Define $G=\mathrm{SU}(2)$ and write down a complete list

$$\left\{V_{n}: n=0,1,2, \ldots\right\}$$

of its continuous finite-dimensional irreducible representations. You should define all the terms you use but proofs are not required. Find the character $\chi_{V_{n}}$ of $V_{n}$. State the Clebsch-Gordan formula.

(a) Stating clearly any properties of symmetric powers that you need, decompose the following spaces into irreducible representations of $G$ :

(i) $V_{4} \otimes V_{3}, V_{3} \otimes V_{3}, S^{2} V_{3}$;

(ii) $V_{1} \otimes \cdots \otimes V_{1}$ (with $n$ multiplicands);

(iii) $S^{3} V_{2}$.

(b) Let $G$ act on the space $M_{3}(\mathbb{C})$ of $3 \times 3$ complex matrices by

$$A: X \mapsto A_{1} X A_{1}^{-1}$$

where $A_{1}$ is the block matrix $\left(\begin{array}{cc}A & 0 \\ 0 & 1\end{array}\right)$. Show that this gives a representation of $G$ and decompose it into irreducible summands.