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Mutation classes of skew-symmetrizable $ 3\times3$ matrices


Author: Ahmet I. Seven
Journal: Proc. Amer. Math. Soc. 141 (2013), 1493-1504
MSC (2010): Primary 05E15; Secondary 15B36, 05C22, 13F60
DOI: https://doi.org/10.1090/S0002-9939-2012-11477-7
Published electronically: September 27, 2012
MathSciNet review: 3020837
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Abstract: Mutation of skew-symmetrizable matrices is a fundamental operation that first arose in Fomin-Zelevinsky's theory of cluster algebras; it also appears naturally in many different areas of mathematics. In this paper, we study mutation classes of skew-symmetrizable $ 3\times 3$ matrices and associated graphs. We determine representatives for these classes using a natural minimality condition, generalizing and strengthening results of Beineke-Brustle-Hille and Felikson-Shapiro-Tumarkin. Furthermore, we obtain a new numerical invariant for the mutation operation on skew-symmetrizable matrices of arbitrary size.


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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/S0002-9939-2012-11477-7
Received by editor(s): March 3, 2011
Received by editor(s) in revised form: August 17, 2011, and August 26, 2011
Published electronically: September 27, 2012
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 2012 American Mathematical Society

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