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

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.