A categorification of Grassmannian cluster algebras

Abstract

We describe a ring whose category of Cohen-Macaulay modules provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. More precisely, there is a cluster character defined on the category which maps the rigid indecomposable objects to the cluster variables and the maximal rigid objects to clusters. This is proved by showing that the quotient of this category by a single projective-injective object is Geiss-Leclerc-Schroer's category Sub $Q_k$, which categorifies the coordinate ring of the big cell in this Grassmannian.

v2: minor change in title, new Sec 9 on categorification, small changes in exposition and some new figures; v3: Sec 2 rearranged and shortened, small changes in exposition, to appear in Proc. Lond. Math. Soc

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• Calabi-Yau properties of Postnikov diagrams
• Tilting theory for Gorenstein rings in dimension one
• Lifting preprojective algebras to orders and categorifying partial flag varieties
• Friezes satisfying higher SL${}_k$-determinants
• Cluster categories from Grassmannians and root combinatorics
• Perfect matching modules, dimer partition functions and cluster characters
• Newton--Okounkov bodies and minimal models for cluster varieties
• Orbifold diagrams
• From frieze patterns to cluster categories
• Consistent Dimer Models on Surfaces with Boundary
• Cluster categories for completed infinity-gons I: Categorifying triangulations
• Derived equivalences of self-injective 2-Calabi--Yau tilted algebras