equation_number":null,"raw_text":"the polynomial–is the same regardless of which approach to the P ̸= NP","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq235","equation_number":null,"raw_text":"Next, let us assume that K = C, as in this paper. Let n be the number","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq236","equation_number":null,"raw_text":"of input parameters. Let m(n) = nlogn (say) be the circuit-size parameter,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq237","equation_number":null,"raw_text":"h(n) = nlogn the height-parameter, and d(n) ≤2h(n) the degree parameter","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq238","equation_number":null,"raw_text":"functions over C computable by circuits of size at most m = m(n), height at","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq239","equation_number":null,"raw_text":"most h = h(n) and degree at most d = d(n) is an algebraically constructible","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq240","equation_number":null,"raw_text":"that h(X) does not belong to S, when m = nlog n. Let ̄S be the closure of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq241","equation_number":null,"raw_text":"Remark 19.2 The triple exponential size of this gap when K = C as","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq242","equation_number":null,"raw_text":"against the exponential size over K = Fp does not mean that the P ̸= NP","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq243","equation_number":null,"raw_text":"conjecture is easier when K = Fp. In fact, it is the other way around. Since","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq244","equation_number":null,"raw_text":"the (nonuniform) P ̸= NP conjecture in characteristic zero (over Z) is a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq245","equation_number":null,"raw_text":"mathematics required for any P-verifiable approach towards the P ̸= NP","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq246","equation_number":null,"raw_text":"Definition 19.3 We say that a P-verifiable proof technique for the P ̸=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq247","equation_number":null,"raw_text":"proof strategies is akin to the relationship between P and NP. The P ̸= NP","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq248","equation_number":null,"raw_text":"P ̸= NP conjecture may, however, seem paradoxical at the surface. Be-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq249","equation_number":null,"raw_text":"the very philosophical essence of the P ̸= NP conjecture that discovery is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq250","equation_number":null,"raw_text":"In this context, C is the class of tuples (n, m(n)), m(n) = nc for any con-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq251","equation_number":null,"raw_text":"stant c > 0, over all n (large enough). The property Q(X), X = (n, m(n)),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq252","equation_number":null,"raw_text":"enough. Here ⟨X⟩= n+m; i.e, we assume that n and m are given in unary.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq253","equation_number":null,"raw_text":"the ones in Definitions 19.1 and 19.3. When m = nc, an obstruction would","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq254","equation_number":null,"raw_text":"always exist, assuming that h(X) is co-(NP)-complete, P ̸= NP and the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq255","equation_number":null,"raw_text":"over C, and hence the P ̸= NP conjecture over C is also P-constructive; the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq256","equation_number":null,"raw_text":"(Hypothesis 7.1) fully to prove P ̸= NP over C. That is, it may suffice to","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq257","equation_number":null,"raw_text":"nique for the P ̸= NP conjecture is the computational complexity of the algo-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq258","equation_number":null,"raw_text":"technique for the P ̸= NP conjecture is the computational complexity class","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq259","equation_number":null,"raw_text":"Now we ask if there is a P-verifiable proof technique towards the P ̸= NP","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq260","equation_number":null,"raw_text":"is P-verifiable and also P-constructive over C as per Hypothesis 7.1, P ̸=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq261","equation_number":null,"raw_text":"tion complexity classes of the P ̸= NP conjecture?","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq262","equation_number":null,"raw_text":"verifiable approach to the P ̸= NP conjecture should be at least as hard as","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq263","equation_number":null,"raw_text":"P ̸= NP conjecture over C. The story over Fp may be similar; cf. [GCT11].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq264","equation_number":null,"raw_text":"T. Baker, J. Gill, R. Soloway, Relativization of the P =?NP ques-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq265","equation_number":null,"raw_text":"R. Impagliazzo, A. Wigderson, P=BPP if E requires exponential","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq266","equation_number":null,"raw_text":"A. Shamir, IP=PSPACE, Journal of the ACM, vol. 39, issue 4","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":210378,"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}}