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The Facial Weak Order on Hyperplane Arrangements

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Abstract

We extend the facial weak order from finite Coxeter groups to central hyperplane arrangements. The facial weak order extends the poset of regions of a hyperplane arrangement to all its faces. We provide four non-trivially equivalent definitions of the facial weak order of a central arrangement: (1) by exploiting the fact that the faces are intervals in the poset of regions, (2) by describing its cover relations, (3) using covectors of the corresponding oriented matroid, and (4) using certain sets of normal vectors closely related to the geometry of the corresponding zonotope. Using these equivalent descriptions, we show that when the poset of regions is a lattice, the facial weak order is a lattice. In the case of simplicial arrangements, we further show that this lattice is semidistributive and give a description of its join-irreducible elements. Finally, we determine the homotopy type of all intervals in the facial weak order.

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Notes

  1. Just like the poset of regions, it is tempting to call this order the poset of faces. However, the facial weak order IS NOT the classical face poset (the poset of faces ordered by inclusion, see Corollary 2.7). We have thus chosen to borrow the name facial weak order from the context of Coxeter groups studied in [8] to the present context of hyperplane arrangements.

  2. This terminology is once again inherited from Coxeter systems, but it should be noted that these roots do not necessarily form root systems.

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Acknowledgements

We thank Nathan Reading and Hugh Thomas for their relevant suggestions on an initial version of this paper. We are also grateful to three anonymous reviewers for useful input on the presentation of the paper.

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Correspondence to Aram Dermenjian.

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AD was supported by a Fonds de recherche du Québec – Nature et Technologies (FRQNT) scholarship. CH was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery grant Geometric and Algebraic Combinatorics of Coxeter Groups. VP was supported by French Agence Nationale de la Recherche (ANR) grants SC3A (15 CE40 0004 01) and CAPPS (17 CE40 0018).

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Dermenjian, A., Hohlweg, C., McConville, T. et al. The Facial Weak Order on Hyperplane Arrangements. Discrete Comput Geom 67, 166–202 (2022). https://doi.org/10.1007/s00454-021-00312-5

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