Universal character and q-difference Painlev\'e 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 Painleve equations of types $A_{2g+1}^{(1)}$ $(g \geq 1)$, $D_5^{(1)}$, 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 tau-functions based on the geometry of certain rational surfaces.