Let $G=\mathrm{SU}(2)$. Let $V_{n}$ be the complex vector space of homogeneous polynomials of degree $n$ in two variables $z_{1}, z_{2}$. Define the usual left action of $G$ on $V_{n}$ and denote by $\rho_{n}: G \rightarrow \operatorname{GL}\left(V_{n}\right)$ the representation induced by this action. Describe the character $\chi_{n}$ afforded by $\rho_{n}$.

Quoting carefully any results you need, show that

(i) The representation $\rho_{n}$ has dimension $n+1$ and is irreducible for $n \in \mathbb{Z}_{\geqslant 0}$;

(ii) Every finite-dimensional continuous irreducible representation of $G$ is one of the $\rho_{n}$;

(iii) $V_{n}$ is isomorphic to its dual $V_{n}^{*}$.