Geodesic
Cluster-additive functions on stable translation quivers
Journal of Algebraic Combinatorics, Tome 36 (2012) no. 3, pp. 475-500.

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Summary: 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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     author = {Ringel, Claus Michael},
     title = {Cluster-additive functions on stable translation quivers},
     journal = {Journal of Algebraic Combinatorics},
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     number = {3},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2012__36_3_a0/}
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Ringel, Claus Michael. Cluster-additive functions on stable translation quivers. Journal of Algebraic Combinatorics, Tome 36 (2012) no. 3, pp. 475-500. http://geodesic.mathdoc.fr/item/JAC_2012__36_3_a0/