Painleve Equations and Complex Reflections

TL;DR

This paper explores how algebraic solutions to the sixth Painleve equation can be derived from complex reflection groups, extending previous work on real reflection groups and highlighting new algebraic structures.

Contribution

It introduces a novel connection between complex reflection groups and algebraic solutions of Painleve VI, expanding the scope beyond real reflection groups.

Findings

New algebraic solutions from complex reflection groups

Extension of Hitchin and Dubrovin-Mazzocco results

Framework for future explicit formula derivations

Abstract

We will explain how some new algebraic solutions of the sixth Painleve equation arise from complex reflection groups, thereby extending some results of Hitchin and Dubrovin-Mazzocco for real reflection groups. The problem of finding explicit formulae for these solutions will be addressed elsewhere.