(i) Let $D_{6}$ denote the symmetry group of rotations and reflections of a regular hexagon. The elements of $D_{6}$ are given by $\left\{e, c, c^{2}, c^{3}, c^{4}, c^{5}, b, b c, b c^{2}, b c^{3}, b c^{4}, b c^{5}\right\}$ with $c^{6}=b^{2}=e$ and $c b=b c^{5}$. The conjugacy classes of $D_{6}$ are $\{e\},\left\{c, c^{5}\right\},\left\{c^{2}, c^{4}\right\},\left\{c^{3}\right\},\left\{b, b c^{2}, b c^{4}\right\}$ and $\left\{b c, b c^{3}, b c^{5}\right\}$.

Show that the character table of $D_{6}$ is

\begin{tabular}{l|rrrrrr} $D_{6}$ & $e$ & $\left\{c, c^{5}\right\}$ & $\left\{c^{2}, c^{4}\right\}$ & $\left\{c^{3}\right\}$ & $\left\{b, b c^{2}, b c^{4}\right\}$ & $\left\{b c, b c^{3}, b c^{5}\right\}$ \ \hline$\chi_{1}$ & 1 & 1 & 1 & 1 & 1 & 1 \ $\chi_{2}$ & 1 & 1 & 1 & 1 & $-1$ & $-1$ \ $\chi_{3}$ & 1 & $-1$ & 1 & $-1$ & 1 & $-1$ \ $\chi_{4}$ & 1 & $-1$ & 1 & $-1$ & $-1$ & 1 \ $\chi_{5}$ & 2 & 1 & $-1$ & $-2$ & 0 & 0 \ $\chi_{6}$ & 2 & $-1$ & $-1$ & 2 & 0 & 0 \end{tabular}

(ii) Show that the character of an $S O(3)$ rotation with angle $\theta$ in the $2 l+1$ dimensional irreducible representation of $S O(3)$ is given by

$$\chi_{l}(\theta)=1+2 \cos \theta+2 \cos (2 \theta)+\ldots+2 \cos ((l-1) \theta)+2 \cos (l \theta)$$

For a hexagonal crystal of atoms find how the degeneracy of the D-wave orbital states $(l=2)$ in the atomic central potential is split by the crystal potential with $D_{6}$ symmetry and give the new degeneracies.

By using the fact that $D_{3}$ is isomorphic to $D_{6} /\left\{e, c^{3}\right\}$, or otherwise, find the degeneracies of eigenstates if the hexagonal symmetry is broken to the subgroup $D_{3}$ by a deformation. The introduction of a magnetic field further reduces the symmetry to $C_{3}$. What will the degeneracies of the energy eigenstates be now?