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.
Cite
@article{arxiv.1309.6524,
title = {Dimer models and cluster categories of Grassmannians},
author = {Karin Baur and Alastair King and Bethany Marsh},
journal= {arXiv preprint arXiv:1309.6524},
year = {2020}
}
Comments
49 pages, no separate figure files. To appear in Proc. London Math. Soc. (The published version is different)