Character of an $\mathbb{R}G$-module constructed from a $\mathbb{CG}$-module I have been reading Representations and Characters of Groups by Gordon James and Martin Liebeck. I encountered the following construction of an $\mathbb{R}G$-module from a $\mathbb{C}G$-module.

Next, there's a proposition related to this on the next page.

The proof of Proposition 23.6(1) is clear from the first image. I don't understand the proof of Proposition 23.6(2). How does the author break down $V_{\mathbb{R}}$ into a direct sum of $U$ and $W$? Are we using the fact that $V$ is irreducible as a $\mathbb{C}G$-module?

Indeed the argument used is not quite clear as it stands, because context is missing. Apparently something is already known about uniqueness of decomposition of characters into irreducibles. What is known here is that we have a $\Bbb R G$-module that is reducible, and whose character is $\chi+\overline\chi$; its character must break up as the sum of at least two characters as submodules. And since $\chi$ and $\overline\chi$ are irreducuble characters (though a priori only characters of $\Bbb C G$-modules) the author concludes that the breaking up of the character can only correspond to its decomposition as $\chi+\overline\chi$. And then this implies that $\chi$ and $\overline\chi$ are actually characters of $\Bbb R G$-modules (indeed of the submodules $U,V$).
