  We consider dimers on the star lattice (aka the 3-12, Fisher, expanded kagome
or triangle-honeycomb lattice). We show that dimer coverings on this lattice
have Z_2 arrow and pseudo-spin representations analogous to those for the
kagome lattice, and use these to construct an exactly solvable quantum dimer
model (QDM) with a Rokhsar-Kivelson (RK) ground state. This QDM, first
discussed by Moessner and Sondhi from a different point of view, is the
star-lattice analogue of a kagome-lattice QDM analyzed by Misguich et al. We
give a detailed analysis of various properties of the classical equal-weight
dimer model on the star lattice, most of which are related to those of the RK
state. Using both the arrow representation and the fermionic path integral
formulation of the Pfaffian method, we calculate the number of dimer coverings,
dimer occupation probabilities, and dimer, vison, and monomer correlation
functions. We show that a dimer is uncorrelated with dimers further away than
on neighboring triangles, that the vison-vison correlation function vanishes,
and that the monomer-monomer correlation function equals 1/4 regardless of the
monomer positions (with one exception). A key result in the fermionic approach
is the vanishing of the two-point Green function beyond a short distance. These
properties are very similar to those of dimers on the kagome lattice. We also
discuss some generalizations involving arrow representations for dimer
coverings on "general Fisher lattices" and their "reduced" lattices, with the
kagome, squagome, and triangular kagome lattice being examples of the latter.
