Timeline for Classification of finite complex reflection groups
Current License: CC BY-SA 2.5
21 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2019 at 4:03 | comment | added | LSpice | I think that your definition of (pseudo-)reflection is non-standard, since it doesn't include the usual finite-order condition. | |
| Jun 24, 2010 at 8:54 | comment | added | Bugs Bunny | Steve: I think so. The reflection rep is determined by the diagram, diagram automorphism does not change the data that determines it. | |
| Jun 23, 2010 at 18:01 | comment | added | GS | Arminius: For non-faithful rep's $G$, T. Chmutova has written this paper: arxiv.org/abs/math/0505653 I think she will have something to say about what you're interested in. | |
| Jun 23, 2010 at 10:19 | comment | added | GS | Bugs: Interesting! Is it obvious that should be so? | |
| Jun 23, 2010 at 7:08 | comment | added | Bugs Bunny | Hey, Steve! I think I have gone slightly bonkers with my comment: twisting by the outer automorphism does not change to reflection representation. I see it is true, in general: twist by a diagram automorphism is not changing the reflection representation of the Coxeter group. | |
| Jun 22, 2010 at 12:15 | history | edited | user717 | CC BY-SA 2.5 |
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| Jun 22, 2010 at 12:09 | history | edited | user717 | CC BY-SA 2.5 |
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| Jun 22, 2010 at 12:00 | history | edited | user717 | CC BY-SA 2.5 |
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| Jun 22, 2010 at 11:54 | history | edited | user717 | CC BY-SA 2.5 |
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| Jun 22, 2010 at 11:13 | comment | added | GS | You're absolutely right! That observation (that reflection groups are precisely those with polynomial invariants) really was important. | |
| Jun 22, 2010 at 11:10 | comment | added | Victor Protsak | I just want to point out that the point of Shephard-Todd was not classification, rather it was the Chevalley-Shephard-Todd theorem. Since their proof is case-by-case, they listed complex reflection groups and so their paper became a convenient reference. | |
| Jun 22, 2010 at 10:24 | comment | added | GS | ...though Shephard-Todd probably gets more credit for completing the classification than is really deserved. Most of the work had been done, I believe, by the time that paper was written---especially for the classification of the primitive groups. | |
| Jun 22, 2010 at 10:17 | comment | added | GS | ..also, if we're identifying things up to automorphisms coming from GL(V), the question is much less interesting (and in fact, Arminius gave the answer in the text of the question). | |
| Jun 22, 2010 at 10:04 | comment | added | GS | Hiya Bugs, For W(B2) at least they are not different. The group of outer automorphisms may well fix rep'ns occasionally. | |
| Jun 22, 2010 at 9:59 | comment | added | Bugs Bunny | The difference between these definitions is in outer automorphisms. Arminius identifies twists by outer automorphisms, while Stephen does not. Say, take the dihedral group with even edge: it has two standard reflection representations, different by outer twist. In Arminius' world they are the same and in Stephen's they are different. | |
| Jun 22, 2010 at 9:50 | answer | added | Bugs Bunny | timeline score: 3 | |
| Jun 22, 2010 at 9:13 | comment | added | GS | I don't have Kane's book to hand, but it seems likely to me that he introduces this definition in order to discuss the classification of subgroups of GL(V) up to conjugacy---which is really a different from classifying representations of a particular group up to isomorphism! The latter is a finer (fewer elements in each equivalence class) classification, and more interesting/difficult. | |
| Jun 22, 2010 at 9:07 | comment | added | user717 | Good question. I don't know. I came across this definition in "Reflection groups and invariant theory" by Richard Kane (p. 156). But he's not really using this concept... | |
| Jun 22, 2010 at 9:04 | comment | added | GS | Your definition of isomorphism of representations is somewhat different from usual no? Normally one would require an elementwise compatibility $f \rho(g) f^{-1}=\rho'(g)$ for all $g \in G$. | |
| Jun 22, 2010 at 8:46 | answer | added | Bruce Westbury | timeline score: 3 | |
| Jun 22, 2010 at 8:38 | history | asked | user717 | CC BY-SA 2.5 |