Littlewood-Richardson Coefficients via Yang-Baxter Equation

Details Littlewood-Richardson Coefficients via Yang-Baxter Equation Littlewood-Richardson Coefficients via Yang-Baxter Equation

1999-09-21

arxiv.org/abs/math/9909124v1

IMRN 2000, No. 14, 741-774

Mathematics

Quantum Algebra

24 pages

Scientific paper

The purpose of this paper is to present an interpretation for the decomposition of the tensor product of two or more irreducible representations of GL(N) in terms of a system of quantum particles. Our approach is based on a certain scattering matrix that satisfies a Yang-Baxter type equation. The corresponding piecewise-linear transformations of parameters give a solution to the tetrahedron equation. These transformation maps are naturally related to the dual canonical bases for modules over the quantum enveloping algebra $U_q(sl_n)$. A byproduct of our construction is an explicit description for the cone of Kashiwara's parametrizations of dual canonical bases. This solves a problem posed by Berenstein and Zelevinsky. We present a graphical interpretation of the scattering matrices in terms of web functions, which are related to honeycombs of Knutson and Tao.

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