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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.11220 |
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| _version_ | 1866911881403301888 |
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| author | Wu, Wenxia Li, Yunnan |
| author_facet | Wu, Wenxia Li, Yunnan |
| contents | The complex representation rings of finite groups are the fundamental class of fusion rings, categorified by the corresponding fusion categories of complex representations. The category of $\mathbb{Z}_+$-modules of finite rank over such a representation ring is also semisimple.
In this paper, we classify the irreducible based modules of rank up to 5 over the complex representation ring $r(S_4)$ of the symmetric group $S_4$. Totally 16 inequivalent irreducible based modules are obtained. Based on such a classification result, we further discuss the categorification of based modules over $r(S_4)$ by module categories over the complex representation category ${\rm Rep}(S_4)$ of $S_4$ arisen from projective representations of certain subgroups of $S_4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11220 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Categorification of based modules over the complex representation ring of $S_4$ Wu, Wenxia Li, Yunnan Representation Theory Rings and Algebras 13C05, 18M20, 19A22 The complex representation rings of finite groups are the fundamental class of fusion rings, categorified by the corresponding fusion categories of complex representations. The category of $\mathbb{Z}_+$-modules of finite rank over such a representation ring is also semisimple. In this paper, we classify the irreducible based modules of rank up to 5 over the complex representation ring $r(S_4)$ of the symmetric group $S_4$. Totally 16 inequivalent irreducible based modules are obtained. Based on such a classification result, we further discuss the categorification of based modules over $r(S_4)$ by module categories over the complex representation category ${\rm Rep}(S_4)$ of $S_4$ arisen from projective representations of certain subgroups of $S_4$. |
| title | Categorification of based modules over the complex representation ring of $S_4$ |
| topic | Representation Theory Rings and Algebras 13C05, 18M20, 19A22 |
| url | https://arxiv.org/abs/2405.11220 |