Relation between $f$-vectors and $d$-vectors in cluster algebras of finite type or rank 2

Abstract

We study $f$-vectors, which are the maximal degree vectors of $F$-polynomials in cluster algebra theory. For a cluster algebra is of finite type, we find that positive $f$-vectors correspond with $d$-vectors, which are exponent vectors of denominators of cluster variables. Furthermore, using this correspondence and properties of $d$-vectors, we prove that cluster variables in a cluster are uniquely determined by their $f$-vectors when the cluster algebra is of finite type or rank $2$.