Tropical friezes and cluster-additive functions via Fock-Goncharov duality and a conjecture of Ringel
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| author | Cao, Peigen Germain, Antoine de Saint Lu, Jiang-Hua |
| author_facet | Cao, Peigen Germain, Antoine de Saint Lu, Jiang-Hua |
| contents | We study tropical friezes and cluster-additive functions associated to symmetrizable generalized Cartan matrices in the framework of Fock-Goncharov duality in cluster algebras. In particular, we generalize and prove a conjecture of C. M. Ringel on cluster-additive functions associated to arbitrary Cartan matrices of finite type. For a Fock-Goncharov dual pair of positive spaces of finite type, we use tropical friezes and cluster-additive functions to explicitly express the Fock-Goncharov pairing between their tropical points and the bijections between global monomials on one and tropical points of the other. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10330 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tropical friezes and cluster-additive functions via Fock-Goncharov duality and a conjecture of Ringel Cao, Peigen Germain, Antoine de Saint Lu, Jiang-Hua Representation Theory Rings and Algebras 13F60 We study tropical friezes and cluster-additive functions associated to symmetrizable generalized Cartan matrices in the framework of Fock-Goncharov duality in cluster algebras. In particular, we generalize and prove a conjecture of C. M. Ringel on cluster-additive functions associated to arbitrary Cartan matrices of finite type. For a Fock-Goncharov dual pair of positive spaces of finite type, we use tropical friezes and cluster-additive functions to explicitly express the Fock-Goncharov pairing between their tropical points and the bijections between global monomials on one and tropical points of the other. |
| title | Tropical friezes and cluster-additive functions via Fock-Goncharov duality and a conjecture of Ringel |
| topic | Representation Theory Rings and Algebras 13F60 |
| url | https://arxiv.org/abs/2405.10330 |