On the minimal dimension of a faithful linear representation of a finite group
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866912909939965952 |
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| author | Moretó, Alexander |
| author_facet | Moretó, Alexander |
| contents | The representation dimension of a finite group $G$ is the minimal dimension of a faithful complex linear representation of $G$. We prove that the representation dimension of any finite group $G$ is at most $\sqrt{|G|}$ except if $G$ is a $2$-group with elementary abelian center of order $8$ and all irreducible characters of $G$ whose kernel does not contain $Z(G)$ are fully ramified with respect to $G/Z(G)$. We also obtain bounds for the representation dimension of quotients of $G$ in terms of the representation dimension of $G$, and discuss the relation of this invariant with the essential dimension of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_01463 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the minimal dimension of a faithful linear representation of a finite group Moretó, Alexander Group Theory Algebraic Geometry Representation Theory Primary 20C15, Secondary 14E07, 12F10 The representation dimension of a finite group $G$ is the minimal dimension of a faithful complex linear representation of $G$. We prove that the representation dimension of any finite group $G$ is at most $\sqrt{|G|}$ except if $G$ is a $2$-group with elementary abelian center of order $8$ and all irreducible characters of $G$ whose kernel does not contain $Z(G)$ are fully ramified with respect to $G/Z(G)$. We also obtain bounds for the representation dimension of quotients of $G$ in terms of the representation dimension of $G$, and discuss the relation of this invariant with the essential dimension of $G$. |
| title | On the minimal dimension of a faithful linear representation of a finite group |
| topic | Group Theory Algebraic Geometry Representation Theory Primary 20C15, Secondary 14E07, 12F10 |
| url | https://arxiv.org/abs/2102.01463 |