Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 302 / 1 / vi

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- Solution vi

Solution

Starting from the highest weight

μ_{1} = 2 α_{1} + α_{2}

, lowering with the simple-root operators and closing under the Weyl group gives

0, \pm α_{1}, \pm (α_{1} + α_{2}), \pm (2 α_{1} + α_{2}) .

Every weight has multiplicity one. The diagram is the hexagon of short roots with one central weight, so this is the seven-dimensional fundamental irreducible representation of

G_{2}

.

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