Frobenius–Schur indicator

In mathematics the Schur indicator, named after Issai Schur, or Frobenius–Schur indicator describes what invariant bilinear forms a given irreducible representation of a compact group on a complex vector space has. It can be used to classify the irreducible representations of compact groups on real vector spaces.

Definition

If a finite-dimensional continuous complex representation of a compact group G has character χ its Frobenius-Schur indicator is defined to be

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

for Haar measure μ with μ(G) = 1. When G is finite it is given by

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

The Frobenius-Schur indicator is always 1, 0, or -1. It provides a criterion for deciding whether an irreducible representation of G is real, complex or quaternionic, in a specific sense defined below. Below we discuss the case of finite groups, but the general compact case is completely analogous.
Real irreducible representations

There are 3 types of irreducible real representations of a finite group on a real vector space V, as the ring of endomorphisms commuting with the group action can be isomorphic to either the real numbers, or the complex numbers, or the quaternions.

If the ring is the real numbers, then V⊗C is an irreducible complex representation with Schur indicator 1, also called a real representation.
If the ring is the complex numbers, then V has two different conjugate complex structures, giving two irreducible complex representations with Schur indicator 0, sometimes called complex representations.
If the ring is the quaternion numbers, then choosing a subring of the quaternions isomorphic to the complex numbers makes V into an irreducible complex representation of G with Schur indicator −1, called a quaternionic representation.

Moreover every irreducible representation on a complex vector space can be constructed from a unique irreducible representation on a real vector space in one of the three ways above. So knowing the irreducible representations on complex spaces and their Schur indicators allows one to read off the irreducible representations on real spaces.

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

Self-dual complex irreducible representation correspond to either real irreducible representation of the same dimension or real irreducible representations of twice the dimension called quaternionic representations (but not both) and non-self-dual complex irreducible representation correspond to a real irreducible representation of twice the dimension. Note for the latter case, both the complex irreducible representation and its dual give rise to the same real irreducible representation. An example of a quaternionic representation would be the four dimensional real irreducible representation of the quaternion group Q8.
Invariant bilinear forms

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

\( V\otimes V\)

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

\( V\otimes_S V\)

and the antisymmetric tensor product

\( V\otimes_A V.\)

It's easy to show that

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

and

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

using a basis set.

\( \frac{1}{|G|}\sum_{g\in G}\chi_{V\otimes_S V}(g)\)

and

\( \frac{1}{|G|}\sum_{g\in G}\chi_{V\otimes_A V}(g)\)

are the number of copies of the trivial representation in

\( V\otimes_S V\)

and

\( V\otimes_A V,\)

respectively. As observed above, if V is an irreducible representation,

V\otimes V\)

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

The Frobenius-Schur indicator of an irreducible representation is always 1, 0, or −1. More precisely:

It is 0 exactly when the irreducible representation has no invariant bilinear form, which is equivalent to saying that its character is not always real.
It is 1 exactly when the irreducible representation has a symmetric invariant bilinear form. These are the representations that can be defined over the reals.
It is −1 exactly when the irreducible representation has a skew symmetric invariant bilinear form. These are the representations whose character is real but that cannot be defined over the reals. They are less common than the other two cases.

Higher Frobenius-Schur indicators

Just as for any complex representation ρ,

\( \frac{1}{|G|}\sum_{g\in G}\rho(g)\)

is a self-intertwiner, for any integer n,

\( \frac{1}{|G|}\sum_{g\in G}\rho(g^n)\)

is also a self-intertwiner. By Schur's lemma, this will be a multiple of the identity for irreducible representations. The trace of this self-intertwiner is called the nth Frobenius-Schur indicator.

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

It resembles the Casimir invariants for Lie algebra irreducible representations. 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

\( \sum_{g\in G}g^n\)

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

Jean-Pierre, Serre (1977). Linear Representations of Finite Groups. Springer-Verlag. ISBN 0-387-90190-6.

Mathematics Encyclopedia