Littlewood-Richardson Coeffients via Yang-Baxter Equations

The purpose of this talk is to present a combinatorial interpretation for the decomposition of the tensor product of two or more irreducible representations of GL(n) in terms of certain scattering R-matrix. This R-matrix satisfies a Yang-Baxter type equation. The corresponding piecewise-linear transformation of parameters maps coincide with the transition maps for Kashiwara parametrizations of canonical bases for modules over type A quantum groups. We provide an explicit description for the cone of such parameterizations, thus solving a problem posed by Berenstein and Zelevinsky. Geometrically, our R-matrix can be interpreted via certain web diagrams, which are similar in appearance to honeycomb tinkertoys of Knutson and Tao. This is a joint work with Oleg Gleizer.