High Quality Content by WIKIPEDIA articles! In the mathematical field of representation theory a real representation is usually a representation on a real vector space U, but it can also mean a representation on a complex vector space V with an invariant real structure. The two viewpoints are equivalent because if U is a real vector space acted on by a group G (say), then V = U?C is a representation on a complex vector space with an antilinear equivariant map given by complex conjugation. Conversely, if V is such a complex representation, then U can be recovered as the fixed point set of j (the eigenspace with eigenvalue 1).

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