Timeline for Smallest dimension of nontrivial representation of a simple Lie algebra over `$\mathbb{C}$`
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Apr 28, 2013 at 22:10 | vote | accept | Jim Humphreys | ||
| Oct 15, 2012 at 22:01 | comment | added | Jim Humphreys |
Yes, this involves the whole story about affine Weyl groups relative to the root system and its dual: see [B1] $\S6.2$ and $\S4.9$ of my book on Coxeter groups along with the papers by Verma and Iwahori-Matsumoto. Everything here is connected to everything else, but for the representation theory it's hard to find a conceptual pathway through the maze.
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| Oct 10, 2012 at 13:09 | comment | added | Allen Knutson | Something a little weird. The set of cosets $X/X_r$ is $Z(G)^*$, where $G$ is the simply-connected group. The set of minuscule representations plus the trivial representation corresponds to the pointiest corners of the Weyl alcove, which in turn correspond to $Z(G)$. So your (1) seems to be asserting a bijection between $Z(G)$ and its dual. Right? | |
| Oct 3, 2012 at 12:38 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
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| Oct 3, 2012 at 7:40 | answer | added | Alexander Premet | timeline score: 7 | |
| Sep 30, 2012 at 13:28 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
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| Sep 30, 2012 at 13:23 | history | asked | Jim Humphreys | CC BY-SA 3.0 |