Schur Polynomials and The Yang-Baxter Equation

Abstract

We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map \({g \mapsto R (g)}\) from \({ \rm{GL}(2, \mathbb{C}) \times \rm{GL}(1, \mathbb{C})}\) to End \({(V \otimes V)}\) , where V is a two-dimensional vector space such that if \({g, h \in G}\) then R 12(g)R 13(gh) R 23(h) = R 23(h) R 13(gh)R 12(g). Here R i j denotes R applied to the i, j components of \({V \otimes V \otimes V}\) . 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 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.