Supersolvable reflection arrangements

Hoge, Torsten; Röhrle, Gerhard

doi:10.1090/S0002-9939-2014-12144-7

Supersolvable reflection arrangements
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by Torsten Hoge and Gerhard Röhrle PDF

Proc. Amer. Math. Soc. 142 (2014), 3787-3799 Request permission

Abstract:

Let $\mathcal {A} = (\mathcal {A},V)$ be a complex hyperplane arrangement and let $L(\mathcal {A})$ denote its intersection lattice. The arrangement $\mathcal {A}$ is called supersolvable, provided its lattice $L(\mathcal {A})$ is supersolvable, a notion due to Stanley. Jambu and Terao showed that every supersolvable arrangement is inductively free, a notion previously 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 $\mathcal {A} = (\mathcal {A}(W), V)$ be the associated hyperplane arrangement of $W$. In a forthcoming paper by the authors, we determine all inductively free reflection arrangements.

The aim of this paper 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.

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