Timeline for Non-inherited symmetries of shadows of point sets
Current License: CC BY-SA 2.5
8 events
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| Dec 6, 2010 at 4:40 | comment | added | Peter LeFanu Lumsdaine | (cont’d) The answer has to depend on which of $e_1,e_2,e_3$ you were sending to which roots of unity! So I would correct the definition by: instead of just defining a relation “$\Omega_1 \mapsto \Omega_2$”, define something like “$f : \Omega_1 \mapsto \Omega_2$”, and then give the later definitions in terms of $f$ as well as $\Omega_i$. (If I’m not misunderstanding something, there is also a second issue with the current “inheritance” definition, which Gerhard brings up in the comments to his answer.) | |
| Dec 6, 2010 at 4:34 | comment | added | Peter LeFanu Lumsdaine | @David: You say “If $\Omega_1 \mapsto \Omega_2$ then one has a natural way to identify [their] permutation symmetries…” I don’t think you have, without specifying which linear map you have in mind witnessing $\Omega_1 \mapsto \Omega_2$. For instance, take $\Omega_1$ to be for example the standard basis $\{e_1,e_2,e_3\}$ in $\mathbb{R}^3$; take $\Omega_2$ to be the cube roots of unity $\omega,\omega^2,1$ in $\mathbb{C}$. Certainly we have $\Omega_1 \mapsto \Omega_2$. Take the symmetry of $\Omega_1$ switching $e_1$, $e_2$. What symmetry does it correspond to on $\Omega_2$? | |
| Dec 6, 2010 at 1:38 | comment | added | David Richter | Peter: After you prompted me to look at it more carefully, I made a correction to my explanation of the notation $\Omega_1\mapsto\Omega_2$. Thanks! (I still don't understand the mistake you are talking about, however. Can you tell me how to correct it?) | |
| Dec 6, 2010 at 1:26 | history | edited | David Richter | CC BY-SA 2.5 |
added 43 characters in body
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| Dec 6, 2010 at 0:43 | comment | added | Peter LeFanu Lumsdaine | Your current definition of $\Omega_1 \mapsto \Omega_2$ just as a binary relation (“there exists a linear map such that…”) isn’t quite right, I think: there could be multiple linear maps sending $\Omega_1$ to $\Omega_2$, and your later definitions (inherited symmetries etc.) depend heavily on which map is under consideration. So surely the definitions need to involve choosing a specific linear map? (Mistakes of this sort have a long pedigree: there is one in Hartshorne, for instance :-P The good news is that they’re generally easily fixed…) | |
| Dec 5, 2010 at 22:34 | answer | added | Gerhard Paseman | timeline score: 3 | |
| Dec 5, 2010 at 22:16 | comment | added | Joseph O'Rourke | Related (but related only) is this MO question asking whether the most symmetric 3D polyhedra can be obtained as projections of the most symmetric 4D polytopes (definitely not "unexpected"): mathoverflow.net/questions/45503/… | |
| Dec 5, 2010 at 21:37 | history | asked | David Richter | CC BY-SA 2.5 |