Supersolvable restrictions 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. For X in L(A), it is known that the restriction AX is supersolvable provided A is.Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let A=(A(W),V) be its associated hyperplane arrangement. Recently, the last two authors classified all supersolvable reflection arrangements. Extending this work, the aim of this note is to determine all supersolvable restrictions of reflection arrangements. It turns out that apart from the obvious restrictions of supersolvable reflection arrangements there are only a few additional instances. Moreover, in a recent paper we classified all inductively free restrictions A(W)X of reflection arrangements A(W). Since every supersolvable arrangement is inductively free, the supersolvable restrictions A(W)X of reflection arrangements A(W) form a natural subclass of the class of inductively free restrictions A(W)X.Finally, we characterize the irreducible supersolvable restrictions of reflection arrangements by the presence of modular elements of dimension 1 in their intersection lattice. This in turn leads to the surprising fact that reflection arrangements as well as their restrictions are supersolvable if and only if they are strictly linearly fibered.