Dimer models and cluster categories of Grassmannians

By Karin Baur, A. King, B. Marsh. 2013

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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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

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