Symplectic Ice

Abstract

In this chapter, we construct an ice model (a six-vertex model) whose partition function equals the product of a deformation of Weyl’s denominator and an irreducible character of the symplectic group $Sp(2n, \mathbb{C})$. Similar results have been obtained by Brubaker et al. (Schur polynomials and the Yang–Baxter equation. Comm. Math. Phys. 308(2):281–301, 2011) (for the general linear group) and by Hamel and King (Symplectic shifted tableaux and deformations of Weyl’s denominator formula for sp(2n), J. Algebraic Combin. 16, 2002, no. 3, 269–300, 2003) (for the symplectic group). The difference between our result and that of (Hamel and King, Symplectic shifted tableaux and deformations of Weyl’s denominator formula for sp(2n), J. Algebraic Combin. 16, 2002, no. 3, 269–300, 2003) is that our Boltzmann weights for cap vertices are different from those in (Hamel and King, Symplectic shifted tableaux and deformations of Weyl’s denominator formula for sp(2n), J. Algebraic Combin. 16, 2002, no. 3, 269–300, 2003). Also, our proof uses the Yang–Baxter equation, while that of Hamel and King does not.