On a family of symmetric rational functions

Abstract

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 the 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. Keywords: Symmetric rational functions