Cluster Algebras and the Positive Grassmannian

Understand

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.