Journal of Algebraic Combinatorics

, Volume 36, Issue 3, pp 475–500 | Cite as

Cluster-additive functions on stable translation quivers

Article

Abstract

Additive functions on translation quivers have played an important role in the representation theory of finite-dimensional algebras, the most prominent ones are the hammock functions introduced by S. Brenner. When dealing with cluster categories (and cluster-tilted algebras), one should look at a corresponding class of functions defined on stable translation quivers, namely the cluster-additive ones. We conjecture that the cluster-additive functions on a stable translation quiver of Dynkin type \(\mathbb{A}_{n}, \mathbb{D}_{n}, \mathbb{E}_{6}, \mathbb {E}_{7}, \mathbb{E}_{8}\) are non-negative linear combinations of cluster-hammock functions (with index set a tilting set). The present paper provides a first study of cluster-additive functions and gives a proof of the conjecture in the case \(\mathbb{A}_{n}\).

Keywords

Translation quiver Additive function Cluster-additive function Hammocks Cluster-hammocks Dynkin quiver Cluster category Cluster-tilted algebra 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.King Abdulaziz UniversityJeddahSaudi Arabia

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