Leclerc and Zelevinsky described quasicommuting families of quantum minors in terms of a certain combinatorial condition, called weak separation. They conjectured that all inclusion‐maximal weakly separated collections of minors have the same cardinality, and that they can be related to each other by a sequence of mutations. Postnikov studied total positivity on the Grassmannian. He described a stratification of the totally non‐negative Grassmannian into positroid strata, and constructed theirparameterization using plabic graphs. In this paper, we link the study of weak separation to plabic graphs. We extend the notion of weak separation to positroids. We generalize the conjectures of Leclerc and Zelevinsky, and related ones of Scott, and prove them. We show that the maximal weakly separated collections in a positroid are in bijective correspondence with the plabic graphs. This correspondence allows us to use the combinatorial techniques of positroids and plabic graphs to prove the (generalized) purity and mutation connectedness conjectures.

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Stanley has conjectured that the h-vector of a matroid complex is a pure O-sequence. We will prove this for cotransversal matroids by using generalized permutohedra. We construct a bijection between lattice points inside a r-dimensional convex polytope and bases of a rank r transversal matroid. arXiv:1005.5586v3 [math.CO]

International audience Develin and Sturmfels showed that regular triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ can be thought of as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with all subdivisions of $\Delta_{n-1} \times \Delta_{d-1}$. In this paper, we show that any triangulation of $\Delta_{n-1} \times \Delta_{d-1}$ encodes a tropical oriented matroid. We also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes. Develin et Sturmfels ont montré que les triangulations de $\Delta_{n-1} \times \Delta_{d-1}$ peuvent être considérées comme des polytopes tropicaux. Les matroïdes orientés tropicaux ont été définis par Ardila et Develin, et ils ont été conjecturés être en bijection avec les subdivisions de $\Delta_{n-1} \times \Delta_{d-1}$. Dans cet article, nous montrons que toute triangulation de $\Delta_{n-1} \times \Delta_{d-1}$ encode un matroïde orienté tropical. De plus, nous proposons une nouvelle classe d'objets combinatoires qui peuvent décrire toutes les subdivisions d'une plus grande classe de polytopes.