Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 19I / b

$SU(2)$ consists of unitary $2\times 2$ matrices of determinant one. Every element is unitarily diagonalizable with eigenvalues $e^{\pm i\theta }$, and two are conjugate exactly when they have the same trace. Thus $t(g)=\mathrm{tr}(g)/2=\cos \theta$ bijects conjugacy classes with $[-1,1]$.

For $-1<x<1$, the centralizer of the diagonal representative is the maximal torus $T$. Hence the orbit is $SU(2)/T$. Identifying $SU(2)$ with $S^3$ makes this quotient the Hopf quotient $S^3/S^1\cong S^2$.

If $V$ is the defining representation, ${\chi }_V=2t$. The character of ${\mathrm{Sym}}^{3}V$ is $(2t{)}^3-2(2t)=8t^3-4t$. Therefore $8t^3$ is the character of ${\mathrm{Sym}}^{3}V\oplus V\oplus V$.