  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.
