Point configurations, Cremona transformations and the elliptic difference Painlev\'e equation

TL;DR

This paper develops a higher-dimensional generalization of the elliptic difference Painlevé equation using birational Weyl group actions on point configurations, introducing elliptic parametrization and a $ au$-function theory.

Contribution

It introduces a novel elliptic Cremona system with a Weyl group realization via Cremona transformations, extending Painlevé equations to higher dimensions.

Findings

Established a framework for higher-dimensional elliptic Painlevé equations.

Developed a $ au$-function theory translating the system into bilinear Hirota-Miwa equations.

Connected point configurations, Cremona transformations, and elliptic functions in a unified setting.

Abstract

A theoretical foundation for a generalization of the elliptic difference Painlev\'e equation to higher dimensions is provided in the framework of birational Weyl group action on the space of point configurations in general position in a projective space. By introducing an elliptic parametrization of point configurations, a realization of the Weyl group is proposed as a group of Cremona transformations containing elliptic functions in the coefficients. For this elliptic Cremona system, a theory of $\tau$-functions is developed to translate it into a system of bilinear equations of Hirota-Miwa type for the $\tau$-functions on the lattice.