[proofplan] We compute $\langle \operatorname{Res}^G_H \chi, \operatorname{Res}^G_H \chi \rangle_H$ in two ways. By orthogonality of irreducible characters of $H$ ([Row Orthogonality](/theorems/2430)), expanding $\operatorname{Res}^G_H \chi = \sum_i c_i \psi_i$ in the orthonormal basis gives $\sum_i c_i^2$. By the definition of the inner product, the same quantity equals $\frac{1}{|H|}\sum_{h \in H} |\chi(h)|^2$. Splitting the $G$-orthonormality $\langle \chi, \chi \rangle_G = 1$ into contributions from $H$ and from $G \setminus H$, and rearranging, gives an exact identity for $\frac{1}{|H|}\sum_{h \in H}|\chi(h)|^2 = |G : H| - \frac{1}{|H|}\sum_{g \in G \setminus H}|\chi(g)|^2$. Since the second term is non-negative, we get $\sum_i c_i^2 \leq |G : H|$, with equality iff $\chi$ vanishes on $G \setminus H$. [/proofplan] [step:Express $\langle \operatorname{Res}^G_H \chi, \operatorname{Res}^G_H \chi \rangle_H$ as $\sum_i c_i^2$ via orthonormality of $\{\psi_i\}$] The restriction $\operatorname{Res}^G_H \chi: H \to \mathbb{C}$ is a (genuine, not virtual) character of $H$, since restriction of a representation $\rho: G \to \operatorname{GL}(V)$ to $H$ yields a representation $\rho\big|_H: H \to \operatorname{GL}(V)$. As a character of $H$, it decomposes uniquely as a non-negative-integer combination of the irreducible characters $\psi_1, \psi_2, \ldots$ of $H$: \begin{align*} \operatorname{Res}^G_H \chi = \sum_{i} c_i \psi_i, \qquad c_i \in \mathbb{Z}_{\geq 0}. \end{align*} By [Row Orthogonality](/theorems/2430), $\langle \psi_i, \psi_j \rangle_H = \delta_{ij}$. Sesquilinearity of the inner product (linear in the first argument, conjugate-linear in the second) and reality of the integers $c_i$ give \begin{align*} \langle \operatorname{Res}^G_H \chi, \operatorname{Res}^G_H \chi \rangle_H = \biggl\langle \sum_i c_i \psi_i, \sum_j c_j \psi_j \biggr\rangle_H = \sum_{i, j} c_i\, \overline{c_j}\, \delta_{ij} = \sum_i c_i^2. \end{align*} [/step] [step:Express the same inner product as $\frac{1}{|H|}\sum_{h \in H} |\chi(h)|^2$ from the definition] By definition of the $H$-inner product on $\mathcal{C}(H)$, applied to the function $\operatorname{Res}^G_H \chi: H \to \mathbb{C}$ paired with itself, \begin{align*} \langle \operatorname{Res}^G_H \chi, \operatorname{Res}^G_H \chi \rangle_H = \frac{1}{|H|} \sum_{h \in H} (\operatorname{Res}^G_H \chi)(h)\, \overline{(\operatorname{Res}^G_H \chi)(h)} = \frac{1}{|H|} \sum_{h \in H} \chi(h)\, \overline{\chi(h)} = \frac{1}{|H|} \sum_{h \in H} |\chi(h)|^2. \end{align*} The middle equality uses that $\operatorname{Res}^G_H \chi$ at $h \in H$ is by definition $\chi(h)$ (where the right-hand $\chi$ is the original $G$-character). [/step] [step:Split $\langle \chi, \chi \rangle_G = 1$ into $H$- and $(G \setminus H)$-contributions] By hypothesis $\chi$ is an irreducible character of $G$, so by [Row Orthogonality](/theorems/2430) (or directly by the [Irreducibility Criterion](/theorems/2426) applied to $\chi$), \begin{align*} \langle \chi, \chi \rangle_G = 1. \end{align*} Expanding the definition of $\langle \chi, \chi \rangle_G$ and partitioning $G = H \sqcup (G \setminus H)$: \begin{align*} 1 = \langle \chi, \chi \rangle_G = \frac{1}{|G|} \sum_{g \in G} |\chi(g)|^2 = \frac{1}{|G|} \biggl( \sum_{h \in H} |\chi(h)|^2 + \sum_{g \in G \setminus H} |\chi(g)|^2 \biggr). \end{align*} Multiplying both sides by $|G|$ and rearranging, \begin{align*} \sum_{h \in H} |\chi(h)|^2 = |G| - \sum_{g \in G \setminus H} |\chi(g)|^2. \end{align*} Dividing by $|H|$ and using $|G|/|H| = |G : H|$, \begin{align*} \frac{1}{|H|} \sum_{h \in H} |\chi(h)|^2 = |G : H| - \frac{1}{|H|} \sum_{g \in G \setminus H} |\chi(g)|^2. \end{align*} [/step] [step:Combine to derive $\sum_i c_i^2 \leq |G : H|$ with equality criterion] By Steps 1 and 2, $\sum_i c_i^2 = \frac{1}{|H|}\sum_{h \in H}|\chi(h)|^2$. By Step 3, this equals $|G : H| - \frac{1}{|H|}\sum_{g \in G \setminus H}|\chi(g)|^2$. Hence \begin{align*} \sum_i c_i^2 = |G : H| - \frac{1}{|H|} \sum_{g \in G \setminus H} |\chi(g)|^2. \end{align*} The second term is a sum of non-negative real numbers $|\chi(g)|^2 \geq 0$, divided by the positive integer $|H|$. So \begin{align*} \frac{1}{|H|} \sum_{g \in G \setminus H} |\chi(g)|^2 \geq 0, \end{align*} and subtracting a non-negative quantity from $|G : H|$ gives an upper bound: \begin{align*} \sum_i c_i^2 \leq |G : H|, \end{align*} as claimed. For the equality criterion, equality holds in $\sum_i c_i^2 \leq |G : H|$ iff the subtracted term equals zero, iff \begin{align*} \sum_{g \in G \setminus H} |\chi(g)|^2 = 0. \end{align*} Since each $|\chi(g)|^2 \geq 0$, the sum vanishes iff every term vanishes, iff $\chi(g) = 0$ for all $g \in G \setminus H$. This proves the equality clause: $\sum_i c_i^2 = |G : H|$ if and only if $\chi$ vanishes on $G \setminus H$. [guided] The proof rests on a single bookkeeping identity: $\langle \chi, \chi \rangle_G$ — which equals $1$ because $\chi$ is irreducible — splits as a sum of an $H$-piece and a $(G \setminus H)$-piece, and the $H$-piece is exactly $\sum_i c_i^2$ rescaled. **Step 1 — what $\sum c_i^2$ measures.** The decomposition $\operatorname{Res}^G_H \chi = \sum_i c_i \psi_i$ is unique by linear independence of the $\psi_i$, and the multiplicities $c_i$ are non-negative integers (since $\operatorname{Res}^G_H \chi$ is a genuine character of $H$, not just a virtual one). Pairing with itself in the $H$-inner product, orthonormality of the $\psi_i$ ([Row Orthogonality](/theorems/2430)) gives the diagonal sum $\sum_i c_i^2$. **Step 2 — what the same inner product is by definition.** The $H$-inner product on $\mathcal{C}(H)$ is the average of $|\chi(h)|^2$ over $h \in H$ (since $\overline{\chi(h)}\chi(h) = |\chi(h)|^2$). So \begin{align*} \sum_i c_i^2 = \frac{1}{|H|}\sum_{h \in H}|\chi(h)|^2. \end{align*} **Step 3 — leveraging $\langle \chi, \chi \rangle_G = 1$.** The $G$-irreducibility of $\chi$ gives $\frac{1}{|G|}\sum_{g \in G}|\chi(g)|^2 = 1$. Splitting $G = H \sqcup (G \setminus H)$: \begin{align*} \frac{1}{|G|}\sum_{h \in H}|\chi(h)|^2 + \frac{1}{|G|}\sum_{g \in G \setminus H}|\chi(g)|^2 = 1. \end{align*} Multiply by $|G|$ and rearrange to isolate $\sum_{h \in H}|\chi(h)|^2$, then divide by $|H|$ and use $|G|/|H| = |G : H|$: \begin{align*} \frac{1}{|H|}\sum_{h \in H}|\chi(h)|^2 = |G : H| - \frac{1}{|H|}\sum_{g \in G \setminus H}|\chi(g)|^2. \end{align*} The $|G : H|$ on the right is the "naive" upper bound from $G$-irreducibility scaled to $H$. The correction is the contribution of values of $\chi$ outside $H$: this is a non-negative quantity (sum of squared moduli, a real-valued non-negative function). **Combining.** $\sum_i c_i^2 = |G : H| - \text{(non-negative correction)} \leq |G : H|$. **Equality criterion.** A non-negative sum is zero iff every summand is zero. So $\sum_i c_i^2 = |G : H|$ iff $|\chi(g)|^2 = 0$ for all $g \in G \setminus H$, iff $\chi(g) = 0$ on $G \setminus H$. **Geometric meaning.** The bound $\sum_i c_i^2 \leq |G : H|$ says the irreducible representation $\chi$ of $G$, when restricted to $H$, "spreads out" among the irreducibles of $H$ — but the spreading is bounded by the index. The smaller the index, the less the irreducible can fragment. Equality (on $G \setminus H$, $\chi$ vanishes) is the rigid case: it means $\chi$ is supported on $H$, and (related to a fact from induction theory) typically signals that $\chi$ is induced from $H$. [/guided] [/step]