(i) Given that the character of an $S U(2)$ transformation in the $(2 l+1)$-dimensional irreducible representation $d_{l}$ is given by

$$\chi_{l}(\theta)=\frac{\sin \left(l+\frac{1}{2}\right) \theta}{\sin \frac{\theta}{2}}$$

show how the direct product representation $d_{l_{1}} \otimes d_{l_{2}}$ decomposes into irreducible $S U(2)$ representations.

(ii) Find the decomposition of the direct product representation $3 \otimes \overline{3}$ of $S U(3)$ into irreducible $S U(3)$ representations.

Mesons consist of one quark and one antiquark. The scalar Meson Octet consists of the following particles: $K^{\pm}\left(Y=\pm 1, I_{3}=\pm \frac{1}{2}\right), K^{0}\left(Y=1, I_{3}=-\frac{1}{2}\right), \bar{K}^{0}(Y=-1$, $\left.I_{3}=\frac{1}{2}\right), \pi^{\pm}\left(Y=0, I_{3}=\pm 1\right), \pi^{0}\left(Y=0, I_{3}=0\right)$ and $\eta\left(Y=0, I_{3}=0\right)^{2}$.

Use the direct product representation $3 \otimes \overline{3}$ of $S U(3)$ to identify the quark-type of the particles in the scalar Meson Octet. Deduce the quark-type of the $S U(3)$ singlet state $\eta^{\prime}$ contained in $3 \otimes \overline{3}$.