Past exam of the groups and representations course of the University of Oxford / 2024 / 2 / b

Use the Hermitian $H_a$ from a to record real weight coordinates: $H_a{e}_j=(w_j{)}_a{e}_j$. Reading off the diagonal entries givesThese are weights in the simple-coroot coordinate system, not coordinates in an orthonormal basis. In the skew-Hermitian convention they mean $(i{H}_a)e_j=i(w_j{)}_a{e}_j$. Intrinsically, the weight on $i\mathrm{diag}(h_1,h_2,h_3,h_4)$ is the functional $h\mapsto h_j$, with ${\sum }_j{h}_j=0$.

The conjugate representation acts by $U^{\ast }$, so a diagonal transformation with entries $e^{i{h}_j}$ becomes one with entries $e^{-i{h}_j}$. Thus its four weights areAll have multiplicity one. In both representations the sum of the weights is zero, consistent with the vanishing trace of every Lie algebra generator.