Arrangements of equal minors in the positive Grassmannian

We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with sorted sets, which earlier appeared in the context of alcoved polytopes and Grobner 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 conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the weakly separated sets. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the positive Grassmannian and the associated cluster algebra.

This is a joint work with Alexander Postnikov.