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Supersolvable reflection arrangements


Authors: Torsten Hoge and Gerhard Röhrle
Journal: Proc. Amer. Math. Soc. 142 (2014), 3787-3799
MSC (2010): Primary 20F55, 52C35, 14N20; Secondary 13N15
Published electronically: August 4, 2014
MathSciNet review: 3251720
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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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Additional Information

Torsten Hoge
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Address at time of publication: Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
Email: hoge@math.uni-hannover.de

Gerhard Röhrle
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Email: gerhard.roehrle@rub.de

DOI: https://doi.org/10.1090/S0002-9939-2014-12144-7
Keywords: Complex reflection groups, reflection arrangements, free arrangements, supersolvable arrangements
Received by editor(s): September 14, 2012
Received by editor(s) in revised form: January 7, 2013
Published electronically: August 4, 2014
Communicated by: Lev Borisov
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.