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.