Timeline for Generation of $\mathrm{SO}(n,\mathbb{Q})$ by coordinate subgroups
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Oct 11, 2023 at 10:40 | comment | added | IJL | Maybe Stefan's question about $\mathbb{Z}_p$ group schemes should be posed as a new question as he suggests. | |
| Oct 11, 2023 at 8:49 | comment | added | Stefan Witzel | I had written in an earlier post that looking modulo $8$ would allow to treat the $n = 5$ case but I was mistaken: while $\operatorname{SO}_4(\mathbb{Q}_2)$ is compact (and strictly larger than $\operatorname{SO}_4(\mathbb{Z}_2)$) the coordinate subgroups in $\operatorname{SO}_5(\mathbb{Q}_2)$ seem to generate a non-compact subgroup. So the $n = 5$ case is intricate precisely for this compact vs. integral question. | |
| Oct 11, 2023 at 8:42 | history | edited | Stefan Witzel | CC BY-SA 4.0 |
Corrected claim.
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| Oct 11, 2023 at 4:45 | history | edited | Stefan Witzel | CC BY-SA 4.0 |
deleted 13 characters in body
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| Oct 10, 2023 at 18:21 | comment | added | Stefan Witzel | @YCor: by $<$ I don't mean ``properly''. Right for the other point, I'll change accordingly. Err, once the problem that you also ran into goes away. | |
| Oct 10, 2023 at 18:11 | comment | added | YCor | How could $\mathbf{G}(\mathbb{Q}_p)$ be properly included in $\mathbf{G}(\mathbb{Z}_p)$? More seriously, beware that $\mathbf{G}(\mathbb{Z}_p)$ is not uniquely defined for a $\mathbb{Q}_p$-group. It depends on a matrix representation here (which, more formally, makes it a group scheme over $\mathbb{Z}_p$). This is often unimportant, but might typically impact whether the inclusion $\mathbf{G}(\mathbb{Q}_p)\le \mathbf{G}(\mathbb{Z}_p)$ holds. | |
| Oct 10, 2023 at 18:05 | history | answered | Stefan Witzel | CC BY-SA 4.0 |