Cluster Algebras and the Positive Grassmannian

TL;DR

This paper explores the relationship between plabic graphs, positroid stratification, and cluster algebras within the positive Grassmannian, revealing a map from positroid strata to cluster subalgebras and implications for scattering amplitudes.

Contribution

It extends the connection between plabic graphs and cluster algebras to lower-dimensional cells, establishing a map from positroid strata to cluster subalgebras.

Findings

Cluster algebras correspond to positroid varieties for all cells.

A map from positroid strata to cluster subalgebras is constructed.

Implications for tree-level scattering amplitudes in super Yang-Mills theory are discussed.

Abstract

Plabic graphs are intimately connected to the positroid stratification of the positive Grassmannian. The duals to these graphs are quivers, and it is possible to associate to them cluster algebras. For the top-cell graph of $Gr_{+}(k,n)$, this cluster algebra is the homogeneous coordinate ring of the corresponding positroid variety. We prove that the same statement holds for plabic graphs describing lower dimensional cells. In this way we obtain a map from the positroid strata onto cluster subalgebras of $Gr_{+}(k,n)$. We explore some of the consequences of this map for tree-level scattering amplitudes in $\mathcal N=4$ super Yang-Mills theory.