Generic bases of skew-symmetrizable affine type cluster algebras
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915664266002432 |
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| author | Mou, Lang Su, Xiuping |
| author_facet | Mou, Lang Su, Xiuping |
| contents | Geiss, Leclerc and Schröer introduced a class of 1-Iwanaga-Gorenstein algebras $H$ associated to symmetrizable Cartan matrices with acyclic orientations, generalizing the path algebras of acyclic quivers. They also proved that indecomposable rigid $H$-modules of finite projective dimension are in bijection with non-initial cluster variables of the corresponding Fomin-Zelevinsky cluster algebra. In this article, we prove in all affine types that their conjectural Caldero-Chapoton type formula on these modules coincide with the Laurent expression of cluster variables. By taking generic Caldero-Chapoton functions on varieties of modules of finite projective dimension, we obtain bases for affine type cluster algebras with full-rank coefficients containing all cluster monomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03954 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generic bases of skew-symmetrizable affine type cluster algebras Mou, Lang Su, Xiuping Representation Theory 13F60 Geiss, Leclerc and Schröer introduced a class of 1-Iwanaga-Gorenstein algebras $H$ associated to symmetrizable Cartan matrices with acyclic orientations, generalizing the path algebras of acyclic quivers. They also proved that indecomposable rigid $H$-modules of finite projective dimension are in bijection with non-initial cluster variables of the corresponding Fomin-Zelevinsky cluster algebra. In this article, we prove in all affine types that their conjectural Caldero-Chapoton type formula on these modules coincide with the Laurent expression of cluster variables. By taking generic Caldero-Chapoton functions on varieties of modules of finite projective dimension, we obtain bases for affine type cluster algebras with full-rank coefficients containing all cluster monomials. |
| title | Generic bases of skew-symmetrizable affine type cluster algebras |
| topic | Representation Theory 13F60 |
| url | https://arxiv.org/abs/2409.03954 |