Real $C$-, $G$-structures and sign-coherence of cluster algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914165966241792 |
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| author | Akagi, Ryota Chen, Zhichao |
| author_facet | Akagi, Ryota Chen, Zhichao |
| contents | We generalize the theory of integer $C$-, $G$-matrices in cluster algebras to the real case. By a skew-symmetrizing method, we can reduce the problem of skew-symmetrizable patterns to the one of skew-symmetric patterns. In this sense, we extend the sign-coherence of integer $C$-, $G$-matrices proved by Gross-Hacking-Keel-Kontsevich to a more general real class called of quasi-integer type. Furthermore, we give a complete classification of this type by a combinatorial method of real weighted quivers. However, the sign-coherence of real $C$-, $G$-matrices does not always hold in general. For this purpose, we classify all the rank $2$ case and the finite type case via the Coxeter diagrams. We also give two conjectures about the real exchange matrices and $C$-, $G$-matrices. Under these conjectures, the dual mutation, $G$-fan structure and synchronicity property hold. As an application, the isomorphism of several kinds of exchange graphs is studied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06486 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Real $C$-, $G$-structures and sign-coherence of cluster algebras Akagi, Ryota Chen, Zhichao Representation Theory Combinatorics Rings and Algebras 13F60, 05E10, 20F55 We generalize the theory of integer $C$-, $G$-matrices in cluster algebras to the real case. By a skew-symmetrizing method, we can reduce the problem of skew-symmetrizable patterns to the one of skew-symmetric patterns. In this sense, we extend the sign-coherence of integer $C$-, $G$-matrices proved by Gross-Hacking-Keel-Kontsevich to a more general real class called of quasi-integer type. Furthermore, we give a complete classification of this type by a combinatorial method of real weighted quivers. However, the sign-coherence of real $C$-, $G$-matrices does not always hold in general. For this purpose, we classify all the rank $2$ case and the finite type case via the Coxeter diagrams. We also give two conjectures about the real exchange matrices and $C$-, $G$-matrices. Under these conjectures, the dual mutation, $G$-fan structure and synchronicity property hold. As an application, the isomorphism of several kinds of exchange graphs is studied. |
| title | Real $C$-, $G$-structures and sign-coherence of cluster algebras |
| topic | Representation Theory Combinatorics Rings and Algebras 13F60, 05E10, 20F55 |
| url | https://arxiv.org/abs/2509.06486 |