Arrangements Of Minors In The Positive Grassmannian And a Triangulation of The Hypersimplex

TL;DR

This paper explores the combinatorial structure of minors in the positive Grassmannian, linking arrangements of equal minors to triangulations of the hypersimplex and revealing new geometric and order-theoretic properties.

Contribution

It extends the understanding of minors arrangements in the positive Grassmannian by connecting them to hypersimplex triangulations and introducing cubical distance and poset structures.

Findings

Second largest minors correspond to facets of the triangulation

Introduces cubical distance relating minors to triangulation structure

Largest minors induce a partial order on all minors

Abstract

The structure of zero and nonzero minors in the Grassmannian leads to rich combinatorics of matroids. In this paper, we investigate an even richer structure of possible equalities and inequalities between the minors in the positive Grassmannian. It was previously shown that arrangements of equal minors of largest value are in bijection with the simplices in a certain triangulation of the hypersimplex that was studied by Stanley, Sturmfels, Lam and Postnikov. Here we investigate the entire set of arrangements and its relations with this triangulation. First, 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 the arrangement of t-th largest minors. Finally, we show that arrangements of largest minors induce a structure of partially ordered sets on the entire collection of minors. We use the Lam and Postnikov circuit triangulation of the hypersimplex to describe a 2-dimensional grid structure of this poset.