A geometric approach to tau-functions of difference Painlev\'e equations

TL;DR

This paper provides a geometric framework for understanding tau-functions of difference Painlevé equations through Weyl group representations and configurations of points in the projective plane, unifying various affine types.

Contribution

It introduces a geometric formulation of tau-functions using point configurations and polynomials, linking Weyl group actions to difference Painlevé equations for affine types.

Findings

Unified geometric description of Weyl group actions

Representation of difference Painlevé equations via point configurations

Connection between tau-functions and algebraic curves

Abstract

We present a unified description of birational representation of Weyl groups associated with T-shaped Dynkin diagrams, by using a particular configuration of points in the projective plane. A geometric formulation of tau-functions is given in terms of defining polynomials of certain curves. If the Dynkin diagram is of affine type ($E_6^{(1)}$, $E_7^{(1)}$ or $E_8^{(1)}$), our representation gives rise to the difference Painlev\'e equations.