Timeline for Reps of $U(n)$ for the bundles of holomorphic and antiholomorphic forms of projective space
Current License: CC BY-SA 2.5
23 events
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Dec 14, 2009 at 15:38 | vote | accept | Jean Delinez | ||
| Dec 11, 2009 at 19:45 | comment | added | José Figueroa-O'Farrill | I've now taken the liberty to edit the question to show the right groups. | |
| Dec 11, 2009 at 19:45 | history | edited | José Figueroa-O'Farrill | CC BY-SA 2.5 |
Reindexed the groups correctly.
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| Dec 11, 2009 at 18:50 | comment | added | José Figueroa-O'Farrill | Yes, but you've relabeled the wrong way. $\mathbb{CP}^n = SU(n+1)/U(n)$ in your notation. Notice that the action of $SU(n+1)$ is basically the linear action on $\mathbb{C}^{n+1}$, thought of as the fundamental representation. Think about the case $n=1$: $\mathbb{CP}^1$ is the 2-sphere acted on by $SU(2)$. | |
| Dec 11, 2009 at 18:23 | history | edited | Jean Delinez | CC BY-SA 2.5 |
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| Dec 11, 2009 at 18:21 | comment | added | Jean Delinez | Yes, of course. | |
| Dec 11, 2009 at 18:01 | comment | added | José Figueroa-O'Farrill | I just noticed that what you have defined is $\mathbb{CP}^{n-1}$. I think that it's best to leave that as $\mathbb{CP}^n$ and relabel the groups. So your question is about $U(n)$. | |
| Dec 11, 2009 at 17:51 | history | edited | Jean Delinez | CC BY-SA 2.5 |
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| Dec 11, 2009 at 17:50 | comment | added | Jean Delinez | yes, that's what I mean. It came from my supervisor who is pretty heavily physics influenced. | |
| Dec 11, 2009 at 17:15 | answer | added | David E Speyer | timeline score: 4 | |
| Dec 11, 2009 at 16:47 | comment | added | David E Speyer | I think you are right. (And some googling of the physics literature agrees with you). I think I am also right. See my longer answer below. | |
| Dec 11, 2009 at 16:32 | comment | added | José Figueroa-O'Farrill | I think that holomorphic and antiholomorphic refer, respectively, to (1,0) and (0,1) forms. This is a typical notation in the Physics literature, at least. | |
| Dec 11, 2009 at 13:15 | comment | added | David E Speyer | Holomorphic forms appears to mean holomorphic 1-forms. At least, that's how Jean Delinez seemed to use this terminology in his previous question. mathoverflow.net/questions/8282/… It also seems that "antiholomorphic forms" is the dual vector bundle to "holomorphic forms", that is to say, it is the holomorphic tangent bundle. I find this terminology very strange, so I worry that I am misunderstanding it. If anyone has run across this terminology before, please leave a helpful comment. | |
| Dec 11, 2009 at 9:47 | comment | added | José Figueroa-O'Farrill | I think that there might be some ambiguity in the notation in the original version of the question and this might be causing some consusion below. (I would edit, but I'm not sure if this is OK.) The projective space is $U(n)/(U(n-1) \times U(1))$ or, if you prefer to have $SU(n)$ on the top, $SU(n)/(S(U(n-1)\times U(1))$. The subgroup in the denominator consists of block diagonal $U(n)$ matrices of the form $(g,1/\det g)$, where $g$ is unitary of size $n-1$ embedded in $U(n)$ in the obvioua way. This specifies the $U(n-1)$ subgroup -- up to conjugation, of course. | |
| Dec 11, 2009 at 7:11 | answer | added | David Bar Moshe | timeline score: 3 | |
| Dec 11, 2009 at 1:58 | comment | added | Kevin H. Lin | For the second question, the first paragraph of the section "Formulation" on this wikipedia page en.wikipedia.org/wiki/Borel%E2%80%93Weil%E2%80%93Bott_theorem might help? | |
| Dec 11, 2009 at 0:20 | comment | added | Kevin H. Lin | What is meant by "bundle of holomorphic forms" and "bundle of antiholomorphic forms"? | |
| Dec 11, 2009 at 0:20 | history | edited | Kevin H. Lin | CC BY-SA 2.5 |
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| Dec 10, 2009 at 22:43 | history | edited | Kevin H. Lin |
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| Dec 10, 2009 at 21:36 | history | edited | Jean Delinez | CC BY-SA 2.5 |
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| Dec 10, 2009 at 21:15 | history | edited | Jean Delinez | CC BY-SA 2.5 |
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| Dec 10, 2009 at 21:08 | history | asked | Jean Delinez | CC BY-SA 2.5 |