# Abstract Nonsense

## 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 $\mathbb{C}$-representation and using it to create another representation which satisfies the real condition. Moreover, when the original $\mathbb{C}$-representation is an irrep which is quaternionic or complex then the resulting representation will have no non-trivial $(\rho,J)$-invariant subspaces.

Motivation

In our last post we have seen that $\mathbb{C}$-representations satisfying the real condition are important since they correspond to real representations in a natural way. Moreover, we have seen that $\mathbb{C}$-representations satisfying the real condition with realizer $J$ with no non-trivial proper $\left(\rho,J\right)$-invariant subspaces are even more interesting since they correspond naturally to irreducible real representations. We shall now show a method which takes a general $\mathbb{C}$-representations and produces a $\mathbb{C}$-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 $(\rho,J)$-invariant subspaces.

April 2, 2011

## 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.

March 30, 2011

## 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.

March 30, 2011