Dimer models and cluster categories of Grassmannians

Abstract

We associate a dimer algebra A to a Postnikov diagram D (in a disk) 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 disk. 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.

49 pages, no separate figure files. To appear in Proc. London Math. Soc. (The published version is different)

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• Total positivity, Grassmannians, and networks
• Calabi-Yau algebras
• A categorification of Grassmannian cluster algebras
• Ice quivers with potentials associated with triangulations and Cohen-Macaulay modules over orders

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