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 (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).
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.
A criterion (for compact groups G) for reality of irreducible representations in terms of character theory is based on the Frobenius-Schur indicator defined by where χ is the character of the representation and μ is the Haar measure with μ(G) = 1.