Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 19H (Representation Theory)

The topological group $S^1=\{z\in \mathbb{C}:|z|=1\}$ has complex multiplication and the subspace topology. It is compact and abelian, so every finite-dimensional irreducible complex representation is one-dimensional. A continuous character lifts along $t\mapsto e^{it}$ to a continuous homomorphism $\mathbb{R}\to {\mathbb{C}}^{\times }$, hence has the form $t\mapsto e^{i\lambda t}$. Periodicity by $2\pi$ forces $\lambda =n\in \mathbb{Z}$, giving precisely $z\mapsto z^n$.

Every element of $SU(2)$ is unitarily conjugate to an element of the maximal torusSince characters are class functions, ${\chi }_V$ is determined on $T$. The restriction decomposes into finitely many torus weights, soA Weyl-group element conjugates $\mathrm{diag}(z,z^{-1})$ to $\mathrm{diag}(z^{-1},z)$, so ${\chi }_V(z)={\chi }_V(z^{-1})$ and $a_n=a_{-n}$.

For the $(n+1)$-dimensional irreducible,The Clebsch--Gordan rule givesThe even total-spin summands are symmetric and the odd ones alternating, hence