Publication Date:
2016-08-06
Description:
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 ${\rm Gr}(k,n)$ . 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 ${\rm Gr}(k,n)$ 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.
Print ISSN:
0024-6115
Electronic ISSN:
1460-244X
Topics:
Mathematics
