Arrangements of equal minors in the positive Grassmannian

TL;DR

This paper explores the structure of equal minors in totally positive matrices within the positive Grassmannian, linking arrangements of largest minors to sorted sets and smallest minors to weakly separated sets, with new examples beyond existing theory.

Contribution

It establishes a bijection between arrangements of largest minors and sorted sets, and shows that arrangements of smallest minors are often weakly separated sets, including new counterexamples.

Findings

Largest minors arrangements correspond to sorted sets and alcoved triangulations.

Smallest minors are typically weakly separated sets, but exceptions exist.

Number of maximal arrangements equals the Eulerian number.

Abstract

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 Gr\"obner 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 exactly 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.