Geometric complexity theory IV : nonstandard quantum group for the Kronecker problem
著者
書誌事項
Geometric complexity theory IV : nonstandard quantum group for the Kronecker problem
（Memoirs of the American Mathematical Society, no. 1109）
American Mathematical Society, 2015, c2014
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注記
"Volume 235, number 1109 (fourth of 5 numbers), May 2015"
Includes bibliographical references (p. 157-160)
内容説明・目次
内容説明
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.
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.
目次
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 $\mathscr{O}(M_q(\check{X}))$
Nonstandard determinant and minors
The nonstandard quantum groups $GL_q(\check{X})$ and $\texttt{U}_q(\check{X})$
The nonstandard Hecke algebra $\check{\mathscr{H}}_r$
Nonstandard Schur-Weyl duality
Nonstandard representation theory in the two-row case
A canonical basis for $\check{Y}_\alpha$
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 ${\mathscr{O}}(M_q(\check{X}))$
Appendix B. The Hopf algebra ${\mathscr{O}}_{q}^\tau$
Bibliography
「Nielsen BookData」 より