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We study Hall-Littlewood polynomials using an integrable lattice model of$t$-deformed bosons. Working with row-to-row transfer matrices, we review theconstruction of Hall-Littlewood polynomials (of the $A_n$ root system) withinthe framework of this model. Introducing appropriate double-row transfermatrices, we extend this formalism to Hall-Littlewood polynomials based on the$BC_n$ root system, and obtain a new combinatorial formula for them. We thenapply our methods to prove a series of refined Cauchy and Littlewood identitiesinvolving Hall-Littlewood polynomials. The last two of these identities arenew, and relate infinite sums over hyperoctahedrally symmetric Hall-Littlewoodpolynomials with partition functions of the six-vertex model on finite domains.

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