On a family of symmetric rational functions

TL;DR

This paper introduces a new family of symmetric rational functions generalizing Hall-Littlewood polynomials, with combinatorial, algebraic, and orthogonality properties, linked to higher spin six-vertex models.

Contribution

It defines new symmetric rational functions with combinatorial formulas, symmetrization, identities, explicit specializations, and orthogonality, connecting to integrable models.

Findings

Combinatorial path ensemble representation

Symmetrization formulas for non-skew functions

Orthogonality relations and explicit formulas

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 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.