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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, i.e., an antilinear equivariant map
which satisfies
The two viewpoints are equivalent because if U is a real vector space acted on by a group G , 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.
In physics, where representations are often viewed concretely in terms of matrices, a real representation is one in which the entries of the matrices representing the group elements are real numbers. These matrices can act either on real or complex column vectors.