Supersolvable simplicial arrangements

TL;DR

This paper provides a complete classification of supersolvable simplicial arrangements across all ranks, revealing that higher-rank irreducible arrangements are crystallographic, and introduces Coxeter graphs as a key analytical tool.

Contribution

It offers the first comprehensive classification of supersolvable simplicial arrangements and uncovers their crystallographic nature in higher ranks, using Coxeter graphs for analysis.

Findings

Almost all known simplicial arrangements are included in the classification.

Supersolvability in higher ranks implies arrangements are crystallographic.

Coxeter graphs are introduced as a new tool for studying arrangements.

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.