For a representation $(V, p_V)$ by a finite group and $W = \bigoplus \limits_{i = 1}^{n} V$ calculate $\dim(\text{Hom}_G(V,W))$

Question

Let $(V, p_V)$ be a vector space with a representation by a finite group G. Assume further that $V$ is irreducible and $W = \bigoplus \limits_{i = 1}^{n} V$ with the direct-sum representation.
Namely: For $(v_1,...,v_n) \in W$ we have $p_{W}(v_1,...,v_n) =p_V(v_1) + ... +p_V(v_n)$.
Now why is $\dim(\text{Hom}_G(V,W)) = n$ ?

Ideas: I understand the case $n = 1$ by Schur's Lemma. But I don't know how to go on from there.
Thanks in advance for any help !

Answer 1

Suppose we have $G$-equivariant map $\phi$ from $V$ to $W = V^{\oplus n}$. For $i = 1, 2, \dots n$ we have a projection map $\rho_i$ from $W$ to $V$ sending $(w_1, w_2, \dots, w_n)$ to $w_i$. The composition $\rho_i \circ \phi$ is a map from $V$ to $V$, so by Schur's lemma it must send $\vec{v}$ to $c_i \vec{v}$. So we see that $\phi$ must just send $\vec{v}$ to $(c_1 \vec{v}, c_2\vec{v}, \dots , c_n \vec{v})$, and moreover it is easy to check that any map of that form is actually $G$-equivariant.

So in order to specify a map from $V$ to $W$ we have to pick these numbers $c_1, c_2, \dots c_n$, which give us our $n$-dimensions.