Details
| Original language | English |
|---|---|
| Number of pages | 85 |
| Journal | Annals of combinatorics |
| Volume | 26 |
| Issue number | 1 |
| Early online date | 8 Nov 2021 |
| Publication status | Published - Mar 2022 |
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.
Keywords
- Congruence normality and uniformity, Covectors, Poset of regions, Shards, Simplicial hyperplane arrangements
ASJC Scopus subject areas
- Mathematics(all)
- Discrete Mathematics and Combinatorics
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In: Annals of combinatorics, Vol. 26, No. 1, 03.2022.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Congruence Normality of Simplicial Hyperplane Arrangements via Oriented Matroids
AU - Cuntz, Michael
AU - Elia, Sophia
AU - Labbé, Jean Philippe
N1 - Funding Information: S. Elia: With the support of the Research Training Group 2434 “Facets of Complexity”. J.-P. Labbé: With the support of the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”.
PY - 2022/3
Y1 - 2022/3
N2 - 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.
AB - 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.
KW - Congruence normality and uniformity
KW - Covectors
KW - Poset of regions
KW - Shards
KW - Simplicial hyperplane arrangements
UR - http://www.scopus.com/inward/record.url?scp=85118635423&partnerID=8YFLogxK
U2 - 10.1007/s00026-021-00555-2
DO - 10.1007/s00026-021-00555-2
M3 - Article
AN - SCOPUS:85118635423
VL - 26
JO - Annals of combinatorics
JF - Annals of combinatorics
SN - 0218-0006
IS - 1
ER -