Seminar

Leclerc and Zelevinsky, motivated by the study of quasi-commuting quantum flag minors, introduced the notions of strongly and weakly separated collections.   These notions are closely related to cluster algebras, double Bruhat cells, and the positive Grassmannian.   A key feature of strongly/weakly separated collections is the purity phenomenon.

We introduce the notion of M-separation, for any oriented matroid M, and investigate its relationship with zonotopal tilings.   We define the class of pure oriented matroids for which the purity phenomenon holds.   It turns out that an oriented matroid of rank 3 is pure if and only if it is isomorphic to a positroid (or, equivalently, positively oriented matroid).   A graphical matroid is pure if and only if it corresponds to an outerplanar graph.   We give a conjectural characterization of pure oriented matroids in terms of forbidden minors and prove it in many cases.

The talk is based on a joint paper with Pavel Galashin.

arxiv.org/abs/1708.01329