Timeline for The normalizer of $\operatorname{Spin}(2N)$ in $\operatorname{U}(2^{N-1})$?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Feb 22, 2021 at 15:24 | vote | accept | wonderich | ||
| Feb 22, 2021 at 11:30 | comment | added | Robert Bryant | I have added an explicit description of each of the embeddings of the $\mathrm{Spin}(2n)$ groups you wanted to study and indicated what the normalizers are. I believe that this answers your original questions. Is there anything else you wanted to know? If not, and you are satisfied with my answer, it would be good for you to accept it so that your question won't continue to come up as open. That won't prevent other people from adding answers of their own, it just clarifies the status of the question. | |
| Feb 16, 2021 at 1:01 | comment | added | Robert Bryant | @wonderich: The careful list of where each of the spin groups is actually embedded is in Harvey's book, as well as the subgroups that Harvey calls the 'classical companions'. I think I know how to formulate your questions, but maybe you should explain what it really is that you want to know. | |
| Feb 16, 2021 at 0:35 | comment | added | wonderich | @Robert Bryant, could you help to modify my Spin(8) example to appropriate? Should I consider any other unitary group to embed Spin(8)? Thank you! | |
| Feb 15, 2021 at 23:09 | history | edited | wonderich | CC BY-SA 4.0 |
added 98 characters in body
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| Feb 15, 2021 at 23:06 | comment | added | wonderich | Thanks very much, I found this mathoverflow.net/questions/295711/is-spinn-a-subgroup-of-sun, I shall fix the question by U(8) $\not \supset$ Spin(8). | |
| Feb 15, 2021 at 21:00 | comment | added | Robert Bryant | There is a minor error in the OP's question. When $N$ is even, the semi-spinor representation of $\mathrm{Spin}(2N)$ on $\mathbb{C}^{2^{N-1}}$ is not faithful. (This is clear when $N=2$, where the kernel is a copy of $\mathrm{SU}(2)$, but even when $N>2$ (and $N$ is even) the semi-spinor representation has a kernel isomorphic to $\mathbb{Z}_2$. | |
| Feb 15, 2021 at 11:01 | answer | added | Robert Bryant | timeline score: 11 | |
| Feb 15, 2021 at 5:21 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
Title fix
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| Feb 15, 2021 at 4:52 | comment | added | LSpice | You also mention that you are interested in more than the Lie algebra of the normaliser (I think 'local' there is redundant). Do you know the Lie algebra of the normaliser? | |
| Feb 15, 2021 at 4:51 | comment | added | LSpice | You state two questions, but the second one seems like the definition of the first; what is the difference? You also give examples, but, aside from your first, solved, example, it's not clear what these illustrate except the values of $2N$ and $2^{N - 1}$ for $N = 4$ and $N = 5$, since you don't state partial results or conjecture an answer. | |
| Feb 15, 2021 at 4:50 | history | edited | LSpice | CC BY-SA 4.0 |
\DeclareMathOperator
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| Feb 15, 2021 at 3:09 | history | asked | wonderich | CC BY-SA 4.0 |