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

Starting from the highest weight ${\mu }_1=2{\alpha }_1+{\alpha }_2$, lowering with the simple-root operators and closing under the Weyl group givesEvery 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 $G_2$.