### Abstract

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.

Original language | English |
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Pages (from-to) | 213-260 |

Number of pages | 48 |

Journal | Proceedings of the London Mathematical Society |

Volume | 113 |

Issue number | 2 |

Early online date | 12 Jul 2016 |

DOIs | |

Publication status | Published - 1 Aug 2016 |

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## Cite this

*Proceedings of the London Mathematical Society*,

*113*(2), 213-260. https://doi.org/10.1112/plms/pdw029