Abstract
A lot of recent activity in the theory of cluster algebras has been directed toward various constructions of “natural” bases in them. One of the approaches to this problem was developed several years ago by Sherman and Zelevinsky who have shown that the indecomposable positive elements form an integer basis in any rank 2 cluster algebra of finite or affine type. It is strongly suspected (but not proved) that this property does not extend beyond affine types. Here, we go around this difficulty by constructing a new basis in any rank 2 cluster algebra that we call the greedy basis. It consists of a special family of indecomposable positive elements that we call greedy elements. Inspired by a recent work of Lee and Schiffler; Rupel, we give explicit combinatorial expressions for greedy elements using the language of Dyck paths.
Similar content being viewed by others
References
Berenstein, A., Fomin, S., Zelevinsky, A.: Cluster algebras III: upper bounds and double Bruhat cells. Duke Math. J. 126(1), 1–52 (2005)
Berenstein, A., Zelevinsky, A.: Triangular bases in quantum cluster algebras. Intern. Math. Res. Notices (to appear).
Fomin, S., Zelevinsky, A.: The Laurent phenomenon. Adv. Appl. Math. 28, 119–144 (2002a)
Fomin, S., Zelevinsky, A.: Cluster algebras I: foundations. J. Am. Math. Soc. 15, 497–529 (2002b)
Fomin, S., Zelevinsky, A.: Cluster algebras IV: coefficients. Compos. Math. 143, 112–164 (2007)
Kac, V.: Infinite Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)
Lee, K., Schiffler, R.: A combinatorial formula for rank 2 cluster variables. J. Algebraic Comb. (to appear). http://www.springerlink.com/content/66v123492t622337
Lee, K., Schiffler, R.: Proof of a positivity conjecture of M. Kontsevich on noncommutative cluster variables. Compos. Math. (to appear). http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8730703&fulltextType=RA&fileId=S0010437X12000425
Plamondon, P.-G.: Generic bases for cluster algebras from the cluster category. Intern. Math. Res. Notices (to appear). http://imrn.oxfordjournals.org/content/early/2012/04/11/imrn.rns102.short?rss=1
Rupel, D.: Proof of the Kontsevich non-commutative cluster positivity conjecture. Comptes Rendus Math. 350(21–22), 929–932 (2012)
Sherman, P., Zelevinsky, A.: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types. Mosc. Math. J. 4(4), 947–974 (2004)
Acknowledgments
This paper owes a lot to Paul Sherman. Definition 1.3 and Proposition 1.5 first appeared in an unpublished follow-up to [11] (with the same authors), and Proposition 1.6, Theorem 1.7, and Proposition 1.8 were stated there as conjectures. This was almost a decade ago when Paul was a Master’s student at Northeastern under the guidance of the third author. After getting his degree, Paul has left academia to pursue other interests. If he ever decides to come back to research in mathematics, he is very welcome!We are grateful to Gregg Musiker, Dylan Rupel, and Ralf Schiffler for valuable discussions and to the anonymous referee for a very thorough reading of the paper and many useful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by NSF grants DMS-0901367 (K. L.) and DMS-1103813 (A. Z.)
Rights and permissions
About this article
Cite this article
Lee, K., Li, L. & Zelevinsky, A. Greedy elements in rank 2 cluster algebras. Sel. Math. New Ser. 20, 57–82 (2014). https://doi.org/10.1007/s00029-012-0115-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-012-0115-1