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

References in corpus (1)

• Ice quivers with potentials associated with triangulations and Cohen-Macaulay modules over orders

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