Skip to main content
SpringerLink
Log in

Cell decompositions for rank two quiver Grassmannians

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We prove that all quiver Grassmannians for exceptional representations of a generalized Kronecker quiver admit a cell decomposition. In the process, we introduce a class of regular representations which arise as quotients of consecutive preprojective representations. Cell decompositions for quiver Grassmannians of these “truncated preprojectives” are also established. We provide two combinatorial labelings for these cells. On the one hand, they are naturally labeled by certain subsets of a so-called 2-quiver attached to a (truncated) preprojective representation. On the other hand, the cells are in bijection with compatible pairs in a maximal Dyck path as predicted by the theory of cluster algebras. The provided bijection between these two labelings gives a geometric explanation for the appearance of Dyck path combinatorics in the theory of quiver Grassmannians.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Assem, I., Simson, D., Skowronski, A.: Elements of the representation theory of associative algebras. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  2. Auslander, M., Reiten, I., Smalo, S.O.: Representation theory of Artin algebras 36. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  3. Białynicki-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math. 98, 480–497 (1973)

    Article  MathSciNet  Google Scholar 

  4. Bialynicki-Birula, A., Carrell, J., McGovern, W.M.: Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action. Encyclopaedia Mathematical Science. Invariant Theory and Algebraic Transformation Groups, vol. 131, pp 83–158. Springer, Berlin (2002)

  5. Bernstein, I.N., Gelfand, I.M., Ponomarev, V.A.: Coxeter functors, and Gabriel’s theorem. Russ. Math. Surv. 28(2), 17–32 (1973)

    Article  MathSciNet  Google Scholar 

  6. Brenner, S., Butler, M.C.R.: The equivalence of certain functors occurring in the representation theory of artin algebras and species. J. Lond. Math. Soc. 2(1), 183–187 (1976)

    Article  MathSciNet  Google Scholar 

  7. Caldero, P., Chapoton, F.: Cluster algebras as Hall algebras of quiver representations. Comment. Math. Helv. 81(3), 595–616 (2006)

    Article  MathSciNet  Google Scholar 

  8. Caldero, P., Keller, B.: From triangulated categories to cluster algebras II. Ann. Sci. de l’Ecole Norm. Super. (4) 39(6), 983–1009 (2006)

    Article  MathSciNet  Google Scholar 

  9. Cerulli Irelli, G.: Quiver Grassmannians associated with string modules. J. Algebra. Comb 33(2), 259–276 (2011)

    Article  MathSciNet  Google Scholar 

  10. Cerulli Irelli, G., Esposito, F.: Geometry of quiver Grassmannians of Kronecker type and applications to cluster algebras. Algebra Number Theory 5(6), 777–801 (2011)

    Article  MathSciNet  Google Scholar 

  11. Cerulli Irelli, G., Esposito, F., Franzen, H., Reineke, M.: Topology of Quiver Grassmannians. Preprint (2018)

  12. Caldero, P., Reineke, M.: On the quiver Grassmannian in the acyclic case. J. Pure Appl. Algebra 212(11), 2369–2380 (2008)

    Article  MathSciNet  Google Scholar 

  13. Gabriel, P.: The universal cover of a finite-dimensional algebra. Representations of algebras. Lect. Notes Math. 903, 68–105 (1981)

    Article  Google Scholar 

  14. Haupt, N.: Euler characteristics of quiver Grassmannians and Ringel–Hall algebras of string algebras. Algebras Represent. Theory 15, 755–793 (2012)

    Article  MathSciNet  Google Scholar 

  15. Lee, K., Li, L., Zelevinsky, A.: Greedy elements in rank 2 cluster algebras. Sel. Math. 20(1), 57–82 (2014)

    Article  MathSciNet  Google Scholar 

  16. Lorscheid, O., Weist, T.: Representation type via Euler characteristics and singularities of quiver Grassmannians. Preprint (2017), arXiv:1706.00860

  17. Ringel, C.M.: Exceptional modules are tree modules. Linear Algebra Appl. 275(276), 471–493 (1998)

    Article  MathSciNet  Google Scholar 

  18. Rupel, D.: Rank two non-commutative laurent phenomenon and pseudo-positivity. Preprint (2017) arXiv:1707.02696

  19. Schofield, A.: General representations of quivers. Proc. Lond. Math. Soc. (3) 65(1), 46–64 (1992)

    Article  MathSciNet  Google Scholar 

  20. Weist, T.: Localization of quiver moduli spaces. Represent. Theory 17(13), 382–425 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank Giovanni Cerulli Irelli, Hans Franzen, Oliver Lorscheid and Markus Reineke for very fruitful discussions related to this project.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Dylan Rupel

  2. Bergische Universität Wuppertal, Gaußstr. 20, 42097, Wuppertal, Germany

    Thorsten Weist

Authors
  1. Dylan Rupel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thorsten Weist
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thorsten Weist.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rupel, D., Weist, T. Cell decompositions for rank two quiver Grassmannians. Math. Z. 295, 993–1038 (2020). https://doi.org/10.1007/s00209-019-02379-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-019-02379-6

Keywords

Mathematics Subject Classification

Search

Navigation