Suho Oh

Positroids and Schubert matroids

Postnikov gave a combinatorial description of the cells in a totally nonnegative Grassmannian. These cells correspond to a special class of matroids called positroids. We prove his conjecture that a positroid is exactly an intersection of cyclically shifted Schubert matroids. This leads to a combinatorial description of positroids that is easily computable.

Weak separation and plabic graphs

Leclerc and Zelevinsky described quasicommuting families of quantum minors in terms of a certain combinatorial condition, called weak separation. They conjectured that all maximal by inclusion weakly separated collections of minors have the same cardinality, and that they can be related to each other by a sequence of mutations. On the other hand, Postnikov studied total positivity on the Grassmannian. He described a stratification of the totally nonnegative Grassmannian into positroid strata, and constructed their parametrization using plabic graphs. In this paper we link the study of weak separation to plabeic 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.

Bruhat order, smooth Schubert varieties, and hyperplane arrangements

The aim of this article is to link Schubert varieties in the flag manifold with hyperplane arrangements. For a permutation, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincare polynomial of the corresponding Schubert variety if and only if the Schubert variety is smooth. We give an explicit combinatorial formula for the Poincare polynomial. Our main technical tools are chordal graphs and perfect elimination orderings.

RAINBOW GRAPHS AND SWITCHING CLASSES

A rainbow graph is a graph that admits a vertex-coloring such that every color appears exactly once in the neighborhood of each vertex. We investigate some properties of rainbow graphs. In particular, we show that there is a bijection between the isomorphism classes of

Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements

n

Triangulations of n 1 d 1 and Tropical Oriented Matroids

-rainbow graphs on

Triangulations of

2n

\Delta_{n-1} \times \Delta_{d-1}

vertices and the switching classes of graphs on

and Matching Ensembles

n

Poset vectors and generalized permutohedra

vertices.