## A Way of Creating C-representations Satisfying the Real Condition With No (rho,J)-invariant Subspaces

**Point of post: **In this post we discuss a way of taking a -representation and using it to create another representation which satisfies the real condition. Moreover, when the original -representation is an irrep which is quaternionic or complex then the resulting representation will have no non-trivial -invariant subspaces.

**Motivation**

In our last post we have seen that -representations satisfying the real condition are important since they correspond to real representations in a natural way. Moreover, we have seen that -representations satisfying the real condition with realizer with no non-trivial proper -invariant subspaces are even more interesting since they correspond naturally to irreducible real representations. We shall now show a method which takes a general -representations and produces a -representation which satisfies the real condition. Moreover, we shall see that if the original irrep happens to be complex or quaternionic irrep then the corresponding representation won’t have any non-trivial proper -invariant subspaces.

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

**Point of post: **This is a continuation of this post.

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

**Point of post: **This post is a continuation of this one.