Simpliciality in Arrangements and Matroids
- Prof. Dr. Michael Cuntz (Leibniz Universität Hannover)
- Prof. Dr. Lukas Kühne (Universität Bielefeld)
- Prof. Dr. Raman Sanyal (Goethe-Universität Frankfurt)
A hyperplane arrangement is a finite collection of linear hyperplanes in some finite dimensional vector space. A real arrangement is simplicial if all its regions are simplicial cones. Simplicial arrangements are central geometric structures underlying many important theories in algebra, geometry, and topology with prominent examples arising from finite reflection groups. From the geometric perspective, simpliciality imposes strong restrictions and it is widely believed that simplicial arrangements are rare. This fuels the question for a classification in terms of geometric and algebraic combinatorics, akin to the theory of finite real reflection groups. From the combinatorial perspective, simpliciality only depends on the matroid underlying the arrangement. This prompts the definition of simplicial (oriented) matroids. In contrast to the geometric side, computer experiments suggest an abundance of simplicial matroids! This project initiates a coherent study of simpliciality in arrangements and matroids with an emphasis on algebra, combinatorics, and geometry. The three main directions of research, Generation and Realization, Algebra and Convexity, and Matroidal and Simplicial Combinatorics are pursued in parallel. Three concrete subprojects interconnect the research directions and optimally utilize the expertise of the project members at the locations Bielefeld, Frankfurt, and Hannover.
The Higher Rank subproject (PI Cuntz/Kühne) focusses on the systematic generation of simplicial (oriented) matroids and their realization as geometric arrangements. Whereas a conjecturally complete catalog of simplicial 3-arrangements is available, considerably less is known in higher ranks. Proven concepts such as wiring diagrams and finite field techniques are combined with more sophisticated ideas to inductively generate simplicial arrangements and matroids in higher ranks.
The subproject Reflection Structures (PI Sanyal/Cuntz) explores connections between inscribable arrangements, reflection groupoids, and reflection arrangements. Zonotopes with vertices inscribed to a sphere give rise to a subclass of simplicial arrangements, called strongly inscribable arrangements. Restrictions of reflection arrangements are strongly inscribable and it is conjectured that there are no further examples. Inscribable arrangements are naturally equipped with a groupoid that is generated by Euclidean reflections and the goal of this subproject is to develop a coherent algebraic and combinatorial theory of Euclidean reflection groupoids by building on the theory of reflection and Weyl groupoids.
In addition to the vast algebraic theory of matroids, simplicial oriented matroids are also amendable to Stanley-Reisner theory. The subproject Quantized Invariants (PI Kühne/Sanyal) develops enumerative invariants of simplicial matroids that intertwine the matroidal and the simplicial perspective. The greater goal is the construction of algebraic structures that incorporate Chow rings and face rings and that shed new light on classical applications of simplicial arrangements.
