Type A quiver loci and Schubert varieties

TL;DR

This paper establishes a geometric correspondence between type A quiver orbit closures and Schubert varieties, enabling the transfer of geometric properties and Frobenius splitting results.

Contribution

It introduces the bipartite Zelevinsky map linking quiver representations to Schubert varieties, extending to arbitrary orientations and recovering key geometric properties.

Findings

Orbit closures are normal, Cohen-Macaulay, and have rational singularities.

Representation spaces admit Frobenius splittings compatible with orbit closures.

The map provides a new geometric perspective on quiver orbit closures.

Abstract

We describe a closed immersion from each representation space of a type A quiver with bipartite (i.e., alternating) orientation to a certain opposite Schubert cell of a partial flag variety. This "bipartite Zelevinsky map" restricts to an isomorphism from each orbit closure to a Schubert variety intersected with the above-mentioned opposite Schubert cell. For type A quivers of arbitrary orientation, we give the same result up to some factors of general linear groups. These identifications allow us to recover results of Bobinski and Zwara; namely we see that orbit closures of type A quivers are normal, Cohen-Macaulay, and have rational singularities. We also see that each representation space of a type A quiver admits a Frobenius splitting for which all of its orbit closures are compatibly Frobenius split.