Supersolvable 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 [Sta72]. Jambu and Terao [JT84, Thm. 4.2] showed that every supersolvable arrangement is inductively free, a notion due to Terao, [Ter80]. 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 [HR12], 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.