Timeline for How does one go about finding real/complex irreducible and faithful representations of PSL(2,7)?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 10, 2021 at 14:34 | history | edited | Guntram | CC BY-SA 4.0 |
improved formatting
|
| Mar 2, 2014 at 15:35 | history | edited | Geoff Robinson | CC BY-SA 3.0 |
Modified text in view of comment
|
| Mar 2, 2014 at 12:19 | comment | added | Alex B. | Dear Geoff, a few corrections/questions: you say "to get the degree 6 irreducible, induce the trivial character of the Sylow 7-normalizer"; but this induction will give you an 8-dimensional orthogonal, so the complement of the trivial will be 7-dimensional, not 6. You probably meant "one of the $S_4$s" instead of "Sylow 7-normaliser". Also, I would like to understand your remark that the 8-dimensional representation is "explicitly given as a real representation"? After all, you obtained it by inducing a linear character of order 3. Thanks in advance! | |
| Oct 28, 2012 at 11:57 | comment | added | Alexander Chervov | PS reading your answers I often come to the thought that they would decorate Wikipedia, it is not good that they will be burried in MO... | |
| Oct 27, 2012 at 23:17 | comment | added | Geoff Robinson | One way is to see hat as it is also ${\rm GL}(3,2)$ it has a transitive permutation representation on the non-zero vevtors in a $3$-dimensional vector space over the field of $2$-elemeents. Another was is to not hat it has subgroups of index $7$ (in fact, two conjugacy classes of subgroups, each isomorphic to $S_{4}).$ | |
| Oct 27, 2012 at 18:45 | comment | added | Alexander Chervov | How to see that this group is transitive perm of deg 7 ? | |
| Oct 27, 2012 at 18:20 | comment | added | Geoff Robinson | Transitive permutation group of degree $p$ for $p$ prime just means transitive subgroup of the symmetric group $S_{p}.$ For the second question, refer to the answer by Jim Humphreys. | |
| Oct 27, 2012 at 18:17 | comment | added | Alexander Chervov | can similar ideas be used for other groups gl/psl(n,q) ? | |
| Oct 27, 2012 at 18:13 | comment | added | Alexander Chervov | sorry what means " transitive permutation of degree p" ? | |
| Oct 27, 2012 at 14:44 | history | edited | Geoff Robinson | CC BY-SA 3.0 |
added 706 characters in body
|
| Oct 27, 2012 at 14:03 | comment | added | Geoff Robinson | ${\rm PSL}(2,7)$ is one of the exceptional ${\rm PSL}(2,p)$'s which has a transitive permutation of degree $p$, and this is exploited in the above explanation. This does not happen for $p>11$ as Galois already knew. Frobenius knew the character table of ${\rm PSL}(2,p),$ and as Jim Humphreys say, it is possible to deduce explicit irreducible complex representations to match the characters, by decomposing various induced representations | |
| Oct 27, 2012 at 13:57 | history | answered | Geoff Robinson | CC BY-SA 3.0 |