  This paper is about a family of symmetric rational functions that form a
one-parameter generalization of the classical Hall-Littlewood polynomials. We
introduce two sets of (skew and non-skew) functions that are akin to P and Q
Hall-Littlewood polynomials. We establish (a) a combinatorial formula that
represents our functions as partition functions for certain path ensembles in
the square grid; (b) symmetrization formulas for non-skew functions; (c)
identities of Cauchy and Pieri type; (d) explicit formulas for principal
specializations; (e) two types of orthogonality relations for non-skew
functions.
  Our construction is closely related to the half-infinite volume, finite
magnon sector limit of the higher spin six-vertex (or XXZ) model, with both
sets of functions representing higher spin six-vertex partition functions
and/or transfer-matrices for certain domains.
