Timeline for Irreducible non-Abelian subgroup of $\mathrm{U}_n(\mathbb{C})$, containing diagonal matrices
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 2, 2020 at 15:12 | vote | accept | Mini | ||
| Jun 2, 2020 at 14:07 | answer | added | YCor | timeline score: 3 | |
| Jun 2, 2020 at 13:10 | comment | added | Najib Idrissi | @GeoffRobinson Thanks for the clarification and sorry for the bother. (English is not my native language, so sometimes there are some misunderstandings, unfortunately...) | |
| Jun 2, 2020 at 13:07 | comment | added | Mini | @YCor If you have any ideas regarding the following post also I would be thankful: mathoverflow.net/questions/361477/… | |
| Jun 2, 2020 at 13:05 | comment | added | Mini | Thanks. Can you please give hint on how these you could derive this property? If possible, I would be grateful if you could provide details in an answer (ignoring the irreducibility condition). | |
| Jun 2, 2020 at 12:43 | comment | added | YCor | The answer (without the unnatural irreducibility assumption) will probably be that such subgroups $H$ are automatically closed, that $H^0$ is the stabilizer of some direct sum decomposition (i.e., consists of block-diagonal matrices for some partition of $\{1,\dots,n\}$) and $H$ is contained in the normalizer of $H^0$ (which contains $H^0$ with finite index). Irreducibility will make the result more complicated and less general. | |
| Jun 2, 2020 at 12:35 | comment | added | Geoff Robinson | Since the question was changed in response to comments, my last comment refers to an earlier version. | |
| Jun 2, 2020 at 12:17 | comment | added | Geoff Robinson | @Najib: No, I had no intention at all to be abrasive. I merely meant that your example pointed out that there are many reducible subgroups which contain all diagonal matrices (actually the previous question at least implicitly excluded reducible subgroups, but this one does not). | |
| Jun 2, 2020 at 12:16 | comment | added | Mini | Thanks everyone. I've modified the question. | |
| Jun 2, 2020 at 12:15 | history | edited | Mini | CC BY-SA 4.0 |
added 15 characters in body; edited title
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| Jun 2, 2020 at 11:52 | comment | added | Geoff Robinson | Because of comments like those of Najib Idrissi, I'd suggest trying to classify irreducible unitary groups containing all diagonal unitary matrices. Even among these, there are many imprimitive groups to be considered which complicate things. | |
| Jun 2, 2020 at 11:47 | comment | added | YCor | Again, I'm suggesting to edit your question with this (describing subgroups of $U(n)$ containing the group of diagonal matrices), which is a good question (rather than the current one which is half far too broad and half immediate). | |
| Jun 2, 2020 at 11:10 | comment | added | Mini | Thanks, you are right. Does there exist any result for the converse? I mean, if a non-abelian subgroup contains group of diagonal matrices, then it should have certain properties. As I mentioned, the aim is to use the result for following question: mathoverflow.net/questions/361477/… | |
| Jun 2, 2020 at 11:10 | comment | added | Geoff Robinson | Every finite group, Abelian or not, is isomorphic to a group of unitary matrices. If $n >1$, there is always a proper non-Abelian subgroup $H$ of the unitary group $U_{n}(\mathbb{C})$ which contains the subgroup $D$ of all diagonal unitary matrices, and has $H/D$ isomorphic to the symmetric group $S_{n}$. | |
| Jun 2, 2020 at 11:08 | comment | added | YCor | The question in the first sentence is extremely broad. One short answer is "yes". The second question is quite trivial anyway (use monomial matrices, assuming the dimension is $\ge 2$). A general natural question is then to classify subgroups containing the diagonal group (I'd suggest to edit accordingly). | |
| Jun 2, 2020 at 11:06 | comment | added | Najib Idrissi | Can't you take block matrices of the form $\begin{pmatrix} A & 0 \\ 0 & \theta \end{pmatrix}$ (where $A$ is a matrix of dimension one less)? | |
| Jun 2, 2020 at 11:03 | history | asked | Mini | CC BY-SA 4.0 |