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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2112.11839 |
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| _version_ | 1866914860095242240 |
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| author | Lin, Feiyang Musiker, Gregg Nakanishi, Tomoki |
| author_facet | Lin, Feiyang Musiker, Gregg Nakanishi, Tomoki |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_11839 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Two Formulas for $F$-Polynomials Lin, Feiyang Musiker, Gregg Nakanishi, Tomoki Combinatorics 05E16 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. |
| title | Two Formulas for $F$-Polynomials |
| topic | Combinatorics 05E16 |
| url | https://arxiv.org/abs/2112.11839 |