Arrangements of complex reflection groups: Geometry and combinatorics

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The theory of hyperplane arrangements has close links with many parts of mathematics. The use of modern computer algebra allows for significant advances in resolving deep and longstanding conjectures concerning the combinatorial and geometric nature of hyperplane arrangements. In a recent joint paper with Hoge, using a computer based proof, we were able to confirm a conjecture by Orlik and Terao from 1992 on the question of freeness of restrictions of reflection arrangements of complex reflection groups. These restrictions are key to an understanding of the underlying arrangement. We intend to further investigate questions of combinatorial and geometric properties of reflection arrangements in this proposal. We have three core research strands we aim to pursue.It was only very recently that Bessis established the K(π, 1)-property for all reflection arrangements which had been conjectured since the late 1980s. Orlik and Terao conjectured in the 1990s that this property also holds for all restrictions. For Coxeter arrangements, this had been known since 1972 due to seminal work of Deligne. Our first aim is to prove this conjecture which will lead to a better understanding of the topological nature of reflection arrangements. Freeness is a fundamental notion due to Saito and plays a pivotal role in understanding general hyperplane arrangements. There is the stronger notion of inductive freeness and the weaker one of recursive freeness. While it is known that not every free arrangement is inductively free, it is still an open conjecture by Orlik and Terao from 1992 that every free arrangement is already recursively free. In recent joint work with Hoge, we determined the class of inductively free reflection arrangements. Secondly, we want to investigate these various notions of freeness for reflection arrangements and their associated restrictions. Specifically, we want to confirm this conjecture by Orlik and Terao for reflection arrangements. In our third project, we look beyond reflection arrangements and consider more generally simplicial arrangements. With the aid of Cuntz's database of simplicial arrangements, our goal is to determine combinatorial invariants which will allow us to distinguish the free and inductively free from the remaining simplicial arrangements. Carrying out these projects will enhance our understanding of complex reflection groups andtheir arrangements, as well as hyperplane arrangements in general.