(a) Let $S^{1}$ be the circle group. Assuming any required facts about continuous functions from real analysis, show that every 1-dimensional continuous representation of $S^{1}$ is of the form

$$z \mapsto z^{n}$$

for some $n \in \mathbb{Z}$.

(b) Let $G=S U(2)$, and let $\rho_{V}$ be a continuous representation of $G$ on a finitedimensional vector space $V$.

(i) Define the character $\chi_{V}$ of $\rho_{V}$, and show that $\chi_{V} \in \mathbb{N}\left[z, z^{-1}\right]$.

(ii) Show that $\chi_{V}(z)=\chi_{V}\left(z^{-1}\right)$.

(iii) Let $V$ be the irreducible 4-dimensional representation of $G$. Decompose $V \otimes V$ into irreducible representations. Hence decompose the exterior square $\Lambda^{2} V$ into irreducible representations.