Tropical representation of Weyl groups associated with certain rational varieties

Abstract

Starting from certain rational varieties blown-up from ( P 1 ) N , we construct a tropical, i.e., subtraction-free birational, representation of Weyl groups as a group of pseudo-isomorphisms of the varieties. We develop an algebro-geometric framework of τ-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 a class of (higher order) q-difference Painlevé equations and its algebraic degree grows quadratically.