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I am stuck on a problem in Fulton's Representation Theory: A First Course. Exercise 3.39 states:

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

I had originally misread the problem and taken $V_0$ to be not only a real $G$-invariant vector space, but a representation of $G$ itself. Proceeding from there, I showed that $V= V_0 \oplus iV_0$, but this is clearly wrong, as these $V_0$ is not complex and these two representations are not conjugate. However, given that $V_0$ is only a real $G$-invariant vector space, I'm not sure how to proceed.

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    $V_0$ *is* a representation of $G$, a real one. I don't understand what you misread.2011-09-28
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    Do an example: take $V_0$ to be an irreducible real representation of degree two of the cyclic group of order three. If you do not know any such representation, leave Ex. 3.39 aside for a while, and find one.2011-09-28
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    @Mariano: Dear Mariano, Perhaps in Fulton's book *real representation* means a complex representation which admits an underlying real structure? (Basing this on the OP's comment that "$V = V_0\otimes \mathbb C$ [is] the corresponding real representation", and nothing more.) Regards,2011-09-28
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    @Mariano: As Fulton describes it, "If a group $G$ acts on a real vector space $V_0$, then we say the corresponding complex representation of $V=V_0 \otimes \Bbb C$ is *real*."2011-09-28
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    Yes: Fulton and Harris use "real representation" to mean "complex representation of the form $V_0\otimes\mathbb C$ with $V_0$ are representation of $G$ over $\mathbb R$".2011-09-28
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    Dear Nathan, Now that the terminology is sorted out, regarding your question: it *could be* (i.e. it will be in some cases, but not in others) that the real representation $V$ *is* irreducible. So you won't be able to automatically decompose it into two subreps. (as you tried to too with your $V = V_0 \oplus i V_0$ gambit). You will have to *assume* that $V$ contains a proper subrep. $W$ (i.e. assume that it is *not* irreducible), and then try to deduce that $V$ is a direct sum of $W$ and $\overline{W}$. For this, you should try to work out how $W$ interacts with $V_0$. (And you should ...2011-09-28
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    ... also consider an example or two, as Mariano suggests.) Regards,2011-09-28
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    @Matt and Mariano: I would just like to fully clarify first. Can I or can I not assume that $V_0$ is a representation of $G$? As for an example, I know that the direct sum of the two nontrivial representations of the cyclic group of order 3 is a real representation, but I don't know how to find $V_0$, so I don't really even know how to prove that fact.2011-09-28
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    Nathan: A) If a group acts on a vector space (over any field) by linear transformations, surely that is a representation. B) Have you ever seen the group $S_3$ identified with isometries of a geometric object of the real plane $\mathbf{R}^2$? That group has a copy of $C_3$ as a subgroup. What kind of (real) linear mappings arise out of that action of $C_3$?2011-09-28
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    Dear Nathan, I'm not really sure what is confusing you, but here is an attempt to explain: $V_0$ is not a representation in the strict sense of your text (I think), since your text defines a representation to have complex coefficients (I think), and $V_0$ is a real vector space, not a complex one. But one *can* define the notion of a representation of $G$ over any field $k$ (just replace $\mathbb C$ by $k$ in all the basic definitions), and then *yes*, $V_0$ is a representation of $G$ over $\mathbb R$ (just because it is a $G$-invariant $\mathbb R$-subspace of $V$). Regards,2011-09-28
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    Does anyone have any suggestions on how to do this problem? Here's what I've come up with so far: Since $V=V_0\otimes_{\mathbb{R}}\mathbb{C}$ is reducible, it has a nontrivial irreducible subrepresentation $W$. Now, our goal is to show that $V=W\oplus\overline{W}$. To do so, note that the spaces $W+\overline{W}$ and $W\cap\overline{W}$ are invariant under conjugation. This should somehow allow us to identify $W+\overline{W}$ and $W\cap\overline{W}$ with subrepresentations of $V_0$ and, using the irreducibility of $V_0$, this should give that $W+\overline{W}=V_0$ and $W\cap\overline{W}=0$. I'm2011-10-05

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