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arXiv · 1307.6261

Type A quiver loci and Schubert varieties

Ryan Kinser·Jenna Rajchgot
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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.

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Ryan Kinser, Jenna Rajchgot. 2013-07-23. Type A quiver loci and Schubert varieties. https://doi.org/10.1216/jca-2015-7-2-265

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