Real representation

In mathematics and theoretical physics, a real representation is a group representation that is equivalent to its complex conjugate and that also allows the matrices representing the group elements to be real — unlike a pseudoreal representation (symplectic representation).

In other words, there exists an antilinear map <math>j:V\to V<math> that commutes with the elements of the group, and that satisfies <math>j^2=+1<math>.

A group representation that is neither real nor pseudoreal is called a complex representation. A criterion (for compact groups G) for reality of representations in terms of character theory is based on the Haar measure μ with μ(G) = 1.

Examples of real representations are the spinors in 7 + 8k, 8 + 8k, and 9 + 8k dimensions for k = 1, 2, 3 ... . This periodicity modulo 8 is known in mathematics not only in the theory of Clifford algebras, but also in algebraic topology, in KO-theory. see Representations of Clifford algebras