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arXiv · 1203.0277

Acyclic cluster algebras revisited

David Speyer·Hugh Thomas
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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 skew-symmetrizable 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 (specifically: Conjectures 1.1-1.4 of [Derksen-Weyman-Zelevinsky]).

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David Speyer, Hugh Thomas. 2012-03-01. Acyclic cluster algebras revisited. https://arxiv.org/abs/1203.0277

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