Let $(\rho, V)$ be a reducible representation of a finite group $G$, and let $W_1, \ldots, W_k$ be the irreducible representations of $G$. Let $V \simeq W^{\oplus n_1}_1 \oplus \cdots \oplus W^{\oplus n_k}_k$ be the decomposition of the representation space $V$ into isotypic components.
Schur's Lemma states that if $f$ is a $G$-linear map on $V$, then $f$ has the following block-matrix form (with rows and columns being labeled by the $W_i$'s):
$$
\left(\begin{array}{c|cc}
& W_1 & \cdots & W_1 & \cdots & W_k & \cdots & W_k \\
\hline
W_1 & \lambda^1_{1,1} I & \cdots & \lambda^1_{1,n_1} I & & 0 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots & \cdots & \vdots & \ddots & \vdots\\
W_1 & \lambda^1_{n_1,1} I & \cdots & \lambda^1_{n_1,n_1} I & & 0 & \cdots & 0 \\
\vdots & & \vdots & & \ddots && \vdots \\
W_k & 0 & \cdots & 0 && \lambda^k_{1,1} I & \cdots & \lambda^k_{1,n_k} \\
\vdots & \vdots & \ddots & \vdots & \cdots & \vdots & \ddots & \vdots\\
W_k & 0 & \cdots & 0 && \lambda^k_{n_k,1} I & \cdots & \lambda^k_{n_k,n_k} \\
\end{array}\right),
$$
where each $\lambda^l_{i,j} I$ is a multiple of the identity, i.e. the restriction of $f$ to an irreducible representation $W_i$ is an homothety (see this question).
Now in Theorem $8$ from Serre's book "Linear representations of Finite Groups", the projection $p_i$ of $V$ to $W^{\oplus n_i}_i$ is given by the formula:
$$
p_i := \frac{\dim W_i}{|G|} \sum_{g \in G} \bar{\chi}_i(g) \, \rho(g).
$$
The proof is the following: the restriction of $p_i$ to an irreducible representation $W_j$ is the identity $I$ if $i = j$ and $0$ if $i \neq j$, it follows that $\mathbf{p_i}$ is the identity on $\mathbf{W^{\oplus n_i}_i}$, and this means that $p_i$ is equal to the projection of $V$ to $W^{\oplus n_i}_i$.
I don't understand the part of the proof in bold.
Using Schur's (and orthogonality of the characters), according to Serre's proof, the restriction of $p_i$ to the isotopic component $W^{\oplus n_i}_i$ should have the following block-matrix form
$$
\left(\begin{array}{c|cc}
& W_i & \cdots & W_i \\
\hline
W_i & I & \cdots & I \\
\vdots & \vdots & \ddots & \vdots &\\
W_i & I & \cdots & I \\
\end{array}\right),
$$
and zero elsewhere, which is not even a projection ($p^2_i \neq p_i$). But more importantly, it is different from
$$
\left(\begin{array}{c|cc}
& W_i & \cdots & W_i \\
\hline
W_i & I & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots &\\
W_i & 0 & \cdots & I \\
\end{array}\right) = I^{\oplus n_i},
$$
the identity on $W^{\oplus n_i}_i$.
So I must have missed something but I don't see what…
Best Answer
To answer my own question, the restriction of $p_i$ to the isotopic component $W^{\oplus n_i}_i$ cannot have the block-matrix form $$ \left(\begin{array}{c|cc} & W_i & \cdots & W_i \\ \hline W_i & I & \cdots & * \\ \vdots & \vdots & \ddots & \vdots &\\ W_i & * & \cdots & I \\ \end{array}\right), $$ with nonzero off-diagonal entries, since $\rho$ is a representation, and for all $g \in G$, the action of $g$ on any $W_i$ is a subspace of this $W_i$, i.e. the map $p_i$ does not cause any interleaving between the representations $W_i$.