Perfect matching modules, dimer partition functions and cluster characters

Alastair King, Matthew Pressland, Ilke Canakci

Research output: Working paper / PreprintPreprint


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.
Original languageEnglish
Number of pages55
Publication statusSubmitted - 30 Jun 2021

Bibliographical note

ArXiv version submitted on 30 Jun 2021 (v1), last revised 26 Oct 2021 (v2)


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