Calabi-Yau properties of Postnikov diagrams
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.
Keywords
Cite
@article{arxiv.1912.12475,
title = {Calabi-Yau properties of Postnikov diagrams},
author = {Matthew Pressland},
journal= {arXiv preprint arXiv:1912.12475},
year = {2022}
}
Comments
29 pages, comments welcome. v2: significantly revised and expanded, new results on cluster characters and comparison to Jensen-King-Su's Grassmannian cluster category. v3: final version, to appear in Forum Math. Sigma