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Arrangements with symmetries

对称排列

基本信息

批准号：
280581905
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：
Research Grants
财政年份：
2015
资助国家：
德国
起止时间：
2014-12-31 至 2018-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/280581905?language=en
关键词：
Arrangements symmetries

项目摘要

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.

超平面的排列在数学的许多领域起着核心作用。这个建议有两个主要目标：理解超平面的特殊简单排列的结构，包括分类，并应用结果来构建一个反例，以证明Terao的长期猜想。最近，Heckenberger和申请人在一系列论文中实现了由一定完整性条件定义的简单排列类的一个大子类的分类，即所谓的晶体排列。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。本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的界，将它们的分类简化为有限问题。使用进一步对称的枚举将提供三维的分类，随后将扩展到任意维度。在反驳Terao猜想方面，最近的突破是Hoge和另一个密切相关的猜想的应用的反例，即自由排列在特征零点是递归自由的。这个例子的构造基于一个简单的安排。利用第一部分的结果，我们将生成一个非归纳自由排列的自由排列数据库。出现的模式应该暴露无限级数，并最终允许对潜在的反例集进行完全分类。将表示理论、计算枚举和传统的几何方法相结合是解决这些问题的一种方法，这是以前没有尝试过的，也是很有前途的。