Cluster-additive functions on stable translation quivers |
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Authors: | Claus Michael Ringel |
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Institution: | 1. Fakult?t f??r Mathematik, Universit?t Bielefeld, P.O. Box 100?131, 33? 501, Bielefeld, Germany 2. King Abdulaziz University, P.O. Box 80200, Jeddah, Saudi Arabia
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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 $\mathbb{A}_{n}$ . |
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