Let $G=S U_{2}$, and $V_{n}$ be the vector space of homogeneous polynomials of degree $n$ in the variables $x$ and $y$.

(i) Define the action of $G$ on $V_{n}$, and prove that $V_{n}$ is an irreducible representation of $G$.

(ii) Decompose $V_{4} \otimes V_{3}$ into irreducible representations of $S U_{2}$. Briefly justify your answer.

(iii) $S U_{2}$ acts on the vector space $M_{3}(\mathbf{C})$ of complex $3 \times 3$ matrices via

$$\rho\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \cdot X=\left(\begin{array}{lll} a & b & 0 \\ c & d & 0 \\ 0 & 0 & 1 \end{array}\right) X\left(\begin{array}{lll} a & b & 0 \\ c & d & 0 \\ 0 & 0 & 1 \end{array}\right)^{-1}, \quad X \in M_{3}(\mathbf{C}) .$$

Decompose this representation into irreducible representations.