The Defining Characteristic Case of the Representations of $\mathrm{GL}_{n}$ and $\mathrm{SL}_{n}$ over Principal Ideal Local Rings
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910621335814144 |
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| author | Monteiro, Nariel |
| author_facet | Monteiro, Nariel |
| contents | Let $W_{r}(\mathbb{F}_{q})$ be the ring of Witt vectors of length $r$ with residue field $\mathbb{F}_{q}$ of characteristic $p$. In this paper, we study the defining characteristic case of the representations of $\mathrm{GL}_{n}$ and $\mathrm{SL}_{n}$ over the principal ideal local rings $W_{r}(\mathbb{F}_{q})$ and $\mathbb{F}_{q}[t]/t^{r}$. Let ${\mathbf{G}}$ be either $\mathrm{GL}_{n}$ or $\mathrm{SL}_{n}$ and $F$ a perfect field of characteristic $p$, we prove that for most $p$ the group algebras $F[{\mathbf{G}}(W_{r}(\mathbb{F}_{q}))]$ and $F[{\mathbf{G}}(\mathbb{F}_{q}[t]/t^{r})]$ are not stably equivalent of Morita type. Thus, the group algebras $F[{\mathbf{G}}(W_{r}(\mathbb{F}_{q}))]$ and $F[{\mathbf{G}}(\mathbb{F}_{q}[t]/t^{r})]$ are not isomorphic in the defining characteristic case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17982 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Defining Characteristic Case of the Representations of $\mathrm{GL}_{n}$ and $\mathrm{SL}_{n}$ over Principal Ideal Local Rings Monteiro, Nariel Representation Theory Group Theory 20C20, 20G05 Let $W_{r}(\mathbb{F}_{q})$ be the ring of Witt vectors of length $r$ with residue field $\mathbb{F}_{q}$ of characteristic $p$. In this paper, we study the defining characteristic case of the representations of $\mathrm{GL}_{n}$ and $\mathrm{SL}_{n}$ over the principal ideal local rings $W_{r}(\mathbb{F}_{q})$ and $\mathbb{F}_{q}[t]/t^{r}$. Let ${\mathbf{G}}$ be either $\mathrm{GL}_{n}$ or $\mathrm{SL}_{n}$ and $F$ a perfect field of characteristic $p$, we prove that for most $p$ the group algebras $F[{\mathbf{G}}(W_{r}(\mathbb{F}_{q}))]$ and $F[{\mathbf{G}}(\mathbb{F}_{q}[t]/t^{r})]$ are not stably equivalent of Morita type. Thus, the group algebras $F[{\mathbf{G}}(W_{r}(\mathbb{F}_{q}))]$ and $F[{\mathbf{G}}(\mathbb{F}_{q}[t]/t^{r})]$ are not isomorphic in the defining characteristic case. |
| title | The Defining Characteristic Case of the Representations of $\mathrm{GL}_{n}$ and $\mathrm{SL}_{n}$ over Principal Ideal Local Rings |
| topic | Representation Theory Group Theory 20C20, 20G05 |
| url | https://arxiv.org/abs/2409.17982 |