Geometric Combinatorics of Weyl Groupoids

TL;DR

This paper generalizes properties of the weak order from finite Coxeter groups to Weyl groupoids with finite root systems, revealing their topological and algebraic structures.

Contribution

It extends the weak order properties to Weyl groupoids, characterizes their topological structure, and introduces a Coxeter complex linking to hyperplane arrangements.

Findings

Intervals with respect to weak order have a specific topological structure.

Set of morphisms with fixed target forms an ortho-complemented meet semilattice.

Coxeter complex coincides with a sphere triangulation from hyperplane arrangements.

Abstract

We extend properties of the weak order on finite Coxeter groups to Weyl groupoids admitting a finite root system. In particular, we determine the topological structure of intervals with respect to weak order, and show that the set of morphisms with fixed target object forms an ortho-complemented meet semilattice. We define the Coxeter complex of a Weyl groupoid with finite root system and show that it coincides with the triangulation of a sphere cut out by a simplicial hyperplane arrangement. As a consequence, one obtains an algebraic interpretation of many hyperplane arrangements that are not reflection arrangements.