Quasi-coincidence of cluster structures on positroid varieties

TL;DR

This paper proves that two prominent cluster algebra structures on positroid varieties in the Grassmannian are quasi-coinciding, meaning they are closely related through Laurent monomials, resolving a conjecture from 2017.

Contribution

It demonstrates the quasi-coincidence of source-labelled and target-labelled cluster structures on positroid varieties, confirming a conjecture and advancing understanding of their relationship.

Findings

The two cluster structures quasi-coincide, with variables related by Laurent monomials.

The left twist map is a quasi-cluster equivalence between the structures.

The proof uses categorification and quasi-equivalences of cluster algebras.

Abstract

By work of a number of authors, beginning with Scott and culminating with Galashin and Lam, the coordinate rings of positroid varieties in the Grassmannian carry cluster algebra structures. In fact, they typically carry many such structures, the two best understood being the source-labelled and target-labelled structures, referring to how the initial cluster is computed from a Postnikov diagram or plabic graph. In this article, we show that these two cluster algebra structures quasi-coincide, meaning in particular that a cluster variable in one structure may be expressed in the other structure as the product of a cluster variable and a Laurent monomial in the frozen variables. This resolves a conjecture attributed to Muller and Speyer from 2017. The proof depends critically on categorification: of the relevant cluster algebra structures by the author, of perfect matchings and twists by the author with \c{C}anak\c{c}{\i} and King, and of quasi-equivalences of cluster algebras by Fraser and Keller. By similar techniques, we also show that Muller and Speyer's left twist map is a quasi-cluster equivalence from the target-labelled structure to the source-labelled structure.