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Littlewood-Richardson Coefficients via Yang-Baxter Equation

Oleg Gleizer·Alexander Postnikov
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Abstract

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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Oleg Gleizer, Alexander Postnikov. 1999-09-21. Littlewood-Richardson Coefficients via Yang-Baxter Equation. https://arxiv.org/abs/math/9909124

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