On the Tits cone of a Weyl groupoid

TL;DR

This paper establishes a correspondence between Weyl groupoids with root systems and Tits arrangements that satisfy an integrality condition called crystallographic property, expanding the understanding of their structure.

Contribution

It introduces a new translation of Weyl groupoid axioms into arrangements and characterizes those with root systems via crystallographic Tits arrangements.

Findings

Weyl groupoids with root systems correspond to crystallographic Tits arrangements.

The paper provides a new framework translating axioms into arrangements.

It broadens the understanding of Weyl groupoid structures.

Abstract

We translate the axioms of a Weyl groupoid with (not necessarily finite) root system in terms of arrangements. The result is a correspondence between Weyl groupoids permitting a root system and Tits arrangements satisfying an integrality condition which we call the crystallographic property.