Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 19F (Representation Theory)

$SU(2)=\{A:A^{\ast }A=I,\det A=1\}$; every element is $\left(\begin{array}{cc}a& b\\ -\bar{b}& \bar{a}\end{array}\right)$ with $|a{|}^2+|b{|}^2=1$, giving $S^3$. A maximal torus is $T=\{\mathrm{diag}(e^{it},e^{-it})\}$. A noncentral conjugacy class is determined by trace $2\cos t$, has stabilizer $T$, and is $SU(2)/T\simeq S^2$; it meets $T$ at the two inverse diagonal elements. Since $SU(2)$ has no nontrivial continuous characters, $\det \rho =1$.

Map the quaternion generators to $i{\sigma }_x,i{\sigma }_y$ (and their products), obtaining the unique two-dimensional irreducible $V$ and an embedding $Q_8\subset SU(2)$. Clebsch-Gordan gives $V\otimes V\simeq {\mathrm{Sym}}^{2}V\oplus 1$ and $V^{\otimes 3}\simeq {\mathrm{Sym}}^{3}V\oplus V\oplus V$. On restriction, ${\mathrm{Sym}}^{2}V$ is the sum of the three nontrivial one-dimensional characters of $Q_8$, while the other summand is trivial.