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JOURNAL OF
ALGEBRAIC
COMBINATORICS
  Editors-in-chief: C. A. Athanasiadis, T. Lam, A. Munemasa, H. Van Maldeghem
ISSN 0925-9899 (print) • ISSN 1572-9192 (electronic)
 

Cluster-additive functions on stable translation quivers

Claus Michael Ringel
Fakultät für Mathematik, Universität Bielefeld, P.O. Box 100 hinspace 131, 33 hinspace 501, Bielefeld, Germany

DOI: 10.1007/s10801-012-0346-4

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 $\mathbb{A}_{n}$ .

Pages: 475–500

Keywords: translation quiver; additive function; cluster-additive function; hammocks; cluster-hammocks; Dynkin quiver; cluster category; cluster-tilted algebra

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References

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