Grassmannian cluster subcategories and positroid varieties

A class of subcategories GP $B$ of the Grassmannian cluster category CM $C_{k, n}$ was constructed by Jensen--King--Su from certain superorders $B$ of $C_{k, n}$, which they showed are in bijection with Grassmannian positroids of type $(k, n)$. We prove that GP $B$ admits a cluster substructure of CM $C_{k, n}$, giving rise to a cluster algebra $A_{clu}$. This naturally raises questions regarding the relationship of $A_{clu}$ to $C[Gr(k, n)]$ and to the coordinate ring of the positroid variety associated to $B$. Using the cluster substructure, we show that the ice Gabriel quiver $Q^\circ_U$ of a cluster tilting object $U\in$ GP $B$, consisting of rank one modules, is a subquiver of $Q^\circ_T$ with $T$ a cluster tilting object in CM $C_{k, n}$ containing $U$ as a summand. We also deduce that $A_{clu}$ is a subalgebra of $C[Gr(k, n)]$. Moreover, applying a result of Canakci--King--Pressland on the Gabriel quiver $Q_U$ in the case where $B$ is connected (i.e., has no repeated direct summands), we deduce that $Q^\circ_U$, for arbitrary $B$, coincides with the quiver constructed by Muller-Speyer from a plabic graph whose face labels agree with the indices of the indecomposable summands of $U$. Consequently, the localised algebra $(A_{clu})_B$ is isomorphic to the cluster algebra $A_{MS}$ of Muller-Speyer. We then construct bases for certain subalgebras and for an ideal of $C[Gr(k, n)]$, and apply these to prove that $(A_{clu})_B$ is naturally isomorphic to the coordinate ring of the open positroid variety. As a consequence, we obtain a new proof of Galashin--Lam's Theorem, identifying $A_{MS}$ with the coordinate ring of the open positroid variety, which was originally conjectured by Muller-Speyer. In the connected case, we note also that Pressland gave a categorification of the cluster structure following Galashin-Lam.