Painlevé equations and complex reflections. (Q1415560)

The author shows that the general PVI equation governs isomonodromic deformations of a \(3\times3\) system having four Fuchsian singularities on the sphere with rank one residue at three of the singularities. Thus, for each finite subgroup of \(GL_3({\mathbb C})\) generated by three pseudo-reflections, there exists a finite braid group generating a finite branching solution of PVI. Since the finite complex reflection groups are classified, it is tempting to describe the corresponding solutions of PVI. The idea is illustrated by a short description of the case of the smallest exceptional three-dimensional complex reflection group of order 336 associated to Klein's simple group of order 168 (for the complete construction, see: the author [Proc. Lond. Math. Soc., III. Ser. 90, 167--208 (2005; Zbl 1070.34123)]).