On inductively free reflection arrangements

TL;DR

This paper classifies all inductively free reflection arrangements, revealing that the concepts of inductive and hereditary inductive freeness coincide for these arrangements, and provides a combinatorial characterization based on exponents.

Contribution

It offers a complete classification of inductively free reflection arrangements and establishes the equivalence of inductive and hereditary inductive freeness for these arrangements.

Findings

Classification of all inductively free reflection arrangements

Proof that inductive and hereditary inductive freeness coincide for these arrangements

A combinatorial criterion based on exponents for inductive freeness

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 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 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 and type D are inductively free. Barakat and Cuntz 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 recent paper, 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.