Recursively free reflection arrangements

TL;DR

This paper classifies all recursively free reflection arrangements of finite complex reflection groups and provides new proofs for non-inductive freeness, confirming a conjecture about divisionally free arrangements.

Contribution

It offers a complete classification of recursively free reflection arrangements and introduces a new proof for the non-inductive freeness of certain arrangements, advancing understanding of their structure.

Findings

Classified all recursively free reflection arrangements.

Proved non-inductive freeness of arrangements involving G_{31}.

Confirmed a conjecture on divisionally free arrangements.

Abstract

Let $\mathcal{A} = \mathcal{A}(W)$ be the reflection arrangement of the finite complex reflection group $W$. By Terao's famous theorem, the arrangement $\mathcal{A}$ is free. In this paper we classify all reflection arrangements which belong to the smaller class of recursively free arrangements. Moreover for the case that $W$ admits an irreducible factor isomorphic to $G_{31}$ we obtain a new (computer free) proof for the non-inductive freeness of $\mathcal{A}(W)$. Since our classification implies the non-recursive freeness of the reflection arrangement $\mathcal{A}(G_{31})$, we can prove a conjecture by Abe about the new class of divisionally free arrangements which he recently introduced.