An affine Weyl group approach to the eight-parameter discrete Painlevé equation

Abstract

We present a geometrical construction of the eight-parameter discrete Painlevé equations. Our starting point is the E⁽¹⁾₈ affine Weyl group. We assume that the multi-dimensional τ-function lives on the vertices of the weight lattice of this group. We derive the bilinear equations related to the discrete Painlevé equation in the form of nonautonomous Hirota-Miwa equations and the elementary Miura transformations. The compatibility condition of the various Miura's that can be written leads to three types of equations: difference, multiplicative (q) and another type where the parameters and the independent variable enter through the arguments of elliptic functions. We write explicitly the discrete equations in the first two cases and produce their degeneration through coalescence of parameters.