Hostname: page-component-7f64f4797f-42qgm Total loading time: 0 Render date: 2025-11-06T09:06:16.415Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE NUMBER OF POINTS OVER FINITE FIELDS ON VARIETIES RELATED TO CLUSTER ALGEBRAS

Published online by Cambridge University Press:  22 December 2010

F. CHAPOTON
F. CHAPOTON*
Affiliation:
Institut Camille Jordan, Université Claude Bernard Lyon 1, Bâtiment Braconnier, 21 Avenue Claude Bernard, F-69622 Villeurbanne Cedex, France e-mail: chapoton@math.univ-lyon1.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We start here the study of some algebraic varieties related to cluster algebras. These varieties are defined as the fibres of the projection map from the cluster variety to the affine space of coefficients. We compute the number of points over finite fields on these varieties, for all simply laced Dynkin diagrams. We also compute the cohomology with compact support in some cases.

Keywords

14R16G20

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 1 , January 2011 , pp. 141 - 151
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Berenstein, A., Fomin, S. and Zelevinsky, A., Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (1) (2005), 1–52.CrossRefGoogle Scholar
2.Buan, A. B., Marsh, R., Reineke, M., Reiten, I. and Todorov, G., Tilting theory and cluster combinatorics, Adv. Math. 204 (2) (2006), 572–618.CrossRefGoogle Scholar
3.Fock, V. V. and Goncharov, A. B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Ec. Norm. Supér. (4) 42, no. 65 (2009), 865–930.Google Scholar
4.Fomin, S., Shapiro, M. and Thurston, D., Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math. 201 (1) (2008), 83–146.Google Scholar
5.Fomin, S. and Zelevinsky, A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2) (2002), 497–529 (electronic).CrossRefGoogle Scholar
6.Fomin, S. and Zelevinsky, A., Cluster algebras. II. Finite type classification, Invent. Math. 154 (1) (2003), 63–121.Google Scholar
7.Fomin, S. and Zelevinsky, A., Cluster algebras. IV. Coefficients, Compos. Math. 143 (1) (2007), 112–164.Google Scholar
8.Keller, B., Algèbres amassées et applications, Séminaire Bourbaki 1014 (November 2009), 1–26.Google Scholar