On hyperfactored and recursively factored arrangements

• Applicant: Professor Dr. Gerhard Röhrle
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 508852336

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.

• DFG Programme: Research Grants