At the beginning of this century, Fomin and Zelevinsky invented a new class of algebras called cluster algebras motivated by total positivity in algebraic groups and canonical bases in quantum groups. Since their introduction, cluster algebras have found application in a diverse variety of settings which include Poisson geometry, Teichmüller theory, tropical geometry, algebraic combinatorics and last not least the representation theory of quivers and finite dimensional algebras. The structure of cluster algebras is to a large extent controlled by a family of integer vectors called denominator vectors. In this talk, we give an interpretation of denominator vectors in terms of representations of finite dimensional algebras. Using this interpretation, we confirm several conjectures about denominator vectors for cluster algebras of type C.
This is joint work with C. J. Fu and S. F. Geng.