  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.
