# How to decompose a group representation which is a direct sum of copies of one irreducible representation?

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?

• @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. Commented May 9, 2019 at 19:06
• @Micah: you are right. I spoke too fast. Commented May 9, 2019 at 20:12

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.

• 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. Commented May 9, 2019 at 20:24
• You're welcome! I'm also not sure how it works in general, which is why I restricted myself to $\mathbb{C}$... Commented May 9, 2019 at 20:31
• 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. Commented May 9, 2019 at 20:39