Let $H \leqslant G$ be finite groups.

(a) Let $\rho$ be a representation of $G$ affording the character $\chi$. Define the restriction, $\operatorname{Res}_{H}^{G} \rho$ of $\rho$ to $H$.

Suppose $\chi$ is irreducible and suppose $\operatorname{Res}_{H}^{G} \rho$ affords the character $\chi_{H}$. Let $\psi_{1}, \ldots, \psi_{r}$ be the irreducible characters of $H$. Prove that $\chi_{H}=d_{1} \psi_{1}+\cdots+d_{r} \psi_{r}$, where the nonnegative integers $d_{1}, \ldots, d_{r}$ satisfy the inequality

$$\sum_{i=1}^{r} d_{i}^{2} \leqslant|G: H|$$

Prove that there is equality in (1) if and only if $\chi(g)=0$ for all elements $g$ of $G$ which lie outside $H$.

(b) Let $\psi$ be a class function of $H$. Define the induced class function, $\operatorname{Ind}_{H}^{G} \psi$.

State the Frobenius reciprocity theorem for class functions and deduce that if $\psi$ is a character of $H$ then $\operatorname{Ind}_{H}^{G} \psi$ is a character of $G$.

Assuming $\psi$ is a character, identify a $G$-space affording the character $\operatorname{Ind}_{H}^{G} \psi$. Briefly justify your answer.

(c) Let $\chi_{1}, \ldots, \chi_{k}$ be the irreducible characters of $G$ and let $\psi$ be an irreducible character of $H$. Show that the integers $e_{1}, \ldots, e_{k}$, which are given by $\operatorname{Ind}_{H}^{G}(\psi)=$ $e_{1} \chi_{1}+\cdots+e_{k} \chi_{k}$, satisfy

$$\sum_{i=1}^{k} e_{i}^{2} \leqslant|G: H|$$