Perfect matching modules, dimer partition functions and cluster characters

Çanakçı, İ., King, A. and Pressland, M. (2024) Perfect matching modules, dimer partition functions and cluster characters. Advances in Mathematics, 443, 109570. (doi: 10.1016/j.aim.2024.109570)

Abstract

Cluster algebra structures for Grassmannians and their (open) positroid strata are controlled by a Postnikov diagram D or, equivalently, a dimer model on the disc, as encoded by either a bipartite graph or the dual quiver (with faces). The associated dimer algebra A, determined directly by the quiver with a certain potential, can also be realised as the endomorphism algebra of a cluster-tilting object in an associated Frobenius cluster category. In this paper, we introduce a class of A-modules corresponding to perfect matchings of the dimer model of D and show that, when D is connected, the indecomposable projective A-modules are in this class. Surprisingly, this allows us to deduce that the cluster category associated to D embeds into the cluster category for the appropriate Grassmannian. We show that the indecomposable projectives correspond to certain matchings which have appeared previously in work of Muller–Speyer. This allows us to identify the cluster-tilting object associated to D, by showing that it is determined by one of the standard labelling rules constructing a cluster of Plücker coordinates from D. By computing a projective resolution of every perfect matching module, we show that Marsh–Scott's formula for twisted Plücker coordinates, expressed as a dimer partition function, is a special case of the general cluster character formula, and thus observe that the Marsh–Scott twist can be categorified by a particular syzygy operation in the Grassmannian cluster category.

• Item Type: Articles
• Additional Information: At different stages of the project, the first author was supported by EPSRC grants EP/K026364/1, EP/N005457/1 and EP/P016014/1, while the third author was supported by a fellowship from the Max-Planck-Gesellschaft and the EPSRC postdoctoral fellowship EP/T001771/1.
• Keywords: Cluster algebra, dimer model, partition function, perfect matching, positroid variety.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Pressland, Dr Matthew
• Authors: Çanakçı, İ., King, A., and Pressland, M.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Advances in Mathematics
• Publisher: Elsevier
• ISSN: 0001-8708
• ISSN (Online): 1090-2082
• Published Online: 13 March 2024
• Copyright Holders: Copyright © 2024 The Authors
• First Published: First published in Advances in Mathematics 443: 109570
• Publisher Policy: Reproduced under a Creative Commons License