Plücker Relations for Quiver Grassmannians

Lorscheid, Oliver; Weist, Thorsten

doi:10.1007/s10468-017-9762-4

Cited by 4 publications

(7 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let β = {2, 4, 5, 6}. Then the Schubert cell C X λ β is open dense in Gr e (X λ ) and we can apply the description of the quiver Grassmannian in terms of homogeneous coordinates from [21].…”

Section: The Kronecker Quivermentioning

confidence: 99%

“…Note that we can simplify the equations of [21] if we make use of the fact that the embedding Gr e (X λ ) → Gr(4, 6) factors through the product Grassmannian Gr(1, 2) × Gr(2, 2) × Gr (1,2), which is isomorphic to P 1 × P 1 with bihomogeneous coordinates [ Δ 1 : Δ 4 | Δ 3 : Δ 6 ]. Then the defining bihomogeneous equation of Gr e (X λ ) inside P 1 × P 1 is…”

Section: The Kronecker Quivermentioning

confidence: 99%

“…Let

β = {2, 4, 5, 6}

. Then the Schubert cell

C_{β}^{X_{λ}}

is open dense in

{prefixGr}_{\underset{̲}{e}} false(X_{λ} false)

and we can apply the description of the quiver Grassmannian in terms of homogeneous coordinates from . Note that we can simplify the equations of if we make use of the fact that the embedding

{prefixGr}_{\underset{̲}{e}} false(X_{λ} false) \to Gr false(4, 6 false)

factors through the product Grassmannian

Gr (1, 2) \times Gr (2, 2) \times Gr (1, 2)

, which is isomorphic to

{double-struckP}^{1} \times {double-struckP}^{1}

with bihomogeneous coordinates

[{normalΔ}_{1} : {normalΔ}_{4} | {normalΔ}_{3} : {normalΔ}_{6}]

.…”

Section: Singular Quiver Grassmannians For Tame Quiversmentioning

confidence: 99%

“…Then the Schubert cell

C_{β}^{X_{λ}}

is open dense in

{prefixGr}_{\underset{̲}{e}} false(X_{λ} false)

and we can apply the description of the quiver Grassmannian in terms of homogeneous coordinates from . Note that we can simplify the equations of if we make use of the fact that the embedding

{prefixGr}_{\underset{̲}{e}} false(X_{λ} false) \to Gr false(4, 6 false)

factors through the product Grassmannian

Gr (1, 2) \times Gr (2, 2) \times Gr (1, 2)

, which is isomorphic to

{double-struckP}^{1} \times {double-struckP}^{1}

with bihomogeneous coordinates

[{normalΔ}_{1} : {normalΔ}_{4} | {normalΔ}_{3} : {normalΔ}_{6}]

. Then the defining bihomogeneous equation of

{prefixGr}_{\underset{̲}{e}} false(X_{λ} false)

inside

{double-struckP}^{1} \times {double-struckP}^{1}

F_{β} false(c, 6, 1 false) = λ {normalΔ}_{4} {normalΔ}_{3} - λ {normalΔ}_{1} {normalΔ}_{6} + {normalΔ}_{1} {normalΔ}_{3} = 0 .

From this, we see that

{prefixGr}_{\underset{̲}{e}} false(X_{λ} false)

forms a flat family over

double-struckC

with respect to the parameter

…”

Section: Singular Quiver Grassmannians 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.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: The Kronecker Quivermentioning

confidence: 99%

Section: The Kronecker Quivermentioning

confidence: 99%

“…Let

β = {2, 4, 5, 6}

. Then the Schubert cell

C_{β}^{X_{λ}}

is open dense in

{prefixGr}_{\underset{̲}{e}} false(X_{λ} false)

and we can apply the description of the quiver Grassmannian in terms of homogeneous coordinates from . Note that we can simplify the equations of if we make use of the fact that the embedding

{prefixGr}_{\underset{̲}{e}} false(X_{λ} false) \to Gr false(4, 6 false)

factors through the product Grassmannian

Gr (1, 2) \times Gr (2, 2) \times Gr (1, 2)

, which is isomorphic to

{double-struckP}^{1} \times {double-struckP}^{1}

with bihomogeneous coordinates

[{normalΔ}_{1} : {normalΔ}_{4} | {normalΔ}_{3} : {normalΔ}_{6}]

.…”

Section: Singular Quiver Grassmannians For Tame Quiversmentioning

confidence: 99%

“…Then the Schubert cell

C_{β}^{X_{λ}}

is open dense in

{prefixGr}_{\underset{̲}{e}} false(X_{λ} false)

and we can apply the description of the quiver Grassmannian in terms of homogeneous coordinates from . Note that we can simplify the equations of if we make use of the fact that the embedding

{prefixGr}_{\underset{̲}{e}} false(X_{λ} false) \to Gr false(4, 6 false)

factors through the product Grassmannian

Gr (1, 2) \times Gr (2, 2) \times Gr (1, 2)

, which is isomorphic to

{double-struckP}^{1} \times {double-struckP}^{1}

with bihomogeneous coordinates

[{normalΔ}_{1} : {normalΔ}_{4} | {normalΔ}_{3} : {normalΔ}_{6}]

. Then the defining bihomogeneous equation of

{prefixGr}_{\underset{̲}{e}} false(X_{λ} false)

inside

{double-struckP}^{1} \times {double-struckP}^{1}

F_{β} false(c, 6, 1 false) = λ {normalΔ}_{4} {normalΔ}_{3} - λ {normalΔ}_{1} {normalΔ}_{6} + {normalΔ}_{1} {normalΔ}_{3} = 0 .

From this, we see that

{prefixGr}_{\underset{̲}{e}} false(X_{λ} false)

forms a flat family over

double-struckC

with respect to the parameter

…”

Section: Singular Quiver Grassmannians For Tame Quiversmentioning

confidence: 99%

See 2 more Smart Citations

Representation type via Euler characteristics and singularities of quiver Grassmannians

Lorscheid

Weist

2019

Bull. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Next, we describe the reduced scheme structure for the quiver Grassmannians corresponding to the representations in U f lat,irr by providing an explicit set of quadratic generators for the ideal describing the Plücker embedding (see also [17]). Our main combinatorial tool is the notion of PBW semi-standard Young tableaux (see [11]), parametrizing a basis in the homogeneous coordinate ring of the PBW degenerate flag varieties.…”

Section: Introductionmentioning

confidence: 99%

Linear degenerations of flag varieties: partial flags, defining equations, and group actions

et al. 2019

View full text Add to dashboard Cite

We continue, generalize and expand our study of linear degenerations of flag varieties from [5]. We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degenerate flag variety. We compute the generating function of the number of orbits in the flat irreducible locus and study the natural family of line bundles on the degenerations from the flat irreducible locus. We also describe explicitly the reduced scheme structure on these degenerations and conjecture that similar results hold for the whole flat locus. Finally, we prove an analogue of the Borel-Weil theorem for the flat irreducible locus.

show abstract

Matroids with coefficients and F1-geometry

Zanoelo Jarra

View full text Add to dashboard Cite

the members of the assessment committe: Omid Amini, Matt Baker, Reimundo Heluani and Jaap Top, for the careful reading and valuable comments on a previous version of this thesis. I would also like to thank Jaap Top for the help regarding the Dutch language. Additionally, I thank CNPq and the University of Groningen for the financial support received during the development of this work.I am grateful to my classmates and colleagues from both IMPA and the University of Groningen for making these last years more enjoyable.

show abstract

Plücker Relations for Quiver Grassmannians

Abstract: In this text, we exhibit the quiver Plücker relations for a quiver Grassmannian and show that they describe the quiver Grassmannian as a closed subscheme of a product of usual Grassmannians.

Cited by 4 publications

References 8 publications

Representation type via Euler characteristics and singularities of quiver Grassmannians

Representation type via Euler characteristics and singularities of quiver Grassmannians

Linear degenerations of flag varieties: partial flags, defining equations, and group actions

Matroids with coefficients and F1-geometry

Contact Info

Product

Resources

About