Inductive freeness of Ziegler's canonical multiplicity

超平面排列的理论在数学中已经有几十年的推动力。它自然地位于代数、组合学、代数几何和拓扑学的十字路口。通常，与反射排列有关的问题，由底层反射群的反射超平面组成，首先出现在对称群中，然后扩展到所有考克斯特群，最后包括整个复反射群类。随后，在复杂反射安排的更广泛的限制范围内进一步研究了所讨论的性质。这种现象的一个主要例子是关于拓扑性质的补充工会的超平面在一个复杂的反射安排的问题。在他的开创性工作从1989年，齐格勒表明，限制任何超平面的自由超平面安排赋予自然的多样性，然后是一个自由multiarrangement。在2018年，Hoge和PI研究了这些正则自由多重排列的归纳自由性的更强自由性，并将所有反射排列分类为具有诱导齐格勒多重性在任何限制下都是归纳自由的性质。这项研究提案的主要目标是证明齐格勒定理的精确模拟，以完全一般性地证明归纳自由性的这种更强性质：如果A是归纳自由的，那么A对A的每个超平面的齐格勒限制也是如此。我们旨在证明的第二个相关猜想如下，其中我们不要求底层安排本身是归纳自由的：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的限制上的齐格勒重数也是如此。

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.