Real representation

In the <a href="/facts/Mathematics/pxTouaz4">mathematical</a> field of <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a> a real representation is usually a <a href="/facts/Group_representation/FsuuDagQ">representation</a> on a <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Vector_space/M1pxMLx2">vector space</a> U, but it can also mean a representation on a <a href="/facts/Complex_number/2w7mqI7e">complex</a> vector space V with an invariant <a href="/facts/Real_structure/vahYuPa6">real structure</a>, i.e., an <a href="/facts/Antilinear/GTXzWMKc">antilinear</a> <a href="/facts/Representation_theory/3ydLtaGH">equivariant map</a>

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 <a href="/facts/Complex_conjugation/7TEaoQDe">complex conjugation</a>. Conversely, if V is such a complex representation, then U can be recovered as the <a href="/facts/Fixed_point_set/vVVddTKc">fixed point set</a> of j (the <a href="/facts/Eigenspace/8TjEoT8u">eigenspace</a> with <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a> 1).
In <a href="/facts/Physics/a7aH2y08">physics</a>, 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 <a href="/facts/Complex_conjugate_representation/2WPGvRL3">complex conjugate representation</a>, but the converse is not true: a representation which is isomorphic to its complex conjugate but which is not real is called a <a href="/facts/Pseudoreal_representation/XrwUqEMl">pseudoreal representation</a>. An irreducible pseudoreal representation V is necessarily a <a href="/facts/Quaternionic_representation/XrwUqEMl">quaternionic representation</a>: it admits an invariant <a href="/facts/Quaternionic_structure/XSbd4YSa">quaternionic structure</a>, i.e., an antilinear equivariant map

j
        :
        V
        →
        V
      
    
    {\displaystyle j\colon V\to V}

which satisfies

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

A <a href="/facts/Direct_sum_of_representations/x3ljVsP6">direct sum</a> 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 <a href="/facts/Dual_representation/HJWTCk5d">dual representation</a> of its complex conjugate. This happens precisely when the representation admits a nondegenerate invariant <a href="/facts/Sesquilinear_form/Kp6qUA8v">sesquilinear form</a>, e.g. a <a href="/facts/Hermitian_form/Kp6qUA8v">hermitian form</a>. Such representations are sometimes said to be complex or (pseudo-)hermitian.

Real representation open-in-new

Real representation