Tohoku Mathematical Journal 2013 September SECOND SERIES VOL. 65, NO. 3

 Tohoku Math. J. 65 (2013), 313-319

Title REFLECTION ARRANGEMENTS ARE HEREDITARILY FREE

Author Torsten Hoge and Gerhard Röhrle

(Received May 25, 2012, revised October 9, 2012)
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.

Mathematics Subject Classification. Primary 20F55; Secondary 52B30, 52C35, 14N20 13N15.

Key words and phrases. Complex reflection groups, Freeness of restrictions of reflection arrangements.