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Bäcklund transformations for certain rational solutions of Painlevé VI

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