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

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 $\mathrm{\Delta }=\pm 1\mathrm{/}2$. 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 ${\mathrm{\Gamma }}_0(12)$. 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.