On Ziegler Extensions of Multiarrangements

• Applicant: Professor Dr. Gerhard Röhrle
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539858198

The significance of combinatorial aspects of hyperplane arrangements underlies much of June Huh's work on the log-concavity of the coefficients of the characteristic polynomial of a hyperplane arrangement or more generally a matroid (the so-called Heron-Rota-Welsh conjecture). The interplay between algebraic and combinatorial structures of hyperplane arrangements has been a driving force in the study of the field for many decades. In turn the classes of free arrangements and free multi-arrangements play pivotal roles in the theories of hyperplane arrangements and multi-arrangements, respectively. Ziegler showed in his seminal work that a free arrangement A gives rise to a canonical free multi-arrangement on any restriction of A with exponents only depending on the exponents of A. In this proposal we refer to this construction as a Ziegler restriction. In 2010, Yoshinaga asked for a reverse procedure, which he coins as extension: given a free multi-arrangement is this the Ziegler restriction of a free arrangement A? In general, this is not the case. In this proposal we call this a Ziegler extension. The ultimate goal of Yoshinaga's work in this context is a complete description of all free intermediate arrangements between the extended Shi and extended Catalan arrangements for simply laced underlying root systems. There is a long history and continued interest in the literature regarding questions of freeness of the families of extended Shi and extended Catalan arrangements. Among other aspects, this proposal is a contribution to this theory. The goal of this proposal is to continue Yoshinaga's investigation about extensions of free multi-arrangements. Firstly, we aim to answer some of the questions and conjectures raised in Yoshinaga's paper. Secondly, we intend to extend the classification from this work to all intermediate free arrangements between extended Shi and extended Catalan arrangements for arbitrary Coxeter types, i.e. to also include non-simply laced ones. Thirdly, we plan to investigate extensions of free multi-arrangements over finite fields, a topic which has not been studied in the literature as of yet. Finally, in a fourth research stand we aim at studying free extensions of various natural free complex multireflection arrangements which were defined and studied in earlier work of the PI and his coauthors. These extensions then might be thought of as complex (non-real) analogues of extended Shi and extended Catalan arrangements. This last is an entirely novel theme. The topic of extensions of free multi-arrangements in general in the sense of Yoshinaga is largely uncharted territory; this seems ideal for a PhD project.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2458: Combinatorial Synergies