Caldero–Keller Approach to the Denominators of Cluster Variables

Abstract

Buan, Marsh, and Reiten proved that if a cluster-tilting object T in a cluster category 𝒞 associated to an acyclic quiver Q satisfies certain conditions with respect to the exchange pairs in 𝒞, then the denominator in its reduced form of every cluster variable in the cluster algebra associated to Q has exponents given by the dimension vector of the corresponding module over the endomorphism algebra of T. In this article, we give an alternative proof of this result using the Caldero–Keller approach to acyclic cluster algebras and the work of Palu on cluster characters.

Key Words:

• Cluster algebras
• Cluster characters
• Denominators of cluster variables

2000 Mathematics Subject Classification:

• 16G20
• 16S99
• 16G70

ACKNOWLEDGMENTS

The author would like to thank Idun Reiten for corrections and interesting remarks concerning the subject. He would also like to thank the coordinators of the Liegrits network for organizing his stay at the NTNU in Trondheim where this article was written.

Notes

Communicated by C. Cibils.