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使得在解决有关超平面排列的组合和几何性质的深层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年关于复杂反射群的反射排列的限制的自由性问题的一个猜想。这些限制是理解基本安排的关键。在这个提议中，我们打算进一步研究反射排列的组合和几何性质问题。我们有三个核心的研究方向。直到最近，Bessis才为所有的反射排列建立了K(π，1)-性质，这一性质自20世纪80年代末以来一直被猜测。Orlik和Terao在20世纪90年代推测，这一财产也适用于所有限制。对于Coxeter的安排，由于Deligne的开创性工作，这一点自1972年以来就已经知道了。我们的第一个目的是证明这一猜想，这将导致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。归纳自由性的概念较强，递归自由性的概念较弱。虽然众所周知，并不是每一种自由排列都是归纳自由的，但Orlik和Terao自1992年以来一直在猜测，每一种自由排列都已经是递归自由的。在最近与Hoge的合作中，我们确定了感应自由反射排列的类。其次，我们想要研究反射安排的自由性的这些不同的概念及其相关限制。具体地说，我们想要证实Orlik和Terao关于反射排列的这个猜想。在我们的第三个项目中，我们不仅着眼于反射安排，而且考虑更一般的简单安排。借助于Cuntz的单纯排列数据库，我们的目标是确定组合不变量，这将使我们能够区分自由和诱导自由与剩余的单纯排列。开展这些项目将增进我们对复杂反射群及其排列以及一般超平面排列的了解。

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.