A Gelfand-MacPherson correspondence for quiver moduli

TL;DR

This paper establishes a new correspondence linking semi-stable quiver representation moduli spaces with GIT quotients of quiver Grassmannians, unifying several classical and recent results in representation theory.

Contribution

It introduces a Gelfand-MacPherson type correspondence for acyclic quivers, connecting moduli spaces with Grassmannian quotients, extending classical and recent frameworks.

Findings

Identifies semi-stable moduli spaces with two GIT quotients of quiver Grassmannians

Recovers classical Gelfand-MacPherson correspondence as a special case

Generalizes Zelevinsky map for Dynkin type A quivers

Abstract

We show that a semi-stable moduli space of representations of an acyclic quiver can be identified with two GIT quotients by reductive groups. One of a quiver Grassmannian of a projective representation, the other of a quiver Grassmannian of an injective representation. This recovers as special cases the classical Gelfand-MacPherson correspondence and its generalization by Hu and Kim to bipartite quivers, as well as the Zelevinsky map for a quiver of Dynkin type A with the linear orientation.