Real representation

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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).

Another formulation of the condition is that 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.

Frobenius-Schur indicator

A criterion (for compact groups G) for reality of representations in terms of character theory is based on the Frobenius-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.

The theory is given here for finite groups; the ideas extend directly to compact groups, using integrals for sums.

Real reps can be complexified to get a complex rep of the same dimension and complex reps can be converted into a real rep of twice the dimension by treating the real and imaginary components separately. Also, since all finite dimensional complex reps can be turned into a unitary rep. for unitary reps, the dual rep is also a (complex) conjugate rep because the Hilbert space norm gives an antilinear bijective map from the rep to its dual rep.

Self dual complex irreps correspond to either real irreps of the same dimension or real irreps of twice the dimension called pseudoreal representations (but not both) and non self dual complex irreps correspond to a real irrep of twice the dimension. Note for the latter case, both the complex irrep and its dual give rise to the same real irrep. An example of a pseudoreal rep would be the four dimensional real irrep of the quaternion group Q₈.

Just as for any complex rep ρ,

<math>\frac{1}{|G|}\sum_{g\in G}\rho(g)</math>

is a self-intertwiner, for any integer n,

<math>\frac{1}{|G|}\sum_{g\in G}\rho(g)^n</math>

is also a self-intertwiner. By Schur's lemma, this will be a multiple of the identity for irreps. The trace of this self-intertwiner is called the n^th Frobenius-Schur indicator.

The original case of the Frobenius-Schur indicator is that for n = 2. The zeroth indicator is the dimension of the irrep, the first indicator would be 1 for the trivial representation and zero for the other irreps.

It resembles the Casimir invariants for Lie algebra irreps. In fact, since any rep of G can be thought of as a module for C[G] and vice versa, we can look at the center of C[G]. This is analogous to looking at the center of the universal enveloping algebra of a Lie algebra. It is simple to check that

<math>\sum_{g\in G}g^n</math>

belongs to the center of C[G], which is simply the subspace of class functions on G .

If V is the underlying vector space of a rep, then

<math>V\otimes V</math>

can be decomposed as the direct sum of two subreps, the symmetric tensor product

<math>V\otimes_S V</math>

and the antisymmetric tensor product

<math>V\otimes_A V</math>.

It's easy to show that

<math>\chi_{V\otimes_S V}(g)=\frac{1}{2}[\chi_V(g)^2+\chi_V(g^2)]</math>

and

<math>\chi_{V\otimes_A V}(g)=\frac{1}{2}[\chi_V(g)^2-\chi_V(g^2)]</math>

using a basis set.

<math>\frac{1}{|G|}\sum_{g\in G}\chi_{V\otimes_S V}(g)</math>

and

<math>\frac{1}{|G|}\sum_{g\in G}\chi_{V\otimes_A V}(g)</math>

are the number of copies of the trivial rep in

<math>V\otimes_S V</math>

and

<math>V\otimes_A V</math>,

respectively. As observed above, if V is an irrep,

<math>V\otimes V</math>

contains exactly one copy of the trivial rep if V is equivalent to its dual rep and no copies otherwise. For the former case, the trivial rep could either lie in the symmetric product, or the antisymmetric product.

Putting all of this together, for an irrep, the second Frobenius-Schur indicator is zero if the irrep isn't self-dual, 1 if it's self-dual and there's a nonzero symmetric intertwiner from <math>V\otimes V</math> to the trivial rep and -1 if it's self-dual and there's a nonzero antisymmetric intertwiner from <math>V\otimes V</math> to the trivial rep; and there are no other possibilities.

Examples

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