GEOMETRIC COMPLEXITY THEORY IV: NONSTANDARD QUANTUM GROUP FOR THE KRONECKER PROBLEM JONAH BLASIAK, KETAN D. MULMULEY, AND MILIND SOHONI Dedicated to Sri Ramakrishna Abstract. The Kronecker coefficient gλµν is the multiplicity of the GL(V ) × GL(W )- irreducible Vλ ⊗ Wµ in the restriction of the GL(X)-irreducible Xν via the natural map GL(V ) × GL(W ) → GL(V ⊗ W ), where V, W are C-vector spaces and X = V ⊗ W . A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients. We construct two quantum objects for this problem, which we call the nonstandard quantum group and nonstandard Hecke algebra. We show that the nonstandard quantum group has a compact real form and its representations are completely reducible, that the nonstandard Hecke algebra is semisimple, and that they satisfy an analog of quantum Schur-Weyl duality. Using these nonstandard objects as a guide, we follow the approach of Adsul, Sohoni, and Subrahmanyam [1] to construct, in the case dim(V ) = dim(W ) = 2, a representation ˇ Xν of the nonstandard quantum group that specializes to ResGL(V )×GL(W )Xν at q = 1. We then define a global crystal basis +HNSTC(ν) of Xˇν that solves the two-row Kronecker problem: the number of highest weight elements of +HNSTC(ν) of weight (λ, µ) is the Kronecker coefficient gλµν . We go on to develop the beginnings of a graphical calculus for this basis, along the lines of the Uq(sl2) graphical calculus from [19], and use this to organize the crystal components of +HNSTC(ν) into eight families. This yields a fairly simple, explicit and positive formula for two-row Kronecker coefficients, generalizing a formula in [15]. As a byproduct of the approach, we also obtain a rule for the decomposition of ResGL2×GL2⋊S2 Xν into irreducibles. Contents 1. Introduction 4 1.1. The Kronecker problem 4 1.2. The basis-theoretic version of the Kronecker problem 7 1.3. Canonical bases connect quantum Schur-Weyl duality with RSK 8 1.4. The nonstandard quantum group and Hecke algebra 10 1.5. Towards an upper canonical basis for Xˇ ⊗r 13 1.6. The approach of Adsul, Sohoni, and Subrahmanyam 13 Key words and phrases. Kronecker problem, complexity theory, canonical basis, quantum group, Hecke algebra, graphical calculus. J. Blasiak was supported by an NSF postdoctoral fellowship. 1 2 JONAHBLASIAK,KETAND.MULMULEY,ANDMILINDSOHONI 1.7. A global crystal basis for Xˇν 14 1.8. Organization 16 2. Basic concepts and notation 16 2.1. General notation 17 2.2. Tensor products 17 2.3. Words and tableaux 17 2.4. Cells 18 2.5. Comodules 19 2.6. Dually paired Hopf algebras 20 3. Hecke algebras and canonical bases 21 3.1. The upper canonical basis of H (W ) 21 3.2. Cells in type A 22 4. The quantum group GLq(V ) 22 4.1. The quantized enveloping algebra Uq(gV ) 22 4.2. FRT-algebras 23 4.3. The quantum coordinate algebra O(Mq(V )) 24 4.4. The quantum determinant and the Hopf algebra O(GLq(V )) 25 4.5. A reduction system for O(Mq(V )) 26 4.6. Compactness, unitary transformations 27 4.7. Representations of GLq(V ) 27 5. Bases for GLq(V ) modules 28 5.1. Gelfand-Tsetlin bases and Clebsch-Gordon coefficients 28 5.2. Crystal bases 29 5.3. Global crystal bases 29 5.4. Projected based modules 31 5.5. Tensor products of based modules 32 6. Quantum Schur-Weyl duality and canonical bases 33 6.1. Commuting actions on T = V ⊗r 33 6.2. Upper canonical basis of T 34 6.3. Graphical calculus for Uq(gl2)-modules 35 7. Notation for GLq(V ) × GLq(W ) 38 8. The nonstandard coordinate algebra O(Mq(Xˇ)) 38 8.1. Definition of O(Mq(Xˇ)) 39 8.2. Nonstandard symmetric and exterior algebras 41 8.3. Explicit product formulae 45 GCT IV: NONSTANDARD QUANTUM GROUP FOR THE KRONECKER PROBLEM 3 8.4. Examples and computations for O(Mq(Xˇ)) 46 9. Nonstandard determinant and minors 48 9.1. Definitions 48 9.2. Nonstandard minors in the two-row case 49 9.3. Symmetry of the determinants and minors 52 9.4. Formulae for nonstandard minors 54 10. The nonstandard quantum groups GLq(Xˇ) and Uq(Xˇ) 56 10.1. Hopf structure 56 10.2. Compact real form 58 10.3. Complete reducibility 59 ˇ 11. The nonstandard Hecke algebra Hr 59 ˇ 11.1. Definition of Hr and basic properties 59 ˇ 11.2. Semisimplicity of KHr 61 2 11.3. Representation theory of S Hr 62 ˇ 11.4. Some representation theory of Hr 63 11.5. The sign representation in the canonical basis 65 ˇ 11.6. The algebra H3 65 ˇ 11.7. A canonical basis of H3 67 ˇ 11.8. The algebra H4 68 12. Nonstandard Schur-Weyl duality 69 12.1. Nonstandard Schur-Weyl duality 69 12.2. Consequences for the corepresentation theory of O(Mq(Xˇ)) 70 12.3. The two-row, r = 3 case 71 13. Nonstandard representation theory in the two-row case 72 τ 13.1. The Hopf algebra Uq 72 Oτ 13.2. The Hopf algebra q 73 τ Oτ 13.3. Representation theory of Uq and q 73 τ 2H 13.4. Schur-Weyl duality between Uq and S r 74 τ 13.5. Upper based Uq -modules 74 13.6. The nonstandard two-row case 75 14. A canonical basis for Yˇα 76 14.1. Nonstandard columns label a canonical basis for Λˇ rXˇ 76 14.2. Nonstandard tabloids label a canonical basis of Yˇα 79 14.3. The action of the Kashiwara operators and τ on NST 82 15. A global crystal basis for two-row Kronecker coefficients 83 4 JONAHBLASIAK,KETAND.MULMULEY,ANDMILINDSOHONI 15.1. Invariants 84 15.2. Two-column moves 85 15.3. Invariant moves 90 15.4. Nonstandard tabloid classes 90 15.5. Justifying the combinatorics 93 15.6. Explicit formulae for nonintegral NST(β, ⊲γ, δ) 98 15.7. A basis for the two-row Kronecker problem 100 16. Straightened NST and semistandard tableaux 105 16.1. Lexicographic order on NST 105 16.2. Bijection with semistandard tableaux 108 16.3. Invariant-free straightened highest weight NST 111 17. A Kronecker graphical calculus and applications 112 17.1. Kronecker graphical calculus 112 17.2. Action of the Chevalley generators on +HNSTC 115 17.3. The action of τ on +HNSTC 122 18. Explicit formulae for Kronecker coefficients 124 18.1. Invariant-free Kronecker coefficients and explicit formulae 124 18.2. The symmetric and exterior Kronecker coefficients 125 18.3. Comparisons with other formulae 128 19. Future work 130 19.1. A canonical basis for Xˇ ⊗r 130 19.2. Defining Xˇν outside the two-row case 136 Acknowledgments 137 Appendix A. Reduction system for O(Mq(Xˇ)) 137 Oτ Appendix B. The Hopf algebra q 140 References 143 1. Introduction 1.1. The Kronecker problem. This is a continuation of the series of articles [47, 48, 44] on geometric complexity theory (GCT), an approach to P vs. NP and related problems using geometry and representation theory. A basic philosophy of this approach is called the flip; see [46, 43, 41] for its detailed exposition. The flip suggests that separating the classes P and NP will require solving difficult positivity problems in algebraic geometry and representation theory. A central positivity problem arising here is the following GCT IV: NONSTANDARD QUANTUM GROUP FOR THE KRONECKER PROBLEM 5 fundamental problem in the representation theory of the symmetric group concerning Kronecker coefficients. Let Sr denote the symmetric group on r letters and let Mν denote the irreducible Sr- module corresponding to the partition ν. Given three partitions λ,µ,ν of r, the Kronecker coefficient gλµν is defined to be the multiplicity of Mν in the tensor product Mλ ⊗ Mµ. It can also be defined in terms of general linear groups. Let n be the maximum of the n n heights of λ and µ. Embed H = GLn(C)×GLn(C) in GL(C ×C )= GLn2 (C) naturally. Let Nα(GLn(C)) denote the Weyl module of GLn(C) corresponding to the partition α. Then gλµν is also the multiplicity of the H-module Nλ(GLn(C)) ⊗ Nµ(GLn(C)) in Nν(GLn2 (C)) considered as an H-module via the embedding in G above. Problem 1.1 (Kronecker problem). Find a positive combinatorial formula for the Kro- necker coefficients gλµν. There are two precise related problems in complexity theory that arise in the flip: (1) find a (positive) #P formula for Kronecker coefficients, and harder, (2) find a polynomial time algorithm to determine whether a Kronecker coefficient is zero. There is also a much deper basis-theoretic version of the Kroncker problem moti- vated by the positivity property of the canonical basis [35] of the tensor product of ir- reducible GLn(C)-modules. This positivity property is related to, but goes far deeper than, the well known Littlewood-Richardson rule that gives a positive combinatorial for- mula for Littlewood-Richardson coefficients. Specifically, embed GLn(C) in GLn(C) × λ GLn(C) diagonally. Then the Littlewood-Richardson coefficient cα,β is the multiplicity of Nλ(GLn(C)) in Nα(GLn(C)) ⊗ Nβ(GLn(C)) considered as a GLn(C)-module via the diagonal embedding. The canonical basis B of Nα(GLn(C)) ⊗ Nβ(GLn(C)) constructed in [35] has the following properties: (1) It is GLn(C)-compatible. This means that B has a filtration B = B0 ⊃ B1 ⊃ · · · such that, letting hBii denote the span of Bi, each hBii/hBi+1i is an irreducible GLn(C)-module.