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Wait a minute! Are you defining a real rep as a rep over a COMPLEX vector space satisfying certain reality conditions and then claiming any rep over COMPLEX vector spaces which is self-dual isn't a complex rep????????

A complex rep is a rep over a complex vector space while a real rep is a rep over a real vector space. It doesn't matter if a complex rep is self-dual. It's still a complex rep and NOT a real rep! Since you're a physicist, take the example of a Dirac spinor. It's a self-dual complex rep of the double cover of the Lorentz group which satisfies your reality conditions. But it's still a complex rep nevertheless. Or how about the Majorana spinor? It's a real rep, and to use your matrix notation in your article on real reps, it doesn't even make sense to talk of antilinear maps over it because a real rep is NOT the same thing as its complexification!!!!!!

Phys 21:58, 6 Aug 2004 (UTC)

Wait another minute, please. It's only a confusion of language. If one hears of a "real representation", one should have in mind that it might be a rep living on a real vector space but also a rep being defined on a complex space and having a real structure (i.e. there is an anti(!)linear mapping whose square equals the identity map). Personally, for the latter I prefer the term "complex rep of real type". --Stefan Neumeier (talk) 23:02, 2 January 2013 (UTC)Reply