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

ISBN-13: 9781470410117 / Angielski

Jonah Blasiak;Ketan D. Mulmuley;Milind Sohoni

Opis

The Kronecker coefficient $g_{lambda mu u}$ is the multiplicity of the $GL(V) imes GL(W)$-irreducible $V_lambda otimes W_mu$ in the restriction of the $GL(X)$-irreducible $X_u$ via the natural map $GL(V) imes GL(W) o GL(V otimes W)$, where $V, W$ are $mathbb$-vector spaces and $X = V otimes W$. A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients. The authors construct two quantum objects for this problem, which they call the nonstandard quantum group and nonstandard Hecke algebra. They 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.

Spis treści

• Introduction
• Basic concepts and notation
• Hecke algebras and canonical bases
• The quantum group $GL_q(V)$
• Bases for $GL_q(V)$ modules
• Quantum Schur-Weyl duality and canonical bases
• Notation for $GL_q(V) \times GL_q(W)$
• The nonstandard coordinate algebra $\mathscr(M_q(\check))$
• Nonstandard determinant and minors
• The nonstandard quantum groups $GL_q(\check)$ and $\texttt_q(\check)$
• The nonstandard Hecke algebra $\check{\mathscr}_r$
• Nonstandard Schur-Weyl duality
• Nonstandard representation theory in the two-row case
• A canonical basis for $\check_\alpha$
• A global crystal basis for two-row Kronecker coefficients
• Straightened NST and semistandard tableaux}
• A Kronecker graphical calculus and applications
• Explicit formulae for Kronecker coefficients
• Future work
• Appendix A. Reduction system for ${\mathscr}(M_q(\check))$
• Appendix B. The Hopf algebra ${\mathscr}_^\tau$
• Bibliography