Universal character and q-difference Painlevé equations with affine Weyl groups

The universal character is a polynomial attached to a pair of partitions and is a generalization of the Schur polynomial. In this paper, we introduce an integrable system of q-difference lattice equations satisfied by the universal character, and call it the lattice q-UC hierarchy. We regard it as generalizing both q-KP and q-UC hierarchies. Suitable similarity and periodic reductions of the hierarchy yield the q-difference Painlevé equations of types A 2g+1 (g ≥ 1), D 5 , and E (1) 6 . As its consequence, a class of algebraic solutions of the q-Painlevé equations is rapidly obtained by means of the universal character. In particular, we demonstrate explicitly the reduction procedure for the case of type E 6 , via the framework of τ-functions based on the geometry of certain rational surfaces. 2000 Mathematics Subject Classification 34M55, 37K10, 39A13.