## 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 -representations of a finite group and show that this subset is in bijective correspondence with the set of all -representations of . Moreover, we shall show that this bijection naturally restricts to a subset of the set of all -reps of to the the -irreps of .

*Motivation*

In our last post we discussed the notion of representations for a finite group . Naturally our first desire would be to see if we could, in some way, connect representations of to the -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 -representations occur. Namely, we shall see that every -representation for which there is a complex conjugate for which for every naturally admits a -representation . We shall show then that in fact the reverse is true–namely that for every -representation there is a natural way to produce a -representation such that . Moreover, we’ll show that if is a -representation satisfying the aforementioned conditions and the added condition that there does not exist such that and is -invariant then we shall see that is an irreducible -representation.