Clusters and seeds in acyclic cluster algebras

Bakke Buan, Aslak; Marsh, Bethany; Reiten, Idun; Todorov, Gordana

doi:10.1090/S0002-9939-07-08801-6

Clusters and seeds in acyclic cluster algebras
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by Aslak Bakke Buan, Bethany R. Marsh, Idun Reiten and Gordana Todorov; with an Appendix coauthored in addition by P. Caldero; B. Keller PDF

Proc. Amer. Math. Soc. 135 (2007), 3049-3060

Abstract:

Cluster algebras are commutative algebras that were introduced by Fomin and Zelevinsky in order to model the dual canonical basis of a quantum group and total positivity in algebraic groups. Cluster categories were introduced as a representation-theoretic model for cluster algebras. In this article we use this representation-theoretic approach to prove a conjecture of Fomin and Zelevinsky, that for cluster algebras with no coefficients associated to quivers with no oriented cycles, a seed is determined by its cluster. We also obtain an interpretation of the monomial in the denominator of a non-polynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra.

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