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

## TL;DR

This paper explores the relationship between $f$-vectors and $d$-vectors in finite type and rank 2 cluster algebras, establishing a correspondence and uniqueness results for cluster variables.

## Contribution

It demonstrates a correspondence between positive $f$-vectors and $d$-vectors and proves the uniqueness of cluster variables determined by their $f$-vectors in specific cases.

## Key findings

- Positive $f$-vectors correspond with $d$-vectors in finite type cluster algebras.
- Cluster variables are uniquely determined by their $f$-vectors in finite type or rank 2 cases.

## 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 $2$.