Refined Littlewood identity for spin Hall-Littlewood symmetric rational functions

TL;DR

This paper introduces a refined Littlewood identity for inhomogeneous spin Hall-Littlewood functions, expressing sums over partitions as Pfaffians and connecting to six vertex models via the Yang-Baxter equation.

Contribution

It derives a new Pfaffian expression for weighted sums of spin Hall-Littlewood functions, linking symmetric functions with integrable models.

Findings

Pfaffian formula for weighted sums of $F_$ functions.

Connection between symmetric functions and six vertex model partition functions.

Proof based on Yang-Baxter equation techniques.

Abstract

Fully inhomogeneous spin Hall-Littlewood symmetric rational functions $F_\lambda$ are multiparameter deformations of the classical Hall-Littlewood symmetric polynomials and can be viewed as partition functions in $\mathfrak{sl}(2)$ higher spin six vertex models. We obtain a refined Littlewood identity expressing a weighted sum of $F_\lambda$'s over all partitions $\lambda$ with even multiplicities as a certain Pfaffian. This Pfaffian can be derived as a partition function of the six vertex model in a triangle with suitably decorated domain wall boundary conditions. The proof is based on the Yang-Baxter equation.