Acyclic Cluster Algebras Revisited
We describe a new way to relate an acyclic, skew-symmetrizable cluster algebra to the representation theory of a finite dimensional hereditary algebra. This approach is designed to explain the c-vectors of the cluster algebra. We obtain a necessary and su
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Abstract We describe a new way to relate an acyclic, skew-symmetrizable cluster algebra to the representation theory of a finite dimensional hereditary algebra. This approach is designed to explain the c-vectors of the cluster algebra. We obtain a necessary and sufficient combinatorial criterion for a collection of vectors to be the c-vectors of some cluster in the cluster algebra associated to a given skewsymmetrizable matrix. Our approach also yields a simple proof of the known result that the c-vectors of an acyclic cluster algebra are sign-coherent, from which Nakanishi and Zelevinsky have showed that it is possible to deduce in an elementary way several important facts about cluster algebras.
1 Introduction 0 be the 2n × n Let B 0 be an acyclic skew-symmetrizable n × n integer matrix. Let B 0 matrix whose top half is B and whose bottom half is an n × n identity matrix. We consider an infinite n-ary tree Tn , with each edge labelled by a number from 1 to n, such that at each vertex, there is exactly one edge with each label. We label 0 to it. one vertex vb , and we associate the matrix B There is an operation called matrix mutation which plays a fundamental role in the construction of cluster algebras. (We recall the definition in Sect. 3.) Using this definition, it is possible to associate a 2n × n matrix to each vertex of Tn , so that if two vertices are joined by an edge labelled i, the corresponding matrices are related by matrix mutation in the i-th position.
Dedicated to Idun Reiten on the occasion of her seventieth birthday. D. Speyer (B) Department of Mathematics, University of Michigan, Ann Arbor, MI, USA e-mail: [email protected] H. Thomas Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, Canada e-mail: [email protected] A.B. Buan et al. (eds.), Algebras, Quivers and Representations, Abel Symposia 8, DOI 10.1007/978-3-642-39485-0_12, © Springer-Verlag Berlin Heidelberg 2013
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v for the associated (2n × n) B-matrix, and B v for its top Let v ∈ Tn . We write B half. The c-vectors for v, denoted c1v , . . . , cnv are by definition the columns of the v . bottom half of B It has recently been understood that the c-vectors play an important role in the behaviour of a cluster algebra associated to B 0 . Nakanishi and Zelevinsky showed in [21] that, once it is established that the c-vectors are sign-coherent, meaning that, for each c-vector, either all the entries are non-negative or all are non-positive, then several fundamental results on the corresponding cluster algebra follow by an elementary argument (specifically, Conjectures 1.1–1.4 of [10]). In this paper, we give a representation-theoretic interpretation of the c-vectors as classes in the Grothendieck group of indecomposable objects in the bounded derived category of a hereditary abelian category. Their sign-coherence is an immediate consequence of this description. We use our representation-theoretic interpretation of c-vectors to give a purely combinatorial description of w
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