Tetrahedral $L$-operators, tensor Schur polynomials and $q$-deformed loop elementary symmetric functions

TL;DR

This paper explores three-dimensional partition functions derived from tetrahedral $L$-operators, connecting them to tensor Schur polynomials, $q$-deformed symmetric functions, and applications in algebraic combinatorics and statistical mechanics.

Contribution

It introduces new classes of partition functions expressed as tensor Schur polynomials and extends identities and formulas related to symmetric functions and integrable models.

Findings

Derived shuffle formula for Schur polynomials as a pushforward

Unified identities of Gustafson-Milne and Fehér–Némethi–Rimányi

Presented $q$-deformed loop elementary symmetric functions

Abstract

We study three-dimensional partition functions constructed from the tetrahedral $L$-operator introduced and studied by Bazhanov-Sergeev and Kuniba-Maruyama-Okado. First, we explore the $q=0$ case, extending the authors' previous results and giving applications by a further analysis on the Zamolodchikov-Faddeev algebra. We introduce a class of partition functions which can be expressed as the tensor Schur polynomials, a class of products of Schur polynomials. As an application, we derive the shuffle formula for the Schur polynomials which is geometrically the pushforward formula by Jo\'zefiak-Pragacz-Lascoux. We also give a derivation and a unification of the Gustafson-Milne and Feh{\'e}r--N{\'e}methi--Rim{\'a}nyi identities, and introduce a family of Laurent polynomials using divided difference operators which imitates the Schubert polynomials from the perspective of our study. We also present an application to the steady state of the multispecies totally asymmetric simple exclusion process. Second, we investigate several classes of partition functions for the generic $q$ case, and determine the explicit forms as deformations of the elementary symmetric functions. One of them can be regarded as (an extension of) a $q$-deformed loop elementary symmetric functions.