On inductively free reflection arrangements

Abstract

Abstract Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let 𝒜 = 𝒜(W) be the associated hyperplane arrangement of W. Terao [J. Fac. Sci. Univ. Tokyo 27 (1980), 293–320] has shown that each such reflection arrangement 𝒜 is free. There is the stronger notion of an inductively free arrangement. In 1992, Orlik and Terao [Arrangements of hyperplanes, Springer-Verlag, Berlin 1992, Conjecture 6.91] conjectured that each reflection arrangement is inductively free. It has been known for quite some time that the braid arrangement as well as the Coxeter arrangements of type B ℓ and type D ℓ are inductively free. Barakat and Cuntz [Adv. Math. 229 (2012), 691–709] completed this list only recently by showing that every Coxeter arrangement is inductively free. Nevertheless, Orlik and Terao's conjecture is false in general. In a paper which will appear in Tôhoku Math. J., we already gave two counterexamples to this conjecture among the exceptional complex reflection groups. In this paper we classify all inductively free reflection arrangements. In addition, we show that the notions of inductive freeness and that of hereditary inductive freeness coincide for reflection arrangements. As a consequence of our classification, we get an easy, purely combinatorial characterization of inductively free reflection arrangements 𝒜 in terms of exponents of the restrictions to any hyperplane of 𝒜.