On hyperfactored and recursively factored 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、代数几何和拓扑学的十字路口。这项建议反过来又涉及组合和几何方面的相互作用。我们在这项建议中提出的研究思路有三个方面。其中每一个都与Hoge和Röhrle关于良好排列的增删定理有关。与著名的Terao关于自由排列的添加-删除定理类似，该定理导致了归纳和递归自由性的更强概念，前述定理提供了归纳因式和递归因式的类似对应。虽然归纳因式分解的概念在一般文献中以及与反射排列有关的情况下已经被研究过，但递归因式分解的概念是全新的；尽管它相当自然，但到目前为止还没有出现在文献中。我们的第一个项目是启动对这类新的排列的研究。一个目的是类比JAMBU和PARIS关于归纳因式的结果，即递归因式需要递归自由。递归的自由性是出了名的难以捉摸，很可能递归的事实证明是相同的。我们希望构造不是归纳因式的递归因式分解安排的例子。自然地，这类新类与规范结构(如乘积和本地化)的兼容性也是自然的。第二，我们打算重温同样由JAMBU和PARIS引入的超因式因式安排族。我们在这里的目的是类似于前面提到的加减定理，以便为这类非常特殊的真实安排提供良好的安排。在他们的论文中，JAMBU和PARIS证明了实归纳因式分解的排列是超因式分解的，并提出了相反的问题。我们希望计算方法将导致不是归纳因式分解的超因式因式排列的例子，从而证明这两类排列实际上是不同的。我们讨论了一些自然的排列类，它们为这种例子提供了检验的基础。仔细阅读JAMBU和PARIS的文章中的论点，人们注意到涉及排列的超因式分解的几个证明并没有充分利用这样一个事实的力量，即分割是因式分解，并且只需要较弱的性质，连同基室的选择，在排列的区域的偏序集和由基础划分纯粹组合定义的偏序集之间诱导双射。这导致了一种潜在的弱于超因式安排的概念。我们的第三个研究方向旨在调查这类新的安排。特别地，这里我们还旨在证明一个增删定理。

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.