Real representation

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There are some small but confusing differences in terminology between mathematicians and physicists when discussing real and complex representations.

In mathematics a real representation is usually a group representation on a real vector space, while in physics a real representation is usually a group representation on a complex vector space that allows the matrices representing the group elements to be real. These two definitions are essentially equivalent, because if U is a real vector space acted on by a group G, then V = U⊗C is a representation on a complex vector space that can be represented by real matrices. The difference in viewpoints is in whether one thinks of the real representation as the action of G on the real vector space U or the action of G on the complex vector space V.

A real representation is equivalent to its complex conjugate but the converse is not true: representations equivalent to their complex conjugate but that are not real are called pseudoreal representations (symplectic representations).

Another formulation of the (physics) definition is that there exists an antilinear map on the complex vector space V

j:V\to V\,

that commutes with the elements of the group, and that satisfies

j^2=+1.\,

The fixed points of j form a real vector space U with V = U⊗C.

In physics, a group representation on a complex vector space that is neither real nor pseudoreal is called a complex representation. (In mathematics, any representation on a complex vector space is called a complex representation.)

[edit] Frobenius-Schur indicator

A criterion (for compact groups G) for reality of irreducible representations in terms of character theory is based on the Frobenius-Schur indicator. If a representation of a compact group G has character χ its Schur indicator is defined to be

\int_{g\in G}\chi(g^2)\,d\mu

which when G is finite is given by

{1\over |G|}\sum_{g\in G}\chi(g^2)

which may take the values 1, 0 or −1, for Haar measure μ with μ(G) = 1. If the indicator is 1, then the representation is real (for the physics definition of real). If the indicator is zero, the representation is complex (for the physics definition of complex), and if the indicator is −1, the representation is pseudoreal.

[edit] Examples

Examples of real representations are the spinors in 8k−1, 8k, and 1 + 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.