Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-02-05T07:57:39.597Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear independence of cluster monomials for skew-symmetric cluster algebras

Part of: Abelian categories Arithmetic rings and other special rings

Published online by Cambridge University Press:  28 August 2013

Giovanni Cerulli Irelli ,
Bernhard Keller ,
Daniel Labardini-Fragoso  and
Pierre-Guy Plamondon
Giovanni Cerulli Irelli
Affiliation:
Mathematisches Institut, Universität Bonn, D-53115 Bonn, Germany email [email protected]
Bernhard Keller
Affiliation:
Université Paris Diderot - Paris 7, UFR de Mathématiques, Case 7012, Bâtiment Chevaleret, 75205 Paris Cedex 13, France email [email protected]
Daniel Labardini-Fragoso
Affiliation:
Mathematisches Institut, Universität Bonn, D-53115 Bonn, Germany email [email protected]
Pierre-Guy Plamondon
Affiliation:
Laboratoire LMNO, Équipe Algèbre, Géométrie et Logique, Université de Caen, F14032 Caen Cedex, France email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Fomin–Zelevinsky conjectured that in any cluster algebra, the cluster monomials are linearly independent and that the exchange graph and cluster complex are independent of the choice of coefficients. We confirm these conjectures for all skew-symmetric cluster algebras.

Keywords

cluster algebrascluster categoriescluster monomials

MSC classification

Secondary: 18E30: Derived categories, triangulated categories 13F60: Cluster algebras
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 10 , October 2013 , pp. 1753 - 1764
Copyright
© The Author(s) 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amiot, C., Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier (Grenoble) 59 (2009), 2525–2590.Google Scholar
Amiot, C., On generalized cluster categories, in ICRA XIV Proceedings (European Mathematical Society, 2011).Google Scholar
Buan, A. B., Marsh, R., Reineke, M., Reiten, I. and Todorov, G., Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572–618.Google Scholar
Buan, A. B., Marsh, R. J., Reiten, I. and Todorov, G., Clusters and seeds in acyclic cluster algebras, Proc. Amer. Math. Soc. 135 (2007), 3049–3060; with an appendix coauthored in addition by P. Caldero and B. Keller.Google Scholar
Caldero, P. and Chapoton, F., Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv. 81 (2006), 595–616.Google Scholar
Caldero, P. and Keller, B., From triangulated categories to cluster algebras. II, Ann. Sci. Éc. Norm. Supér. (4) 39 (2006), 983–1009.Google Scholar
Caldero, P. and Keller, B., From triangulated categories to cluster algebras, Invent. Math. 172 (2008), 169–211.Google Scholar
Cerulli Irelli, G., Positivity in skew-symmetric cluster algebras of finite type, Preprint (2011), arXiv:1102.3050 [math.RA].Google Scholar
Cerulli Irelli, G. and Labardini-Fragoso, D., Quivers with potentials associated to triangulated surfaces, part III: Tagged triangulations and cluster monomials, Compositio Math. 148 (2012), 1833–1866.Google Scholar
Dehy, R. and Keller, B., On the combinatorics of rigid objects in 2-Calabi–Yau categories, Int. Math. Res. Not. IMRN (2008), doi:10.1093/imrn/rnn029.Google Scholar
Demonet, L., Mutations of group species with potentials and their representations. Applications to cluster algebras, Preprint (2010), arXiv:1003.5078 [math.RT].Google Scholar
Demonet, L., Categorification of skew-symmetrizable cluster algebras, Algebr. Represent. Theory 14 (2011), 1087–1162.Google Scholar
Derksen, H., Weyman, J. and Zelevinsky, A., Quivers with potentials and their representations. I. Mutations, Selecta Math. (N.S.) 14 (2008), 59–119.Google Scholar
Derksen, H., Weyman, J. and Zelevinsky, A., Quivers with potentials and their representations II: applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), 749–790.Google Scholar
Fomin, S., Shapiro, M. and Thurston, D., Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math. 201 (2008), 83–146.Google Scholar
Fomin, S. and Zelevinsky, A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497–529.Google Scholar
Fomin, S. and Zelevinsky, A., Cluster algebras II: finite type classification, Invent. Math. 154 (2003), 63–121.Google Scholar
Fomin, S. and Zelevinsky, A., Cluster algebras: notes for the CDM-03 conference, in Current developments in mathematics (Int. Press, Somerville, MA, 2003), 1–34.Google Scholar
Fomin, S. and Zelevinsky, A., Cluster algebras. IV. Coefficients, Compositio Math. 143 (2007), 112–164.CrossRefGoogle Scholar
Geiss, C., Leclerc, B. and Schröer, J., Generic bases for cluster algebras and the Chamber Ansatz, J. Amer. Math. Soc. 25 (2012), 21–76.CrossRefGoogle Scholar
Gekhtman, M., Shapiro, M. and Vainshtein, A., On the properties of the exchange graph of a cluster algebra, Math. Res. Lett. 15 (2008), 321–330.Google Scholar
Gekhtman, M., Shapiro, M. and Vainshtein, A., Cluster algebras and Poisson geometry, Amer. Math. Soc., Math. Surv. Monogr. 167 (2010).Google Scholar
Keller, B., Cluster algebras and derived categories, Preprint (2012), arXiv:1202.4161v3 [math.RT].Google Scholar
Keller, B. and Reiten, I., Cluster-tilted algebras are Gorenstein and stably Calabi–Yau, Adv. Math. 211 (2007), 123–151.Google Scholar
Keller, B. and Yang, D., Derived equivalences from mutations of quivers with potential, Adv. Math. 226 (2011), 2118–2168.CrossRefGoogle Scholar
Musiker, G., Schiffler, R. and Williams, L., Bases for cluster algebras from surfaces, Compositio Math. 149 (2013), 217–263.Google Scholar
Palu, Y., Cluster characters for 2-Calabi–Yau triangulated categories, Ann. Inst. Fourier (Grenoble) 58 (2008), 2221–2248.Google Scholar
Plamondon, P.-G., Cluster algebras via cluster categories with infinite-dimensional morphism spaces, Compositio Math. 147 (2011), 1921–1954.Google Scholar
Plamondon, P.-G., Cluster characters for cluster categories with infinite-dimensional morphism spaces, Adv. Math. 227 (2011), 1–39.Google Scholar
Reiten, I., Tilting theory and cluster algebras, Proc. Trieste Workshop, Preprint (2008), arXiv: 1012.6014.Google Scholar
Reiten, I., Proceedings of the International Congress of Mathematicians, Volume I (Hindustan Book Agency, New Delhi, 2010), 558–594.Google Scholar