Quotients of uniform positroids

Abstract

Flag matroids are a rich family of Coxeter matroids that can be characterized using pairs of matroids that form a quotient. We consider a class of matroids called positroids, introduced by Postnikov, and utilize their combinatorial representations to explore characterizations of flag positroids.Given a uniform positroid, we give a purely combinatorial characterization of a family of positroids that form quotients with it. We state this in terms of their associated decorated permutations. In proving our characterization we also fully describe the circuits of this family.

Positivity for partial tropical flag varieties

“…Remark 3.1. In [BCT19] the term flag positroid is used for positroids which form a matroid quotient. However one can easily construct positroids which form a quotient but do not consitute a flag positroid in our terms, for example the flag matroid with bases 1, 2, 3, 12, 13 (see [JLLO23,Example 3.3].…”

The positive Grassmannian, the amplituhedron, and cluster algebras

“…1 For example, the four permutations (3, 1, 4, 2), (2, 4, 1, 3), (4, 3, 1, 2), (3, 4, 2, 1) labeling the positroid tilings of ∆ 2,4 in Figure 7 are loopless. Their T-dual images are (2, 3, 1, 4), (3,2,4,1), (2,4,3,1), and (1, 3, 4, 2) -precisely the permutations labeling the positroid tilings of A 4,1,2 (Z) in Figure 8! The T-duality map appears in [35,41,9,21], and is a version of an m = 4 map from [3]. One can also describe T-duality as a map on reduced plabic graphs G; we say G is black-trivalent (white-trivalent) if all of its interior black (white) vertices are trivalent.…”