Timeline for Should the Dynkin diagrams of types $A_1$ and $B_2$ be labelled $C_1$ and $C_2$?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jun 14, 2016 at 6:16 | comment | added | Skip | @fherzig : The Dynkin index of $G$ is the gcd of the Dynkin indexes of its representations (wikipedia). For simple simply connected groups, see e.g. section 2 of arxiv.org/abs/alg-geom/9507002 . For groups that are not simply connected, see arxiv.org/abs/0712.3764 | |
| Jun 14, 2016 at 0:45 | comment | added | André Henriques | Closely related to (A): The universal central extension of $G(k)$ (where $k$ is a field) is Milnor's $K_2(k)$ when $G$ is not of type $A_1$, $B_2$, or $C_n$. | |
| Jun 13, 2016 at 23:11 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
added 351 characters in body; edited title
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| Jun 13, 2016 at 23:10 | comment | added | Jim Humphreys | @Skip: Your points are well taken. Usually $B_2, C_2$ are considered interchangeable, but the behavior of fundamental weights and the highest root makes $C_2$ a better label. | |
| Oct 2, 2014 at 8:10 | comment | added | Bugs Bunny | I just wanted to ask precisely the same question but good old Jim Humphreys already asked it! | |
| Oct 27, 2012 at 21:01 | comment | added | fherzig | @Skip: can you please define what you mean by Dynkin index? | |
| Jan 19, 2011 at 3:48 | comment | added | Skip | Another reason to start the series B at 3 and C at 2 (i.e., call the root system C2 instead of B2, even though both mean the same thing) is that the Dynkin index of the (split) simply connected group of type $C_n$---i.e., $Sp_{2n}$---is 1 for all $n$, but the Dynkin index of $B_n$---i.e., $Spin_{2n+1}$---is 2 for $n \ge 3$. | |
| Jan 17, 2011 at 15:05 | comment | added | Skip | A small comment on your first paragraph that is in the theme of the rest of your question: one might prefer to start the series with B3 and C2 (instead of B2 and C3) because the extended Dynkin diagram for B2=C2 is of the same form as the rest of the C-series, but looks different than for B3, B4, etc. | |
| Jan 17, 2011 at 14:50 | history | asked | Jim Humphreys | CC BY-SA 2.5 |