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

About this Title

Jonah Blasiak, Department of Mathematics, Drexel University, Philadelphia, PA 19104, Ketan D. Mulmuley, The University of Chicago and Milind Sohoni, I.I.T., Mumbai

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 235, Number 1109
ISBNs: 978-1-4704-1011-7 (print); 978-1-4704-2227-1 (online)
DOI: https://doi.org/10.1090/memo/1109
Published electronically: October 24, 2014
Keywords: Kronecker problem, complexity theory, canonical basis, quantum group, Hecke algebra, graphical calculus
MSC: Primary 33D80, 20C30, 05E10.; Secondary 16S80, 11Y16

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Table of Contents

Chapters

• 1. Introduction
• 2. Basic concepts and notation
• 3. Hecke algebras and canonical bases
• 4. The quantum group $GL_q(V)$
• 5. Bases for $GL_q(V)$ modules
• 6. Quantum Schur-Weyl duality and canonical bases
• 7. Notation for $GL_q(V) \times GL_q(W)$
• 8. The nonstandard coordinate algebra $\mathscr {O}(M_q(\check {X}))$
• 9. Nonstandard determinant and minors
• 10. The nonstandard quantum groups $GL_q(\check {X})$ and $\texttt {U}_q(\check {X})$
• 11. The nonstandard Hecke algebra $\check {\mathscr {H}}_r$
• 12. Nonstandard Schur-Weyl duality
• 13. Nonstandard representation theory in the two-row case
• 14. A canonical basis for $\check {Y}_\alpha$
• 15. A global crystal basis for two-row Kronecker coefficients
• 16. Straightened NST and semistandard tableaux
• 17. A Kronecker graphical calculus and applications
• 18. Explicit formulae for Kronecker coefficients
• 19. Future work
• A. Reduction system for ${\mathscr {O}}(M_q(\check {X}))$
• B. The Hopf algebra ${\mathscr {O}}_{q}^\tau$

Abstract

The Kronecker coefficient $g_{\lambda \mu \nu }$ is the multiplicity of the $GL(V)\times GL(W)$-irreducible $V_\lambda \otimes W_\mu$ in the restriction of the $GL(X)$-irreducible $X_\nu$ via 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 $\check {X}_\nu$ of the nonstandard quantum group that specializes to $\operatorname {Res}_{GL(V) \times GL(W)} X_\nu$ at $q=1$. We then define a global crystal basis $+\mathrm {HNSTC}(\nu )$ of $\check {X}_\nu$ that solves the two-row Kronecker problem: the number of highest weight elements of $+\mathrm {HNSTC}(\nu )$ 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 $U_q(\mathfrak {sl}_2)$ graphical calculus and use this to organize the crystal components of $+\mathrm {HNSTC}(\nu )$ 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 $\operatorname {Res}_{GL_2 \times GL_2 \rtimes {\mathcal {S}}_2} X_\nu$ into irreducibles.