Type A quiver loci and Schubert varieties

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 Bobi\'nski 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.