  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.
