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

Abstract

AustraliaThe inherent self-consistency properties of the sixpoint multi-ratio equation allow it to be considered on a domain associated with a T-shaped Coxeter-Dynkin diagram. This extends the Kadomtsev-Petviashvili lattice, which has A N symmetry, and incorporates also Korteweg-de Vries-type dynamics on a sub-domain with D N symmetry, and Painlevé dynamics on a subdomain withẼ 8 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.

Integrable two-component systems of difference equations

“…The connection of Yang-Baxter maps with integrable quad-equations was originated in [6,33] and complemented in [8][9][10][11]34] where also the connection with higher degree quad-relations was established. Moreover, the interplay between Yang-Baxter maps and discrete integrable systems led to fruitful results [30,[35][36][37][38][39][40][41][42][43][44][45][46][47][48]…”

Reflection groups and discrete integrable systems

“…While there have been several examples associating discrete integrable systems with regular polytopes of the Weyl groups in the literature, they have either not been space-filling or have been restricted to hypercubes. For example, Hirota's d-KP equation is a six-point equation associated with the vertices of an octahedron [11,22,29,10,4]; the ABS classification of four-point partial difference equations is associated with quadrilaterals [1], while six-point equations are associated with the octahedron (octahedron-equations) [2]; and the quadrilateral Yang-Baxter maps have variables associated with the edges of a 3-cube [26,6]. In contrast, our approach extends to infinite-dimensional affine Weyl groups, which give rise to spacefilling polytopes, given by translations of their Voronoi cells.…”

“…Systems of quad-equations in the literature have been mainly defined on the vertices of the n-dimensional hypercube (n-cube) [1], or constructed as a consistent system of the same equations [4]. In contrast, the systems we consider have a wide variety of symmetries, related via reductions to the types of the discrete Painlevé equations in Sakai's classification: 19 types of the discrete Painlevé equations, which follows a degeneration pattern of the affine Weyl symmetry groups [28]: from the root system of type E This means that we have on our hands a corresponding class of quad-equations with immensely rich combinatorial/geometrical structures.…”

Coble's group and the integrability of the Gosset-Elte polytopes and tessellations

“…Add one further point to the figure, p {1,2} , chosen freely on C. Lines connecting this point with the determining points of the Pascal line (8), intersect C at three further points:…”

“…The six-point multi-ratio equation appears in the theory of integrable systems on a discrete domain with octahedral cells [1,2,3,4,5,6,7,8]. It has the following geometric meaning, which has not been considered previously in this area.…”