Bounds on multiplicities of symmetric pairs of finite groups
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910704512008192 |
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| author | Avni, Nir Aizenbud, Avraham |
| author_facet | Avni, Nir Aizenbud, Avraham |
| contents | Let $Γ$ be a finite group, let $θ$ be an involution of $Γ$, and let $ρ$ be an irreducible complex representation of $Γ$. We bound $\dim ρ^{Γ^θ}$ in terms of the smallest dimension of a faithful $\mathbb{F}_p$-representation of $Γ/Rad_p(Γ)$, where $p$ is any odd prime and $Rad_p(Γ)$ is the maximal normal $p$-subgroup of $Γ$.
This implies, in particular, that if $\mathbf{G}$ is a group scheme over $\mathbb{Z}$ and $θ$ is an involution of $\mathbf{G}$, then the multiplicity of any irreducible representation in $C^\infty \left( \mathbf{G}(\mathbb{Z}_p)/ \mathbf{G} ^θ(\mathbb{Z}_p) \right)$ is bounded, uniformly in $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_12217 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bounds on multiplicities of symmetric pairs of finite groups Avni, Nir Aizenbud, Avraham Representation Theory Group Theory 20C15, 20G25, 43A85 Let $Γ$ be a finite group, let $θ$ be an involution of $Γ$, and let $ρ$ be an irreducible complex representation of $Γ$. We bound $\dim ρ^{Γ^θ}$ in terms of the smallest dimension of a faithful $\mathbb{F}_p$-representation of $Γ/Rad_p(Γ)$, where $p$ is any odd prime and $Rad_p(Γ)$ is the maximal normal $p$-subgroup of $Γ$. This implies, in particular, that if $\mathbf{G}$ is a group scheme over $\mathbb{Z}$ and $θ$ is an involution of $\mathbf{G}$, then the multiplicity of any irreducible representation in $C^\infty \left( \mathbf{G}(\mathbb{Z}_p)/ \mathbf{G} ^θ(\mathbb{Z}_p) \right)$ is bounded, uniformly in $p$. |
| title | Bounds on multiplicities of symmetric pairs of finite groups |
| topic | Representation Theory Group Theory 20C15, 20G25, 43A85 |
| url | https://arxiv.org/abs/2202.12217 |