Calabi-Yau properties of Postnikov diagrams

TL;DR

This paper demonstrates that the dimer algebra derived from a connected Postnikov diagram exhibits bimodule internally 3-Calabi-Yau properties, leading to a categorification of associated cluster algebras and connections to positroid varieties.

Contribution

It establishes the bimodule internally 3-Calabi-Yau property for dimer algebras of Postnikov diagrams and links this to cluster algebra categorification and positroid varieties.

Findings

Dimer algebra of a Postnikov diagram is bimodule internally 3-Calabi-Yau.

Provides an additive categorification of the associated cluster algebra.

Connects categorification to Grassmannian cluster categories and positroid varieties.

Abstract

We show that the dimer algebra of a connected Postnikov diagram in the disc is bimodule internally 3-Calabi-Yau in the sense of the author's earlier work. As a consequence, we obtain an additive categorification of the cluster algebra associated to the diagram, which (after inverting frozen variables) is isomorphic to the homogeneous coordinate ring of a positroid variety in the Grassmannian by a recent result of Galashin and Lam. We show that our categorification can be realised as a full extension closed subcategory of Jensen-King-Su's Grassmannian cluster category, in a way compatible with their bijection between rank 1 modules and Pl\"ucker coordinates.