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2013 Reflection arrangements are hereditarily free
Torsten Hoge, Gerhard Röhrle
Tohoku Math. J. (2) 65(3): 313-319 (2013). DOI: 10.2748/tmj/1378991017

Abstract

Suppose that $W$ is a finite, unitary, reflection group acting on the complex vector space $V$. Let ${\mathcal A} = {\mathcal A}(W)$ be the associated hyperplane arrangement of $W$. Terao has shown that each such reflection arrangement ${\mathcal A}$ is free. Let $L({\mathcal A})$ be the intersection lattice of ${\mathcal A}$. For a subspace $X$ in $L({\mathcal A})$ we have the restricted arrangement ${\mathcal A}^X$ in $X$ by means of restricting hyperplanes from ${\mathcal A}$ to $X$. In 1992, Orlik and Terao conjectured that each such restriction is again free. In this note we settle the outstanding cases confirming the conjecture.

In 1992, Orlik and Terao also conjectured that every reflection arrangement is hereditarily inductively free. In contrast, this stronger conjecture is false however; we give two counterexamples.

Citation

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Torsten Hoge. Gerhard Röhrle. "Reflection arrangements are hereditarily free." Tohoku Math. J. (2) 65 (3) 313 - 319, 2013. https://doi.org/10.2748/tmj/1378991017

Information

Published: 2013
First available in Project Euclid: 12 September 2013

zbMATH: 1287.51006
MathSciNet: MR3102536
Digital Object Identifier: 10.2748/tmj/1378991017

Subjects:
Primary: 20F55
Secondary: 13N15, 14N20, 52B30, 52C35

Rights: Copyright © 2013 Tohoku University

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