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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2511.17289 |
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| _version_ | 1866908668980625408 |
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| author | Tanimoto, Ryuji |
| author_facet | Tanimoto, Ryuji |
| contents | Let $k$ be an algebraically closed field of positive characteristic $p$ and let $\mathbb{G}_a$ denote the additive group of $k$. Let $n \geq 1$ and let ${\rm Mat}(n, k[T])^E$ denote the set of all exponential matrices of ${\rm Mat}(n, k[T])$. Let $\mathbb{E}_{\geq 0}(n, k)$ denote the set of all group homomorphisms from $(\mathbb{Z}/p\mathbb{Z})^r$ to ${\rm GL}(n, k)$, where $r$ ranges over all non-negative integers. In the first, we show that there exists a one-to-one correspondence between the set ${\rm Mat}(n, k[T])^E$ and the set of all $\mathbb{G}_a$-actions on $\mathbb{P}^{n - 1}$. In the second, we show that there exists a one-to-one correspondence between $\mathbb{E}_{\geq 0}(n, k)$ and the set ${\rm Mat}(n, k[T])^E \times \mathbb{Z}_{\geq 0}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17289 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exponential matrices, $\mathbb{G}_a$-actions on projective spaces and modular representations of elementary abelian $p$-groups Tanimoto, Ryuji Representation Theory Primary 15A54, Secondary 14L30, 20C20 Let $k$ be an algebraically closed field of positive characteristic $p$ and let $\mathbb{G}_a$ denote the additive group of $k$. Let $n \geq 1$ and let ${\rm Mat}(n, k[T])^E$ denote the set of all exponential matrices of ${\rm Mat}(n, k[T])$. Let $\mathbb{E}_{\geq 0}(n, k)$ denote the set of all group homomorphisms from $(\mathbb{Z}/p\mathbb{Z})^r$ to ${\rm GL}(n, k)$, where $r$ ranges over all non-negative integers. In the first, we show that there exists a one-to-one correspondence between the set ${\rm Mat}(n, k[T])^E$ and the set of all $\mathbb{G}_a$-actions on $\mathbb{P}^{n - 1}$. In the second, we show that there exists a one-to-one correspondence between $\mathbb{E}_{\geq 0}(n, k)$ and the set ${\rm Mat}(n, k[T])^E \times \mathbb{Z}_{\geq 0}$. |
| title | Exponential matrices, $\mathbb{G}_a$-actions on projective spaces and modular representations of elementary abelian $p$-groups |
| topic | Representation Theory Primary 15A54, Secondary 14L30, 20C20 |
| url | https://arxiv.org/abs/2511.17289 |