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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/2407.08116 |
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| _version_ | 1866916319905972224 |
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| author | Hirai, Takeshi Mikami, Itsumi Tsurii, Tatsuya Yamanaka, Satoe |
| author_facet | Hirai, Takeshi Mikami, Itsumi Tsurii, Tatsuya Yamanaka, Satoe |
| contents | Schur multiplier $M(G)$ of a finite group $G$ has been studied heavily. To proceed further to the study of projective (or spin) representations of $G$ and their characters (called spin characters), it is necessary to construct explicitly a representation group $R(G)$ of $G$, a certain central extension of $G$ by $M(G)$, since projective representations of $G$ correspond bijectively to linear representations of $R(G)$. We propose here a practical method to construct $R(G)$ by repetition of one-step efficient central extensions according to a certain choice of a series of elements of $M(G)$. This method is also helpful for constructing linear representations of $R(G)$ and accordingly for calculating spin characters. Actually, we will apply this method to several examples of $G$ with prime number 3 in $M(G)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08116 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projective (or spin) representations of finite groups. I Hirai, Takeshi Mikami, Itsumi Tsurii, Tatsuya Yamanaka, Satoe Representation Theory Schur multiplier $M(G)$ of a finite group $G$ has been studied heavily. To proceed further to the study of projective (or spin) representations of $G$ and their characters (called spin characters), it is necessary to construct explicitly a representation group $R(G)$ of $G$, a certain central extension of $G$ by $M(G)$, since projective representations of $G$ correspond bijectively to linear representations of $R(G)$. We propose here a practical method to construct $R(G)$ by repetition of one-step efficient central extensions according to a certain choice of a series of elements of $M(G)$. This method is also helpful for constructing linear representations of $R(G)$ and accordingly for calculating spin characters. Actually, we will apply this method to several examples of $G$ with prime number 3 in $M(G)$. |
| title | Projective (or spin) representations of finite groups. I |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2407.08116 |