Universal character and q-difference Painlevé equations

Abstract

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 Painleve equations of types \({(A_{2g+1}+A_1)^{(1)}(g \geq 1)}\) , \({D_5^{(1)}}\) , and \({E_6^{(1)}}\) . As its consequence, a class of algebraic solutions of the q-Painleve equations is rapidly obtained by means of the universal character. In particular, we demonstrate explicitly the reduction procedure for the case of type \({E_6^{(1)}}\) via the framework of τ based on the geometry of certain rational surfaces.