Dimer models and cluster categories of Grassmannians

We associate a dimer algebra A to a Postnikov diagram D (in a disc) corresponding to a cluster of minors in the cluster structure of the Grassmannian Gr(k,n). We show that A is isomorphic to the endomorphism algebra of a corresponding Cohen–Macaulay module T over the algebra B used to categorify the...

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Bibliographic Details
Published in:Proceedings of the London Mathematical Society 2016-08, Vol.113 (2), p.213-260
Main Authors: Baur, Karin, King, Alastair D., Marsh, Robert J.
Format: Article
Language:English
Subjects:
Algebra
Boundaries
Categories
Clusters
Dimers
Disks
Formulas (mathematics)
Mathematical models
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Summary:We associate a dimer algebra A to a Postnikov diagram D (in a disc) corresponding to a cluster of minors in the cluster structure of the Grassmannian Gr(k,n). We show that A is isomorphic to the endomorphism algebra of a corresponding Cohen–Macaulay module T over the algebra B used to categorify the cluster structure of Gr(k,n) by Jensen–King–Su. It follows that B can be realised as the boundary algebra of A, that is, the subalgebra eAe for an idempotent e corresponding to the boundary of the disc. The construction and proof uses an interpretation of the diagram D, with its associated plabic graph and dual quiver (with faces), as a dimer model with boundary. We also discuss the general surface case, in particular computing boundary algebras associated to the annulus.
ISSN:0024-6115
1460-244X
DOI:10.1112/plms/pdw029