Noncommutative Schur polynomials and the crystal limit of the U_q sl(2)-vertex model

TL;DR

This paper explores the connection between noncommutative Schur polynomials, integrable models, and the crystal limit of the U_q sl(2)-vertex model, revealing their algebraic and geometric significance.

Contribution

It establishes that generating functions for noncommutative symmetric polynomials satisfy Baxter's TQ-equation in the crystal limit, linking them to solutions of the Yang-Baxter equation.

Findings

Demonstrated the TQ-equation for noncommutative symmetric polynomials in the crystal limit

Connected the algebraic structures to the fusion ring of the sl(n)_k-WZNW model

Provided a new perspective on the combinatorial and geometric applications of these polynomials

Abstract

Starting from the Verma module of U_q sl(2) we consider the evaluation module for affine U_q sl(2) 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 Sch\"{u}tzenberger. 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 U_q sl(2) solutions of the Yang-Baxter equation. The TQ-equation amounts to the well-known Jacobi-Trudy 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 sl(n)_k -WZNW model whose structure constants are the dimensions of spaces of generalized theta-functions over the Riemann sphere with three punctures.