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We give a birational realization of affine Weyl group of type A(1)m−1 × A(1)n−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.

Discrete dynamical systems with W(A(m-1)((1)) x A(n-1)((1))) symmetry