Two Formulas for $F$-Polynomials

TL;DR

This paper presents a product formula for F-polynomials in cluster algebras, providing two proofs and an explicit combinatorial expansion that aids in their computation.

Contribution

It introduces a new product formula for F-polynomials with two different proofs and an explicit sum expansion for practical computation.

Findings

Provides a product formula for F-polynomials

Offers two proofs: inductive and based on Fock-Goncharov decomposition

Includes an explicit combinatorial expansion for calculations

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 $\mathbf{c}$-vectors and $\mathbf{g}$-vectors along a finite sequence of mutations from the initial seed to the given seed.