Green-to-Red Sequences for Positroids

TL;DR

This paper investigates the quivers derived from Le-diagrams and positroid cells, demonstrating that they admit a specific mutation sequence called a green-to-red sequence, which has implications for cluster algebras and positroid varieties.

Contribution

It shows that quivers from Le-diagrams and reduced plabic graphs always admit a green-to-red mutation sequence, extending understanding of their combinatorial and algebraic structures.

Findings

Quivers from Le-diagrams can be constructed from Grassmannian quivers by deleting and merging vertices.

All quivers from Le-diagrams and reduced plabic graphs admit a green-to-red mutation sequence.

The results connect combinatorial objects with cluster algebra mutation properties.

Abstract

Le-diagrams are combinatorial objects that parametrize cells of the totally nonnegative Grassmannian, called positroid cells, and each Le-diagram gives rise to a cluster algebra which is believed to be isomorphic to the coordinate ring of the corresponding positroid variety. We study quivers arising from these diagrams and show that they can be constructed from the well-behaved quivers associated to Grassmannians by deleting and merging certain vertices. Then, we prove that quivers coming from arbitrary Le-diagrams, and more generally reduced plabic graphs, admit a particular sequence of mutations called a green-to-red sequence.