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Are highest root and highest short root the same? Are there some example to show that the highest root and the highest short are not the same? Are there some example to show that the highest root and the highest short are the same?

Are there some example for the highest weight of a representation of an Lie algebra is (respectively, is not) the highest root? Many thanks.

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In an irreducible root system with two root lengths, the highest root is long, hence distinct from the highest short root. For example, in the root system $B_2$, the highest root is $2\alpha+\beta$, whereas the highest short root is $\alpha+\beta$. Of course, if the root system is simply laced (all roots are of the same length) then the two notions coincide.

Regarding your last question, for every integral dominant weight $\lambda$ there exists an irreducible representation with $\lambda$ as its highest weight. In particular, the highest weight of a representation need not be a root at all.

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