Geometric reductions of ABS equations on an n-cube to discrete Painlevé systems*

Abstract

In this paper, we show how to relate n-dimensional cubes on which ABS equations hold to the symmetry groups of discrete Painlevé equations. We here focus on the reduction from the four-dimensional cube to the q-discrete third Painlevé equation, which is a dynamical system on a rational surface of type with the extended affine Weyl group . We provide general theorems to show that this reduction also extends to other discrete Painlevé equations at least of type A.