The matrix is the <Cartan matrix of type A3>. The positive roots are
$$
\alpha_1,
\quad\alpha_2,
\quad\alpha_3,
\quad\alpha_1+\alpha_2,
\quad\alpha_2+\alpha_3,
\quad\alpha_1+\alpha_2+\alpha_3.
$$
Together with their negatives they form
$$
\Phi=\{\pm\alpha_1,\pm\alpha_2,\pm\alpha_3,
\pm(\alpha_1+\alpha_2),\pm(\alpha_2+\alpha_3),
\pm(\alpha_1+\alpha_2+\alpha_3)\},
$$
so $|\Phi|=12$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 302 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-302/3/c/solution