Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



Cluster algebras and symmetric matrices

Author: Ahmet I. Seven
Journal: Proc. Amer. Math. Soc. 143 (2015), 469-478
MSC (2010): Primary 05E15; Secondary 13F60
Published electronically: October 23, 2014
MathSciNet review: 3283637
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Michael Barot, Christof Geiss, and Andrei Zelevinsky, Cluster algebras of finite type and positive symmetrizable matrices, J. London Math. Soc. (2) 73 (2006), no. 3, 545-564. MR 2241966 (2007i:05190),
  • [2] M. Barot and R. Marsh, Reflection group presentations arising from cluster algebras. arXiv:1112.2300v1 (2011).
  • [3] Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten, Cluster-tilted algebras, Trans. Amer. Math. Soc. 359 (2007), no. 1, 323-332 (electronic). MR 2247893 (2007f:16035),
  • [4] Aslak Bakke Buan, Idun Reiten, and Ahmet I. Seven, Tame concealed algebras and cluster quivers of minimal infinite type, J. Pure Appl. Algebra 211 (2007), no. 1, 71-82. MR 2333764 (2008f:16039),
  • [5] Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky, Quivers with potentials and their representations II: applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), no. 3, 749-790. MR 2629987 (2012c:16044),
  • [6] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), no. 1, 112-164. MR 2295199 (2008d:16049),
  • [7] Martin Herschend and Osamu Iyama, Selfinjective quivers with potential and 2-representation-finite algebras, Compos. Math. 147 (2011), no. 6, 1885-1920. MR 2862066,
  • [8] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219 (92k:17038)
  • [9] T. Nakanisihi and A. Zelevinsky, On tropical dualities in acyclic cluster algebras, Proceedings of the Representation Theory of Algebraic Groups and Quantum Groups, 10, Contemp. Math. 565 (2012), 217-226.
  • [10] M. J. Parsons, Companion bases for cluster-tilted algebras, Algebras and Representation Theory, June 2014, volume 17, issue 3, pp. 775-808.
  • [11] N. Reading and D. Speyer, Combinatorial frameworks for cluster algebras, arXiv:1111.2652v1 (2011).
  • [12] Ahmet I. Seven, Cluster algebras and semipositive symmetrizable matrices, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2733-2762. MR 2763735 (2012c:13056),
  • [13] D. Speyer and H. Thomas, Acyclic cluster algebras revisited, Algebras, quivers and representations, Abel Symp., 8, Springer, Heidelberg, 2013, pp. 275-298. MR 3183889

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 05E15, 13F60

Retrieve articles in all journals with MSC (2010): 05E15, 13F60

Additional Information

Ahmet I. Seven
Affiliation: Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey

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.

American Mathematical Society