Crystal Structure of Upper Cluster Algebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910744054857728 |
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| author | Fei, Jiarui |
| author_facet | Fei, Jiarui |
| contents | We describe the upper seminormal crystal structure for the $μ$-supported $δ$-vectors for any quiver with potential with reachable frozen vertices, or equivalently for the tropical points of the corresponding cluster $\mc{X}$-variety. We show that the crystal structure can be algebraically lifted to the generic basis of the upper cluster algebra. This can be viewed as an additive categorification of the crystal structure arising from cluster algebras. We introduce the biperfect bases in the cluster algebra setting and give a description of all biperfect bases, which are parametrized by lattice points in a product of polyhedral sets. We illustrate this theory from classical examples and new examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_08326 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Crystal Structure of Upper Cluster Algebras Fei, Jiarui Representation Theory Combinatorics Rings and Algebras Primary 13F60, Secondary 05E10, 16G10 We describe the upper seminormal crystal structure for the $μ$-supported $δ$-vectors for any quiver with potential with reachable frozen vertices, or equivalently for the tropical points of the corresponding cluster $\mc{X}$-variety. We show that the crystal structure can be algebraically lifted to the generic basis of the upper cluster algebra. This can be viewed as an additive categorification of the crystal structure arising from cluster algebras. We introduce the biperfect bases in the cluster algebra setting and give a description of all biperfect bases, which are parametrized by lattice points in a product of polyhedral sets. We illustrate this theory from classical examples and new examples. |
| title | Crystal Structure of Upper Cluster Algebras |
| topic | Representation Theory Combinatorics Rings and Algebras Primary 13F60, Secondary 05E10, 16G10 |
| url | https://arxiv.org/abs/2309.08326 |