### Abstract

Starting from certain rational varieties blown-up from (P^1)^N, we construct a tropical (or subtraction-free birational) representation of Weyl groups as a group of pseudo isomorphisms of the varieties. Furthermore, we introduce a geometric framework of tau-functions as defining functions of exceptional divisors on the varieties. In the case where the corresponding root system is of affine type, our construction yields (higher order) q-difference Painleve equations. This is a joint work with Tomoyuki Takenawa.