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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/2412.08539 |
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| _version_ | 1866917866420305920 |
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| author | Monteiro, Nariel Stasinski, Alexander |
| author_facet | Monteiro, Nariel Stasinski, Alexander |
| contents | The conjugation representation of a finite group $G$ is the complex permutation module defined by the action of $G$ on itself by conjugation. Addressing a problem raised by Hain motivated by the study of a Hecke action on iterated Shimura integrals, Tiep proved that for $G=\operatorname{SL}_{2}(\mathbb{Z}/p^{r})$, where $r\geq1$ and $p\geq5$ is a prime, any irreducible representation of $G$ that is trivial on the centre of $G$ is contained in the conjugation representation. Moreover, Tiep asked whether this can be generalised to $p=2$ or $3$. We answer the Hain--Tiep question in the affirmative and also prove analogous statements for $\operatorname{SL}_{2}$ and $\operatorname{GL}_{2}$ over any finite local principal ideal ring with residue field of odd characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08539 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The conjugation representation of $\operatorname{GL}_{2}$ and $\operatorname{SL}_{2}$ over finite local rings Monteiro, Nariel Stasinski, Alexander Representation Theory The conjugation representation of a finite group $G$ is the complex permutation module defined by the action of $G$ on itself by conjugation. Addressing a problem raised by Hain motivated by the study of a Hecke action on iterated Shimura integrals, Tiep proved that for $G=\operatorname{SL}_{2}(\mathbb{Z}/p^{r})$, where $r\geq1$ and $p\geq5$ is a prime, any irreducible representation of $G$ that is trivial on the centre of $G$ is contained in the conjugation representation. Moreover, Tiep asked whether this can be generalised to $p=2$ or $3$. We answer the Hain--Tiep question in the affirmative and also prove analogous statements for $\operatorname{SL}_{2}$ and $\operatorname{GL}_{2}$ over any finite local principal ideal ring with residue field of odd characteristic. |
| title | The conjugation representation of $\operatorname{GL}_{2}$ and $\operatorname{SL}_{2}$ over finite local rings |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2412.08539 |