Schur Polynomials and The Yang-Baxter Equation

We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map g bar right arrow R(g) from GL(2, C) x GL(1, C) to End(V circle times V), where V is a two-dimensional vector space such that if g, h is an element of G then R-12(g)R-13(gh) R-23(h) = R-23(h) R-13(gh) R-12(g). Here R-ij denotes R applied to the i, j components of V circle times V circle times V. The image of this map consists of matrices whose nonzero coefficients a(1), a(2), b(1), b(2), c(1), c(2) are the Boltzmann weights for the non-field-free six-vertex model, constrained to satisfy a(1)a(2) + b(1)b(2) - c(1)c(2) = 0. This is the exact center of the disordered regime, and is contained within the free fermionic eight-vertex models of Fan andWu. As an application, we show that with boundary conditions corresponding to integer partitions lambda the six-vertex model is exactly solvable and equal to a Schur polynomial s(lambda) times a deformation of the Weyl denominator. This generalizes and gives a new proof of results of Tokuyama and Hamel and King.