Supersolvable simplicial arrangements

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, top...

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Bibliographic Details
Published in:Advances in applied mathematics 2019-06, Vol.107, p.32-73
Main Authors: Cuntz, Michael, Mücksch, Paul
Format: Article
Language:English
Subjects:
Coxeter graph
Hyperplane arrangements
Reflection arrangements
Root system
Simplicial arrangements
Supersolvable arrangements
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Summary: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.
ISSN:0196-8858
1090-2074
DOI:10.1016/j.aam.2019.02.008