Skew-symmetrizable cluster algebras from surfaces and symmetric quivers
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911377503813632 |
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| author | Ciliberti, Azzurra |
| author_facet | Ciliberti, Azzurra |
| contents | We study skew-symmetrizable cluster algebras $\mathcal{A}$ associated with unpunctured surfaces $\tilde{\mathbf{S}}$ endowed with an orientation-preserving involution $σ$. We give a geometric realization of such cluster algebras by showing that cluster variables of $\mathcal{A}$ correspond to $σ$-orbits of arcs of $\tilde{\mathbf{S}}$, while clusters are given by admissible $σ$-invariant triangulations. We establish a ring homomorphism from $\mathcal{A}$ to a skew-symmetric cluster algebra of the same rank, which is combinatorially derived from $\mathcal{A}$. We use this result to provide a cluster expansion formula for any $σ$-orbit $[γ]$ in terms of perfect matchings of some labeled modified snake graphs constructed from the arcs of $[γ]$. Then, we associate a symmetric finite-dimensional algebra $A$ to any seed of $\mathcal{A}$, such that non-initial cluster variables bijectively correspond to orthogonal indecomposable $A$-modules. Finally, we exhibit a purely representation-theoretic map from the category of orthogonal $A$-modules to $\mathcal{A}$, providing a Caldero-Chapoton map in this setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12247 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Skew-symmetrizable cluster algebras from surfaces and symmetric quivers Ciliberti, Azzurra Representation Theory Combinatorics Rings and Algebras 13F60, 16G20 We study skew-symmetrizable cluster algebras $\mathcal{A}$ associated with unpunctured surfaces $\tilde{\mathbf{S}}$ endowed with an orientation-preserving involution $σ$. We give a geometric realization of such cluster algebras by showing that cluster variables of $\mathcal{A}$ correspond to $σ$-orbits of arcs of $\tilde{\mathbf{S}}$, while clusters are given by admissible $σ$-invariant triangulations. We establish a ring homomorphism from $\mathcal{A}$ to a skew-symmetric cluster algebra of the same rank, which is combinatorially derived from $\mathcal{A}$. We use this result to provide a cluster expansion formula for any $σ$-orbit $[γ]$ in terms of perfect matchings of some labeled modified snake graphs constructed from the arcs of $[γ]$. Then, we associate a symmetric finite-dimensional algebra $A$ to any seed of $\mathcal{A}$, such that non-initial cluster variables bijectively correspond to orthogonal indecomposable $A$-modules. Finally, we exhibit a purely representation-theoretic map from the category of orthogonal $A$-modules to $\mathcal{A}$, providing a Caldero-Chapoton map in this setting. |
| title | Skew-symmetrizable cluster algebras from surfaces and symmetric quivers |
| topic | Representation Theory Combinatorics Rings and Algebras 13F60, 16G20 |
| url | https://arxiv.org/abs/2512.12247 |