Arrangements with symmetries

超平面的排列在数学的许多领域中起着核心作用。这个建议有两个主要目标：理解超平面的特殊单纯排列的结构，包括分类，并应用这些结果构造一个反例来反驳Terao的长期猜想。最近，Heckenberger和申请者在一系列论文中实现了对由某种完整性条件定义的单纯排列的一个大子类的分类，即所谓的结晶学排列。这是Nichols代数理论的一个基本结果，但同时很明显，它将对离散几何产生更大的影响。这个建议的第一个目标是使用类似的技术来证明更大类有限反射群胚的界，将它们的分类归结为一个有限问题。使用进一步的对称性的计数将提供第三维的分类，该分类随后将被扩展到任意维。最近在反驳Terao猜想方面的突破是Hoge和申请者对密切相关的猜想的反例，即自由排列在特征零递归自由。这个例子的构造是基于一个简单的安排。利用第一部分的结果，我们将产生一个不是归纳自由的自由排列的数据库。新出现的模式应该揭示无穷级数，并最终允许对潜在反例集进行完整的分类。结合表示理论、计算枚举和传统的几何方法是解决这些问题的一种方法，以前从未尝试过，而且非常有前途。

Arrangements of hyperplanes play a central role in many areas of mathematics. This proposal has two major goals: to understand the structure of the special simplicial arrangements of hyperplanes, including classifications, and to apply the results to construct a counterexample to the longstanding conjecture of Terao.The classification of a large subclass of the class of simplicial arrangements defined by a certain integrality condition, the so-called crystallographic arrangements, was achieved recently by Heckenberger and the applicant in a series of papers. This was a fundamental result for the theory of Nichols algebras, but meanwhile it is clear that it will have an even greater impact on discrete geometry. The first goal of this proposal is to use similar techniques to prove bounds for the larger class of finite reflection groupoids reducing their classification to a finite problem. An enumeration using further symmetries will provide a classification in dimension three which will be extended to arbitrary dimensions subsequently.The most recent breakthrough towards disproving Terao's conjecture is the counterexample by Hoge and the applicant to the closely related conjecture that free arrangements are recursively free in characteristic zero. The construction of this example is based on a simplicial arrangement. Using the results of the first part, we will produce a database of free arrangements which are not inductively free. The emerging patterns should expose infinite series and eventually allow to classify the set of potential counterexamples completely.Combining representation theory, computational enumerations, and traditional methods of geometry is an approach to these problems which has not been attempted before and is very promising.