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For every cluster-tilted algebra of simply-laced Dynkin type we
provide a companion basis which is strong, i.e., gives the set of dimension
vectors of the finitely generated indecomposable modules for the cluster-tilted
algebra. This shows in particular that every companion basis of a clustertilted
algebra of simply-laced Dynkin type is strong. Thus we give a proof of
Parsonsâs conjecture.
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