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.
The Complexification of a Real Inner Product Space
Before we get to the main part of this post we must discuss the complexification of a real inner product space. Namely, we define the complexification of the real inner product space , denoted , to be the set with addition defined coordinatewise, scalar multiplication defined , and inner product defined by . It is easy to verify that is indeed a pre-Hilbert space and moreover that . What we now claim is that the map is a complex conjugate. Indeed:
Theorem: Let be a real inner product space. Then the map is a complex conjugate on .
Proof: It’s clear that . Next we next claim that is antilinear. Indeed, for any and we have that
and thus and since is trivially linear antilinearity follows. Lastly, we claim with the proposed inner product that is antiunitary. Indeed, let be arbitrary then
and since and were arbitrary antiunitarity follows. Thus, combining these three attributes let us conclude.
In this post we define an inverse (in a sense soon made to be rigorous) to our map discussed previously. Namely:
Theorem: Let be a finite group and be a -representation on the real inner product space . Then the mapping by is a -representation of which satisfies the real condition with realizer . Moreover, has no non-trivial subspaces such that which are -invariant if and only if is an irreducible -representation.
Proof: We first show that really is a homomorphism . We first verify that is actually a unitary linear transformation on for each . Indeed, since it’s clear that is linear it suffices to show that for each is unitary. So, let , then one has that
from where unitarity of follows, and since was arbitrary it follows that really is a mapping .
We now show that satisfies the real condition with realizer . But, this is evident since
Therefore, it remains to verify our last claim. Suppose first that wasn’t reducible so that there existed some such that is -invariant. Note then that is a subspace of and clearly it is and invariant. Conversely, suppose that then by assumption for every we have that and so and for every we have that and similarly for every one has that . Thus, we may conclude that is a subspace of which is clearly -invariant by assumption since for every and we have that and thus by assumption so that . The conclusion follows.
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.