The Facial Weak Order on Hyperplane Arrangements
| Autor: | Vincent Pilaud, Thomas McConville, Christophe Hohlweg, Aram Dermenjian |
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| Přispěvatelé: | Laboratoire de combinatoire et d'informatique mathématique [Montréal] (LaCIM), Université du Québec à Montréal = University of Québec in Montréal (UQAM)-Centre de Recherches Mathématiques [Montréal] (CRM), Université de Montréal (UdeM)-Université de Montréal (UdeM), Massachusetts Institute of Technology (MIT), Centre National de la Recherche Scientifique (CNRS), Laboratoire d'informatique de l'École polytechnique [Palaiseau] (LIX), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), ANR-15-CE40-0004,SC3A,Surfaces, Catégorification et Combinatoire des Algèbres Amassées(2015), ANR-17-CE40-0018,CAPPS,Analyse Combinatoire de Polytopes et de Subdivisions Polyédrales(2017), Centre de Recherches Mathématiques [Montréal] (CRM), Université de Montréal (UdeM)-Université de Montréal (UdeM)-Université du Québec à Montréal = University of Québec in Montréal (UQAM), Pilaud, Vincent, Surfaces, Catégorification et Combinatoire des Algèbres Amassées - - SC3A2015 - ANR-15-CE40-0004 - AAPG2015 - VALID, Analyse Combinatoire de Polytopes et de Subdivisions Polyédrales - - CAPPS2017 - ANR-17-CE40-0018 - AAPG2017 - VALID |
| Rok vydání: | 2021 |
| Předmět: |
Hyperplane arrangements
zonotopes 0102 computer and information sciences Lattice (discrete subgroup) poset of regions 01 natural sciences Theoretical Computer Science Combinatorics [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] FOS: Mathematics 52C35 03G10 Mathematics - Combinatorics Discrete Mathematics and Combinatorics Order (group theory) 0101 mathematics Mathematics Homotopy 010102 general mathematics Coxeter group [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO] Oriented matroid Computational Theory and Mathematics Cover (topology) Hyperplane 010201 computation theory & mathematics Combinatorics (math.CO) Geometry and Topology Partially ordered set |
| Zdroj: | FPSAC 2020-32nd International Conference on Formal Power Series and Algebraic Combinatorics FPSAC 2020-32nd International Conference on Formal Power Series and Algebraic Combinatorics, Jul 2020, online, France |
| ISSN: | 1432-0444 0179-5376 |
| Popis: | 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. 34 pages, 12 figures |
| Databáze: | OpenAIRE |
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