Timeline for Does $SL_3(R)$ embed in $SL_2(R)$?
Current License: CC BY-SA 2.5
21 events
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| Dec 6, 2010 at 1:32 | history | edited | Frogger | CC BY-SA 2.5 |
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| Dec 6, 2010 at 1:07 | comment | added | Tim Dokchitser | @Frogger: Ah, right, I did not realize you wanted a local one. But ${\mathbb F}_2[[e,x,y]]/(e^2,x^2+ex+y^2+y)$ still looks like a problem. | |
| Dec 6, 2010 at 1:00 | comment | added | Frogger | @Tim, Sorry, I see that I forgot to say that one had to map to a local F_2-algebra. (But your example might still turn out to be a problem none the less. | |
| Dec 6, 2010 at 0:59 | history | edited | Frogger | CC BY-SA 2.5 |
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| Dec 6, 2010 at 0:55 | comment | added | Frogger | @Tim, You may be completely correct, but I'm a little confused. What is the maximal ideal here? I wonder if you can do the following computation: Is there an injective map from S_4 to SL_2(F_2[x]/x^3) which injects under the composition to SL_2(F_2[x]/x^2)? | |
| Dec 6, 2010 at 0:50 | comment | added | Frogger | @Jack, your original comment (now deleted) was also rude, and wrong. That was what provoked the response. You can only get away with being rude if you are correct. Of course, by that reckoning, my response was far more rude, and I was also wrong. So, the egg is on my face. | |
| Dec 6, 2010 at 0:44 | history | edited | Frogger | CC BY-SA 2.5 |
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| Dec 6, 2010 at 0:13 | comment | added | Tim Dokchitser | @Frogger: I tried your "bash it out" approach, and unfortunately I seem to have cooked up a counterexample. Take the affine curve over ${\mathbb F}_2[e]/e^2$ given by $x^2+ex=y^2+y$, let $R$ be its ring of regular functions and map $S_4\to SL_2(R)$ by $$ (1234)\mapsto \begin{pmatrix}x+1&y+1+e\cr y+e&x+1+e\end{pmatrix}, \quad (12)\mapsto \begin{pmatrix}0&1\cr 1&0\end{pmatrix}. $$ (Hope I've got this right.) | |
| Dec 5, 2010 at 23:09 | history | edited | Frogger | CC BY-SA 2.5 |
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| Dec 5, 2010 at 22:40 | history | edited | Frogger | CC BY-SA 2.5 |
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| Dec 5, 2010 at 22:35 | history | edited | Frogger | CC BY-SA 2.5 |
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| Dec 5, 2010 at 21:04 | comment | added | Jonathan Kiehlmann | @Jack: good question, but one that, slightly more out of desire to keep these comments from growing too convoluted and get it better attention than to create as many spin-offs from my first question as possible, I think you should ask as its own question :) | |
| Dec 5, 2010 at 20:40 | history | edited | Frogger | CC BY-SA 2.5 |
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| Dec 5, 2010 at 20:03 | comment | added | Jack Schmidt | This argument does point out some strange (to me) things in the Artinian local case: Let G(n) be SL(2,ZZ[x]/(n,xx)). I would have expected G(4) to be more flexible in all ways than G(2), but in fact they each appear to have strengths. G(2) contains dihedral groups of order 6, 8, and 12, and the symmetric group S4, while G(4) contains none of those. On the other hand, G(4) contains elements of 8 and elementary abelian groups of rank 6, while G(2) only has elements of order 4 and subgroups of rank 3. Is there some sense in which G(4) is more abelian? | |
| Dec 5, 2010 at 19:21 | comment | added | Jack Schmidt | There is no need to be rude. The claim is still false, and several of the steps are quite vague. S4 embeds in SL(2,Z[x]/(6,xx)) as (1,2) = [ 1, 3 ; 0, 1 ], (2,3) = [ 4, 3 ; 3, 4 ], (3,4) = [ 1 + 3x, 3 ; 0, 1 + 3x ]. Check that AA = BB = CC = 1 mod (6,xx), ABA-BAB = BCB-CBC = AC-CA = 0 mod (6,xx), and that AC ≠ 1 mod (6,xx) (and check that the determinants are all 1 mod 6 :-). This does not seem like a serious problem, but it requires the proof to be reorganized a bit. | |
| Dec 5, 2010 at 18:37 | comment | added | Kevin Buzzard | @frogger: my girlfriend is on call this weekend and I've got all the kids. I noticed the post but didn't yet read it. I'm really pleased the question is answered but didn't just want to mindlessly upvote until I'd read what you had done. [which I will, soon...] | |
| Dec 5, 2010 at 18:10 | comment | added | Jonathan Kiehlmann | Frogger, I think the lack of interest may be to do with your answer being posted on a Saturday night, and a (incorrect, it emerges) refutation provided soon after. I wouldn't be surprised to hear Sunday being a key part to the lack of response, yet. | |
| Dec 5, 2010 at 15:59 | history | edited | Frogger | CC BY-SA 2.5 |
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| Dec 4, 2010 at 21:04 | comment | added | Jack Schmidt | Oops, mixup between determinant and reduced determinant (the representing matrices in GL(4,Z/4Z) has det 1, but that was the square of the determinant in GL(2,Z[x]/(4,xx))). In fact SL(2,Z[x]/(4,xx)) contains no copy of S4. I deleted the first comment, since we can't edit it. | |
| Dec 4, 2010 at 19:06 | comment | added | Tim Dokchitser | @Jack: I think your group lands in $GL$ rather than $SL$, the generators have determinant $-1=3$. | |
| Dec 4, 2010 at 15:06 | history | answered | Frogger | CC BY-SA 2.5 |