Arrangements with symmetries

超平面的排列在数学的许多领域中起着核心作用。这项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，并应用结果构建Terao长期猜想的反例。由一定的完整性条件定义的单纯排列类的一个大子类的分类，所谓的晶体排列，最近由Heckenberger和申请人在一系列论文中实现。这是尼科尔斯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的有限反射广群的界限，减少他们的分类有限的问题。一个枚举使用进一步的对称性将提供一个分类在三维这将被扩展到任意dimensions.The最近的突破对证伪寺尾的猜想是反例霍格和申请人的密切相关的猜想，自由安排递归自由的特征零。这个例子的结构是基于单纯排列的。使用第一部分的结果，我们将产生一个数据库的自由安排，这不是归纳自由。新兴的模式应该揭露无穷级数，并最终允许分类的潜在的counterexamples.Combining表示理论，计算枚举，和传统的几何方法是一种方法，这些问题还没有尝试过，是非常有前途的。

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.