## Tohoku Mathematical Journal

### Reflection arrangements are hereditarily free

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 12 September 2013

https://projecteuclid.org/euclid.tmj/1378991017

Digital Object Identifier
doi:10.2748/tmj/1378991017

Mathematical Reviews number (MathSciNet)
MR3102536

Zentralblatt MATH identifier
1287.51006

#### Citation

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

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