CLUSTER ALGEBRAS AND SYMMETRIC MATRICES

2015-02-01
In the structural theory of cluster algebras, a crucial role is played by a family of integer vectors, called c-vectors, which parametrize the coefficients. It has recently been shown that each 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 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.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY

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Citation Formats
A. İ. Seven, “CLUSTER ALGEBRAS AND SYMMETRIC MATRICES,” PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, pp. 469–478, 2015, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/55361.