Characterizing positroid quotients of uniform matroids
Zhixing Chen, Yumou Fei, Jiyang Gao, Yuxuan Sun, Yuchong Zhang
TL;DR
This work provides a complete criterion for when a two-step flag positroid $(M,N)$ with $M$ an elementary quotient of $N$ contains a uniform matroid, formulated via CW-arrows: $(M,U_{k,n})$ is a flag positroid iff the union of any $r+1$ CW-arrows of $M$ has cardinality at least $k+1$, where $rk(M)=k-r$. It also establishes a precise connection between necklace containment and cyclic shifts of decorated permutations, resolving a conjecture for elementary quotients, and contrasts nonlocal quotient phenomena in general positroids with the locality that holds for lattice path matroids (LPMs). The results show that Grassmann necklace/conecklace containment suffices to characterize quotients for LPMs but not for general positroids, highlighting intrinsic nonlocality in the broader class. Overall, the paper provides a computationally efficient (polynomial-time) criterion, clarifies the relationship between cyclic shifts and quotient structure, and delineates the limits of locality-based language for flag positroids.
Abstract
We study two-step flag positroids $(P_1, P_2)$, where $P_1$ is a quotient of $P_{2}$. We provide a complete characterization of all two-step flag positroids that contain a uniform matroid, extending and completing a partial result by Benedetti, Chávez, and Jiménez. To contrast general positroids with the special case of lattice path matroids, we show that the containment relations of Grassmann necklaces and conecklaces fully characterize flag lattice path matroids, but are insufficient for general flag positroids. Additionally, we prove that the decorated permutations of any elementary quotient pair are related by a cyclic shift, resolving a conjecture of Benedetti, Chávez and Jiménez.