Two Littlewood identities for fully inhomogeneous spin Hall-Littlewood symmetric rational functions

TL;DR

This paper introduces two new Littlewood identities for fully inhomogeneous spin Hall-Littlewood functions, linking them to Robbins polynomials and providing combinatorial insights.

Contribution

It establishes two novel Littlewood identities for these functions, generalizing classical results and connecting to Robbins polynomials and combinatorial models.

Findings

Derived two new Littlewood identities involving $F_____$ functions.

Connected the functions to Robbins polynomials and alternating sign matrices.

Provided a bijection for the case of strictly decreasing partitions.

Abstract

Fully inhomogeneous spin Hall-Littlewood symmetric rational functions $F_\lambda$ arise as partition functions of certain path configurations in the $\mathfrak{sl}_2$ higher spin six vertex models. They are multiparameter generalizations of the classical Hall-Littlewood symmetric polynomials. We establish two new generalizations of the classical Littlewood identity, where we express a weighted sum of $F_\lambda$'s over all partitions $\lambda$ as a product of the Littlewood kernel and another simple product in one case, and a product of the Littlewood kernel and a Pfaffian in the other case. As a corollary we obtain a novel Littlewood identity for Hall-Littlewood symmetric polynomials. We also elaborate on the newly established connection between the fully inhomogeneous spin Hall-Littlewood symmetric rational functions $F_\lambda$ and the modified Robbins polynomials, the latter being multivariate generating functions for alternating sign matrices. This connection allowed us to discover the two generalizations of the Littlewood identity and we provide a bijection between the underlying combinatorial models in the case where $\lambda$ is strictly decreasing.