On Generalized Minors and Quiver Representations
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.