Special polynomials related to the supersymmetric eight-vertex model. I. Behaviour at cusps

Abstract

We study certain symmetric polynomials, which as very special cases include polynomials related to the supersymmetric eight-vertex model, and other elliptic lattice models with $\Delta=\pm 1/2$. In this paper, which is the first part of a series, we study the behaviour of the polynomials at special parameter values, which can be identified with cusps of the modular group $\Gamma_0(12)$. In subsequent papers, we will show that the polynomials satisfy a non-stationary Schr\"odinger equation related to the Knizhnik--Zamolodchikov--Bernard equation and that they give a four-dimensional lattice of tau functions of Painlev\'e VI.