  The cluster algebra of any acyclic quiver can be realized as the coordinate
ring of a subvariety of a Kac-Moody group -- the quiver is an orientation of
its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat
cell. We use this realization to connect representations of the quiver with
those of the group. We show that cluster variables of preprojective (resp.
postinjective) quiver representations are realized by generalized minors of
highest-weight (resp. lowest-weight) group representations, generalizing
results of Yang-Zelevinsky in finite type. In type $A_n^{\!(1)}$ and finitely
many other affine types, we show that cluster variables of regular quiver
representations are realized by generalized minors of group representations
that are neither highest- nor lowest-weight; we conjecture this holds more
generally.
