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

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

The representation with highest weight $[0,0,1]$ is the dual representation of $d_1$. Its weights are the negatives of those of $d_1$:Thus $\dim d_2=4$.

For $B_3\cong {\mathfrak{so}}_7(\mathbb{C})$, the highest weight $[0,0,1]$ is the spinor representation. Its eight weights, all of multiplicity one, areEquivalently, in an orthonormal basis they are $(\pm e_1\pm e_2\pm e_3)/2$ with independent signs. Hence $\dim d_3=8$.