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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.14727 |
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| _version_ | 1866910578151260160 |
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| author | Yamanaka, Satoe Tsurii, Tatsuya Mikami, Itsumi Hirai, Takeshi |
| author_facet | Yamanaka, Satoe Tsurii, Tatsuya Mikami, Itsumi Hirai, Takeshi |
| contents | In previous papers I and II under the same title, we proposed a practical method called Efficient stairway up to the Sky, and apply it to some typical finite groups $G$, with Schur multiplier $M(G)$ containing prime number 3, to construct explicitly their representation groups $R(G)$, and then, to construct a complete set of representatives of linear IRs of $R(G)$, which gives naturally, through sectional restrictions, a complete set of representatives of spin IRs of $G$. In the present paper, we are concerned mainly with group $G=G_{39}$ of order 27 in a list of Tahara's paper, with $M(G)={\mathbb Z}_3\times {\mathbb Z}_3$. In this case, to arrive up to the Sky, we have two steps of one-step efficient central extensions. By the 1st step, we obtain a covering group of order 81, and by the 2nd step we arrive to $R(G)$ of order 243. At the 1st step, to construct explicitly a complete set of representatives of IRs of the group, we apply Mackey's induced representations, and at the 2nd step, with a help of this result, we apply so-called classical method for semidirect product groups given by Hirai and arrive to a complete list of IRs of $R(G)$. Then, using explicit realization of these IRs, we can compute their characters (called spin characters). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14727 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projective (or spin) representations of finite groups. III Yamanaka, Satoe Tsurii, Tatsuya Mikami, Itsumi Hirai, Takeshi Representation Theory In previous papers I and II under the same title, we proposed a practical method called Efficient stairway up to the Sky, and apply it to some typical finite groups $G$, with Schur multiplier $M(G)$ containing prime number 3, to construct explicitly their representation groups $R(G)$, and then, to construct a complete set of representatives of linear IRs of $R(G)$, which gives naturally, through sectional restrictions, a complete set of representatives of spin IRs of $G$. In the present paper, we are concerned mainly with group $G=G_{39}$ of order 27 in a list of Tahara's paper, with $M(G)={\mathbb Z}_3\times {\mathbb Z}_3$. In this case, to arrive up to the Sky, we have two steps of one-step efficient central extensions. By the 1st step, we obtain a covering group of order 81, and by the 2nd step we arrive to $R(G)$ of order 243. At the 1st step, to construct explicitly a complete set of representatives of IRs of the group, we apply Mackey's induced representations, and at the 2nd step, with a help of this result, we apply so-called classical method for semidirect product groups given by Hirai and arrive to a complete list of IRs of $R(G)$. Then, using explicit realization of these IRs, we can compute their characters (called spin characters). |
| title | Projective (or spin) representations of finite groups. III |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2408.14727 |