Littlewood-Richardson Coefficients via Yang-Baxter Equation

TL;DR

This paper introduces a novel interpretation of Littlewood-Richardson coefficients using Yang-Baxter equations, linking quantum scattering matrices to representation theory and providing explicit descriptions of canonical bases.

Contribution

It offers a new approach connecting tensor product decompositions with Yang-Baxter equations and dual canonical bases, solving a problem posed by Berenstein and Zelevinsky.

Findings

Explicit description of the cone of Kashiwara's parametrizations

Graphical interpretation of scattering matrices via web functions

Solution to the problem of describing dual canonical bases

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.