## Cluster algebras and symmetric matrices

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- 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

- 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**, DOI 10.1112/S0024610706022769 - M. Barot and R. Marsh,
*Reflection group presentations arising from cluster algebras*. arXiv:1112.2300v1 (2011). - Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten,
*Cluster-tilted algebras*, Trans. Amer. Math. Soc.**359**(2007), no. 1, 323–332. MR**2247893**, DOI 10.1090/S0002-9947-06-03879-7 - 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**, DOI 10.1016/j.jpaa.2006.12.007 - 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**, DOI 10.1090/S0894-0347-10-00662-4 - Sergey Fomin and Andrei Zelevinsky,
*Cluster algebras. IV. Coefficients*, Compos. Math.**143**(2007), no. 1, 112–164. MR**2295199**, DOI 10.1112/S0010437X06002521 - Martin Herschend and Osamu Iyama,
*Selfinjective quivers with potential and 2-representation-finite algebras*, Compos. Math.**147**(2011), no. 6, 1885–1920. MR**2862066**, DOI 10.1112/S0010437X11005367 - Victor G. Kac,
*Infinite-dimensional Lie algebras*, 3rd ed., Cambridge University Press, Cambridge, 1990. MR**1104219**, DOI 10.1017/CBO9780511626234 - 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. - M. J. Parsons,
*Companion bases for cluster-tilted algebras*, Algebras and Representation Theory, June 2014, volume 17, issue 3, pp. 775–808. - N. Reading and D. Speyer,
*Combinatorial frameworks for cluster algebras*, arXiv:1111.2652v1 (2011). - Ahmet I. Seven,
*Cluster algebras and semipositive symmetrizable matrices*, Trans. Amer. Math. Soc.**363**(2011), no. 5, 2733–2762. MR**2763735**, DOI 10.1090/S0002-9947-2010-05255-9 - David Speyer and Hugh Thomas,
*Acyclic cluster algebras revisited*, Algebras, quivers and representations, Abel Symp., vol. 8, Springer, Heidelberg, 2013, pp. 275–298. MR**3183889**, DOI 10.1007/978-3-642-39485-0_{1}2

## 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