Journal of Commutative Algebra

Type A quiver loci and Schubert varieties

Ryan Kinser and Jenna Rajchgot

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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.

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J. Commut. Algebra Volume 7, Number 2 (2015), 265-301.

First available in Project Euclid: 14 July 2015

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Kinser, Ryan; Rajchgot, Jenna. Type A quiver loci and Schubert varieties. J. Commut. Algebra 7 (2015), no. 2, 265--301. doi:10.1216/JCA-2015-7-2-265.

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