Induced representation

Process of extending a representation of a subgroup to the parent group

In group theory, the induced representation is a representation of a group, G, which is constructed using a known representation of a subgroup H. Given a representation of H, the induced representation is, in a sense, the "most general" representation of G that extends the given one. Since it is often easier to find representations of the smaller group H than of G, the operation of forming induced representations is an important tool to construct new representations.

Induced representations were initially defined by Frobenius, for linear representations of finite groups. The idea is by no means limited to the case of finite groups, but the theory in that case is particularly well-behaved.

Let G be a finite group and H any subgroup of G. Furthermore let (π, V) be a representation of H. Let n = [G : H] be the index of H in G and let g1, ..., gn be a full set of representatives in G of the left cosets in G/H. The induced representation IndG
H
π
can be thought of as acting on the following space:

This latter formula can also be used to define IndG
H
π
for any group G and subgroup H, without requiring any finiteness.[1]

For any group, the induced representation of the trivial representation of the trivial subgroup is the right regular representation. More generally the induced representation of the trivial representation of any subgroup is the permutation representation on the cosets of that subgroup.

An induced representation of a one dimensional representation is called a monomial representation, because it can be represented as monomial matrices. Some groups have the property that all of their irreducible representations are monomial, the so-called monomial groups.

If H is a subgroup of the group G, then every K-linear representation ρ of G can be viewed as a K-linear representation of H; this is known as the restriction of ρ to H and denoted by Res(ρ). In the case of finite groups and finite-dimensional representations, the Frobenius reciprocity theorem states that, given representations σ of H and ρ of G, the space of H-equivariant linear maps from σ to Res(ρ) has the same dimension over K as that of G-equivariant linear maps from Ind(σ) to ρ.[2]

The Frobenius formula states that if χ is the character of the representation σ, given by χ(h) = Tr σ(h), then the character ψ of the induced representation is given by

where the sum is taken over a system of representatives of the left cosets of H in G and

If G is a locally compact topological group (possibly infinite) and H is a closed subgroup then there is a common analytic construction of the induced representation. Let (π, V) be a continuous unitary representation of H into a Hilbert space V. We can then let:

Here φ∈L2(G/H) means: the space G/H carries a suitable invariant measure, and since the norm of φ(g) is constant on each left coset of H, we can integrate the square of these norms over G/H and obtain a finite result. The group G acts on the induced representation space by translation, that is, (g.φ)(x)=φ(g−1x) for g,xG and φ∈IndG
H
π
.

This construction is often modified in various ways to fit the applications needed. A common version is called normalized induction and usually uses the same notation. The definition of the representation space is as follows:

Here ΔG, ΔH are the modular functions of G and H respectively. With the addition of the normalizing factors this induction functor takes unitary representations to unitary representations.

One other variation on induction is called compact induction. This is just standard induction restricted to functions with compact support. Formally it is denoted by ind and defined as:

Suppose G is a topological group and H is a closed subgroup of G. Also, suppose π is a representation of H over the vector space V. Then G acts on the product G × V as follows:

In the case of unitary representations of locally compact groups, the induction construction can be formulated in terms of systems of imprimitivity.

In Lie theory, an extremely important example is parabolic induction: inducing representations of a reductive group from representations of its parabolic subgroups. This leads, via the philosophy of cusp forms, to the Langlands program.