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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Cluster algebras and symmetric matrices
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by Ahmet I. Seven PDF
Proc. Amer. Math. Soc. 143 (2015), 469-478 Request permission

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.
References
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Additional Information
  • Ahmet I. Seven
  • Affiliation: Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey
  • MR Author ID: 764933
  • Email: aseven@metu.edu.tr
  • 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
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 3283637