Tetrahedron equation and Schur functions

TL;DR

This paper explores solutions to the tetrahedron equation using $q$-oscillator models, connecting them to Schur functions and symmetric functions through algebraic and lattice model constructions.

Contribution

It introduces a new family of partition functions related to Schur functions based on $q=0$-oscillator models and extends the framework to inhomogeneous models.

Findings

Partition functions expressed via Schur functions

Inhomogeneous models linked to loop elementary symmetric functions

Explicit algebraic presentation of solutions to the tetrahedron equation

Abstract

The tetrahedron equation introduced by Zamolodchikov is a three-dimensional generalization of the Yang-Baxter equation. Several types of solutions to the tetrahedron equation that have connections to quantum groups can be viewed as $q$-oscillator valued vertex models with matrix elements of the $L$-operators given by generators of the $q$-oscillator algebra acting on the Fock space. Using one of the $q=0$-oscillator valued vertex models introduced by Bazhanov-Sergeev, we introduce a family of partition functions that admits an explicit algebraic presentation using Schur functions. Our construction is based on the three-dimensional realization of the Zamolodchikov-Faddeev algebra provided by Kuniba-Okado-Maruyama. Furthermore, we investigate an inhomogeneous generalization of the three-dimensional lattice model. We show that the inhomogeneous analog of (a certain subclass of) partition functions can be expressed as loop elementary symmetric functions.