Two Formulas for F-Polynomials

Abstract

Abstract We discuss a product formula for $F$-polynomials in cluster algebras and provide two proofs. One proof is inductive and uses only the mutation rule for $F$-polynomials. The other is based on the Fock–Goncharov decomposition of mutations. We conclude by expanding this product formula as a sum and illustrate applications. This expansion provides an explicit combinatorial computation of $F$-polynomials in a given seed that depends only on the $\textbf {c}$-vectors and $\textbf {g}$-vectors along a finite sequence of mutations from the initial seed to the given seed.