Three Lectures on Quiver Grassmannians

 

SLIDES:

 Generalities on quiver Grassmannians pdf

 Quiver Grassmannians and degenerate flag varieties pdf

 Cellular decomposition and Property (S) pdf

 Desingularizations 

LECTURE NOTES (In preparation)

EXERCISES pdf


BIBLIOGRAPHY:

1) Generalities:  

G. Cerulli Irelli. Geometry of quiver Grassmannians of Dynkin type with applications to cluster algebras. Representation Theory-Current trends and perspectives. EMS-Series of congress reports. 2017. pdf

Lievan Le Bruyn post on quiver Grassmannians on his blog.

M. Reineke. Every projective variety is a quiver Grassmannian. ART 16 (2013). pdf

C.M. Ringel. Quiver Grassmannians for wild acyclic quivers. Proc. AMS 146 (2018). pdf

A. Hubery. Irreducible components of quiver Grassmannians. Trans. AMS. 369 (2017). pdf

A. Schofield. General Representations of Quivers.  Proc. London Math. Soc. 65 (1992). pdf

Connection with degenerate flag varieties (Quiver Grassmannians of Dynkin type)


[CFR] G. Cerulli Irelli, E. Feigin, M. Reineke. Quiver Grassmannians and degenerate flag varieties. ANT 6 (2012), 165-194. pdf

[CFFFR] G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, M. Reineke, . Linear degenerations of flag varieties. Math. Z. 287 (2017), no 1-2, 615-654. pdf
3) Cellular decomposition and property (S)

[CEFR] G. Cerulli Irelli, F. Esposito, H. Franzen, M. Reineke. Cellular decomposition and algebraicity of cohomolofy for quiver Grassmannians. Preprint 2018. pdf

D. Rupel, T. Weist. Cell decompositions for rank two quiver Grassmannians. 
arXiv: 1803.06590. pdf

O. Lorscheid, T. Weist. Quiver Grassmannians of type extended Dynkin type D. 
arXiv: 1507.00392, 1507.00395. pdf1. pdf2.
O. Lorscheid. Schubert decomposition for quiver Grassmannians of tree modules. 
ANT 9 (2015), no. 6, 1337-1362. pdf


Desingularization

G. Cerulli Irelli, E. Feigin, M. Reineke. Desingularization of quiver Grassmannians for Dynkin quivers. Adv. Math. 245 (2013), 182-207. pdf

G. Cerulli Irelli, E. Feigin, M. Reineke. Homological approach to the Hernandez-Leclerc construction and quiver varieties. Representation Theory AMS 18 (2014). pdf

W. Crawley-Boevey, J. Sauter. On quiver Grassmannians and orbit closures for representation-finite algebras. Math. Z. 285 (2017), no. 1-2, 367-395. pdf

B. Keller, S. Scherotzke. Desingularization of quiver Grassmannians via graded quiver varieties. Adv. Math. 256 (2014), 318-347. pdf

S. Scherotzke. Desingularization of quiver Grassmannians via Nakajima categories. ART 20 (2017), no 1, 231-243. pdf


The cluster multiplication formula

P. Caldero, F. Chapoton. Cluster algebras as Hall algebras of quiver representaitons. Comm. Math. Helv. 81 (2006), 595-616. pdf

P. Caldero, B. Keller. From triangulated categories to cluster algebras. Inv. Math. 172 (2008), 169-211.  pdf

P. Caldero, B. Keller. From triangulated categories to cluster algebras II. Ann. Sc. Ec. Norm. Sup. (2006), 983-1009.  pdf


SLIDES:


  1. Generalities on quiver Grassmannians pdf

  2. Quiver Grassmannians and degenerate flag varieties pdf

  3. Cellular decomposition and Property (S) pdf

  4. Desingularizations


LECTURE NOTES (In preparation)


EXERCISES pdf



BIBLIOGRAPHY:


1) Generalities:


G. Cerulli Irelli. Geometry of quiver Grassmannians of Dynkin type with applications to cluster algebras. Representation Theory-Current trends and perspectives. EMS-Series of congress reports. 2017. pdf


Lievan Le Bruyn post on quiver Grassmannians on his blog.


M. Reineke. Every projective variety is a quiver Grassmannian. ART 16 (2013). pdf


C.M. Ringel. Quiver Grassmannians for wild acyclic quivers. Proc. AMS 146 (2018). pdf


A. Hubery. Irreducible components of quiver Grassmannians. Trans. AMS. 369 (2017). pdf


A. Schofield. General Representations of Quivers.  Proc. London Math. Soc. 65 (1992). pdf



  1. 2)Connection with degenerate flag varieties (Quiver Grassmannians of Dynkin type)



[CFR] G. Cerulli Irelli, E. Feigin, M. Reineke. Quiver Grassmannians and degenerate flag varieties. ANT 6 (2012), 165-194. pdf


[CFFFR] G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, M. Reineke, . Linear degenerations of flag varieties. Math. Z. 287 (2017), no 1-2, 615-654. pdf


3) Cellular decomposition and property (S)


[CEFR] G. Cerulli Irelli, F. Esposito, H. Franzen, M. Reineke. Cellular decomposition and algebraicity of cohomolofy for quiver Grassmannians. Preprint 2018. pdf


D. Rupel, T. Weist. Cell decompositions for rank two quiver Grassmannians.

arXiv: 1803.06590. pdf


O. Lorscheid, T. Weist. Quiver Grassmannians of type extended Dynkin type D.

arXiv: 1507.00392, 1507.00395. pdf1. pdf2.


O. Lorscheid. Schubert decomposition for quiver Grassmannians of tree modules.

ANT 9 (2015), no. 6, 1337-1362. pdf



  1. 4)Desingularization


G. Cerulli Irelli, E. Feigin, M. Reineke. Desingularization of quiver Grassmannians for Dynkin quivers. Adv. Math. 245 (2013), 182-207. pdf


G. Cerulli Irelli, E. Feigin, M. Reineke. Homological approach to the Hernandez-Leclerc construction and quiver varieties. Representation Theory AMS 18 (2014). pdf


W. Crawley-Boevey, J. Sauter. On quiver Grassmannians and orbit closures for representation-finite algebras. Math. Z. 285 (2017), no. 1-2, 367-395. pdf


B. Keller, S. Scherotzke. Desingularization of quiver Grassmannians via graded quiver varieties. Adv. Math. 256 (2014), 318-347. pdf


S. Scherotzke. Desingularization of quiver Grassmannians via Nakajima categories. ART 20 (2017), no 1, 231-243. pdf



  1. 5)The cluster multiplication formula


P. Caldero, F. Chapoton. Cluster algebras as Hall algebras of quiver representaitons. Comm. Math. Helv. 81 (2006), 595-616. pdf

P. Caldero, B. Keller. From triangulated categories to cluster algebras. Inv. Math. 172 (2008), 169-211. pdf


P. Caldero, B. Keller. From triangulated categories to cluster algebras II. Ann. Sc. Ec. Norm. Sup. (2006), 983-1009. pdf


Series of Lectures given in Prague, August 7-11, during the workshop of the ICRA 2018