Links in the complex of weakly separated collections

TL;DR

This paper explores the structure of weakly separated collections in plabic graphs, showing that mutations between maximal collections can be performed while preserving common face labels, enhancing understanding of their combinatorial properties.

Contribution

It demonstrates that mutations between maximal weakly separated collections can be executed without altering shared face labels, revealing new structural insights.

Findings

Mutations can be performed while freezing common face labels.

Maximal weakly separated collections have flexible mutation pathways.

The results connect plabic graph moves with cluster algebra mutations.

Abstract

Plabic graphs are interesting combinatorial objects used to study the totally nonnegative Grassmannian. Faces of plabic graphs are labeled by $k$-element sets of positive integers, and a collection of such $k$-element sets are the face labels of a plabic graph if that collection forms a maximal weakly separated collection. There are moves that one can apply to plabic graphs, and thus to maximal weakly separated collections, analogous to mutations of seeds in cluster algebras. In this short note, we show that if two maximal weakly separated collections can be mutated from one to another, then one can do so while freezing the face labels they have in common.