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 Pl\"{u}cker coordinates from D. By computing a
projective resolution of every perfect matching module, we show that
Marsh-Scott's formula for twisted Pl\"{u}cker 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.Comment: 55 pages; v2: reworked Section 7, new material on Muller-Speyer
twists; v3: final version, Adv. Mat
