Congruence Normality of Simplicial Hyperplane Arrangements via Oriented Matroids

authored by

Michael Cuntz, Sophia Elia, Jean Philippe Labbé

Abstract

A catalogue of simplicial hyperplane arrangements was first given by Grünbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and also the weak order through the poset of regions. The weak order is known to be a congruence normal lattice, and congruence normality of lattices of regions of simplicial arrangements can be determined using polyhedral cones called shards. In this article, we update Grünbaum’s catalogue by providing normals realizing all known simplicial arrangements with up to 37 lines and key invariants. Then we add structure to this catalogue 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. We also show that lattices of regions coming from finite Weyl groupoids of any rank are always congruence normal.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

External Organisation(s)

Freie Universität Berlin (FU Berlin)

Type

Article

Journal

Annals of combinatorics

Volume

26

No. of pages

85

ISSN

0218-0006

Publication date

03.2022

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

Electronic version(s)

https://arxiv.org/abs/2009.14152 (Access: Open)
https://doi.org/10.1007/s00026-021-00555-2 (Access: Open)