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 → 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 ∈ 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 ⊗ V ⊗ 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.Baxter's method of solving lattice models in statistical mechanics is based on the star-triangle relation, which is the identitywhere R, S, T are endomorphisms of V ⊗ V for some vector space V . Here R ij is the endomorphism of V ⊗ V ⊗ V in which R is applied to the i-th and j-th copies of V 1

Introduction: Multiple Dirichlet Series

“…Scalar Rmatrices may be added trivially, so the actual group is SL(2, C)×GL 1 (C). This nonabelian parametrized Yang-Baxter equation was rediscovered in slightly greater generality by Brubaker, Bump and Friedberg [15], who found a parametrized Yang-Baxter equation for the entire set of R-matrices in the free-fermionic six-vertex model, with parameter group GL 2 (C) × GL 1 (C). It is an interesting question how to formulate this in terms of a Hopf algebra, analogous to the field-free case.…”

“…The multiplicativities (15) and (16) together imply that the function H is determined by its values on prime powers. In other words if we specify H(p k1 , · · · , p kr ; p l1 , · · · , p lr ) for prime elements p, the function is determined.…”

“…Tokuyama's formula may be given another interpretation, as evaluating the partition function of a statistical mechanical system of the free-fermionic six-vertex model. For this, see [15], Chapter 19 of [14], and the paper [8] in this volume. Since more details may be found in these references, we will be brief.…”

“…Hamel and King [38] and Brubaker, Bump and Friedberg [15] showed that one may exhibit a free-fermionic six-vertex model whose partition function is exactly the Tokuyama expression (33). In [8] in this volume, this is generalized to a system whose partition function is the metaplectic spherical Whittaker function (35).…”

“…Brubaker, Bump and Friedberg [15] showed that a system may be found, with free-fermionic Boltzmann weights, whose partition function is precisely (33). This fact was generalized Bump, McNamara and Nakasuji [22], who showed that one may replace the character on the left-hand side by a factorial Schur function.…”