TITLE: How to decompose a group representation which is a direct sum of copies of one irreducible representation?
QUESTION [5 upvotes]: Let $G$ be a finite group. Let $F$ be a field of characteristic zero. Let $V$ and $W$ be finite dimensional representations of $G$. Assume that $V$ is irreducible and $W$ is isomorphic to $nV$ for some $n$ (this is the external direct sum of $n$ copies of $V$).
I'm looking for a procedure to find elements $w_1,\dotsc,w_n\in W$ such that $w_i$ generates a subrepresentation $V_i$ of $W$ which is isomorphic to $V$, and such that the internal direct sum of the $V_i$ is  $W$.
I'm not even sure how to just find $w_1\in W$ that generates a subrepresentation isomorphic to $V$.
EDIT: The accepted answer tells us what to do if $F=\mathbb{C}$. What if $F=\mathbb{R}$?. Can we use extension of scalars?

REPLY [5 votes]: Let's try this again!
Let $I$ be the two-sided ideal of $\mathbb{C}G$ corresponding to the representation $V$. Then:
1) $I$ contains an element which acts as the identity on $V$, and
2) $I$ is isomorphic as a left $\mathbb{C}G$-module to $mV$, where $m=\dim(V)$. Write $I=V_1\oplus V_2\oplus\dots \oplus V_m$ for some explicit submodules $V_1,\dots,V_m$ of $\mathbb{C}G$.
Now, fix $w \in W$. By 1), $I$ acts nontrivially on $w$: that is, the span of $Iw$ is nonzero. By 2), there is then some $V_i$ which acts nontrivially on $w$. But $V_iw$ is a nontrivial image of the irreducible $V_i$. So it is isomorphic to $V$ by Schur's lemma. This gives you your first factor; induct via Maschke's theorem.