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Greedy elements in rank 2 cluster algebras

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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.

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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 Definition 1.3 and Proposition 1.5 first appeared in an unpublished follow-up to [11] (with the same authors), and Proposition 1.6, Theorem 1.7, and Proposition 1.8 were stated there as conjectures.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.

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Correspondence to Andrei Zelevinsky.

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Research supported in part by NSF grants DMS-0901367 (K. L.) and DMS-1103813 (A. Z.)

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Lee, K., Li, L. & Zelevinsky, A. Greedy elements in rank 2 cluster algebras. Sel. Math. New Ser. 20, 57–82 (2014).

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