Lattice path matroids and quotients

TL;DR

This paper characterizes quotients among lattice path matroids using diagrams, establishes a graded poset structure with Narayana number coefficients, and proves a conjecture relating lattice path flag matroids to the nonnegative flag variety.

Contribution

It provides a diagram-based characterization of LPM quotients, introduces a graded poset structure, and proves a conjecture connecting lattice path flag matroids with the nonnegative flag variety.

Findings

Quotients among LPMs form a graded poset with Narayana number coefficients.

Full lattice path flag matroids correspond to points in the nonnegative flag variety.

The paper proves a conjecture relating LPM quotients to positroid quotients.

Abstract

We characterize the quotients among lattice path matroids (LPMs) in terms of their diagrams. This characterization allows us to show that ordering LPMs by quotients yields a graded poset, whose rank polynomial has the Narayana numbers as coefficients. Furthermore, we study full lattice path flag matroids and show that -- contrary to arbitrary positroid flag matroids -- they correspond to points in the nonnegative flag variety. At the basis of this result lies an identification of certain intervals of the strong Bruhat order with lattice path flag matroids. A recent conjecture of Mcalmon, Oh, and Xiang states a characterization of quotients of positroids. We use our results to prove this conjecture in the case of LPMs.