Greedy elements in rank 2 cluster algebras

By Kyungyong Lee, Li Li, Andrei Zelevinsky. 2012

51 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ed87a1a79403310925adc84699181a300fec625a

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Citations in the corpus

Introduction to Cluster Algebras (2018, background): “[36] For k ∈ Z \ {1, 2}, the cluster variable xk is given by xk = ∑”
Canonical bases for cluster algebras (2014, unknown): “ +2A −2 2 +A −2 1 A −2 2 +2A −2 1 +A1 A 2 2. In [CGMMRSW], it was shown that for any b,c, with Qlying in the ﬁrst quadrant, the ϑQ,mwith mranging over all elements of mcoincides with the greedy basis [LLZ13]. 44 MARKGROSS,PAULHACKING,SEANKEEL,ANDMAXIMKONTSEVICH Q A−1 1 A2 1+A2 2 A1A −1 A 2 −1 1 A2 1”
Triangular bases in quantum cluster algebras and monoidal categorification conjectures (2015, background): “of leading degrees for every seed t of a cluster algebra together with its dominance order ≺t. We then deﬁne and study pointed elements in the quantum torus T(t), 6 FANQIN whose name is borrowed from [LLZ14]. We recall the tropical transformation expected by [FG09]. We end the section by a natural d”
Positivity for cluster algebras (2013, background): “The main tools are the rank 2 formula from [22] (in the parametrization of [19]) and the rank 2 formula from [18].”