COMs: Complexes of oriented matroids

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:156

Article

Bandelt, Hans-Juergen¹  Chepoi, Victor^2,3  Knauer, Kolja^2,3 

[1] Univ Hamburg, Fachbereich Math, Bundesstr 55, D-20146 Hamburg, Germany

[2] Aix Marseille Univ, Lab Informat Fondamentale, F-13288 Marseille 9, France

[3] Fac Sci Luminy, CNRS, F-13288 Marseille 9, France

关键词: Oriented matroid;    Lopsided set;    Cell complex;    Tope graph;    Cocircuit;    Coxeter zonotope;   

DOI  :  10.1016/j.jcta.2018.01.002

来源: Elsevier

【 摘 要 】

In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured them as asymmetric counterparts of oriented matroids, both sharing the key property of strong elimination. Moreover, symmetry of faces holds in both structures as well as in the so-called aline oriented matroids. These two fundamental properties (formulated for covectors) together lead to the natural notion of conditional oriented matroid (abbreviated COM). These novel structures can be characterized in terms of three cocircuits axioms, generalizing the familiar characterization for oriented matroids. We describe a binary composition scheme by which every COM can successively be erected as a certain complex of oriented matroids, in essentially the same way as a lopsided set can be glued together from its maximal hypercube faces. A realizable COM is represented by a hyperplane arrangement restricted to an open convex set. Among these are the examples formed by linear extensions of ordered sets, generalizing the oriented matroids corresponding to the permutohedra. Relaxing realizability to local realizability, we capture a wider class of combinatorial objects: we show that non positively curved Coxeter zonotopal complexes give rise to locally realizable COMs. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free