Talk:Symplectic representation
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I guess that for the criterion involving j, one should say that the representation is taken to be irreducible.
In fact, if one wants the term 'pseudoreal' to be the same as 'quaternionic', that might be imposed from the start (I don't know what is standard usage). Otherwise a direct sum of real and quaternionic representations would come out as pseudoreal.
Charles Matthews 05:05, 5 Jun 2004 (UTC)
Dear Charles, I don't really know the difference between "pseudoreal" and "quaternionic" in this context, but what is more certain is that your last sentence sounds incorrect to me. On the direct sum of a real and a quaternionic representation, there is no way to define "j" that squares to minus one, unless the real representation is pseudoreal, too. For example, the direct sum 2+3 of representations of SU(2) is a sum of a real and a quaternionic representation, and the criterion will tell you that it is neither real, not pseudoreal, because there exists no antilinear "j" that would square to +1 exactly or -1 exactly. Instead, the natural "j" "partly" squares to +1, and partly to "-1" (a block diagonal matrix). On the other hand, 2+2 of SU(2) is both real and pseudoreal, according to the j-criterion, which I think is the correct answer. A direct sum of two representations that are complex conjugate to one another is always real, and a sum of many pseudoreal reps is always pseudoreal.
I didn't perhaps make myself completely clear. Anyway, I have added further material and terminology. There still needs to be more discussion on the page, in the 'Schur's lemma' direction, before it is satisfactory.
The example 'real+symplectic' direct sum representation would have a j which squares to +1 on one subspace, -1 on another (so isn't a scalar ... that's what is let in by reducible representations). Of course this kind of point isn't so useful.
I intend to move the page to 'symplectic representation', since this is the more accepted mathematical term, and is more precise. Why there is an invariant 2-form is one thing that has to be added to the page, though.
Charles Matthews 17:01, 8 Jun 2004 (UTC)
