The facial weak order on hyperplane arrangements

TL;DR

This paper generalizes the facial weak order from finite Coxeter groups to central hyperplane arrangements, providing multiple equivalent definitions and analyzing its lattice and topological properties.

Contribution

It introduces four equivalent definitions of the facial weak order for central arrangements and explores its lattice and homotopy properties.

Findings

Facial weak order is a lattice when the poset of regions is a lattice.

For simplicial arrangements, the lattice is semidistributive.

The homotopy type of all intervals in the facial weak order is determined.

Abstract

We extend the facial weak order from finite Coxeter groups to central hyperplane arrangements. The facial weak order extends the poset of regions of a hyperplane arrangement to all its faces. We provide four non-trivially equivalent definitions of the facial weak order of a central arrangement: (1) by exploiting the fact that the faces are intervals in the poset of regions, (2) by describing its cover relations, (3) using covectors of the corresponding oriented matroid, and (4) using certain sets of normal vectors closely related to the geometry of the corresponding zonotope. Using these equivalent descriptions, we show that when the poset of regions is a lattice, the facial weak order is a lattice. In the case of simplicial arrangements, we further show that this lattice is semidistributive and give a description of its join-irreducible elements. Finally, we determine the homotopy type of all intervals in the facial weak order.