Details
| Original language | English |
|---|---|
| Pages (from-to) | 32-73 |
| Number of pages | 42 |
| Journal | Advances in applied mathematics |
| Volume | 107 |
| Early online date | 5 Mar 2019 |
| Publication status | Published - Jun 2019 |
Abstract
Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane arrangements with particularly nice geometric, algebraic, topological, and combinatorial properties are the supersolvable arrangements. In this paper we give a complete classification of supersolvable simplicial arrangements (in all ranks). For each fixed rank, our classification already includes almost all known simplicial arrangements. Surprisingly, for irreducible simplicial arrangements of rank greater than three, our result shows that supersolvability imposes a strong integrality property; such an arrangement is called crystallographic. Furthermore we introduce Coxeter graphs for simplicial arrangements which serve as our main tool of investigation.
Keywords
- Coxeter graph, Hyperplane arrangements, Reflection arrangements, Root system, Simplicial arrangements, Supersolvable arrangements
ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics
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In: Advances in applied mathematics, Vol. 107, 06.2019, p. 32-73.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Supersolvable simplicial arrangements
AU - Cuntz, Michael
AU - Mücksch, Paul
PY - 2019/6
Y1 - 2019/6
N2 - Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane arrangements with particularly nice geometric, algebraic, topological, and combinatorial properties are the supersolvable arrangements. In this paper we give a complete classification of supersolvable simplicial arrangements (in all ranks). For each fixed rank, our classification already includes almost all known simplicial arrangements. Surprisingly, for irreducible simplicial arrangements of rank greater than three, our result shows that supersolvability imposes a strong integrality property; such an arrangement is called crystallographic. Furthermore we introduce Coxeter graphs for simplicial arrangements which serve as our main tool of investigation.
AB - Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane arrangements with particularly nice geometric, algebraic, topological, and combinatorial properties are the supersolvable arrangements. In this paper we give a complete classification of supersolvable simplicial arrangements (in all ranks). For each fixed rank, our classification already includes almost all known simplicial arrangements. Surprisingly, for irreducible simplicial arrangements of rank greater than three, our result shows that supersolvability imposes a strong integrality property; such an arrangement is called crystallographic. Furthermore we introduce Coxeter graphs for simplicial arrangements which serve as our main tool of investigation.
KW - Coxeter graph
KW - Hyperplane arrangements
KW - Reflection arrangements
KW - Root system
KW - Simplicial arrangements
KW - Supersolvable arrangements
UR - http://www.scopus.com/inward/record.url?scp=85062290556&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1712.01605
DO - 10.48550/arXiv.1712.01605
M3 - Article
AN - SCOPUS:85062290556
VL - 107
SP - 32
EP - 73
JO - Advances in applied mathematics
JF - Advances in applied mathematics
SN - 0196-8858
ER -