Arrangement of minors in the positive Grassmannian

Abstract

This thesis consists of three parts. In the first chapter we discuss arrangements of equal minors of totally positive matrices. More precisely, we investigate the structure of equalities and inequalities between the minors. We show that arrangements of equal minors of largest value are in bijection with sorted sets, which earlier appeared in the context of alcoved polytopes and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the Eulerian number. On the other hand, we prove in many cases that arrangements of equal minors of smallest value are weakly separated sets. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the positive Grassmannian and the associated cluster algebra. However, we also construct examples of arrangements of smallest minors which are not weakly separated using chain reactions of mutations of plabic graphs. In the second chapter, we investigate arrangements of tth largest minors and their relations with alcoved triangulation of the hypersimplex. We show that second largest minors correspond to the facets of the simplices. We then introduce the notion of cubical distance on the dual graph of the triangulation, and study its relations with these arrangements. In addition, we show that arrangements of largest minors induce a structure of a partially ordered set on the entire collection of minors. We use this triangulation of the hypersimplex to describe a 2-dimensional grid structure on this poset. In the third chapter, we obtain new families of quadratic Schur function identities, via examination of several types of networks and the usage of Lindstrdm-Gessel- Viennot lemma. We generalize identities obtained by Kirillov, Fulmek and Kleber and also prove a conjecture suggested by Darij Grinberg.