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Cluster algebras and symmetric matrices


Author: Ahmet I. Seven
Journal: Proc. Amer. Math. Soc. 143 (2015), 469-478
MSC (2010): Primary 05E15; Secondary 13F60
DOI: https://doi.org/10.1090/S0002-9939-2014-12252-0
Published electronically: October 23, 2014
MathSciNet review: 3283637
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Abstract: In the structural theory of cluster algebras, a crucial role is played by a family of integer vectors, called $ \mathbf {c}$-vectors, which parametrize the coefficients. It has recently been shown that each $ \mathbf {c}$-vector with respect to an acyclic initial seed is a real root of the corresponding root system. In this paper, we obtain an interpretation of this result in terms of symmetric matrices. We show that for skew-symmetric cluster algebras, the $ \mathbf {c}$-vectors associated with any seed defines a quasi-Cartan companion for the corresponding exchange matrix (i.e. they form a companion basis), and we establish some basic combinatorial properties. In particular, we show that these vectors define an admissible cut of edges in the associated quivers.


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Additional Information

Ahmet I. Seven
Affiliation: Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey
Email: aseven@metu.edu.tr

DOI: https://doi.org/10.1090/S0002-9939-2014-12252-0
Received by editor(s): April 11, 2012
Received by editor(s) in revised form: February 1, 2013, and April 10, 2013
Published electronically: October 23, 2014
Additional Notes: The author’s research was supported in part by the Scientific and Technological Research Council of Turkey (TUBITAK) grant #110T207
Communicated by: Harm Derksen
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.