User mcampo - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T22:56:39Z http://mathoverflow.net/feeds/user/18478 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/124577/char-p-representations-of-sl-2-mathbbf-p-and-gl-2-mathbbf-p Char $p$ representations of $SL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$ mcampo 2013-03-15T01:27:26Z 2013-03-16T13:09:09Z <p>It is a well known fact that the irreducible representations of $SL_2(\mathbb{F}_p)$ over $\overline{\mathbb{F_p}}$ are given by the symmetric powers $Symm^k(V)$, where $V = \overline{\mathbb{F}_p}^2$, for $k$ ranging in $0,\ldots,p-1$. In the case of $GL_2(\mathbb{F}_p)$ irreducible representations are exhausted by:</p> <p>$$\sigma_{k,n} = Symm^k(V) \otimes det^n$$</p> <blockquote> <p>My question is then if there is a systematic way to know which of these irreducible representations appear in higher symmetric powers ($k>p-1$), either for $SL_2$ or $GL_2$?</p> </blockquote> <p>And what about tensor products $Symm^k(V) \otimes Symm^l(V)$? </p> http://mathoverflow.net/questions/77950/cohomology-of-sl-2-mathbbf-p-acting-on-trace-zero-matrices-over-mathbbf Cohomology of $SL_2(\mathbb{F}_p)$ acting on trace zero matrices over $\mathbb{F}_p$ mcampo 2011-10-12T19:07:03Z 2011-11-11T01:12:00Z <p>I've been reading a paper by R.Ramakrishna about lifting Galois representations and at some point he uses the fact that a representation with big image has trivial first cohomology group.</p> <p>In other words, what i've been trying to prove (and failing at) is that given $p>5$ a prime, if $M$ is the group of trace zero $2\times 2$ matrices over $\mathbb{F}_p$ then:</p> <p>$H^1(SL_2(\mathbb{F}_p),M) = 0$</p> <p>where the action is given by conjugation.</p> <p>My first approach was calculating the group $H^1(S_p,M)$, where $S_p$ is a $p$-sylow subgroup of $SL_2(\mathbb{F}_p)$. If it were trivial then the one that I want should be, because $M$ is a $p$-group, but it turns out it's not! (I think it's one dimensional).</p> <p>Maybe there is some cohomological-theoretic reason I'm missing, thanks.</p> http://mathoverflow.net/questions/124577/char-p-representations-of-sl-2-mathbbf-p-and-gl-2-mathbbf-p/124607#124607 Comment by mcampo mcampo 2013-03-19T16:12:00Z 2013-03-19T16:12:00Z Thanks for the answer! The kind of description that appears in the paper of Glover seems to be enough for the cases I needed. http://mathoverflow.net/questions/82176/groups-with-a-representation-of-degree-n-for-each-n-1-2-3 Comment by mcampo mcampo 2011-11-29T13:55:06Z 2011-11-29T13:55:06Z an irreducible representation? http://mathoverflow.net/questions/77950/cohomology-of-sl-2-mathbbf-p-acting-on-trace-zero-matrices-over-mathbbf/78021#78021 Comment by mcampo mcampo 2011-10-14T21:47:20Z 2011-10-14T21:47:20Z Thanks for the reference, it's nice to have another way to think about these things. I'll have a look at it. http://mathoverflow.net/questions/77950/cohomology-of-sl-2-mathbbf-p-acting-on-trace-zero-matrices-over-mathbbf Comment by mcampo mcampo 2011-10-14T21:47:05Z 2011-10-14T21:47:05Z I think the suggestion works out well, I have to check my calculations but it seems that the restriction to the Borel is trivial. Also added the group theory tag. Thanks. http://mathoverflow.net/questions/77950/cohomology-of-sl-2-mathbbf-p-acting-on-trace-zero-matrices-over-mathbbf/78007#78007 Comment by mcampo mcampo 2011-10-13T21:18:45Z 2011-10-13T21:18:45Z But in this case all the elements which lie in the center act trivially on M, since they are diagonal matrices and the action is by conjugation. I think the lemma doesn't help here.