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

TL;DR

This paper explores the geometric and algebraic structures underlying the six-point multi-ratio equation, revealing its connections to Coxeter-Dynkin diagrams, integrable systems, and Cremona groups, thus broadening the understanding of its symmetries and dynamics.

Contribution

It introduces a novel geometric framework for the six-point multi-ratio equation, linking it to Coxeter-Dynkin diagrams and Cremona groups, and extends known integrable lattice systems.

Findings

Connects the six-point multi-ratio equation to T-shaped Coxeter-Dynkin diagrams.

Shows the equation's relation to KP, KdV, and Painleve dynamics on different sub-domains.

Identifies the equation as a representation of Coble's Cremona group.

Abstract

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.