Geometrical description of equations related to the E8 affine Weyl group

Abstract

We present a method for the construction of the trajectory of a discrete Painlev\'e equation associated with the affine Weyl group E$_8^{(1)}$ on the weight lattice of said group. The method is based on the geometrical description of the lattice and the construction of the fundamental Miura relation. To this end we introduce the relation between the nonlinear variables and the corresponding $\tau$ functions. Our approach is heuristic and makes use of some simple rules of thumb in order to derive the result. Once the latter is obtained, verifying that it does indeed correspond to the equation at hand is elementary. We apply our approach to the explicit construction of the trajectory of well-known, E$_8^{(1)}$ associated, discrete Painlev\'e equations derived in previous works of ours. For each of them we investigate the possibility of defining an evolution by periodically skipping up to four intermediate points in the trajectory and identifying the resulting equation to one previously obtained, whenever the latter exists.