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arXiv · 1309.6524

Dimer models and cluster categories of Grassmannians

Karin Baur·Alastair King·Bethany Marsh
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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.

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Karin Baur, Alastair King, Bethany Marsh. 2016-06-15. Dimer models and cluster categories of Grassmannians. https://doi.org/10.1112/plms%2Fpdw029

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