Schur Polynomials and the and the Yang-Baxter Equation

Tokuyama [31] proved a deformation of the Weyl character formula for GL (C). A substantial generalization of Tokuyama's deformation was found by Hamel and King [8]. The formula of Hamel and King expresses the Schur polynomial times a deformation of the Weyl denominator as a sum over states of the two-dimensional ice or six-vertex model in statistical mechanics. It turns out that there are two fundamentally distinct ways of doing this. We will call these Gamma ice and Delta ice. The Delta model is essentially that given by Hamel and King. In statistical physics, the partition function is the sum of certain Boltzmann weights over all states of the system. The six-vertex model is an example that is much studied in the literature. If the Boltzmann weights are invariant under sign reversal the system is called field-free, corresponding to the physical assumption of the absence of an external field. For field-free weights, the six-vertex model was solved by Lieb [19] and Sutherland [30], in the sense that the partition function can be exactly computed. A very interesting treatment based on the " star-triangle relation " or Yang-Baxter equation ([13], [21]) was given by Baxter [1] and [2], Chapter 9. The papers of Lieb, Sutherland and Baxter assume periodic boundary conditions, but non-periodic boundary conditions were treated by Korepin [14] and Izergin [12]. Much of the literature assumes that the model is field free, but Baxter asserts that the six-vertex model can be solved even in the presence of fields. We do not know whether this has been carried out using the method of [1] and [2]. We will exhibit two particular choices of Boltzmann weights and boundary conditions in the six-vertex model giving systems S Γ λ and S ∆ λ for every partition λ of length. We will study the system by the method of [1] and [2]. The 1