Acyclic cluster algebras revisited
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]).
Cite
@article{arxiv.1203.0277,
title = {Acyclic cluster algebras revisited},
author = {David Speyer and Hugh Thomas},
journal= {arXiv preprint arXiv:1203.0277},
year = {2012}
}
Comments
23 pages