Deformations of the Weyl Character Formula for $SO(2n+1,\mathbb{C})$ via Ice Models

Abstract

We explore combinatorial formulas for deformations of highest weight characters of the odd orthogonal group $SO(2n+1)$. Our goal is to represent these deformations of characters as partition functions of statistical mechanical models -- in particular, two-dimensional solvable lattice models. In Cartan type $A$, Hamel and King [8] and Brubaker, Bump, and Friedberg [3] gave square ice models on a rectangular lattice which produced such a deformation. Outside of type $A$, ice-type models were found using rectangular lattices with additional boundary conditions that split into two classes -- those with `nested' and `non-nested bends.' Our results fill a gap in the literature, providing the first such formulas for type $B$ with non-nested bends. In type $B$, there are many known combinatorial parameterizations of highest weight representation basis vectors as catalogued by Proctor [19]. We show that some of these permit ice-type models via appropriate bijections (those of Sundaram [21] and Koike-Terada [15]) while other examples due to Proctor do not.