Lifting preprojective algebras to orders and categorifying partial flag varieties

Abstract

We describe a categorification of the cluster algebra structure of multihomogeneous coordinate rings of partial flag varieties of arbitrary Dynkin type using Cohen–Macaulay modules over orders. This completes the categorification of Geiss, Leclerc and Schröer by adding the missing coefficients. To achieve this, for an order $A$ and an idempotent $e\in A$, we introduce a subcategory ${\mathsf{CM}}_{e}A$ of $\mathsf{CM}A$ and study its properties. In particular, under some mild assumptions, we construct an equivalence of exact categories $\left({\mathsf{CM}}_{e}A\right)∕\left[Ae\right]\cong \mathsf{Sub}Q$ for an injective $B$-module $Q$, where $B:=A∕\left(e\right)$. These results generalize work by Jensen, King and Su concerning the cluster algebra structure of the Grassmannian ${\mathsf{Gr}}_m\left({ℂ}^n\right)$.