eBook ISBN:  9781470422271 
Product Code:  MEMO/235/1109.E 
List Price:  $86.00 
MAA Member Price:  $77.40 
AMS Member Price:  $51.60 
eBook ISBN:  9781470422271 
Product Code:  MEMO/235/1109.E 
List Price:  $86.00 
MAA Member Price:  $77.40 
AMS Member Price:  $51.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 235; 2015; 160 ppMSC: 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 SchurWeyl duality.

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 SchurWeyl 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 SchurWeyl duality

13. Nonstandard representation theory in the tworow case

14. A canonical basis for $\check {Y}_\alpha $

15. A global crystal basis for tworow 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 $


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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 SchurWeyl 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 SchurWeyl 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 SchurWeyl duality

13. Nonstandard representation theory in the tworow case

14. A canonical basis for $\check {Y}_\alpha $

15. A global crystal basis for tworow 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 $