Abstract Nonsense

A Bijection Between A Subset of the Complex Reps of a Finite Group and the Real Reps (Pt. I)

Point of post: In this post we describe a certain subset of the set of all $\mathbb{C}$-representations of a finite group $G$ and show that this subset is in bijective correspondence with the set of all $\mathbb{R}$-representations of $G$. Moreover, we shall show that this bijection naturally restricts to a subset of the set of all $\mathbb{C}$-reps of $G$ to the the $\mathbb{R}$-irreps of $G$.

Motivation

In our last post we discussed the notion of $\mathbb{R}$representations for a finite group $G$. Naturally our first desire would be to see if we could, in some way, connect $\mathbb{R}$representations of $G$ to the $\mathbb{C}$-representations  which has held the center of our attention for so long. We begin this process in this post by showing that there is a natural place for which these $\mathbb{R}$-representations occur. Namely, we shall see that every $\mathbb{C}$-representation $\rho:G\to\mathcal{U}\left(\mathscr{V}\right)$ for which there is a complex conjugate $J$ for which $J\rho(g)J=\rho(g)$ for every $g\in G$ naturally admits a $\mathbb{R}$-representation $\rho_{\Re}$. We shall show then that in fact the reverse is true–namely that for every $\mathbb{R}$-representation $\psi$ there is a natural way to produce a $\mathbb{C}$-representation $\rho$ such that $\rho_{\Re}=\psi$. Moreover, we’ll show that if $\rho:G\to\mathcal{U}\left(\mathscr{V}\right)$ is a $\mathbb{C}$-representation satisfying the aforementioned conditions and the added condition that there does not exist $\mathscr{W}\leqslant \mathscr{V}$ such that $J\left(\mathscr{W}\right)=\mathscr{W}$ and $\mathscr{W}$ is $\rho$-invariant then we shall see that $\rho_{\Re}$ is an irreducible $\mathbb{R}$-representation.