Representation problem: I don't understand the setting of the question! (From Serre's book) Ex 2.8 of Serre's book "Linear Representations of Finite Groups" says: Let $\rho:G\to V$ be a representation ($G$ finite and $V$ is complex, finite dimensional) and $V=W_1\oplus W_1 \oplus \dotsb \oplus W_2 \oplus \dotsb W_2\oplus \dotsb \oplus W_k$ be its explicit decomposition into irreducible subrepresentations. We know $W_i$ is only determined up to isomorphism, but $V_i := W_i\oplus \dotsb \oplus W_i$ is uniquely determined.
The question asks: Let $H_i$ be the vector space of linear mappings $h:W_i\to V_i$ such that $\rho_s h = h\rho_s$ for all $s\in G$. Show that $\dim H_i = \dim V_i / \dim W_i$ etc...
But I don't even understand what $\rho_s h = h\rho_s$ means in this case: to make sense of $h\rho_s$, don't we need to first fix some decomposition $W_i\oplus \dotsb \oplus W_i$, and consider $\rho$ restricted to one of these $W_i$? Is that what the question wants? But then how do I make sense of $\rho_s h$?

I'm not sure I understand what is worrying you, but each $W_{i}$ is a $G$-submodule, for any $w \in W_{i}, w\rho_{s} \in W_{i}.$ Then we can apply the map $h,$ s $(w\rho_{s})h$ is an element of $V_{i}.$ On the other hand, $wh \in V_{i}.$ We know that $V_{i}$ is also a $G$-submodule, so $(wh)\rho_{s}$ is an element of $V_{i}.$ The question asks you to consider those maps $h$ such that these two resulting elements of $V_{i}$ coincide for each $w \in W_{i}$ and each $s \in G.$ ( I am assuming that $\rho_{s}$ means the linear transformation associated to $s \in G$ by the representation $\rho).$
Clarification: Suppose that $V_{i}$ is a direct sum of $n_{i}$ isomorphic irreducible submodules. Strictly speaking, these could be labelled $W_{i_{1}},\ldots ,W_{i_{n_{i}}}.$ The intention is to look at maps from $W_{i_{1}}$ to $V_{i},$ but the resulting dimensions would be the same if any $W_{i_{j}}$ was used.
