Nice reflection arrangements

TL;DR

This paper classifies all nice and inductively factored reflection arrangements, revealing that only certain monomial groups produce such arrangements, and connects these classifications with supersolvable and inductively free arrangements.

Contribution

It provides a complete classification of nice and inductively factored reflection arrangements, linking these classes to supersolvable and inductively free arrangements.

Findings

Only monomial groups G(r,r,3) for r ≥ 3 yield nice reflection arrangements.

Inductively factored reflection arrangements coincide with supersolvable arrangements.

Classifications extend to hereditarily factored and hereditarily inductively factored arrangements.

Abstract

The aim of this note is a classification of all nice and all inductively factored reflection arrangements. It turns out that apart from the supersolvable instances only the monomial groups $G(r,r,3)$ for $r \ge 3$ give rise to nice reflection arrangements. As a consequence of this and of the classification of all inductively free reflection arrangements from our earlier work, we deduce that the class of all inductively factored reflection arrangements coincides with the supersolvable reflection arrangements. Moreover, we extend these classifications to hereditarily factored and hereditarily inductively factored reflection arrangements.