A Geometrical Description¶of the Discrete Painlevé VI and V Equations

Abstract:

We present a geometrical approach for the discrete Painlevé equations based on Weyl groups. The method relies on the bilinear formalism and assumes that the multidimensional τ-function lives on the weight lattice of the appropriate affine Weyl group. The equations for the τ-function, a system of nonautonomous Hirota–Miwa equations, govern the evolution along the independent variable and the parameters of the equation (the latter evolution induced by the Schlesinger transformations). In the present paper we analyse the case of the E⁽¹⁾ ₇ group. Using the geometrical description we derive the nonlinear discrete equations. We find that in the case of the E⁽¹⁾ ₇ group these are the “asymmetric”q-P_VI and d-P_V that were recently proposed.