CLUSTER ALGEBRAS AND SYMMETRIC MATRICES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, vol.143, pp.469-478, 2015 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 143
- Publication Date: 2015
- Journal Name: PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Page Numbers: pp.469-478
- Middle East Technical University Affiliated: Yes
Abstract
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.