Type A quiver loci and Schubert varieties

@article{Kinser2013TypeAQ,
  title={Type A quiver loci and Schubert varieties},
  author={Ryan Kinser and Jenna Rajchgot},
  journal={Journal of Commutative Algebra},
  year={2013},
  volume={7},
  pages={265-301}
}
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… 
View PDF on arXiv
10 Citations
Highly Influential Citations
1
Background Citations
2
Methods Citations
1
Results Citations
2

Figures from this paper

10 Citations

Type D quiver representation varieties, double Grassmannians, and symmetric varieties

Singularities of zero sets of semi-invariants for quivers

Let Q be a quiver with dimension vector α prehomogeneous under the action of the product of general linear groups GL(α) on the representation variety Rep(Q,α). We study geometric properties of zero…

Free resolutions of orbit closures of Dynkin quivers

We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolutions of some orbit closures of quivers. As a consequence, we obtain that for Dynkin quivers orbit…

K-polynomials of type A quiver orbit closures and lacing diagrams

This article contains an overview of the author’s joint work with Allen Knutson and Jenna Rajchgot on K-polynomials of orbit closures for type A quivers. It is written to an audience interested in…

REPRESENTATION VARIETIES OF ALGEBRAS WITH NODES

Abstract We study the behaviour of representation varieties of quivers with relations under the operation of node splitting. We show how splitting a node gives a correspondence between certain closed…

Singularities of zero sets of semi-invariants for quivers

  • A. Lorincz
  • Mathematics
    Journal of Commutative Algebra
  • 2021
Let $Q$ be a quiver with dimension vector $\alpha$ prehomogeneous under the action of the product of general linear groups $\operatorname{GL}(\alpha)$ on the representation variety…

Minimal free resolutions of ideals of minors associated to pairs of matrices

Consider the affine space consisting of pairs of matrices $(A,B)$ of fixed size, and its closed subvariety given by the rank conditions $\operatorname{rank} A \leq a$, $\operatorname{rank} B \leq b$…

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

We provide combinatorial formulas for the multidegree and K-polynomial of an (arbitrarily oriented) type A quiver locus embedded inside of its representation space. These formulas are generalizations…

Schubert geometry and combinatorics

This chapter combines an introduction and research survey about Schubert varieties. The theme is to combinatorially classify their singularities using a family of polynomial ideals generated by…

Bernstein-Sato Polynomials for Quivers

References

SHOWING 1-10 OF 49 REFERENCES

Degeneracy schemes, quiver schemes, and Schubert varieties

abstract. A result of Zelevinsky states that an orbit closure in the space of representations of the equioriented quiver of type A h is in bijection with the opposite cell in a Schubert variety of a…

Chern class formulas for quiver varieties

In this paper a formula is proved for the general degeneracy locus associated to an oriented quiver of type A_n. Given a finite sequence of vector bundles with maps between them, these loci are…

Quiver coefficients of Dynkin type

We study the Grothendieck classes of quiver cycles, i.e. invariant closed subvarieties of the representation space of a quiver. For quivers without oriented loops we show that the class of a quiver…

Orbit closures and rank schemes

Let A be a finitely generated associative algebra over an algebraically closed field k, and consider the variety mod A.k/ of A-module structures on k d . In case A is of finite representation type,…

GOVERNING SINGULARITIES OF SCHUBERT VARIETIES

Positivity in the Grothendieck group of complex flag varieties

Projections of Richardson Varieties

While the projections of Schubert varieties in a full generalized flag manifold G/B to a partial flag manifold $G/P$ are again Schubert varieties, the projections of Richardson varieties…

Flags, Schubert polynomials, degeneracy loci, and determinantal formulas

Under appropriate conditions on the rank function r, which guarantee that, for generic h, f,(h) is irreducible, we prove a formula for the class [f,(h)] of this locus in the Chow or cohomology ring…

Grothendieck classes of quiver varieties

We prove a formula for the structure sheaf of a quiver variety in the Grothendieck ring of its embedding variety. This formula generalizes and gives new expressions for Grothendieck polynomials. We…

On the B-canonical splittings of flag varieties