Refined Cauchy/Littlewood identities and six-vertex model partition functions: III. Deformed bosons

Abstract

We study Hall–Littlewood polynomials using an integrable lattice model of t-deformed bosons. Working with row-to-row transfer matrices, we review the construction of Hall–Littlewood polynomials (of the An root system) within the framework of this model. Introducing appropriate double-row transfer matrices, we extend this formalism to Hall–Littlewood polynomials based on the BCn root system, and obtain a new combinatorial formula for them. We then apply our methods to prove a series of refined Cauchy and Littlewood identities involving Hall–Littlewood polynomials. The last two of these identities are new, and relate infinite sums over hyperoctahedrally symmetric Hall–Littlewood polynomials with partition functions of the six-vertex model on finite domains.