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 that commutes with the elements of the group, and that satisfies j² = + 1.

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 Schur indicator . It involves the integral over G of

χ(g²)

which may take the values 1, 0 or −1, for 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