Résume | The representation theories of both quivers and of Kac-Moody groups are described by certain ternary classifications. On one hand, indecomposable quiver representations are classified as either preprojective, preinjective, or regular according to the action of the Auslander-Reiten translation. On the other, irreducible representations of affine Kac-Moody groups are classified as highest-weight, lowest-weight, or level zero according to their central character (similarly for non-affine types, though the term ``level zero'' is less apt). In both classifications the first two classes are well understood and dual to each other in a suitable sense, while the third is much more mysterious. The theme of the talk is that these two classifications are in fact directly related to one another. We formulate a general conjecture to this effect, which we prove in affine type A and finitely many other types. The conjecture is couched in terms of cluster algebras -- on one hand these are repositories for certain generating functions (called cluster characters) associated to quiver representations, and on the other they are coordinate rings of certain subvarieties of Kac-Moody groups. The conjecture states that the cluster character of a rigid indecomposable quiver representation is a generalized minor of a specific Kac-Moody representation, and that this relationship intertwines the classifications described above. This is joint work with Dylan Rupel and Salvatore Stella. |