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Littlewood-Richardson coefficients via Yang-Baxter equation. (English) Zbl 0987.20023
The authors present an interpretation for the Littlewood-Richardson coefficients in terms of a system of quantum particles. Their 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 \(U_q({\mathfrak{sl}}_n)\). A byproduct of their construction is an explicit description for Kashiwara’s parametrizations of dual canonical bases. This solves a problem posed by Berenstein and Zelevinsky. Also, they present a graphical interpretation of the scattering matrices in terms of web functions, which are related to honeycombs of Knutson and Tao. The aim of this paper is to further investigate the Grothendieck ring \(K_N\) of polynomial representations of the general linear group \(\text{GL}(N)\). The structure constants \(c_{\lambda\mu}^\nu\) of the Grothendieck ring in the basis of irreducible representations are given by \[ V_\lambda\otimes V_\mu=\sum_\nu c_{\lambda\mu}^\nu V_\nu. \] They present a new interpretation of the Grothendieck ring \(K_N\) and the Littlewood-Richardson coefficients \(c_{\lambda\mu}^\nu\). They also present a graphical (or “pseudophysical”) interpretation of the scattering matrices and their compositions in the language of web functions and “systems of quantum particles”. Web functions are closely related to Knutson and Tao’s honeycombs and Berenstein-Zelevinsky’s triangles. It is shown by Knutson and Tao that integral honeycombs are in one-to-one correspondence with Berenstein-Zelevinsky patterns. They establish a simple “dual” correspondence between integral web functions and Berenstein-Zelevinsky patterns. This reveals the “hidden duality” of the Littlewood-Richardson coefficients under the conjugation of partitions.

MSC:
20G05 Representation theory for linear algebraic groups
05E15 Combinatorial aspects of groups and algebras (MSC2010)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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