## 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.

*Relation Between The Two Maps*

* *

What we now claim is that the above construction and the construction we previously made for turning -representations into -representations are intimately related. Indeed:

**Theorem: ***Let be a -representation. Then, .*

**Proof: **We have that for the constructed with the distinguished inner product , . So, we can clearly define evidently this is invertible and for every . The conclusion follows.

Similarly:

**Theorem: ***Let be a -representation satisfying the real condition with realizer . Then, .*

**Proof: **We have by definition that where . So, define by . We claim that this is an isomorphism. Indeed, it’s clearly additive, so let’s now prove that for each . To see this merely note

Note though that is surjective. Indeed, for any we have that there exists such that where is the basis guaranteed by our characterization of complex conjugates such that . Clearly then and since we have that and evidently . Thus, is an epimorphism but since we may conclude that is an isomorphism. That said, note that for any and

where we used the fact that . The conclusion follows.

As a last matter

So why does this matter? Because since satisfying the real condition is a attribute constant on equivalency classes of irreps we may conclude by our previous work that

given by is a bijection. Moreover, since having no non-trivial proper subspaces which are -invariant (this just means that the subspace is -invariant and ) is also constant on equivalency classes of irreps we may conclude again that

given by is a bijection. Thus, if we can identify all the which make up the domain of this second function and calculate , or more likely their characters, we will have a lot of information on the irreducible -representations of . That shall be our focus for the next few posts.

**References:**

1. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

[…] our last post we have seen that -representations satisfying the real condition are important since they […]

Pingback by Representation Theory: A Way of Creating C-representations Satisfying the Real Condition With No (rho,J)-invariant Subspaces « Abstract Nonsense | April 2, 2011 |

[…] about the latter for a specific group what could we say about the former. We began this search by showing that there was a bijection between equivalency classes of -representations satisfying the real […]

Pingback by Representation Theory: A Classification of C-representations With No (rho,J)-invariant Subspaces « Abstract Nonsense | April 4, 2011 |