On Schubert decompositions of quiver Grassmannians

2015

Algebra Number Theory

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ABSTRACT. Let Q be a quiver, M a representation of Q with an ordered basis B and e a dimension vector for Q. In this note we extend the methods of [7] to establish Schubert decompositions of quiver Grassmannians Gr e (M) into affine spaces to the ramified case, i.e. the canonical morphism F : T → Q from the coefficient quiver T of M w.r.t. B is not necessarily unramified.In particular, we determine the Euler characteristic of Gr e (M) as the number of extremal successor closed subsets of T 0 , which extends the results of Cerulli Irelli ([4]) and Haupt ([6]) (under certain additional assumptions on B).

“…One requirement of [7] is that the morphism F : T → Q is unramified. It is the purpose of this note to extend the methods of [7] to ramified F : T → Q.…”

Section: Schubert Decompositions and Ramified Tree Modules Caldero Amentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Schubert decompositions for quiver Grassmannians of tree modules

2015

Algebra Number Theory

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“…In particular, we like to mention that the general case is proven analogously to the special case where Q is the Kronecker quiver and X is a preprojective representation (cf. [17,Example 4.5]) for the former method and [19, Proposition 3.1] for the latter method.…”

Section: String Modules and Extended Dynkin Type Amentioning

confidence: 99%

“…For type

E

, this is made explicit in . Therefore, the closures of the Schubert cells form an additive basis of the cohomology ring of the quiver Grassmannian (see [, Section 6] for more details).…”

Section: Introductionmentioning

confidence: 99%

Representation type via Euler characteristics and singularities of quiver Grassmannians

Bull. London Math. Soc.

Weist

2019

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In this paper, we characterize the representation type of an acyclic quiver by the properties of its associated quiver Grassmannians. This characterization utilizes and extends known results about singular quiver Grassmannians and cell decompositions into affine spaces. While all quiver Grassmannians for indecomposable representations of quivers of finite representation type are smooth and admit cell decompositions, it turns out that all quiver Grassmannians for indecomposable representations of tame quivers admit cell decompositions, but some of these quiver Grassmannians are singular (even as varieties). A quiver is wild if and only if there exists a quiver Grassmannian with negative Euler characteristic.

“…In this case, the classes of the closures of the non-empty Schubert cells in Gr e (M) form an additive basis for the singular cohomology ring of Gr e (M), and the cohomology is concentrated in even degree, cf. [29,Cor. 6.2].…”

Section: Introductionmentioning

confidence: 99%

Quiver Grassmannians of Extended Dynkin type 𝐷 Part 1: Schubert Systems and Decompositions Into Affine Spaces

Weist

2019

Memoirs of the AMS

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Let Q be a quiver of extended Dynkin type D n . In this first of two papers, we show that the quiver Grassmannian Gr e (M) has a decomposition into affine spaces for every dimension vector e and every indecomposable representation M of defect −1 and defect 0, with exception of the non-Schurian representations in homogeneous tubes. We characterize the affine spaces in terms of the combinatorics of a fixed coefficient quiver for M. The method of proof is to exhibit explicit equations for the Schubert cells of Gr e (M) and to solve this system of equations successively in linear terms. This leads to an intricate combinatorial problem, for whose solution we develop the theory of Schubert systems.In the sequel [30], we extend the result of this paper to all indecomposable representations M of Q and determine explicit formulae for their F-polynomials. CONTENTS27 4. Schubert decompositions for type D n 30 5. Proof of Theorem 4.4 37 Appendix A. Representations for quivers of type D n 58 Appendix B. Bases for representations of type D n 61 References 72 arXiv:1507.00392v1 [math.RT]