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 \({g \mapsto R (g)}\) 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 R12(g)R13(gh) R23(h) = R23(h) R13(gh)R12(g). Here Rij 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 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 sλ times a deformation of the Weyl denominator. This generalizes and gives a new proof of results of Tokuyama and Hamel and King.