Abstract
AbstractWe initiate the axiomatic study of affine oriented matroids (AOMs) on arbitrary ground sets, obtaining fundamental notions such as minors, reorientations and a natural embedding into the frame work 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 $$\mathbb {R}^n$$
R
n
. 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 enumerative properties of toric arrangements to a combinatorially defined class of arrangements of submanifolds. This answers partially a question by Ehrenborg and Readdy.
Funder
SUPSI - University of Applied Sciences and Arts of Southern Switzerland
Publisher
Springer Science and Business Media LLC
Reference35 articles.
1. Aguiar, M., Petersen, T.K.: The Steinberg torus of a Weyl group as a module over the Coxeter complex. J. Algebraic Combin. 42, 1135–1175 (2015)
2. Albenque, M., Knauer, K.: Convexity in partial cubes: the hull number. Discrete Math. 339, 866–876 (2016)
3. Ardila, F.: Semimatroids and their Tutte polynomials. Rev. Colombiana Mat. 41, 39–66 (2007)
4. Bandelt, H.-J., Chepoi, V., Knauer, K.: COMs: complexes of oriented matroids. J. Comb. Theory Ser. A 156, 195–237 (2018)
5. Bandelt, H.-J., Mulder, H.M.: Infinite median graphs, (0,2)-graphs, and hypercubes. J. Graph Theory 7, 487–497 (1983)