Bäcklund transformations for certain rational solutions of Painlevé VI

Abstract

We introduce certain Bäcklund transformations for rational solutions of the Painlevé VI equation. These transformations act on a family of Painlevé VI tau functions. They are obtained from reducing the Hirota bilinear equations that describe the relation between certain points in the 3 component polynomial KP Grassmannian. In this way we obtain transformations that act on the root lattice of A 5 . We also show that this A 5 root lattice can be related to the F (1) 4 root lattice. We thus obtain Bäcklund transformations that relate Painlevé VI tau functions, parametrized by the elements of this F (1) 4 root lattice.