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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn029, 17 pages, doi:10.1093/imrn/rnn029 published on April 25, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

On the Combinatorics of Rigid Objects in 2–Calabi–Yau Categories

Raika Dehy1 and Bernhard Keller2

1 Universitée Cergy–Pontoise/Saint–Martin, Déepartment de Mathéematiques, UMR 8088 du CNRS, 2 avenue Adolphe Chauvin, 95302 Cergy–Pontoise Cedex, France
2 UFR de Mathéematiques, Universitée Denis Diderot – Paris 7, Institut de Mathéematiques, UMR 7586 du CNRS, 2, place Jussieu, 75251 Paris Cedex 05, France

Correspondence: Correspondence to be sent to: keller{at}math.jussieu.fr

Given a triangulated 2-Calabi–Yau category C and a cluster-tilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposables of a cluster-tilting subcategory T' form a basis of the Grothendieck group of T and that, if T and T' are related by a mutation, then the indices with respect to T and T' are related by a certain piecewise linear transformation introduced by Fomin and Zelevinsky in their study of cluster algebras with coefficients. This allows us to give a combinatorial construction of the indices of all rigid objects reachable from the given cluster-tilting subcategory T. Conjecturally, these indices coincide with Fomin–Zelevinsky's g-vectors.



References

  1. Berenstein A., Fomin S., Zelevinsky A. "Cluster algebras III: Upper bounds and double Bruhat cells." Duke Mathematical Journal (2005) 126(1):1–52.[CrossRef][ISI]
  2. Buan A. B., Iyama O., Reiten I., Scott J. "Cluster structures for 2-Calabi–Yau categories and unipotent groups." (2007) preprint arXiv:math.RT/0701557.
  3. Buan A. B., Marsh R. J. "Cluster-tilting theory." In: Trends in representation theory of algebras and related topics (2006) Contemp. Math. 406. Providence, RI: Amer. Math. Soc. 1–30.
  4. Buan A. B., Marsh R. J., Reineke M., Reiten I., Todorov G. "Tilting theory and cluster combinatorics." Advances in Mathematics (2006) 204(2):572–618.[CrossRef][ISI]
  5. Derksen H., Weymann J., Zelevinsky A. "Quivers with potentials and their representations I: Mutations." (2007) preprint arXiv:0704.0649v2.
  6. Fomin S., Zelevinsky A. "Cluster algebras I: Foundations." Journal of the American Mathematical Society (2002) 15(2):497–529. (electronic).[CrossRef][ISI]
  7. Fomin S., Zelevinsky A. "Cluster algebras II: Finite type classification." Inventiones Mathematicae (2003) 154(1):63–121.[ISI]
  8. Fomin S., Zelevinsky A. "Cluster algebras IV: Coefficients." Compositio Mathematica (2007) 143:112–164.[CrossRef][ISI]
  9. Fu C., Keller B. "On cluster algebras with coefficients and 2–Calabi–Yau categories." (2007) preprint arXiv:0710.3152.
  10. Geiss C., Leclerc B., Schröer J. "Rigid modules over preprojective algebras II: The Kac--Moody case." (2007) preprint arXiv:math.RT/0703039.
  11. Geiss C., Leclerc B., Schröer J. "Semicanonical bases and preprojective algebras." Annales Scientifiques de l'Ecole Normale Superieure (4) (2005) 38(2):193–253.[CrossRef]
  12. Geiss C., Leclerc B., Schröer J. "Rigid modules over preprojective algebras." Inventiones Mathematicae (2006) 165(3):589–632.[CrossRef][ISI]
  13. Iyama O. "Higher dimensional Auslander -Reiten theory on maximal orthogonal subcategories." Proceedings of the 37th Symposium on Ring Theory and Representation Theory (2005) 24–30.
  14. Iyama O., Reiten I. "Fomin–Zelevinsky mutation and tilting modules over Calabi–Yau algebras." American Journal of Mathematics (2006) preprint arXiv:math.RT/0605136.
  15. Iyama O., Yoshino Y. "Mutations in triangulated categories and rigid Cohen–Macaulay modules." Inventiones Mathematicae (2006) preprint arXiv:math.RT/0607736.
  16. Keller B., Reiten I. "Acyclic Calabi–Yau categories are cluster categories." Computational Mathematics (2006) preprint arXiv:math.RT/0610594.
  17. Keller B., Reiten I. "Cluster-tilted algebras are Gorenstein and stably Calabi–Yau." Advances in Mathematics (2007) 211:123–51.[CrossRef][ISI]
  18. Marsh R., Reineke M., Zelevinsky A. "Generalized associahedra via quiver representations." Transactions of the American Mathematical Society (2003) 355(10):4171–86. (electronic).[CrossRef][ISI]
  19. Palu Y. "On cluster characters for triangulated categories." Annales De L'Institut Fourier (2007) preprint arXiv:math/0703540v1 [math.RT].

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This Article
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