Inductive freeness of Ziegler's canonical multiplicity

几十年来，超平面排列理论一直是数学中的推动力。它自然地处于代数、组合学、代数几何和拓扑学的十字路口，与反射排列有关的问题，包括一个基础反射群的反射超平面，首先出现在对称群上，然后推广到所有的Coxeter群，最后包括整个复反射群。随后，在复杂反射排列的更广泛的限制类别内进一步研究了所讨论的性质。这个现象的一个主要例子是关于复杂反射排列中超平面并补的拓扑性的问题。在1989年的开创性工作中，Ziegler证明了对具有自然多重性的自由超平面排列的任何超平面的限制是自由多重排列。2018年，Hoge和PI研究了这些正则自由多重排列的诱导自由性的强自由性，并对所有反射排列进行了分类，这些反射排列的诱导Ziegler重数在任何限制下都是诱导自由的。这项研究的主要目的是在完全一般的情况下证明Ziegler定理的精确模拟：如果A是诱导自由的，那么A对A的每个超平面的Ziegler限制也是诱导自由的。我们要证明的第二个相关猜想如下如果我们不要求基础排列本身是归纳自由的：如果A和关于某个超平面H的删除A‘都是自由的，并且A到H本身的限制是归纳自由的，那么A到H的限制的Ziegler多重数也是归纳自由的。

The theory of hyperplane arrangements has been a driving force in mathematics for many decades. It naturally lies at the crossroads of algebra, combinatorics, algebraic geometry, and topology.Frequently, questions relating to reflection arrangements, consisting of the reflecting hyperplanes of an underlying reflection group, arose first for symmetric groups, then were extended to all Coxeter groups and finally embraced the entire class of complex reflection groups. Subsequently, the properties in question were investigated further within the wider class of restrictions of complex reflection arrangements. A prime example of this phenomenon is the question about the topological nature of the complement of the union of the hyperplanes in a complex reflection arrangement.In his seminal work from 1989, Ziegler showed that the restriction to any hyperplane of a free hyperplane arrangement endowed with the natural multiplicity is then a free multiarrangement. In 2018 Hoge and the PI investigated the stronger freeness property of inductive freeness for these canonical free multiarrangements and classified all reflection arrangements with the property that the induced Ziegler multiplicity is inductively free on any restriction.The principal goal of this research proposal is to prove the precise analogue of Ziegler's theorem for this stronger property of inductive freeness in complete generality: If A is inductively free, then so is the Ziegler restriction of A to every hyperplane of A.A second related conjecture we aim to show is the following, where we do not require that the underlying arrangement itself is inductively free: if A and the deletion A' with respect to some hyperplane H are both free and the restriction of A to H itself is inductively free, then so is the Ziegler multiplicity on the restriction of A to H.