On Generalized Minors and Quiver Representations

TL;DR

This paper connects cluster algebras from acyclic quivers to generalized minors in Kac-Moody groups, extending known results and proposing new conjectures for affine types.

Contribution

It generalizes the realization of cluster variables via generalized minors to broader classes of quiver representations and affine types.

Findings

Cluster variables of preprojective and postinjective representations are realized by highest- and lowest-weight minors.

In certain affine types, regular representations correspond to non-highest/lowest weight minors.

The work extends finite type results to more general Kac-Moody group settings.

Abstract

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.