COMs: Complexes of Oriented Matroids

Abstract : In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured them as asym-metric 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 affine 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 permuto-hedra. 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.
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Hans-Jürgen Bandelt, Victor Chepoi, Kolja Knauer. COMs: Complexes of Oriented Matroids. 2017. ⟨hal-01457780⟩

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