Starting from $[1,0,0]$ and applying the simple-root <lowering operator>[lowerings] gives the four weights
$$
[1,0,0],\quad[-1,1,0],\quad[0,-1,1],\quad[0,0,-1].
$$
Each has <weight multiplicity> one, so $d_1$ is the four-dimensional defining representation of $A_3\cong\mathfrak{sl}_4(\mathbb C)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 4 / c / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-302/4/c/i/solution