B\"acklund transformations for certain rational solutions of Painlev\'e VI

We introduce certain B\"acklund transformations for rational solutions of the Painlev\'e VI equation. These transformations act ona family of Painlev\'e 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 sl(6). We also show that this sl(6) root lattice can be related to the $F_4^{(1)}$ root lattice. We thus obtain B\"acklund transformations that relate Painlev\'e VI tau functions, parametrized by the elements of this $F_4^{(1)}$ root lattice.