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

Jonah Blasiak Drexel University, Philadelphia, PA
Ketan D. Mulmuley University of Chicago, Chicago, IL
Milind Sohoni Indian Institute of Technology, Mumbai, India
Available Formats:
Electronic ISBN: 978-1-4704-2227-1
Product Code: MEMO/235/1109.E
List Price: $86.00 MAA Member Price:$77.40
AMS Member Price: $51.60 Click above image for expanded view Geometric Complexity Theory IV: Nonstandard Quantum Group for the Kronecker Problem Jonah Blasiak Drexel University, Philadelphia, PA Ketan D. Mulmuley University of Chicago, Chicago, IL Milind Sohoni Indian Institute of Technology, Mumbai, India Available Formats:  Electronic ISBN: 978-1-4704-2227-1 Product Code: MEMO/235/1109.E  List Price:$86.00 MAA Member Price: $77.40 AMS Member Price:$51.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 2352015; 160 pp
MSC: Primary 33; 20; 05; Secondary 16; 11;

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.

• 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 ${\mathscr {O}}_{q}^\tau$
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2352015; 160 pp
MSC: Primary 33; 20; 05; Secondary 16; 11;

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.

• 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 ${\mathscr {O}}_{q}^\tau$
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
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