Starting from the highest weight $\mu_1=2\alpha_1+\alpha_2$, lowering with the simple-root operators and closing under the <Weyl group> gives
$$
\boxed{0,\quad
\pm\alpha_1,\quad
\pm(\alpha_1+\alpha_2),\quad
\pm(2\alpha_1+\alpha_2)}.
$$
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 $G_2$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-302/1/vi/solution