Timeline for Common basis for permutation matrices
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| S Mar 19, 2020 at 16:41 | history | bounty ended | as2457 | ||
| S Mar 19, 2020 at 16:41 | history | notice removed | as2457 | ||
| Mar 15, 2020 at 14:09 | answer | added | Denis Serre | timeline score: 2 | |
| Mar 13, 2020 at 16:57 | comment | added | Nick Gill | The angled backed notation means "the group generated by $A$ and $B$" -- i.e. all matrices obtained by taking finite length products of $A$, $B$ and their inverses. | |
| Mar 13, 2020 at 16:56 | answer | added | Nick Gill | timeline score: 2 | |
| S Mar 13, 2020 at 14:15 | history | bounty started | as2457 | ||
| S Mar 13, 2020 at 14:15 | history | notice added | as2457 | Draw attention | |
| Mar 10, 2020 at 15:28 | comment | added | as2457 | Clearly if $AB$ is not unitarily similar to a permutation matrix then there is no such $U$. If $AB$ is similar to a permutation as well as $A$ and $B$ separately, can I say anything? Can I say something in the particular case that $A$ and $B$ are representations of the generators of the modular group? | |
| Mar 10, 2020 at 8:58 | comment | added | as2457 | I don't know if it helps (or if I should add to the question), but in the case I'm interested in $A$ and $B$ are representations of the generators of the modular group (90deg rotation and Dehn twist). They are also symmetric in the original basis. | |
| Mar 10, 2020 at 7:20 | comment | added | YCor | @MarkSapir and all the subgroup they generate. This is why I added the group tag. For instance one can wonder, if the product of, say all $2^n$ $n$-fold products ($n$ being the size) of $A,B$ are conjugate to permutation matrices, whether it's enough to ensure that they can be simultaneously conjugate into the subgroup of permutation matrices. (Already one can wonder about an algorithm to check if two matrices generate a finite group.) | |
| Mar 9, 2020 at 17:48 | comment | added | as2457 | Yes, what I am imagining is being given two matrices and trying to determine whether there exists such a U. This could either be using the properties of A and B, or it could be determined through some algorithm designed to find U. Could you also clarify what you mean by the angled bracket notation please? | |
| Mar 9, 2020 at 14:53 | comment | added | Nick Gill | When you say "how can I check", are you imagining that you have been given $A$ and $B$ explicitly, and you want to do a computation... Or are you asking for theoretical conditions that are equivalent? One obvious necessary condition is that $\langle A, B\rangle$ is isomorphic to a subgroup of $S_n$... I'm not sure how close it is to being sufficient. | |
| Mar 9, 2020 at 14:19 | comment | added | YCor | Another restatement: is there a subset of $\mathbf{C}^n$ that forms an orthonormal basis, and which is invariant by both $A$ and $B$? (the existence of such a subset for each of $A$ and $B$ being the assumption) | |
| Mar 9, 2020 at 14:16 | history | edited | YCor |
edited tags; edited tags
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| Mar 9, 2020 at 14:10 | history | asked | as2457 | CC BY-SA 4.0 |