One root basis is
$$
\alpha_1=\varepsilon_1-\varepsilon_2,
\qquad
\alpha_2=\varepsilon_2-\varepsilon_3,
\qquad
\alpha_3=\varepsilon_2+\varepsilon_3.
$$
All roots have the same length. The nonzero inner products are $(\alpha_1,\alpha_2)=(\alpha_1,\alpha_3)=-1$, so the labeled <Dynkin diagram> is
$$
\alpha_2\;\text{---}\;\alpha_1\;\text{---}\;\alpha_3.
$$
It is the three-node $A_3$ diagram.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 3 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-102/3/b/ii/solution