Noncommutative Schur polynomials and the crystal limit of the -vertex model

Abstract

Starting from the Verma module of we consider the evaluation module for affine and discuss its crystal limit (q → 0). There exists an associated integrable statistical mechanics model on a square lattice defined in terms of vertex configurations. Its transfer matrix is the generating function for noncommutative complete symmetric polynomials in the generators of the affine plactic algebra, an extension of the finite plactic algebra first discussed by Lascoux and Schützenberger. The corresponding noncommutative elementary symmetric polynomials were recently shown to be generated by the transfer matrix of the so-called phase model discussed by Bogoliubov, Izergin and Kitanine. Here we establish that both generating functions satisfy Baxter's TQ-equation in the crystal limit by tying them to special solutions of the Yang–Baxter equation. The TQ-equation amounts to the well-known Jacobi–Trudi formula leading naturally to the definition of noncommutative Schur polynomials. The latter can be employed to define a ring which has applications in conformal field theory and enumerative geometry: it is isomorphic to the fusion ring of the Wess–Zumino–Novikov–Witten model whose structure constants are the dimensions of spaces of generalized θ-functions over the Riemann sphere with three punctures.