Abstract
We compute the Euler characteristics of quiver Grassmannians and quiver flag varieties of tree and band modules and prove their positivity. This generalizes some results by G. Cerulli Irelli (2010). As an application we consider the Ringel-Hall algebra \({\mathcal C}(A)\) of some string algebras A and compute in combinatorial terms the products of arbitrary functions in \({\mathcal C}(A)\). These results are transferred to covering theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Assem, I., Simson, D., Skowroński, A.: Elements of the representation theory of associative algebras, 1: techniques of representation theory. In: London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006)
Bialynicki-Birula, A.: On fixed point schemes of actions of multiplicative and additive groups. Topology 12, 99–102 (1973)
Borel, A.: Linear Algebraic Groups, 2nd edn. Graduate Texts in Mathematics, vol. 126. Springer, New York (1991)
Bridgeland, T., Taledano-Laredo, V.: Stability Conditions and Stokes Factors. arXiv:0801.3974 (2010)
Caldero, P., Chapoton, F.: Cluster algebras as Hall algebras of quiver representations. Comment. Math. Helv. 81(3), 595–616 (2006)
Caldero, P., Keller, B.: From triangulated categories to cluster algebras. Invent. Math. 172(1), 169–211 (2008)
Caldero, P., Keller, B.: From triangulated categories to cluster algebras. II. Ann. Sci. École Norm. Sup. (4) 39(6), 983–1009 (2006)
Cerulli Irelli, G.: Quiver Grassmannians associated with string modules. arXiv:0910.2592 (2010)
Crawley-Boevey, W.W.: Maps between representations of zero-relation algebras. J. Algebra 126, 259–263 (1989)
Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations. II: applications to cluster algebras. J. Am. Math. Soc. 23, 749–790 (2010)
Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497–529 (electronic) (2002)
Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63–121 (2003)
Fomin, S., Zelevinsky, A.: Cluster algebras. IV. Coefficients. Compos. Math. 143(1), 112–164 (2007)
Gabriel, P.: The universal cover of a representation finite algebra. In: Representations of Algebras. Lecture Notes in Mathematics, vol. 903. Springer, Berlin (1981)
Joyce, D.: Configurations in abelian categories. II. Ringel-Hall algebras. Adv. Math. 210, 635–706 (2007)
Kapranov, M., Vasserot, E.: Kleinian singularities, derived categories and Hall algebras. Math. Ann. 316, 565–576 (2000)
Krause, H.: Maps between tree and band modules. J. Algebra 137, 186–194 (1991)
Lusztig, G.: Quivers, perverse sheaves and quantized enveloping algebras. J. Am. Math. Soc. 4, 365–421 (1991)
Riedtmann, C.: Lie algebras generated by indecomposables. J. Algebra 170(2), 526–546 (1994)
Schiffmann, O.: Lectures of Hall algebras. Preprint. arXiv:math/0611617 (2006)
Schofield, A.: General representations of quivers. Proc. Lond. Math. Soc. (3) 65(1), 46–64 (1992)
Simson, D., Skowroński, A.: Elements of the representation theory of associative algebras, 2: tubes and concealed algebras of euclidean type. In: London Mathematical Society Student Texts, vol. 71. Cambridge University Press, Cambridge (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by BIGS-Mathematics, Bonn and Mathematical Institute of the University Bonn.
Rights and permissions
About this article
Cite this article
Haupt, N. Euler Characteristics of Quiver Grassmannians and Ringel-Hall Algebras of String Algebras. Algebr Represent Theor 15, 755–793 (2012). https://doi.org/10.1007/s10468-010-9264-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-010-9264-0
Keywords
- Covering theory
- Euler characteristic
- Quiver Grassmannian
- Quiver flag variety
- Quiver representation
- Ringel-Hall algebra
- String algebra
- Band module
- String module
- Tree module