Congruence normality for simplicial hyperplane arrangements

The normal fan of a simple zonotope is a simplicial hyperplane arrangement. In rank 3 there is still much to learn about these arrangements. We provide an updated catalogue of the currently known simplicial hyperplane arrangements of rank 3, and compute normals and invariants of the arrangements. Additionally, we determine whether the associated posets of regions possess the combinatorial property of "congruence normality," which hints at potential geometric interpretations. We use methods from oriented matroids that make the computations possible. This refines the structure of the catalogue, breaking it into three separate combinatorial categories. In particular, we show that arrangements stemming from finite Weyl groupoids have congruence normal posets of regions. This is joint work with Michael Cuntz and Jean-Philippe Labbé.