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arXiv · 2112.11839

Two Formulas for $F$-Polynomials

Feiyang Lin·Gregg Musiker·Tomoki Nakanishi
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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.

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Feiyang Lin, Gregg Musiker, Tomoki Nakanishi. 2023-05-17. Two Formulas for $F$-Polynomials. https://doi.org/10.1093/imrn%2Frnad074

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