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Wich representations of $F_{4}$, $E_{8}$ and $G_{2}$ are quasi-minuscule?

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yes but we don't know to which highest weight they correspond. –  prochet May 7 '13 at 15:16
    
And for these representations, what is the characteristic polynomial? More generally where can we find the polynomial invariants for exceptional groups? –  prochet May 7 '13 at 19:34

2 Answers 2

As a quasi-minuscule representation has a zero weight, its weights must be in the root lattice. Since every non-zero dominant weight is positive on at least one simple root, the quasi-minuscule condition implies that non-zero weights are in fact roots. There is only one orbit of them, so in case there are different root lengths, the long roots are excluded. In summary, for every simple type there is a unique quasi-minuscule representation, and the unique dominant short root is its highest weight.

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@Marc: This is an efficient short summary (more helpful than what's on Wikipedia too, which by the way has an offbeat reference ignoring Bourbaki's introduction of the term minuscule). To attach the precise weight labels is an easy exercise, given the data on each root system in Bourbaki. In the simply-laced cases the representation is of course just the adjoint representation. –  Jim Humphreys May 7 '13 at 17:47

There is a list here.

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