Fractal phenomenon in $c$- and $g$-vectors of the Markov quiver
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914552072896512 |
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| author | Akagi, Ryota Chen, Zhichao |
| author_facet | Akagi, Ryota Chen, Zhichao |
| contents | We study the $C$- and $G$-patterns associated with rank $3$ skew-symmetrizable matrices of $B$-invariant type, including the Markov quiver. Motivated by the self-contained simple mutations in Markov-type cluster algebras, we prove that large classes of subpatterns of modified $c$- and $g$-vectors are linearly isomorphic, yielding a fractal structure of the corresponding $G$-fan. We further derive explicit recursive formulas for all modified $c$- and $g$-vectors in terms of integer pairs satisfying a recursion analogous to the Calkin-Wilf tree, which leads to a parameterization by coprime integers. As an application, we describe all connected components of the complement of the support of the $G$-fan, and show that they are generated recursively by three kinds of linear maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10072 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fractal phenomenon in $c$- and $g$-vectors of the Markov quiver Akagi, Ryota Chen, Zhichao Representation Theory Combinatorics 13F60, 05E10, 11D09 We study the $C$- and $G$-patterns associated with rank $3$ skew-symmetrizable matrices of $B$-invariant type, including the Markov quiver. Motivated by the self-contained simple mutations in Markov-type cluster algebras, we prove that large classes of subpatterns of modified $c$- and $g$-vectors are linearly isomorphic, yielding a fractal structure of the corresponding $G$-fan. We further derive explicit recursive formulas for all modified $c$- and $g$-vectors in terms of integer pairs satisfying a recursion analogous to the Calkin-Wilf tree, which leads to a parameterization by coprime integers. As an application, we describe all connected components of the complement of the support of the $G$-fan, and show that they are generated recursively by three kinds of linear maps. |
| title | Fractal phenomenon in $c$- and $g$-vectors of the Markov quiver |
| topic | Representation Theory Combinatorics 13F60, 05E10, 11D09 |
| url | https://arxiv.org/abs/2605.10072 |