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OverviewThe 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. Full Product DetailsAuthor: Jonah Blasiak , Ketan D. Mulmuley , Milind SohoniPublisher: American Mathematical Society Imprint: American Mathematical Society Volume: 235/1109 Weight: 0.259kg ISBN: 9781470410117ISBN 10: 1470410117 Pages: 160 Publication Date: 30 May 2015 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: Out of stock  The supplier is temporarily out of stock of this item. It will be ordered for you on backorder and shipped when it becomes available. Table of ContentsIntroduction 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 ${\mathscr{O}}_{q}^\tau$ BibliographyReviewsAuthor InformationJonah Blasiak, Drexel University, Philadelphia, PA, USA. Ketan D. Mulmuley, The University of Chicago, IL, USA. Milind Sohoni, Indian Institute of Technology, Mumbai, India. Tab Content 6Author Website:Countries AvailableAll regions |
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