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 {mapping} R(g) from GL(2,ℂ) × GL(1,ℂ) to End (V ⊗ V), where V is a two-dimensional vector space such that if g,h ε G then R₁₂(g)R₁₃(gh) R₂₃(h) = R₂₃(h) R₁₃(gh)R₁₂(g). Here R_ij denotes R applied to the i, j components of V ⊗ V ⊗ V. The image of this map consists of matrices whose nonzero coefficients a₁, a₂, b₁, b₂, c₁, c₂ are the Boltzmann weights for the non-field-free six-vertex model, constrained to satisfy a₁a₂ + b₁b₂ - c₁c₂ = 0. This is the exact center of the disordered regime, and is contained within the free fermionic eight-vertex models of Fan and Wu. As an application, we show that with boundary conditions corresponding to integer partitions λ, the six-vertex model is exactly solvable and equal to a Schur polynomial s_λ times a deformation of the Weyl denominator. This generalizes and gives a new proof of results of Tokuyama and Hamel and King.