Induced representation

In group theory, the induced representation is a representation of a group, Template:Mvar, which is constructed using a known representation of a subgroup Template:Mvar. Given a representation of Template:Mvar, the induced representation is, in a sense, the "most general" representation of Template:Mvar that extends the given one. Since it is often easier to find representations of the smaller group Template:Mvar than of Template:Mvar, 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.

Constructions

Algebraic

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

$W=\underset{i=1}{\overset{n}{⨁}}{g}_{i}V.$

Here each g_i V is an isomorphic copy of the vector space V whose elements are written as g_i v with v ∈ V. For each g in Template:Mvar and each g_i there is an h_i in Template:Mvar and j(i) in {1, ..., n} such that g g_i = g_j(i) h_i . (This is just another way of saying that g₁, ..., g_n is a full set of representatives.) Via the induced representation Template:Mvar acts on Template:Mvar as follows:

$g\cdot \sum _{i=1}^n{g}_i{v}_i=\sum _{i=1}^n{g}_{j(i)}\pi (h_i)v_i$

where $v_i\in V$ for each i.

Alternatively, one can construct induced representations by extension of scalars: any K-linear representation $\pi$ of the group H can be viewed as a module V over the group ring K[H]. We can then define

${\mathrm{Ind}}_H^G\pi =K[G]{\otimes }_{K[H]}V.$

This latter formula can also be used to define Ind π for any group Template:Mvar and subgroup Template:Mvar, without requiring any finiteness.

Examples

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.

Properties

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

The universal property of the induced representation, which is also valid for infinite groups, is equivalent to the adjunction asserted in the reciprocity theorem. If $(\sigma ,V)$ is a representation of H and $(\mathrm{Ind}(\sigma ),\hat{V})$ is the representation of G induced by $\sigma$, then there exists a Template:Mvar-equivariant linear map $j:V\to \hat{V}$ with the following property: given any representation (ρ,W) of Template:Mvar and Template:Mvar-equivariant linear map $f:V\to W$, there is a unique Template:Mvar-equivariant linear map $\hat{f}:\hat{V}\to W$ with $\hat{f}j=f$. In other words, $\hat{f}$ is the unique map making the following diagram commute:

File:Universal property of the induced representation 2.svg

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

$\psi (g)=\sum _{x\in G/H}\hat{\chi }\left(x^{-1}gx\right),$

where the sum is taken over a system of representatives of the left cosets of Template:Mvar in Template:Mvar and

$\hat{\chi }(k)=\{\begin{array}{ll}\chi (k)& \text{if }k\in H\\ 0& \text{otherwise}\end{array}$

Analytic

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

${\mathrm{Ind}}_H^G\pi =\{\varphi :G\to V:\varphi (g{h}^{-1})=\pi (h)\varphi (g)\text{ for all }h\in H,g\in G\text{ and }\varphi \in L^2(G/H)\}.$

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 Template:Mvar acts on the induced representation space by translation, that is, (g.φ)(x)=φ(g⁻¹x) for g,x∈G and φ∈Ind 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:

${\mathrm{Ind}}_H^G\pi =\{\varphi :G\to V:\varphi (g{h}^{-1})={\mathrm{\Delta }}_G^{-\frac{1}{2}}(h){\mathrm{\Delta }}_H^{\frac{1}{2}}(h)\pi (h)\varphi (g)\text{ and }\varphi \in L^2(G/H)\}.$

Here Δ_G, Δ_H are the modular functions of Template:Mvar and Template:Mvar 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:

${\mathrm{ind}}_H^G\pi =\{\varphi :G\to V:\varphi (g{h}^{-1})=\pi (h)\varphi (g)\text{ and }\varphi \text{ has compact support mod }H\}.$

Note that if G/H is compact then Ind and ind are the same functor.

Geometric

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

$g.(g^{\prime },x)=(g{g}^{\prime },x)$

where g and g′ are elements of Template:Mvar and x is an element of V.

Define on G × V the equivalence relation

$(g,x)\sim (gh,\pi (h^{-1})(x))\text{ for all }h\in H.$

Denote the equivalence class of $(g,x)$ by $[g,x]$. Note that this equivalence relation is invariant under the action of Template:Mvar; consequently, Template:Mvar acts on (G × V)/~ . The latter is a vector bundle over the quotient space G/H with H as the structure group and V as the fiber. Let W be the space of sections $\varphi :G/H\to (G\times V)/\sim$ of this vector bundle. This is the vector space underlying the induced representation ${\mathrm{Ind}}_H^G\pi :W\to {{ℒ}}_W$. The group Template:Mvar acts on a section $\varphi :G/H\to (G\times V)/\sim$ given by $gH\mapsto [g,{\varphi }_g]$ as follows:

$(g\cdot \varphi )(g^{\prime }H)=[g^{\prime },{\varphi }_{g^{-1}{g}^{\prime }}]\text{ for }g,g^{\prime }\in G.$

Systems of imprimitivity

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

Lie theory

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.

See also

• Restricted representation
• Nonlinear realization
• Frobenius character formula
• Frobenius reciprocity, an important result that relates induced representations to restricted representations