Arrangements of complex reflection groups: Geometry and combinatorics

复反射群的排列：几何与组合学

• 批准号: 239469709
• 负责人: Professor Dr. Michael Cuntz
• 金额: --
• 依托单位: Lehrstuhl Algebra/Zahlentheorie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 2017-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/239469709?language=en
• 关键词: Arrangements; complex; reflection; groups; Geometry

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.

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允许在解决关于超平面安排的组合和几何性质的深刻和长期的问题方面取得重大进展。在最近的一份联合文件与霍格，使用计算机为基础的证明，我们能够证实一个猜想Orlik和寺尾从1992年的问题自由的限制反射安排复杂的反射群。这些限制是理解潜在安排的关键。我们打算进一步调查的组合和几何性质的反射安排在这个建议的问题。我们有三个核心的研究线索，我们的目标是追求。这只是最近，Bessis建立的K（π，1）-属性的所有反射安排已被证明自20世纪80年代末。Orlik和Terao在20世纪90年代证实，这一属性也适用于所有限制。对于Coxeter安排，这是自1972年以来由于Deligne的开创性工作而已知的。我们的第一个目标是证明这个猜想，这将导致更好地理解反射安排的拓扑性质。自由度是斋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。有较强的归纳自由概念和较弱的递归自由概念。虽然我们知道不是每一个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，每一个自由排列都是递归自由的。在最近的联合工作与霍格，我们确定了类归纳自由反射安排。其次，我们要研究反射安排的自由度的各种概念及其相关的限制。具体地说，我们想证实Orlik和Terao关于反射安排的这一猜想。 在我们的第三个项目中，我们超越反射安排，考虑更一般的单纯安排。借助Cuntz的单纯安排数据库，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归纳自由的剩余单纯安排。 开展这些项目将提高我们对复杂反射群及其排列，以及一般超平面排列的理解。