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

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.

💡 Research Summary

The paper begins by constructing the evaluation representation of the affine quantum group (U_q(\widehat{\mathfrak{sl}}_2)) from the Verma module of (U_q(\mathfrak{sl}_2)). By introducing a spectral parameter (z) the authors obtain an infinite‑dimensional module on which the universal (R)-matrix acts, satisfying the Yang–Baxter equation. They then specialize to the crystal (or “(q\to0)”) limit. In this limit the quantum algebra collapses to a combinatorial structure: the basis vectors become Kashiwara crystal elements and the generators act as simple 0‑1 operators. This reduction allows the definition of a two‑dimensional vertex model on a square lattice whose local states are binary and obey the ice‑rule. The row‑to‑row transfer matrix (T(u)) (with spectral parameter (u)) is shown to be the generating function of non‑commutative complete symmetric polynomials \