Geometric Complexity Theory IV: nonstandard quantum group for the Kronecker problem

Abstract

The Kronecker coefficient g_{λμν} is the multiplicity of the GL(V)\times GL(W)-irreducible V_λ\otimes W_μin the restriction of the GL(X)-irreducible X_νvia the natural map GL(V)\times GL(W) \to GL(V \otimes W), where V, W are \mathbb{C}-vector spaces and X = V \otimes 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 \check{X}_νof the nonstandard quantum group that specializes to Res_{GL(V) \times GL(W)} X_νat q=1. We then define a global crystal basis +HNSTC(ν) of \check{X}_νthat solves the two-row Kronecker problem: the number of highest weight elements of +HNSTC(ν) of weight (λ,μ) is the Kronecker coefficient g_{λμν}. 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(ν) into eight families. This yields a fairly simple, explicit and positive formula for two-row Kronecker coefficients, generalizing a formula of Brown, van Willigenburg, and Zabrocki. As a byproduct of the approach, we also obtain a rule for the decomposition of Res_{GL_2 \times GL_2 \rtimes §_2} X_νinto irreducibles.