Arrangements of equal minors in the positive Grassmannian
Abstract
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 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 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 \textitpositive Grassmannian and the associated cluster algebra.
Keywords
Totally positive matrices
minors
the positive Grassmannian
Plücker coordinates
matrix completion problem
weakly separated sets
cluster algebras
plabic graphs
sorted sets
triangulations
thrackles
alcoved polytopes
affine Coxeter arrangements
hypersimplices
Eulerian numbers
Gröbner bases
Schur positivity.
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