On the Cohomology of complements of complex reflection arrangements

几十年来，超平面排列理论一直是数学中的推动力。它自然位于代数、组合学、代数几何、表示论和拓扑学的十字路口。这项提议反过来又是这些主题事项的核心。多年来，人们发现了排列A的补M(A)的拓扑、A的导子模D(A)的自由性和A的交格L(A)的组合学之间有着深刻而显著的联系。对复杂超平面排列的补的研究有很长的丰富的历史。直到今天，它仍然是一个非常活跃的研究领域。通常，超平面排列中与反射排列A有关的问题首先出现在对称群上，然后扩展到剩余的有限Coxeter群，最后包含整个复反射群。这一现象的一个最好的例子是关于反射排列A中超平面并的补M(A)的拓扑性质的问题，这个问题是由Bessis在2015年经过50多年的发展而解决的。在这个研究方案中，我们穿越了关于复杂反射排列的补语的上同调问题的类似路线。在最近与DouGlass和Pfeiffer的合作中，我们改进了Brieskorn对Coxeter排列A(W)的补语的上同调的研究。作为我们研究的结果，我们从2005年得到了关于有限Coxeter群W的Orlik-所罗门代数的W-不变量的结构的一个猜想。这一猜想的目的之一是研究对于更一般的复反射群的情况，类似于费尔德和Veselov的猜想。1986年，Lehrer和所罗门将W在W的Orlik-所罗门代数上的表示描述为当W是对称群时，由W中元素的中心化子的线性特征所导出的表示之和。他们猜想一般的Coxeter群都有这样的分解。实际上，在与DouGlass和Pfeiffer的一系列联合论文中，对于对称群和所有到8阶的不可约Coxeter群W，我们建立了这个猜想的精化版本。在我们的第二个研究线索中，我们的目标是在更一般的复反射群中研究与Lehrer-所罗门猜想类似的情形。

The theory of hyperplane arrangements has been a driving force in mathematics over many decades. It naturally lies at the crossroads of algebra, combinatorics, algebraic geometry, representation theory and topology. This proposal in turn lies at the very heart of these subject matters. Deep and remarkable connections have been discovered over the years between the topology of the complement M(A) of an arrangement A, the freeness of the module of derivations D(A) of A and the combinatorics of the intersection lattice L(A) of A consisting of the subspaces arising as intersections of hyperplanes from A.The study of the complement of complex hyperplane arrangements has a long and remarkably rich history. To this day it is a very active field of research. Frequently, questions in hyperplane arrangements relating to reflection arrangements A, where A consists of the reflecting hyperplanes of an underlying reflection group, arose first for symmetric groups, then were extended to the remaining finite Coxeter groups and finally embraced the entire class of complex reflection groups. A prime example of this phenomenon is the question about the topological nature of the complement M(A) of the union of the hyperplanes in the reflection arrangement A which was settled after a development streching more than 50 years by Bessis in 2015.In this research proposal we traverse a similar route concerning questions on the cohomology of the complement of a complex reflection arrangement. In recent joint work with Douglass and Pfeiffer we refined Brieskorn's study of the cohomology of the complement of a Coxeter arrangement A(W). As a result of our study we derive a conjecture due to Felder and Veselov from 2005 on the structure of the W-invariants of the Orlik-Solomon algebra of a finite Coxeter group W.One of the aims of this proposal is to investigate an analogue of the conjecture of Felder and Veselov for the more general case of complex reflection groups.In 1986 Lehrer and Solomon have described the representation of W on the Orlik-Solomon algebra of W as a sum of representations induced from linear characters of centralizers of elements in W when W is a symmetric group. They have conjectured that there is such a decomposition for general Coxeter groups. Indeed a refined version of this conjecture was established in a series of joint papers with Douglass and Pfeiffer for symmetric groups and all irreducible Coxeter groups W up to rank 8. In our second research strand we aim to investigate an analogue of the Lehrer-Solomon Conjecture for the more general class of complex reflection groups.