Mathematics Colloquia and Seminars

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The aim of the talk is to present an interpretation for the Littlewood-Richardson coefficients in terms of a system of quantum particles. Recall that the L-R coefficients are the coefficients of irreducible components in the tensor product of irreducible representations of $GL(N)$. 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 reminiscent of honeycombs of Knutson and Tao. This talk is based on a joint work with O. Gleizer.