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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.05560 |
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| _version_ | 1866909091709845504 |
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| author | Li, Wenqi |
| author_facet | Li, Wenqi |
| contents | The fusion rules in $\mathrm{Rep}_f D(G)$ for a finite group $G$ can be computed in terms of character inner products. Using an explicit formula for these fusion rules, we show that $\mathrm{Rep}_f D(G)$ is multiplicity free for two infinite families of finite groups: the Dihedral groups and the Dicyclic groups. In fact, we will compute all fusion rules in these categories. Multiplicity freeness is a desired property for modular tensor categories, since it greatly simplifies the computation of $F$-matrices. Furthermore, we observe that the fusion rules for Dihedral groups $D_{2n}$ with $n$ odd are extremely similar to the fusion rules of Type $B$ level $2$ fusion algebras of Wess-Zumino-Witten conformal field theories. Moreover, we give a proof of the fusion rule formula by using Mackey theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_05560 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fusion Rings over Drinfeld Doubles Li, Wenqi Quantum Algebra Representation Theory 16T05 (Primary) 20C35, 18D10, 81R05 (Secondary) The fusion rules in $\mathrm{Rep}_f D(G)$ for a finite group $G$ can be computed in terms of character inner products. Using an explicit formula for these fusion rules, we show that $\mathrm{Rep}_f D(G)$ is multiplicity free for two infinite families of finite groups: the Dihedral groups and the Dicyclic groups. In fact, we will compute all fusion rules in these categories. Multiplicity freeness is a desired property for modular tensor categories, since it greatly simplifies the computation of $F$-matrices. Furthermore, we observe that the fusion rules for Dihedral groups $D_{2n}$ with $n$ odd are extremely similar to the fusion rules of Type $B$ level $2$ fusion algebras of Wess-Zumino-Witten conformal field theories. Moreover, we give a proof of the fusion rule formula by using Mackey theory. |
| title | Fusion Rings over Drinfeld Doubles |
| topic | Quantum Algebra Representation Theory 16T05 (Primary) 20C35, 18D10, 81R05 (Secondary) |
| url | https://arxiv.org/abs/2306.05560 |