Type A quiver loci and Schubert varieties

Abstract

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 Bobiński 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.

Three combinatorial formulas for type A quiver polynomials and K-polynomials

“…As in (2.2), we label all sink vertices by x j , 1 ≤ j ≤ n, all source vertices by y i , 0 ≤ i ≤ n, left-pointing arrows by α j , 1 ≤ j ≤ n, and right-pointing arrows by β i , 1 ≤ i ≤ n. Subscripts increase from right to left. Since the matrices in our quiver representations act on row vectors instead of column vectors in this paper, this notation slightly differs from [KR15]. Until §5, we work with a fixed bipartite type A quiver Q and dimension vector d unless explicitly stated otherwise; hence, these will be omitted from the notation.…”

“…Zelevinsky permutations. In this section, we recall the bipartite Zelevinsky permutation v(Ω) from [KR15,§4]. The definition we give here is more direct than the one in [KR15, §4] to make this paper more self-contained.…”

“…From another viewpoint, quiver loci generalize some classically studied varieties such as determinantal varieties and varieties of complexes. This is because Bongartz's work [Bon96] implies that Dynkin quiver loci are defined, at least up to radical, by minors of certain block form matrices (see the introduction and §3 of [KR15], or [RZ13]).…”

“…sink-source) orientation; this comprises the bulk of the paper. We then extend the bipartite results to all orientations using the method of [KR15,§5]: if Q is a type A quiver of arbitrary orientation, there is an associated bipartite type A quiver Q and a bijection between orbit closures for Q and a certain subset of orbit closures for Q. We show that our formulas for the quiver loci of Q can be obtained by a simple substitution into the formulas for the corresponding quiver loci of Q (Proposition 5.7), followed by simplification to make them intrinsic to Q (making minimal reference to the associated bipartite quiver Q).…”

Free resolutions of orbit closures of Dynkin quivers

“…Several results are known in the Dynkin case. It has been shown (see [3,5,6,12,14]) that for quivers of type A and D orbit closures have rational singularities (in particular, are normal and Cohen-Macaulay). Furthermore, for equioriented type A quivers it was shown in [14] that singularities of orbit closures are identical to singularities of Schubert varieties.…”