Mathematics > Representation Theory
arXiv:2309.08326 (math)
[Submitted on 15 Sep 2023 (v1), last revised 14 Jun 2026 (this version, v3)]
Abstract:We describe the upper seminormal crystal structure for the $\mu$-supported $\delta$-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.
| Comments: | 65 pages, comments are welcome; v2. minor corrections, 10.3 deleted; v3. minor corrections, more details are given |
| Subjects: | Representation Theory (math.RT); Combinatorics (math.CO); Rings and Algebras (math.RA) |
| MSC classes: | Primary 13F60, Secondary 05E10, 16G10 |
| Cite as: | arXiv:2309.08326 [math.RT] |
| (or arXiv:2309.08326v3 [math.RT] for this version) | |
| https://doi.org/10.48550/arXiv.2309.08326 arXiv-issued DOI via DataCite |
Submission history
From: JiaRui Fei [view email]
[v1]
Fri, 15 Sep 2023 11:29:24 UTC (73 KB)
[v2]
Sun, 15 Dec 2024 08:09:02 UTC (71 KB)
[v3]
Sun, 14 Jun 2026 08:38:18 UTC (75 KB)
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