The positive roots generated by the two simple roots are
$$
\alpha_1,\quad\alpha_2,\quad\alpha_1+\alpha_2,\quad
2\alpha_1+\alpha_2,\quad3\alpha_1+\alpha_2,\quad
3\alpha_1+2\alpha_2.
$$
The <Adjoint representation of a Lie algebra> has one weight space for each positive and negative root and a rank-two zero-weight space. Its nonzero weights are therefore
$$
\begin{array}{c|c}
\text{root}&\text{coordinate}\\ \hline
\alpha_1&(1,0)\\
\alpha_1+\alpha_2&(-\tfrac12,\tfrac{\sqrt3}2)\\
2\alpha_1+\alpha_2&(\tfrac12,\tfrac{\sqrt3}2)\\
\alpha_2&(-\tfrac32,\tfrac{\sqrt3}2)\\
3\alpha_1+\alpha_2&(\tfrac32,\tfrac{\sqrt3}2)\\
3\alpha_1+2\alpha_2&(0,\sqrt3)
\end{array}
$$
together with their negatives. Thus the diagram consists of a hexagon of six long roots, a hexagon of six short roots, and the origin with multiplicity two. The representation has dimension $12+2=14$.

Solution

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

 Solution

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