On the lattice-geometry and birational group of the six-point multi-ratio equation

The inherent self-consistency properties of the six-point multi-ratio equation allow it to be considered on a domain associated with a T-shaped Coxeter-Dynkin diagram. This extends the KP lattice, which has A_N symmetry, and incorporates also KdV-type dynamics on a sub-domain with D_N symmetry, and Painleve dynamics on a sub-domain with affine-E8 symmetry. More generally, it can be seen as a distinguished representation of Coble’s Cremona group associated with invariants of point sets in projective space.