Timeline for Reduction mod $n$ of symplectic group
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Feb 13, 2018 at 13:23 | history | edited | Martin Sleziak | CC BY-SA 3.0 |
added eudml link
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| Feb 9, 2018 at 5:37 | vote | accept | CommunityBot | ||
| Feb 9, 2018 at 5:05 | comment | added | Igor Rivin | I did NOT know that (the last part, that is - of course I did know they were from ND). | |
| Feb 9, 2018 at 4:54 | comment | added | Andy Putman | @IgorRivin: Hahn and O'Meara is a great book (by two Notre Dame people, so I have to like it!). O'Meara was the first lay provost of the university. | |
| Feb 9, 2018 at 4:39 | comment | added | Igor Rivin | Actually (and this is not relevant to the Newman reference), there IS a good reference - Hahn and O'Meara do it in their section 6.2, in fairly great generality. | |
| Feb 9, 2018 at 3:54 | comment | added | Andy Putman | @IgorRivin: I don't have that book handy, but if I recall correctly he only considers the classical groups over fields (or, more generally, skew fields). It is not at all routine to generalize things to more general rings, even to $\mathbb{Z}$. For an elementary example, it is trivial to show that $\text{SL}(n,k)$ is generated by elementary matrices for $k$ a field. That proof actually gives a version of bounded generation. But for $k=\mathbb{Z}$, things are more complicated and you have to work harder (precisely because bounded generation is false for $n=2$). | |
| Feb 9, 2018 at 3:24 | comment | added | Igor Rivin | I theorize that this is in "La geometrie des groupes classiques" by Dieudonne (1955), but I am having trouble opening my electronic copy.... | |
| Feb 9, 2018 at 2:16 | comment | added | Andy Putman | (even today it is nontrivial to find a good source for the fact that $\text{Sp}(2g,\mathbb{Z})$ is generated by transvections -- the proof that works for fields and appears in many places does not generalize in a simple way) | |
| Feb 9, 2018 at 2:13 | comment | added | Andy Putman | @IgorRivin: The result is basically equivalent to knowing generators for $\text{Sp}(2g,\mathbb{Z}/n)$. For $n$ prime, these were certainly known earlier (e.g. it was known since symplectic groups were first studied that symplectic groups over fields are generated by transvections). But I've search extensively, and I can't find much of anything in print before this about symplectic groups over rings like $\mathbb{Z}/n$ for $n$ not a prime (which are not even integral domains). I am morally certain that nothing equivalent to this appeared earlier in the written literature. | |
| Feb 8, 2018 at 23:18 | comment | added | Igor Rivin | It's hard to believe this was not known to people who cared before then... | |
| Feb 8, 2018 at 21:27 | history | answered | Andy Putman | CC BY-SA 3.0 |