Reflection groups and discrete integrable systems
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 spacefilling polytopes arise from the geometric representation of the Weyl groups in the $n$dimensional real Euclidean space $\mathbb{R}^n$. The "multidimensional 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 wellknown discrete systems such as the multidimensional consistent systems of quadequations \cite{abs:03} and discrete Painlevé 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2016
 DOI:
 10.48550/arXiv.1605.01171
 arXiv:
 arXiv:1605.01171
 Bibcode:
 2016arXiv160501171J
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems