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Congruence Normality of Simplicial Hyperplane Arrangements

Algebra & Discrete Mathematics

Speaker: Sophia Elia, Freie Univ. Berlin
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Location: zoom
Start time: Mon, Jan 10 2022, 2:10PM

Simplicial hyperplane arrangements are the normal fans of simple zonotopes, and in rank 3 there is still much to learn about them. Branko Grünbaum provided the first catalogue of rank 3 simplicial arrangements in 1971 with 3 infinite families and 90 sporadic arrangements. We provide an update to this catalogue 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 has 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.

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