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.
In mathematics and theoretical physics, a realrepresentation 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).
A criterion (for compact groups G) for reality of representations in terms of character theory is based on the Schur indicator.
Examples of realrepresentations are the spinors in 7 + 8k, 8 + 8k, and 9 + 8k dimensions for k = 1, 2, 3...
In mathematics and theoretical physics, a pseudoreal representation is a group representation that is equivalent to its complex conjugate, but that is not a realrepresentation.
A symplectic representation is a particular kind of pseudoreal representation, of a finite group or more generally of a compact group, that is an essentially quaternionic representation.
When it comes to representations that are not irreducible, one could give an example of a direct sum of a realrepresentation and a symplectic representation, as satisfying the stated condition to be pseudoreal; this however is less useful.
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