## Representation Theory: A Way of Creating C-representations Satisfying the Real Condition With No (rho,J)-invariant Subspaces (Pt. II)

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

Conversely, let be a complex or quaternionic irreducible -representation and suppose that is -invariant. Let then if we claim that is -invariant. Indeed, let and be arbitrary, then and so by assumption, since is -invariant, we have that but evidently it’s also a member of since . It then follows that and thus as desired. It follows then by assumption (since is an irrep and thus is an irrep) that or –or equivalently or . Suppose that the latter is true, then let be arbitrary, then by assumption and since is -invariant we have that . Thus, since was arbitrary we have that . Thus, we must have (since both of the following are contained in the subspace ) that so that contradictory to assumption. Thus, we may assume that . Using the exact same methodology we can show that is a -invariant subspace and thus by the same reasoning we must have that is either or . We clearly must have though that it is the latter.

Thus, for every we know that there exists some such that . Moreover, we know that is the only element of which appears as a second coordinate after . Indeed, if then but by the first part of the above paragraph we know that or . Thus, we can unambiguously define the map by is the unique element of such that . Note that evidently is linear since and so by definition . Note though that is -invariant since if then and so so that or . But, evidently otherwise but a quick check shows then that is not -invariant in this case. Thus, and thus . Note though that for any and

so that (since is -invariant) or for every –and by an earlier theorem we may conclude that is self-conjugate. Thus, there exists some unitary such that for every . Thus, we have that

for every and thus (since and so is an irrep) by Schur’s lemma we may conclude that and so . Note though that since is -invariant one has that for every

so that . Or using the fact that we have that so that and so itself is unitary. Finally, let then is antilinear, antiunitary, and and so is a complex conjugate. Note though that

for every . Thus, by one of our previous characterizations of real irreps we may conclude that is real–contradictory to assumption. The conclusion follows.

**References:**

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

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