Timeline for Why are the Dynkin diagrams E6, E7 and E8 always drawn the way they are drawn?
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12 events
| when toggle format | what | by | license | comment | |
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| Jul 7, 2010 at 2:05 | history | edited | Jose Brox |
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| Apr 9, 2010 at 18:14 | answer | added | Jim Humphreys | timeline score: 6 | |
| Apr 9, 2010 at 17:48 | answer | added | S. Carnahan♦ | timeline score: 5 | |
| Apr 9, 2010 at 17:20 | comment | added | Mariano Suárez-Álvarez | Also, I have seen the non-$A$ diagrams drawn as a rooted tree with the root the ramified vertex set to the extreme left. I do not recall the context, but I do recall there was a point in drawing them like that. | |
| Apr 9, 2010 at 17:14 | comment | added | Guntram | Actually, the Dynkin diagrams aren't always drawn in the standard way: See en.wikipedia.org/wiki/Root_system for a really nice compact way of organizing all irreducible diagrams into a single picture. | |
| Apr 9, 2010 at 17:11 | comment | added | Matthew Stover | Of course, you're right; one can figure out the automorphisms without the classical picture. I was merely saying that they're always drawn as they are because that's the picture for which the automorphisms stare you in the face. I only meant to argue for aesthetic optimization, not deeper mathematical content. | |
| Apr 9, 2010 at 16:55 | comment | added | José Figueroa-O'Farrill | @Matthew: I do not understand what you are saying. There is no more information in the Dynkin diagram than in the Cartan matrix. Symmetries of a graph are permutations of the vertices which preserve the incidence relations. How you draw the diagram in the plane is inconsequential. | |
| Apr 9, 2010 at 16:21 | comment | added | Qiaochu Yuan | @Jose: Again, I don't necessary know what I'm talking about, but in the Week I linked to, I think John Baez draws all his Dynkin diagrams such that the vertices are arranged in order of the dimension of the corresponding quotient G/P by a maximal parabolic. | |
| Apr 9, 2010 at 16:18 | comment | added | Matthew Stover | I would disagree that only the connections between the nodes are important. The symmetries of the diagram are also fundamental, and the way we typically draw the diagrams exhibits that symmetry, and in the case of the $E_n$ diagrams, how the cases $n = 7, 8$ extend the more symmetric $E_6$ diagram. | |
| Apr 9, 2010 at 16:09 | comment | added | José Figueroa-O'Farrill | The only information in the Dynkin diagram is the incidence relations: which node connects to which node. How you choose to embed that graph in the plane is of no consequence. | |
| Apr 9, 2010 at 16:01 | comment | added | Qiaochu Yuan | I don't necessarily know what I'm talking about, but the interpretation of the vertices of Dynkin diagrams given by John Baez here might be a clue: math.ucr.edu/home/baez/week180.html | |
| Apr 9, 2010 at 15:48 | history | asked | nnn | CC BY-SA 2.5 |