On hyperfactored and recursively factored arrangements

超平面排列的理论在过去的几十年里一直是数学的推动力。它自然位于代数、组合学、代数几何和拓扑学的交叉点。这一建议反过来又涉及到组合和几何方面的相互作用。我们在这一建议中提出的研究思路有三个方面。他们每个人都涉及到添加删除定理的尼斯安排由于霍格和Röhrle。类似于著名的添加删除定理自由安排由于寺尾，导致更强的概念归纳和递归的自由，上述定理提供了类似的副本归纳factoredness和递归factoredness。虽然归纳的因式的概念已经在一般文献中研究，以及与反射安排，递归因式的概念是全新的，虽然相当自然，它还没有出现在文献中的日期。一个目标是一个类似的结果，由于Jambu和巴黎的归纳factoredness，即递归factoredness需要递归的自由。递归的自由性是出了名的难以捉摸，而递归的因式性很可能也是如此。我们的希望是构造递归因式分解的安排，不归纳factored.Natural的例子也是兼容性的结果，在这个新的类与规范的建设，如产品和localizations。其次，我们打算重温家庭的hyperfactored安排，也介绍了Jambu和巴黎。我们的目标是在这里是一个类似的上述添加删除定理的好安排，这一类非常特殊的真实的安排。在他们的论文中，Jambu和巴黎证明了真实的归纳因子化的安排是超因子化的，并提出了关于匡威的问题。我们希望计算方法将导致不归纳因子化的超因子化安排的例子，证明这两类实际上是不同的。我们讨论了一些自然类的安排，这些安排为这样的例子提供了一个试验场。仔细阅读论文中的论点Jambu和巴黎的文章中，人们注意到，涉及到一个安排的超因式分解的几个证明并没有利用划分是一个因式分解这一事实的全部力量，而只需要一个较弱的性质，即该划分连同一个基腔的选择，诱导了该安排的区域的偏序集和由底层划分纯粹组合地定义的偏序集之间的双射。这导致了一个潜在的比超因子安排更弱的概念。我们的第三个研究链旨在研究这类新的安排。特别地，这里我们还旨在证明一个添加-删除定理。

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, and topology. This proposal in turn is concerned with the interplay of combinatorial and geometric aspects.The research strands we are putting forward in this proposal are threefold. Each of them is related to the Addition-Deletion Theorem for nice arrangements due to Hoge and Röhrle. In analogy to the celebrated Addition-Deletion Theorem for free arrangements due to Terao which leads to the stronger notions of inductive and recursive freeness, the aforementioned theorem affords the analogous counterparts of inductive factoredness and recursive factoredness. While the concept of inductive factoredness has been studied in the literature in general as well as in connection with reflection arrangements, the notion of recursive factoredness is entirely new; though rather natural, it has not appeared in the literature to date.Our first project is to initialize a study of this new class of arrangements. One aim is an analogue of a result due to Jambu and Paris for inductive factoredness, namely that recursive factoredness entails recursive freeness. Recursive freeness is notoriously elusive, and quite likely recursive factoredness turns out to be the same. Our hope is to construct examples of recursively factored arrangements that are not inductively factored.Natural are also results about compatibility within this new class with canonical constructions such as products and localizations.Secondly, we intend to revisit the family of hyperfactored arrangements, introduced also by Jambu and Paris. Our aim here is an analogue of the aforementioned Addition-Deletion Theorem for nice arrangements for this very special class of real arrangements. In their paper, Jambu and Paris showed that real inductively factored arrangements are hyperfactored and raised the question about the converse.We hope that a computational approach will lead to examples of hyperfactored arrangements that are not inductively factored, proving that these two classes actually differ.We discuss a number of natural classes of arrangements which provide a testing ground for such examples.Carefully perusing the arguments in the paper of Jambu and Paris, one observes that several of the proofs involving a hyperfactorization of an arrangement do not utilize the full force of the fact that the partition is a factorization and only require the weaker property that the partition, together with a choice of a base chamber, induces a bijection between the poset of regions of the arrangement and a poset defined purely combinatorially by the underlying partition. This leads to a potentially weaker notion than that of a hyperfactored arrangement. Our third research strand aims to investigate this new class of arrangements. In particular, here we also aim to prove an Addition-Deletion Theorem.