Congruence Normality of Simplicial Hyperplane Arrangements via Oriented Matroids

TL;DR

This paper updates the catalogue of simplicial hyperplane arrangements, analyzing their congruence normality using oriented matroids and shards, and identifies which arrangements always or sometimes produce congruence normal lattices.

Contribution

It introduces a new method using oriented matroids to determine congruence normality of simplicial arrangements and extends the classification to sporadic and Weyl groupoid arrangements.

Findings

Lattices of regions of sporadic arrangements of rank 3 are always congruence normal.

Lattices of regions from finite Weyl groupoids are always congruence normal.

The approach recasts shards as covectors to analyze congruence normality.

Abstract

A catalogue of simplicial hyperplane arrangements was first given by Gr\"unbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and the weak order through the poset of regions. For simplicial arrangements, posets of regions are in fact lattices. We update Gr\"unbaum's catalogue, providing normals and invariants for all known sporadic simplicial arrangements with up to 37 lines. The weak order is known to be congruence normal, and congruence normality for simplicial arrangements can be determined using polyhedral cones called shards. In this article, we provide additional structure to the catalogue of simplicial hyperplane arrangements by determining which arrangements always/sometimes/never lead to congruence normal lattices of regions. To this end, we use oriented matroids to recast shards as covectors to determine congruence normality of large hyperplane arrangements. As a consequence of this approach we derive in particular which lattices of regions of sporadic simplicial arrangements of rank 3 are always congruence normal. We also show that lattices of regions from finite Weyl groupoids of any rank are congruence normal.