Arrangements of complex reflection groups: Geometry and combinatorics

超平面排列理论与数学的许多部分有着密切的联系。现代计算机代数的使用在解决关于超平面排列的组合和几何性质的深刻和长期猜想方面取得了重大进展。在最近与Hoge的一篇联合论文中，我们使用基于计算机的证明，证实了Orlik和Terao在1992年提出的关于复杂反射群反射排列的限制自由问题的猜想。这些限制是理解底层安排的关键。我们打算在这个提议中进一步研究反射排列的组合和几何性质问题。我们有三个核心研究方向。直到最近，贝西斯才建立了K(π, 1)-性质，适用于自20世纪80年代末以来一直推测的所有反射排列。Orlik和Terao在20世纪90年代推测，这一性质也适用于所有的限制条件。对于考克斯特的安排，自1972年以来，由于德列涅的开创性工作，人们已经知道了这一点。我们的第一个目标是证明这一猜想，这将有助于更好地理解反射排列的拓扑性质。自由是齐藤提出的一个基本概念，在理解一般超平面排列中起着关键作用。有较强的归纳自由和较弱的递归自由。虽然我们知道并非所有的自由排列都是归纳自由的，但Orlik和Terao在1992年提出的一个公开猜想是，所有的自由排列都是递归自由的。在最近与Hoge的联合工作中，我们确定了归纳自由反射排列的类别。其次，我们想研究反射安排的这些不同的自由概念及其相关的限制。具体来说，我们想要证实Orlik和Terao对反射安排的推测。在我们的第三个项目中，我们超越了反射安排，考虑了更一般的简单安排。借助Cuntz的简单排列数据库，我们的目标是确定组合不变量，这将使我们能够从剩余的简单排列中区分自由和归纳自由。实施这些项目将增强我们对复杂反射群及其排列，以及一般超平面排列的理解。

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.