Timeline for Generation of the symplectic by involutions
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| Aug 1, 2016 at 17:46 | comment | added | Nick Gill | @UriBader, that's an interesting fact about infinite fields -- and one I should have known already! Thanks for filling in a gap in my knowledge. | |
| Aug 1, 2016 at 9:03 | vote | accept | Oliver | ||
| Aug 1, 2016 at 6:31 | comment | added | Uri Bader | @NickGill thanks for this correction. I put an answer based on it. Projective special linear and symplectic groups are always simple over infinite fields. | |
| Aug 1, 2016 at 6:20 | answer | added | Uri Bader | timeline score: 5 | |
| Jul 31, 2016 at 23:50 | comment | added | Nick Gill | Note that if the field has characteristic $2$, then transvections are involutions so the answer is YES immediately (at least for finite fields -- I'm not sure about simplicity results for infinite fields).... In particular (with reference to the comments above) note that ${\rm SL}_2(2^a)$ is generated by involutions | |
| Jul 31, 2016 at 15:31 | history | edited | Oliver | CC BY-SA 3.0 |
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| Jul 31, 2016 at 13:27 | review | Suggested edits | |||
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| Jul 31, 2016 at 12:51 | comment | added | Uri Bader | Just to justify my comment above and finish the discussion: $\text{Sp}(4)$ does contain a copy of $\text{GL}_2$ embedded by $A\mapsto$ a block diagonal matrix with $A$ and the transpose inverse of $A$ on the 2x2 diagonal blocks. Here the form is given by $I$ and $-I$ on the of-diagonal blocks. | |
| Jul 31, 2016 at 12:32 | comment | added | Uri Bader | So the answer should be: no in dim=2 (but yes for the projective group) and yes in higher dimensions. | |
| Jul 31, 2016 at 12:29 | comment | added | Uri Bader | Yes, I just noticed that. $\text{PSL}_2$ is generated by involution, but $\text{SL}_2$ does not, I was too quick to answer... | |
| Jul 31, 2016 at 12:28 | comment | added | Oliver | I am not sure. In $SL_2$ the only nontrivial symplectic involution is $-1$. | |
| Jul 31, 2016 at 12:18 | comment | added | Uri Bader | Note that on a two dimensional space the form $(a,c),(b,d)\mapsto ad-bc$ is just the determinant, so $\text{Sp}=\text{SL}$. Thus also in higher dimensions the symplectic group contains many copies of $\text{SL}_2$. In fact, it is generated by these (this, again, could be seen by simplicity, but in fact it holds in a complete generality (I think)). | |
| Jul 31, 2016 at 12:06 | comment | added | Oliver | May I ask you to be more specific? | |
| Jul 31, 2016 at 12:04 | comment | added | Uri Bader | Alternatively: use the fact that it is generated by $\text{SL}_2$'s. | |
| Jul 31, 2016 at 11:56 | comment | added | Uri Bader | Note that the projective symplectic group is simple (unless you're in some very special cases, see below). In this case (clearly having a non-trivial involution) it must be generated by involutions, as these generate a normal subgroup. See groupprops.subwiki.org/wiki/… | |
| Jul 31, 2016 at 11:53 | history | edited | Oliver | CC BY-SA 3.0 |
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| Jul 31, 2016 at 11:52 | comment | added | Uri Bader | Is $b$ non-degenerate? What about $F=\mathbb{R}$, $V$ 1-dimensional, $b=0$? | |
| Jul 31, 2016 at 11:21 | history | asked | Oliver | CC BY-SA 3.0 |