Greedy elements in rank 2 cluster algebras. (English) Zbl 1295.13031

Cluster algebras, invented by S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)] in order to study total positivity in algebraic groups and canonical bases in quantum groups, are a class of commutative algebras endowed with a distinguished set of generators, the cluster variables. The cluster variables are grouped into finite subsets, called clusters, and are defined recursively from initial variables through mutation on the clusters. A lot of recent activity in the field has been directed towards various constructions of “natural” bases in cluster algebras. One of the approaches to this problem was developed by P. Sherman and A. Zelevinsky [Mosc. Math. J. 4, No. 4, 947–974 (2004; Zbl 1103.16018)] who have shown that the indecomposable positive elements form an integer basis in any rank \(2\) cluster algebra of finite or affine type and suspected that this property does not extend beyond affine types. Sherman and Zelevinsky introduced a special family of {greedy elements} in (the completion of) an arbitrary rank 2 cluster algebra \(\mathcal{A}\), and made several conjectures about them, including the claim that all these elements are indecomposable positive elements, and that they form a \(\mathbb{Z}\)-basis in \(\mathcal{A}\).
In the present paper, the authors study greedy elements and proved all the conjectures mentioned above. In particularly, the authors constructed a new basis in any rank 2 cluster algebra called the greedy basis which consists of a special family of indecomposable positive elements called greedy elements. The key new ingredient is an explicit combinatorial expression for greedy elements inspired by an expression for cluster variables given in [K. Lee and R. Schiffler, J. Algebr. Comb. 37, No. 1, 67–85 (2013; Zbl 1266.13017); Compos. Math. 148, No. 6, 1821–1832 (2012; Zbl 1266.16027); D. Rupel, C. R., Math., Acad. Sci. Paris 350, No. 21–22, 929–932 (2012; Zbl 1266.16028)].


13F60 Cluster algebras
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