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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.16895 |
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| _version_ | 1866913486228946944 |
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| author | Ali, Abid Carbone, Lisa Murray, Scott H. |
| author_facet | Ali, Abid Carbone, Lisa Murray, Scott H. |
| contents | Let $G(\mathbb{Q})$ be a simply connected Chevalley group over $\mathbb{Q}$ corresponding to a simple Lie algebra $\mathfrak g$ over $\mathbb{C}$. Let $V$ be a finite dimensional faithful highest weight $\mathfrak g$-module and let $V_\mathbb{Z}$ be a Chevalley $\mathbb{Z}$-form of $V$. Let $Γ(\mathbb{Z})$ be the subgroup of $G(\mathbb{Q})$ that preserves $V_{\mathbb{Z}}$ and let $G(\mathbb{Z})$ be the group of $\mathbb{Z}$-points of $G(\mathbb{Q})$. Then $G(\mathbb{Q})$ is \emph{integral} if $G(\mathbb{Z})=Γ(\mathbb{Z})$. Chevalley's original work constructs a scheme-theoretic integral form of $G(\mathbb{Q})$ which equals $Γ(\mathbb{Z})$. Here we give a representation-theoretic proof of integrality of $G(\mathbb{Q})$ using only the action of $G(\mathbb{Q})$ on $V$, rather than the language of group schemes. We discuss the challenges and open problems that arise in trying to extend this to a proof of integrality for Kac-Moody groups over $\mathbb{Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16895 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chevalley groups over $\Z$: A representation-theoretic approach Ali, Abid Carbone, Lisa Murray, Scott H. Representation Theory 17B10, 17B67 Let $G(\mathbb{Q})$ be a simply connected Chevalley group over $\mathbb{Q}$ corresponding to a simple Lie algebra $\mathfrak g$ over $\mathbb{C}$. Let $V$ be a finite dimensional faithful highest weight $\mathfrak g$-module and let $V_\mathbb{Z}$ be a Chevalley $\mathbb{Z}$-form of $V$. Let $Γ(\mathbb{Z})$ be the subgroup of $G(\mathbb{Q})$ that preserves $V_{\mathbb{Z}}$ and let $G(\mathbb{Z})$ be the group of $\mathbb{Z}$-points of $G(\mathbb{Q})$. Then $G(\mathbb{Q})$ is \emph{integral} if $G(\mathbb{Z})=Γ(\mathbb{Z})$. Chevalley's original work constructs a scheme-theoretic integral form of $G(\mathbb{Q})$ which equals $Γ(\mathbb{Z})$. Here we give a representation-theoretic proof of integrality of $G(\mathbb{Q})$ using only the action of $G(\mathbb{Q})$ on $V$, rather than the language of group schemes. We discuss the challenges and open problems that arise in trying to extend this to a proof of integrality for Kac-Moody groups over $\mathbb{Q}$. |
| title | Chevalley groups over $\Z$: A representation-theoretic approach |
| topic | Representation Theory 17B10, 17B67 |
| url | https://arxiv.org/abs/2408.16895 |