= Solution
The <restriction of a representation> $\rho:G\to\operatorname{GL}(V)$ to $H$ is the same <vector space> and operators with the domain limited to $H$; its <character> is $\chi_H(h)=\chi(h)$. Averaging a positive Hermitian <inner product> over $H$ makes it invariant, so invariant subspaces have invariant orthogonal complements. Repeating gives a direct sum of irreducible $H$-spaces and hence $\chi_H=\sum_i d_i\psi_i$ with nonnegative integer multiplicities.
By <character inner product> <orthogonality>,
$$
\sum_i d_i^2=\langle\chi_H,\chi_H\rangle_H
=\frac1{|H|}\sum_{h\in H}|\chi(h)|^2
\le\frac1{|H|}\sum_{g\in G}|\chi(g)|^2=\boxed{[G:H]},
$$
since $\chi$ is irreducible and has normalized squared <norm> one on $G$. Equality holds precisely when the omitted nonnegative terms vanish, namely $\boxed{\chi(g)=0\text{ for every }g\notin H}$. This is the <restriction multiplicity bound>.
Back to article page
