An algorithm to construct the Le diagram associated to a Grassmann necklace

TL;DR

This paper presents an algorithm to convert Le diagrams into Grassmann necklaces, facilitating the translation between two combinatorial objects that index positroids in the nonnegative Grassmannian.

Contribution

It provides the first explicit algorithm for constructing a Grassmann necklace from a Le diagram, complementing existing methods.

Findings

The algorithm enables efficient translation between Le diagrams and Grassmann necklaces.

It enhances understanding of the combinatorial structures underlying positroids.

The method simplifies computations involving positroids in the nonnegative Grassmannian.

Abstract

Le diagrams and Grassmann necklaces both index the collection of positroids in the nonnegative Grassmannian $Gr_{\geq 0}(k,n)$, but they excel at very different tasks: for example, the dimension of a positroid is easily extracted from its Le diagram, while the list of bases of a positroid is far more easily obtained from its Grassmann necklace. Explicit bijections between the two are therefore desirable. An algorithm for turning a Le diagram into a Grassmann necklace already exists; in this note we give the reverse algorithm.