Wich representations of $F_{4}$, $E_{8}$ and $G_{2}$ are quasiminuscule?
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As a quasiminuscule representation has a zero weight, its weights must be in the root lattice. Since every nonzero dominant weight is positive on at least one simple root, the quasiminuscule condition implies that nonzero 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 quasiminuscule representation, and the unique dominant short root is its highest weight. 


There is a list here. 

