Research output: Contribution to journal › Journal article
|Journal publication date||2013|
|Number of pages||16|
In recent work with S. Launois, we introduced a framework for integer gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the cluster variables in an initial seed subject to a compatibility with the initial exchange matrix, and then one extends this to all cluster variables by mutation.
In this note, we classify gradings on cluster algebras of finite type A, D and E, in the coefficient-free setting and also for give some examples with coefficients. We find a close connection to properties of the associated cluster category and see that gradings on cluster algebras yield tropical frieze patterns on the Auslander-Reiten quiver of the cluster category.