Discrete dynamical systems with $W(A_{m-1}^{(1)}\times A_{n-1}^{(1)})$ symmetry

Abstract

We give a birational realization of affine Weyl group of type $A_{m-1}^{(1)}\times A_{n-1}^{(1)}$. We apply this representation to construct some discrete integrable systems and discrete Painlevé equations. Our construction has a combinatorial counterpart through the ultra-discretization procedure.

10 pages