Restriction multiplicity bound

For an irreducible complex character $\chi$ of a finite group $G$ and subgroup $H$, write its restriction as ${\sum }_i{d}_i{\psi }_i$. Orthogonality gives ${\sum }_i{d}_i^2=|H{|}^{-1}{\sum }_{h\in H}|\chi (h){|}^2\le |G|/|H|$, because $|G{|}^{-1}{\sum }_G|\chi {|}^2=1$. Equality holds exactly when $\chi$ vanishes outside $H$, since the omitted sum consists of nonnegative terms.