Tetrahedron equation and Schur functions

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–Maruyama–Okado. 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.