On hyperfactored and recursively factored arrangements

关于超分解和递归分解安排

• 批准号: 508852336
• 负责人: Professor Dr. Gerhard Röhrle
• 金额: --
• 依托单位: Lehrstuhl Algebra/Zahlentheorie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/508852336?language=en
• 关键词: hyperfactored; recursively; factored; arrangements

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.

几十年来，超平面排列理论一直是数学的推动力。它自然地处于代数、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。我们在这项建议中提出的研究方向有三个方面。它们中的每一个都与Hoge和Röhrle提出的关于良好排列的加减定理有关。根据Terao，著名的自由排列的加-删定理引出了归纳自由和递归自由的更强概念，与此类似，上述定理提供了归纳可因性和递归可因性的类似对应。虽然归纳可因性的概念已经在一般文献中以及与反射排列相关的文献中进行了研究，但递归可因性的概念是全新的；虽然它很自然，但迄今为止还没有在文献中出现过。我们的第一个项目是对这类新的排列进行初步研究。一个目的是类比Jambu和Paris对归纳可因式性的结果，即递归可因式性需要递归自由。众所周知，递归自由是难以捉摸的，而递归因子性很可能也是如此。我们的希望是构建递归因子排列的例子，而不是归纳因子排列。关于这个新类与规范结构（如产品和本地化）的兼容性的结果也很自然。第二，我们打算重新审议Jambu和Paris也提出的超因素安排系列。我们在这里的目的是一个类似于前面提到的加法-删除定理的很好的排列对于这类非常特殊的实排列。在他们的论文中，Jambu和Paris证明了真实的归纳因子排列是超因子的，并提出了相反的问题。我们希望通过一种计算方法可以得到一些不被归纳分解的超因子排列的例子，从而证明这两类实际上是不同的。我们讨论了一些自然类别的排列，为这些例子提供了一个试验场。仔细审阅本文的论点即和巴黎,一个观察到的几个证明涉及的hyperfactorization安排不利用这一事实的全部力量的分区是分解,只需要分区的较弱的房地产,一起选择一个基础室,诱发一个双射偏序集之间的区域的安排和偏序集定义纯粹的组合由底层分区。这可能导致一种比超因子排列更弱的概念。我们的第三个研究方向旨在调查这类新的排列。特别地，这里我们还旨在证明一个加减定理。