Schubert decompositions for quiver Grassmannians of tree modules

Lorscheid, Oliver

doi:10.2140/ant.2015.9.1337

Cited by 6 publications

(29 citation statements)

References 14 publications

(34 reference statements)

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“…If this was not the case, we can use the reverse order of

scriptB

, that is, exchange

i \in B

r n + k + l - i

, and relabel the vertices of

Q

correspondingly to exchange the roles of

q_{k}

and

q_{l}

, so that our assumption is satisfied. First proof : Theorem 4.1 of provides a cell decomposition of

{prefixGr}_{\underset{̲}{e}} false(X false)

into affine spaces provided that

X

admits an ordered polarization (cf. [, Section 3.3]) such that every relevant pair (cf.…”

Section: Cell Decomposition For Tame Quiversmentioning

confidence: 99%

“…[, Section 3.3]) such that every relevant pair (cf. [, Section 2.3]) is maximal for at most one arrow of

Q

(cf. [, Section 3.4]).…”

Section: Cell Decomposition For Tame Quiversmentioning

confidence: 99%

“…The same theorem states that the cells

C_{β}^{X}

are labelled by the extremal successor closed subsets

β

Γ_{0}

(cf. [, Section 3.1]). Since

π : Γ \to Q

is unramified (cf.…”

Section: Cell Decomposition For Tame Quiversmentioning

confidence: 99%

“…Since

π : Γ \to Q

is unramified (cf. [, Section 3.2]), a subset

β

Γ_{0}

is extremal successor closed if and only if it is successor closed (cf. [, Section 3.1]).…”

Section: Cell Decomposition For Tame Quiversmentioning

confidence: 99%

“…[, Section 3.2]), a subset

β

Γ_{0}

is extremal successor closed if and only if it is successor closed (cf. [, Section 3.1]). We indicate why these hypotheses are satisfied for the chosen ordered basis

scriptB

.…”

Section: Cell Decomposition For Tame Quiversmentioning

confidence: 99%

See 4 more Smart Citations

Representation type via Euler characteristics and singularities of quiver Grassmannians

Lorscheid

Weist

2019

Bull. London Math. Soc.

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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.

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“…If this was not the case, we can use the reverse order of

scriptB

, that is, exchange

i \in B

r n + k + l - i

, and relabel the vertices of

Q

correspondingly to exchange the roles of

q_{k}

and

q_{l}

, so that our assumption is satisfied. First proof : Theorem 4.1 of provides a cell decomposition of

{prefixGr}_{\underset{̲}{e}} false(X false)

into affine spaces provided that

X

admits an ordered polarization (cf. [, Section 3.3]) such that every relevant pair (cf.…”

Section: Cell Decomposition For Tame Quiversmentioning

confidence: 99%

“…[, Section 3.3]) such that every relevant pair (cf. [, Section 2.3]) is maximal for at most one arrow of

Q

(cf. [, Section 3.4]).…”

Section: Cell Decomposition For Tame Quiversmentioning

confidence: 99%

“…The same theorem states that the cells

C_{β}^{X}

are labelled by the extremal successor closed subsets

β

Γ_{0}

(cf. [, Section 3.1]). Since

π : Γ \to Q

is unramified (cf.…”

Section: Cell Decomposition For Tame Quiversmentioning

confidence: 99%

“…Since

π : Γ \to Q

is unramified (cf. [, Section 3.2]), a subset

β

Γ_{0}

is extremal successor closed if and only if it is successor closed (cf. [, Section 3.1]).…”

Section: Cell Decomposition For Tame Quiversmentioning

confidence: 99%

“…[, Section 3.2]), a subset

β

Γ_{0}

is extremal successor closed if and only if it is successor closed (cf. [, Section 3.1]). We indicate why these hypotheses are satisfied for the chosen ordered basis

scriptB

.…”

Section: Cell Decomposition For Tame Quiversmentioning

confidence: 99%

See 3 more Smart Citations

Representation type via Euler characteristics and singularities of quiver Grassmannians

Lorscheid

Weist

2019

Bull. London Math. Soc.

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show abstract

Plücker Relations for Quiver Grassmannians

Lorscheid

Weist

2018

Algebr Represent Theor

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Cell decompositions and algebraicity of cohomology for quiver Grassmannians

Irelli

Esposito

Franzen

et al. 2021

Advances in Mathematics

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We show that the cohomology ring of a quiver Grassmannian associated with a rigid quiver representation has property (S): there is no odd cohomology and the cycle map is an isomorphism; moreover, its Chow ring admits explicit generators defined over any field. From this we deduce the polynomial point count property. By restricting the quiver to finite or affine type, we are able to show a much stronger assertion: namely, that a quiver Grassmannian associated with an indecomposable (not necessarily rigid) representation admits a cellular decomposition. As a corollary, we establish a cellular decomposition for quiver Grassmannians associated with representations with rigid regular part. Finally, we study the geometry behind the cluster multiplication formula of Caldero and Keller, providing a new proof of a slightly more general result.

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Schubert decompositions for quiver Grassmannians of tree modules

Cited by 6 publications

References 14 publications

Representation type via Euler characteristics and singularities of quiver Grassmannians

Representation type via Euler characteristics and singularities of quiver Grassmannians

Plücker Relations for Quiver Grassmannians

Cell decompositions and algebraicity of cohomology for quiver Grassmannians

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