Y-frieze patterns
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911751093616640 |
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| author | Germain, Antoine de Saint |
| author_facet | Germain, Antoine de Saint |
| contents | Motivated by cluster ensembles, we introduce a new variant of frieze patterns associated to acyclic cluster algebras, which we call ${\bf Y}\textit{-frieze patterns}$. Using the mutation rules for ${\bf Y}$-variables, we define a large class of ${\bf Y}$-frieze patterns called $\textit{unitary }{\bf Y}\textit{-frieze patterns}$, and show that the ensemble map induces a map from (unitary) frieze patterns to (unitary) ${\bf Y}$-frieze patterns. In rank 2, we show that ${\bf Y}$-frieze patterns are (associated to) friezes of generalised cluster algebras. In finite type (not necessarily rank 2), we show that ${\bf Y}$-frieze patterns share the same symmetries as frieze patterns, and prove that their number is finite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_03073 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Y-frieze patterns Germain, Antoine de Saint Combinatorics Commutative Algebra Rings and Algebras 13F60 Motivated by cluster ensembles, we introduce a new variant of frieze patterns associated to acyclic cluster algebras, which we call ${\bf Y}\textit{-frieze patterns}$. Using the mutation rules for ${\bf Y}$-variables, we define a large class of ${\bf Y}$-frieze patterns called $\textit{unitary }{\bf Y}\textit{-frieze patterns}$, and show that the ensemble map induces a map from (unitary) frieze patterns to (unitary) ${\bf Y}$-frieze patterns. In rank 2, we show that ${\bf Y}$-frieze patterns are (associated to) friezes of generalised cluster algebras. In finite type (not necessarily rank 2), we show that ${\bf Y}$-frieze patterns share the same symmetries as frieze patterns, and prove that their number is finite. |
| title | Y-frieze patterns |
| topic | Combinatorics Commutative Algebra Rings and Algebras 13F60 |
| url | https://arxiv.org/abs/2311.03073 |