Finitary affine oriented matroids

Abstract

We 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 $\mathbb{R}^n$. 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.

Figures (11)

• Figure 1: An arrangement of hyperplanes in $\mathbb R^2$ with some cells labeled by the respective sign vector.
• Figure 2: A non-stretchable line arrangement with an action of $\mathbb Z^2$ defined by letting a lattice basis act as translations by the two sides of the shaded rectangle. (The picture should be thought of as being repeated in vertical and horizontal direction). Any orientation of it gives rise to a FAOM.
• Figure 3: The quotient of the poset of covectors of the pseudoarrangement in Figure \ref{['fig2']} is the face category of a (pseudo)arrangement in the $2$-dimensional torus (e.g., obtained by identifying opposite sides of the shaded rectangle), whose cells are counted by the Tutte polynomial of the group action on the underlying semimatroid.
• Figure 4: Covector axioms illustrated in the example from Figure \ref{['fig:arrangement']}.
• Figure 5: An example of an infinite (but periodic) pseudoline arrangement. The point highlighted by a hollow bullet corresponds to the covector $X_B$ described in Example \ref{['ex:xbwhite']}.

Theorems & Definitions (175)

• Definition 1.2
• Definition 2.1
• Definition 2.2: AOM, following Kar-92Bau-16
• Remark 2.3
• Proposition 2.4: AOM
• proof
• Example 2.5
• Corollary 2.6
• Corollary 2.7
• proof