Reflection groups and discrete integrable systems

TL;DR

This paper introduces a method to construct discrete integrable systems with Weyl group symmetries, linking their geometric polytopes to system properties and unifying various known systems through geometric reduction.

Contribution

It provides a geometric framework connecting discrete integrable systems with crystallographic reflection groups, clarifying their symmetry-based relationships and inheritance of integrability properties.

Findings

Discrete systems are associated with space-filling polytopes from Weyl group representations.

Multi-dimensional consistency is derived from polytope combinatorics.

Connections between known systems are established via geometric reduction.

Abstract

We present a method of constructing discrete integrable systems with crystallographic reflection group (Weyl) symmetries, thus clarifying the relationship between different discrete integrable systems in terms of their symmetry groups. Discrete integrable systems are associated with space-filling polytopes arise from the geometric representation of the Weyl groups in the $n$-dimensional real Euclidean space $\mathbb{R}^n$. The "multi-dimensional consistency" property of the discrete integrable system is shown to be inherited from the combinatorial properties of the polytope; while the dynamics of the system is described by the affine translations of the polytopes on the weight lattices of the Weyl groups. The connections between some well-known discrete systems such as the multi-dimensional consistent systems of quad-equations \cite{abs:03} and discrete Painlev\'e equations \cite{sak:01} are obtained via the geometric constraints that relate the polytope of one symmetry group to that of another symmetry group, a procedure which we call geometric reduction.