# 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.