Tohoku Mathematical Journal

Reflection arrangements are hereditarily free

Torsten Hoge and Gerhard Röhrle

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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.

Article information

Tohoku Math. J. (2), Volume 65, Number 3 (2013), 313-319.

First available in Project Euclid: 12 September 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]
Secondary: 52B30 52C35: Arrangements of points, flats, hyperplanes [See also 32S22] 14N20: Configurations and arrangements of linear subspaces 13N15: Derivations

Complex reflection groups Freeness of restrictions of reflection arrangements


Hoge, Torsten; Röhrle, Gerhard. Reflection arrangements are hereditarily free. Tohoku Math. J. (2) 65 (2013), no. 3, 313--319. doi:10.2748/tmj/1378991017.

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