Exercise 3.39 of Fulton & Harris

I would like some help with exercise 3.39 from Fulton & Harris’ ‘Representation Theory A First Course’:

Let $V_0$ be a real vector space on which $G$ acts irreducibly, $V=V_0 \otimes \mathbb{C}$ is a complex representation of $G$. Show that if $V$ is not irreducible, then it has two irreducible factors, and they are conjugate complex representations of $G$.

First of all I would like to remark that conjugate complex representations were not covered before in the book, but after a bit of research I believe the statement in the exercise is equivalent to: $V=W \oplus W^*$ for some (complex) irreducible sub-representation $W \subset V$. If I am mistaken please do correct me!

This question was asked here: Question about Real Representations, but the post got bogged down with notations and since it was so old I decided to ask it again here. It was suggested in the comments of that post that we consider a irreducible (proper) sub-representation $W \subset V$, let $V=W \oplus U$, and show that $U=W^*$, but I don’t really see how we can go on…

The only way I have learnt to compute the amount of irreducible factors inside a representation $V$ is by computing $(\chi_V,\chi_V)=\frac{1}{|G|} \sum_{g \in G} |\chi_V(g)|^2$, but since we don’t have a corresponding formula (not one that I am aware of) in the real case, I’m not sure how we can prove there are 2 irreducible factors in $V$ here. I am also failing to see how the second statement about the relation between the two factors can follow from the first part.

Any help is appreciated!

Solutions Collecting From Web of "Exercise 3.39 of Fulton & Harris"

In the context of Fulton and Harris’ book, here is how I would explain what is going on:

Suppose $V$ is not irreducible as a $\mathbf{C}[G]$-module and let $0 \neq W \neq V$ be a proper non-zero sub-module. By restricting to $\mathbf{R}[G]$ we see that $W$ is a proper non-zero $\mathbf{R}[G]$-submodule of $V_0^{\oplus 2}$ (here we are using the relation between the real numbers and the complex numbers).

If $W_0$ is an irreducible $\mathbf{R}[G]$-submodule of $W$ then by Schur’s lemma, $W_0 \cong V_0$ as $\mathbf{R}[G]$-modules. By comparing dimensions we must have $W=W_0$. We have proved that every proper non-zero $\mathbf{C}[G]$-submodule of $V$ is isomorphic, upon restriction to $\mathbf{R}[G]$, to $V_0$. In particular the dimension of $W$ is half the dimension of $V$.

Now as in the discussion on page 40, a positive definite $G$-invariant inner product on $V_0$ induces a non-degenerate symmetric $\mathbf{C}$-bilinear form $(\cdot,\cdot)$ on $V$. Using what we proved in the previous paragraph there are only two possibilies: $W \cap W^\perp=0$ or $W=W^\perp$ (in the second case, $W$ is “Lagrangian”).

In the first case $W \cong W^*$ and $W^\perp \cong (W^\perp)^*$, while in the second case the pairing induces an isomorphism $W^* \cong W’$ for any $G$-stable complement $W’$ to $W$ in $V$. It remains only to observe that in the first case the characters of $W$ and $W^\perp$ are real, implying that they are both equal to half the character of $V_0$ and hence $W^\perp \cong W \cong W^*$ as desired.