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

${\displaystyle j\colon V\to V}$

which satisfies

${\displaystyle j^{2}=+1.}$

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. 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 real representation on a complex vector space is isomorphic to its complex conjugate representation, but the converse is not true: a representation which is isomorphic to its complex conjugate but which is not real is called a pseudoreal representation. An irreducible pseudoreal representation V is necessarily a quaternionic representation: it admits an invariant quaternionic structure, i.e., an antilinear equivariant map

${\displaystyle j\colon V\to V}$

which satisfies

${\displaystyle j^{2}=-1.}$

A direct sum of real and quaternionic representations is neither real nor quaternionic in general.

A representation on a complex vector space can also be isomorphic to the dual representation of its complex conjugate. This happens precisely when the representation admits a nondegenerate invariant sesquilinear form, e.g. a hermitian form. Such representations are sometimes said to be complex or (pseudo-)hermitian.