Timeline for Does the Fano plane "embed" in the complex projective plane?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 14, 2016 at 20:28 | comment | added | benblumsmith | @QiaochuYuan - to be honest, I did not give the question a great deal of thought before asking it here. I am studying the geometry of $PSL(2,7)$'s action(s) on $\mathbb{P}_\mathbb{C}^2$. I am better acquainted with the action on the Fano plane, so I wondered if I might be able to use it to help me think about the complex projective action. It seemed like a bit of a long shot but I thought somebody here might already know the answer. Geoff's answer is so straightforward it makes me feel a little silly for having asked rather than just tried to decide for myself, but that was the motivation. | |
| Mar 12, 2016 at 19:43 | vote | accept | benblumsmith | ||
| Mar 12, 2016 at 19:43 | vote | accept | benblumsmith | ||
| Mar 12, 2016 at 19:43 | |||||
| Mar 12, 2016 at 18:43 | history | edited | user9072 | CC BY-SA 3.0 |
changed quotation marks as "tex-style" is not rendered nicely
|
| Mar 12, 2016 at 18:13 | answer | added | Geoff Robinson | timeline score: 8 | |
| Mar 12, 2016 at 16:54 | comment | added | Derek Holt | @GeoffRobinson I agree entirely. I am afraid that my instinct these days is to try the computer first and think later if necessary. As you observed, it is also very easy to see that $S_4$ has no faithful complex representation of degree $2$. You should probablymake your comment into an answer. | |
| Mar 12, 2016 at 15:39 | comment | added | Geoff Robinson | @Derek Holt : It strikes me that Magma is overkill for that particular calculation, but I appreciate the double ( or triple ) checking :) | |
| Mar 12, 2016 at 9:31 | comment | added | Derek Holt | I confirmed Geoff's comment with a character table computation in Magma. The degree $3$ characters of $L_2(7)$ both restrict to irreducible characters of $S_4$. But of course this is an easy calculation! | |
| Mar 12, 2016 at 9:20 | comment | added | Geoff Robinson | I think the answer is "no". ${\rm PSL}(2,7)$ has two conjugacy classes of subgroups of index $7$, and all subgroups of index $7$ are isomorphic to the symmetric group $S_{4}$. Each of its non-trivial $3$-dimensional complex representations restrict irreducibly to all subgroups isomorphic to $S_{4}$,, so no such subgroup stabilizes a one-dimensional subspace ( to see irreducibility of $S_{4}$ note that the normal Klein $4$-subgroup acts trivially in the irreducible representations of $S_{4}$ of degree less than $3$- or look at the character). | |
| Mar 12, 2016 at 6:18 | comment | added | მამუკა ჯიბლაძე | Restricting the action to the Klein quartic in ${\mathbb C}{\mathbf P}^2$ might help to invoke some geometric intuition. | |
| Mar 12, 2016 at 1:46 | comment | added | Qiaochu Yuan | What have you tried? The permutation action on the Fano plane is transitive, so is determined by its stabilizer. So it's necessary and sufficient to find an element of the complex projective plane with the same stabilizer. | |
| Mar 12, 2016 at 0:43 | history | asked | benblumsmith | CC BY-SA 3.0 |