Quotients of Uniform Positroids

Abstract

Two matroids $M$ and $N$ are said to be concordant if there is a strong map from $N$ to $M$. This also can be stated by saying that each circuit of $N$ is a union of circuits of $M$. In this paper, we consider a class of matroids called positroids, introduced by Postnikov, and utilize their combinatorics to determine concordance among some of them. More precisely, given a uniform positroid, we give a purely combinatorial characterization of a family of positroids that is concordant with it. We do this by means of their associated decorated permutations. As a byproduct of our work, we describe completely the collection of circuits of this particular subset of positroids.