Simplicial arrangements on convex cones

TL;DR

This paper introduces Tits arrangements on convex open cones, extending the concept of simplicial arrangements and Coxeter group representations to a more general, infinite setting with preserved structural properties.

Contribution

It generalizes the notion of simplicial arrangements to convex cones and demonstrates that standard subarrangement and restriction constructions apply in this broader context.

Findings

Established a new framework for Tits arrangements on convex cones

Extended known properties of finite hyperplane arrangements to infinite cases

Showed that geometric structures similar to Coxeter groups are present in this setting

Abstract

We introduce the notion of a Tits arrangement on a convex open cone as a special case of (infinite) simplicial arrangements. Such an object carries a simplicial structure similar to the geometric representation of Coxeter groups. The standard constructions of subarrangements and restrictions, which are known in the case of finite hyperplane arrangements, work as well in this more general setting.