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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?

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    $\begingroup$ @AbdelmalekAbdesselam: In fact it is not necessarily an irreducible representation. Let $G=S_3$, and take $V$ to be the representation spanned by $\{u,v\}$ with $(12)u=v$, $(12)v=u$, $(123)u=v-u$, $(123)v=-u$. If we let $W$ be $\Bbb{R}^4$ with the $V$-action on the first two coordinates and the last two coordinates, then the span of $\{w,(12)w,(13)w,(123)w\}$ is in general the entire space. $\endgroup$
    – Micah
    Commented May 9, 2019 at 19:06
  • $\begingroup$ @Micah: you are right. I spoke too fast. $\endgroup$
    – Abdelmalek Abdesselam
    Commented May 9, 2019 at 20:12

1 Answer 1

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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.

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  • $\begingroup$ Thanks! This looks correct to me. It requires to find a decomposition of $I$, which is a special case of the original problem. This special case can be solved using the Fourier inversion formula (over $\mathbb{C}$). In my question, I did not assume that $F$ is algebraically closed, and I'm not sure how the Fourier inversion formula works when not over $\mathbb{C}$. In any case, the problem of decomposing $I$ needs to be addressed. $\endgroup$
    – American Igor
    Commented May 9, 2019 at 20:24
  • $\begingroup$ You're welcome! I'm also not sure how it works in general, which is why I restricted myself to $\mathbb{C}$... $\endgroup$
    – Micah
    Commented May 9, 2019 at 20:31
  • $\begingroup$ Over $\mathbb{R}$, I think we can extend scalars to $\mathbb{C}$, find $w_1$ there, and take the real part of $w_1$. I think that should work. $\endgroup$
    – American Igor
    Commented May 9, 2019 at 20:39

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