Real representation
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 j : V → V {\displaystyle j\colon V\to V} which satisfies j 2 = + 1. {\displaystyle j^{2}=+1.} 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.