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Journal of the London Mathematical Society Advance Access published online on March 16, 2009
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn082
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© 2009 London Mathematical Society

Denominators of cluster variables

Aslak Bakke Buan

Institutt for Matematiske Fag
Norges Teknisk-Naturvitenskapelige Universitet
N-7491 Trondheim
Norway
aslakb@math.ntnu.no

Robert J. Marsh

School of Mathematics
University of Leeds
Leeds
LS2 9JT
United Kingdom

Idun Reiten

Institutt for Matematiske Fag
Norges Teknisk-Naturvitenskapelige Universitet
N-7491 Trondheim
Norway
idunr@math.ntnu.no

Associated to any acyclic cluster algebra is a corresponding triangulated category known as the cluster category. It is known that there is a one-to-one correspondence between cluster variables in the cluster algebra and exceptional indecomposable objects in the cluster category, inducing a correspondence between clusters and cluster-tilting objects.

Fix a cluster-tilting object T and a corresponding initial cluster. By the Laurent phenomenon, every cluster variable can be written as a Laurent polynomial in the initial cluster. We give conditions on T that are equivalent to the fact that the denominator in the reduced form for every cluster variable in the cluster algebra has exponents given by the dimension vector of the corresponding module over the endomorphism algebra of T.

The authors were supported by Storforsk grant no. 167130 from the Norwegian Research Council. The second author was supported by the EPSRC, grant no. EP/C01040X.

2000 Mathematics Subject Classification 16G20, 16S99 (primary), 16G70, 16E99, 17B99 (secondary).

Received October 23, 2007; revised August 19, 2008;
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