Two Formulas for $F$-Polynomials
Combinatorics
2024-07-09 v2
Abstract
We discuss a product formula for -polynomials in cluster algebras, and provide two proofs. One proof is inductive and uses only the mutation rule for -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 -polynomials in a given seed that depends only on the -vectors and -vectors along a finite sequence of mutations from the initial seed to the given seed.
Cite
@article{arxiv.2112.11839,
title = {Two Formulas for $F$-Polynomials},
author = {Feiyang Lin and Gregg Musiker and Tomoki Nakanishi},
journal= {arXiv preprint arXiv:2112.11839},
year = {2024}
}