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Reflection group relations arising from cluster algebras


Author: Ahmet I. Seven
Journal: Proc. Amer. Math. Soc. 144 (2016), 4641-4650
MSC (2010): Primary 05E15
Published electronically: May 6, 2016
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Abstract: There is a well-known analogy between cluster algebras and Kac-moody algebras: roughly speaking, Kac-Moody algebras are associated with symmetrizable generalized Cartan matrices while cluster algebras correspond to skew-symmetrizable matrices. In this paper, we study an interplay between these two classes of matrices. We obtain relations in the Weyl groups of Kac-Moody algebras that come from mutation classes of skew-symmetrizable matrices. More precisely, we establish a set of relations satisfied by the reflections of the so-called companion bases; these include c-vectors, which parametrize coefficients in a cluster algebra with principal coefficients. These relations generalize the relations obtained by Barot and Marsh for finite type. For affine type, we also show that the reflections of the companion bases satisfy the relations obtained by Felikson and Tumarkin. As an application, we obtain some combinatorial properties of the mutation classes of skew-symmetrizable matrices.


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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/proc/13157
Received by editor(s): April 9, 2014
Received by editor(s) in revised form: January 13, 2016
Published electronically: May 6, 2016
Additional Notes: The author’s research was supported in part by the Scientific and Technological Research Council of Turkey (TUBITAK) grant # 113F138
Communicated by: Harm Derksen
Article copyright: © Copyright 2016 American Mathematical Society