Lifting preprojective algebras to orders and categorifying partial flag varieties

TL;DR

This paper develops a categorification framework for the cluster algebra structures of partial flag varieties using Cohen-Macaulay modules over orders, extending previous work by adding missing coefficients and generalizing to arbitrary Dynkin types.

Contribution

It introduces a new subcategory of Cohen-Macaulay modules over orders and establishes an equivalence of categories, completing the categorification of partial flag varieties.

Findings

Categorification of cluster algebra structures for partial flag varieties.

Introduction of the subcategory CM_e A and its properties.

Generalization of cluster algebra categorification for Grassmannians.

Abstract

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