The University of Michigan Combinatorics Seminar
Winter 2009
February 13, 4:10-5:00, 3866 East Hall

Skew-symmetric cluster algebras of finite mutation type

Michael Shapiro

Michigan State University


Fomin and Zelevinsky obtained a Cartan-Killing type classification of cluster algebras of finite type, i.e. cluster algebras with finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type, which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In a joint work with Anna Felikson and Pavel Tumarkin, we classify all cluster algebras of finite mutation type with skew-symmetric exchange matrices. We show that besides cluster algebras of rank 2 and cluster algebras associated with triangulations of surfaces, there are exactly 11 exceptional skew-symmetric cluster algebras of finite mutation type. To be specific, 9 of them are associated with finite, affine, and elliptic root systems of exceptional types E6, E7, and E8, and the remaining two were recently found by Derksen and Owen. We also describe a criterion which determines if a skew-symmetric cluster algebra is of finite mutation type, and discuss growth rates of cluster algebras.