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. Wikipedia