(i) Let $h: G \rightarrow G^{\prime}$ be a surjective homomorphism between two groups, $G$ and $G^{\prime}$. If $D^{\prime}: G^{\prime} \rightarrow G L\left(\mathbb{C}^{n}\right)$ is a representation of $G^{\prime}$, show that $D(g)=D^{\prime}(h(g))$ for $g \in G$ is a representation of $G$ and, if $D^{\prime}$ is irreducible, show that $D$ is also irreducible. Show further that $\widetilde{D}(\widetilde{g})=D^{\prime}(\widetilde{h}(\widetilde{g}))$ is a representation of $G / \operatorname{ker}(h)$, where $\tilde{h}(\widetilde{g})=h(g)$ for $g \in G$ and $\widetilde{g} \in G / \operatorname{ker}(h)$ (with $g \in \widetilde{g}$ ). Deduce that the characters $\chi, \widetilde{\chi}, \chi^{\prime}$ of $D, \widetilde{D}, D^{\prime}$, respectively, satisfy

$$\chi(g)=\tilde{\chi}(\widetilde{g})=\chi^{\prime}(h(g))$$

(ii) $D_{4}$ is the symmetry group of rotations and reflections of a square. If $c$ is a rotation by $\pi / 2$ about the centre of the square and $b$ is a reflection in one of its symmetry axes, then $D_{4}=\left\{e, c, c^{2}, c^{3}, b, b c, b c^{2}, b c^{3}\right\}$. Given that the conjugacy classes are $\{e\}\left\{c^{2}\right\},\left\{c, c^{3}\right\}$ $\left\{b, b c^{2}\right\}$ and $\left\{b c, b c^{3}\right\}$ derive the character table of $D_{4}$.

Let $H_{0}$ be the Hamiltonian of a particle moving in a central potential. The $S O(3)$ symmetry ensures that the energy eigenvalues of $H_{0}$ are the same for all the angular momentum eigenstates in a given irreducible $S O(3)$ representation. Therefore, the energy eigenvalues of $H_{0}$ are labelled $E_{n l}$ with $n \in \mathbb{N}$ and $l \in \mathbb{N}_{0}, l<n$. Assume now that in a crystal the symmetry is reduced to a $D_{4}$ symmetry by an additional term $H_{1}$ of the total Hamiltonian, $H=H_{0}+H_{1}$. Find how the $H_{0}$ eigenstates in the $S O(3)$ irreducible representation with $l=2$ (the D-wave orbital) decompose into irreducible representations of $H$. You may assume that the character, $g(\theta)$, of a group element of $S O(3)$, in a representation labelled by $l$ is given by

$$\chi\left(g_{\theta}\right)=1+2 \cos \theta+2 \cos (2 \theta)+\ldots+2 \cos (l \theta)$$

where $\theta$ is a rotation angle and $l(l+1)$ is the eigenvalue of the total angular momentum, $\mathbf{L}^{2}$.