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.
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.
We now describe the aforementioned method and show that it does what it claims. Namely:
Theorem: Let be a -representation and a complex conjugate on . Then, by is a complex conjugate on (the direct sum) and is a -representation (where, as usual, this is the direct sum representation which satisfies the real condition with realizer . Moreover, will have no non-trivial proper -invariant if and only if is a complex or quaternionic irrep.
Proof: To see that is a complex conjugate we merely note that for every –
so that by the arbitrariness of we may conclude that . Next,
from where the fact that is complex conjugate on follows. Now, to see that is a realizer for note that for any and one has
so that in fact does satisfy the real condition with realizer .
Now, to verify our second claim we show firstly that if is -invariant then can’t possibly hope to have no non-trivial proper -invariant subspaces. Indeed, suppose that is such a subspace and define , clearly . Moreover, it’s clear that since evidently is the identity map on ! Lastly, it’s clear that it is -invariant since for every and one has . Thus, to find when has non non-trivial proper -invariant we may restrict our attention to irreps. That said, suppose that is a real irrep so that there exists some unitary such that for every . We claim then that is -invariant. Indeed, we note that for every and one has that
but, by definition we have that commutes with for each so that the above can be written which is doubtlessly in so that by the arbitrariness of the above calculation we see that is -invariant. Similarly, for every we have that but evidently so that . Conversely, if then and so so that as desired. Clearly then since we may conclude that real irreps cannot produce -representations with no -invariant subspaces we may conclude that the ‘only if’ part of the theorem is true.
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.