Finite Groups and Three Dimensional Pseudo Real Representations: An Exploration

[rkrsnan]

Can you please give an example for a finite group with a three dimensional pseudo real representation? I can find examples of finite groups with 2, 6 and 8 dimensional pseudo real representations, but couldn't find any with a three dimensional pseudo real rep. Is there some theorem that states that it doesn't exist? Thanks!

 

[morphism]

What do you mean by a "pseudo" real representation?

 

[rkrsnan]

My apologies for the confusing terminology. What I meant was a complex representation which has real characters. It is also called a quaternionic representation. I also understand that for a pseudo real representation the Frobenius–Schur indicator will be -1. From wikipedia: "It is −1 exactly when the irreducible representation has a skew symmetric invariant bilinear form. These are the representations whose character is real but that cannot be defined over the reals."

I am guessing that we can not make a skew symmetric invariant bilinear form using three dimensional vectors. That explains why I can't find a three dimensional pseudoreal representation.

 

[morphism]

Careful - notice that the result you quoted about the Frobenius-Schur indicator applies only to irreducible representations. Are you only interested in these? If this is the case, then your guess that there is no nondegenerate skew symmetric bilinear form on a 3-dimensional vector space is correct and answers your question. Indeed, it's not too hard to show that if a finite-dimensional vector space V admits a nondegenerate skew symmetric bilinear form, then dimV must be even.

One piece of the puzzle remains: you have to also show that the form you get from F-S is nondegenerate. This follows from the invariance of the form and the irreducibility of the representation.

If you need any help filling out any of the details, be sure to post back.

 

[rkrsnan]

Yes, I am only concerned with irreducible representations. I didn't know about the subtlety of the degenerate/nondegenerate bilinear forms though. Thanks very much for the reply and the help.