Discrete integrable systems and special polynomials arising from affine root system

Abstract

I will give an overview of my recent joint work with Yasuhiko YAMADA (Kobe University) on discrete integrable systems and special polynomials associated with affine root systems. We present a general method to realize the Weyl group defined by an arbitrary root system (or generalized Cartan matrix) as a group of canonical birational transformations.  In this framework, we define a class of "special polynomials" associated with root systems.  In the cases of affine root systems, this construction provides a class of discrete integrable systems which can be regarded as the symmetry of certain reduction of the Drinfeld-Sokolov hierarchies.  The special polynomials for the root systems of type $A$ are expressed in terms of Macdonald's nineth variations of Schur functions.  They also describe the generic rational solutions of the discrete integrable systems of type $A$.   This work has its origin in the study of B\"acklund transformations of the Painlev\'e equations.  We will also give some applications to the Painlev\'e equations.