Title: Purity and separation for oriented matroids,
Abstract: Leclerc and Zelevinsky, motivated by the study of quasi-commuting quantum
flag minors, introduced the notions of strongly separated and weakly separated
collections. These notions are closely related to the theory of cluster
algebras, to the combinatorics of the double Bruhat cells, and to the totally
positive Grassmannian.
A key feature, called the purity phenomenon, is that every maximal by
inclusion strongly (resp., weakly) separated collection of subsets in $[n]$ has
the same cardinality.
In this paper, we extend these notions and define $\mathcal{M}$-separated
collections, for any oriented matroid $\mathcal{M}$.
We show that maximal by size $\mathcal{M}$-separated collections are in
bijection with fine zonotopal tilings (if $\mathcal{M}$ is a realizable
oriented matroid), or with one-element liftings of $\mathcal{M}$ in general
position (for an arbitrary oriented matroid).
We introduce the class of pure oriented matroids for which the purity
phenomenon holds: an oriented matroid $\mathcal{M}$ is pure if
$\mathcal{M}$-separated collections form a pure simplicial complex, i.e., any
maximal by inclusion $\mathcal{M}$-separated collection is also maximal by
size.
We pay closer attention to several special classes of oriented matroids:
oriented matroids of rank $3$, graphical oriented matroids, and uniform
oriented matroids. We classify pure oriented matroids in these cases. An
oriented matroid of rank $3$ is pure if and only if it is a positroid (up to
reorienting and relabeling its ground set). A graphical oriented matroid is
pure if and only if its underlying graph is an outerplanar graph, that is, a
subgraph of a triangulation of an $n$-gon.
We give a simple conjectural characterization of pure oriented matroids by
forbidden minors and prove it for the above classes of matroids (rank $3$,
graphical, uniform).