Geometric Complexity Theory: Volume IV: Nonstandard Quantum Group for the Kronecker Problem: 4 (Memoirs of the American Mathematical Society) - Softcover

Jonah Blasiak; Ketan D. Mulmuley; Milind Sohoni

Softcover

ISBN 10: 1470410117 ISBN 13: 9781470410117

Publisher: American Mathematical Society, 2015

Synopsis

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.

About this title

About the Author

Jonah Blasiak, Drexel University, Philadelphia, PA, USA.

Ketan D. Mulmuley, The University of Chicago, IL, USA.

Milind Sohoni, Indian Institute of Technology, Mumbai, India.

Edition details

• Publisher: American Mathematical Society
• Publication date: 2015
• ISBN 10: 1470410117
• ISBN 13: 9781470410117
• Binding: Paperback
• Language: English
• Number of pages: 160