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

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 $A_n$ 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 $BC_n$ 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.