Séminaire Lotharingien de Combinatoire, 85B.48 (2021), 12 pp.

Emanuele Delucchi and Kolja Knauero

Finitary Affine Oriented Matroids

Abstract. We initiate the study of affine oriented matroids (AOMs) on arbitrary ground sets, extending classical structural features of Oriented Matroids and a natural embedding into the framework of Complexes of Oriented Matroids. The restriction to the finitary case (FAOMs) allows us to study tope graphs and covector posets, as well as to view FAOMs as oriented finitary semimatroids. We show shellability of FAOMs and single out the FAOMs that are affinely homeomorphic to Rⁿ. Finally, we study group actions on AOMs, whose quotients in the case of FAOMs are a stepping stone towards a general theory of affine and toric pseudoarrangements. Our results include applications of the multiplicity Tutte polynomial of group actions of semimatroids, generalizing properties of arithmetic Tutte polynomials of toric arrangements.

Received: December 1, 2020. Accepted: March 1, 2021. Final version: April 29, 2021.

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