Combinatorics related to the totally nonnegative Grassmannian

Abstract

In this thesis we study the combinatorial objects that appear in the study of nonnegative part of the Grassmannian. The classical theory of total positivity studies matrices such that all minors are nonnegative. Lustzig extended this theory to arbitrary reductive groups and flag varieties. Postnikov studied the nonnegative part of the Grassmannian, showed that it has a nice cell decomposition using matroid strata, and introduced several combinatorial objects that encode such cells. In this thesis, we focus on the combinatorial aspects of such associated objects. In chapter 1, we review the definition of the cells in the totally nonnegative part of the Grassmannian, and the associated combinatorial objects. Each cell corresponds to a certain matroid called positroid. There are numerous combinatorial objects that can represent a positroid, such as a J-diagram, a Grassmann necklace or a decorated permutation. We will go over the definitions of such objects and check some of their properties. And for decorated permutations, there are certain planar graphs called plabic graphs, that plays the role of wiring diagrams for permutations, and this would serve as the main tool for our result in chapter 3. In chapter 2, we prove a conjecture by Postnikov, that allows us to give a purely combinatorial definition of positroids without relying on its realizability. We will show that positroids can be defined as certain collections that satisfy some cyclic inequalities. In other words, we express positroids using cyclically shifted Schubert matroids. Postnikov showed that each positroid cell is an intersection of the totally nonnegative Grassmannian and cyclically shifted Schubert cells. Combinatorially, this result implies that each positroid is included in an intersection of cyclically shifted Schubert matroids. We extend this result: each positroid is exactly an intersection of certain cyclically shifted Schubert niatroids. In chapter 3, we study maximal weakly separated collections. Weak separation is a condition on pair of sets that first appeared in Leclerc and Zelevinsky's work describing quasicommuting families of quantum minors. They conjectured that all maximal by inclusion weakly separated collections of minors have the same cardinality (the purity conjecture), and that they can be related to each other by a sequence of mutations. We link the study of weak separation with the totally nonnegative Grassmannian, by extending the notion of weak separation to positroids. By using plabic graphs, we generalize the results and conjectures of Leclerc and Zelevinsky, and prove them in this more general setup. This part of the thesis is based on joint work with Alexander Postnikov and David Speyer. In chapter 4, we prove a property on h-vector of positroids. The h-vector of a matroid is an interesting Tutte polynomial evaluation, which is originally defined as the h-vector of the corresponding independent complex of a matroid. Stanley conjectured that h-vector of any matroid is a pure O-sequence, which is a sequence coming froi a Hilbert function of a monomial Artinian level algebra. We show that the conjecture holds for positroids: that is, the h-vector of a positroid is a pure O-sequence.