(a) Let $G=\mathrm{SU}_{2}$, and let $V_{n}$ be the space of homogeneous polynomials of degree $n$ in the variables $x$ and $y$. Thus $\operatorname{dim} V_{n}=n+1$. Define the action of $G$ on $V_{n}$ and show that $V_{n}$ is an irreducible representation of $G$.

(b) Decompose $V_{3} \otimes V_{3}$ into irreducible representations. Decompose $\wedge^{2} V_{3}$ and $\mathrm{S}^{2} V_{3}$ into irreducible representations.

(c) Given any representation $V$ of a group $G$, define the dual representation $V^{*}$. Show that $V_{n}^{*}$ is isomorphic to $V_{n}$ as a representation of $\mathrm{SU}_{2}$.

[You may use any results from the lectures provided that you state them clearly.]