Combinatorial simpliciality of arrangements of hyperplanes

TL;DR

This paper provides a combinatorial characterization of simplicial hyperplane arrangements, establishes bounds in finite fields, explores arrangements with symmetry, and classifies those associated with complex reflection groups.

Contribution

It introduces a new combinatorial criterion for simpliciality, bounds hyperplanes in finite fields, and classifies arrangements linked to complex reflection groups.

Findings

Combinatorial simpliciality coincides with inductive freeness for most complex reflection groups.

Established a sharp upper bound for hyperplanes in projective planes over finite fields.

Enumerated arrangements with specific symmetry groups.

Abstract

We introduce a combinatorial characterization of simpliciality for arrangements of hyperplanes. We then give a sharp upper bound for the number of hyperplanes of such an arrangement in the projective plane over a finite field, and present some series of arrangements related to the known arrangements in characteristic zero. We further enumerate simplicial arrangements with given symmetry groups. Finally, we determine all finite complex reflection groups affording combinatorially simplicial arrangements. It turns out that combinatorial simpliciality coincides with inductive freeness for finite complex reflection groups except for the Shephard-Todd group $G_{31}$.