Calculating entries of unitary $SL_3$-friezes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917639850295296 |
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| author | Surmann, Lucas |
| author_facet | Surmann, Lucas |
| contents | In this article we consider tame $ SL_3 $-friezes that arise by specializing a cluster of Plücker variables in the coordinate ring of the Grassmannian $ \mathscr{G}(3,n) $ to $ 1 $. We show how to calculate arbitrary entries of such friezes from the cluster in question. Let $ \mathscr{F} $ be such a cluster. We study the set $ \mathscr{F}_x $ of cluster variables in $ \mathscr{F} $ that share a given index $ x $ and derive a structure Theorem for $ \mathscr{F}_x $. These sets prove central to calculating the first and last non-trivial rows of the frieze. After that, simple recursive formulas can be used to calculate all remaining entries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09811 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Calculating entries of unitary $SL_3$-friezes Surmann, Lucas Combinatorics 13F60, 14M15, 05C99, 05E99, 51M20 In this article we consider tame $ SL_3 $-friezes that arise by specializing a cluster of Plücker variables in the coordinate ring of the Grassmannian $ \mathscr{G}(3,n) $ to $ 1 $. We show how to calculate arbitrary entries of such friezes from the cluster in question. Let $ \mathscr{F} $ be such a cluster. We study the set $ \mathscr{F}_x $ of cluster variables in $ \mathscr{F} $ that share a given index $ x $ and derive a structure Theorem for $ \mathscr{F}_x $. These sets prove central to calculating the first and last non-trivial rows of the frieze. After that, simple recursive formulas can be used to calculate all remaining entries. |
| title | Calculating entries of unitary $SL_3$-friezes |
| topic | Combinatorics 13F60, 14M15, 05C99, 05E99, 51M20 |
| url | https://arxiv.org/abs/2404.09811 |