On supersolvable reflection arrangements

TL;DR

This paper classifies supersolvable reflection arrangements, a special subclass of free arrangements, and characterizes irreducible cases via modular elements in their intersection lattice.

Contribution

It provides a complete classification of supersolvable reflection arrangements and characterizes irreducible ones using modular elements of rank 2.

Findings

All supersolvable reflection arrangements are classified.

Irreducible arrangements are characterized by modular elements of rank 2.

The classification extends understanding of the structure of reflection arrangements.

Abstract

Let A = (A,V) be a complex hyperplane arrangement and let L(A) denote its intersection lattice. The arrangement A is called supersolvable, provided its lattice L(A) is supersolvable, a notion due to Stanley. Jambu and Terao showed that every supersolvable arrangement is inductively free, a notion due to Terao. So this is a natural subclass of this particular class of free arrangements. Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let A = (A(W), V) be the associated hyperplane arrangement of W. In a recent paper, we determined all inductively free reflection arrangements. The aim of this note is to classify all supersolvable reflection arrangements. Moreover, we characterize the irreducible arrangements in this class by the presence of modular elements of rank 2 in their intersection lattice.