Two Formulas for F-Polynomials

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 su...

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Bibliographic Details
Published in:International mathematics research notices 2024-01, Vol.2024 (1), p.613-634
Main Authors: Lin, Feiyang, Musiker, Gregg, Nakanishi, Tomoki
Format: Article
Language:English
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Summary: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.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnad074