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Geometric Complexity Theory IV: Nonstandard Quantum Group for the Kronecker Problem
Journal article   Open access

Geometric Complexity Theory IV: Nonstandard Quantum Group for the Kronecker Problem

Jonah Blasiak, Ketan D. Mulmuley and Milind Sohoni
Memoirs of the American Mathematical Society, v 235(1109)
01 Jan 2015
DOI: https://doi.org/10.1090/memo/1109
url
https://doi.org/10.1090/memo/1109View
Published, Version of Record (VoR)Maybe Open Access (Publisher Bronze),  Open

Abstract

Mathematics Physical Sciences Science & Technology
The Kronecker coefficient g(lambda mu nu) is the multiplicity of the GL(V) x GL(W)irreducible V lambda circle times W-mu in the restriction of the GL(X)-irreducible X-nu via the natural map GL(V) x GL(W) -> GL(V circle times W), where V, W are C-vector spaces and X = V circle times 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 to construct, in the case dim(V) = dim(W) = 2, a representation. (sic)(nu) of the nonstandard quantum group that specializes to Res(GL(V) x GL(W)) X-nu at q = 1. We then define a global crystal basis + HNSTC(nu) of (sic)(nu) that solves the two-row Kronecker problem: the number of highest weight elements of + HNSTC(nu) of weight (lambda, mu) is the Kronecker coefficient g(lambda mu nu). We go on to develop the beginnings of a graphical calculus for this basis, along the lines of the U-q(sl(2)) graphical calculus and use this to organize the crystal components of + HNSTC(nu) into eight families. This yields a fairly simple, positive formula for two-row Kronecker coefficients, generalizing a formula of Brown, Willigenburg, and Zabrocki. As a byproduct of the approach, we also obtain a rule for the decomposition of Res(GL2) (x) (GL2) (sic) (S2) X-nu into irreducibles.

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Domestic collaboration
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Mathematics
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