Induced representation

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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.

ConstructionsEdit

AlgebraicEdit

Template:See also Let Template:Mvar be a finite group and Template:Mvar any subgroup of Template:Mvar. Furthermore let Template:Math be a representation of Template:Mvar. Let Template:Math be the index of Template:Mvar in Template:Mvar and let Template:Math be a full set of representatives in Template:Mvar of the left cosets in Template:Math. The induced representation Template:Math can be thought of as acting on the following space:

<math>W=\bigoplus_{i=1}^n g_i V.</math>

Here each Template:Math is an isomorphic copy of the vector space V whose elements are written as Template:Math with Template:Math. 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 Template:Math . (This is just another way of saying that Template:Math is a full set of representatives.) Via the induced representation Template:Mvar acts on Template:Mvar as follows:

where <math> v_i \in V</math> for each i.

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

<math>\operatorname{Ind}_H^G\pi= K[G]\otimes_{K[H]} V.</math>

This latter formula can also be used to define Template:Math for any group Template:Mvar and subgroup Template:Mvar, without requiring any finiteness.<ref>Brown, Cohomology of Groups, III.5</ref>

ExamplesEdit

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.

PropertiesEdit

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 Template:Math. 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 Template:Math has the same dimension over K as that of Template:Mvar-equivariant linear maps from Template:Math to Template:Mvar.<ref>Template:Cite book</ref>

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 <math>(\sigma,V)</math> is a representation of H and <math>(\operatorname{Ind}(\sigma),\hat{V})</math> is the representation of G induced by <math>\sigma</math>, then there exists a Template:Mvar-equivariant linear map <math>j:V\to\hat{V}</math> with the following property: given any representation Template:Math of Template:Mvar and Template:Mvar-equivariant linear map <math>f:V\to W</math>, there is a unique Template:Mvar-equivariant linear map <math>\hat{f}: \hat{V}\to W</math> with <math>\hat{f}j=f</math>. In other words, <math>\hat{f}</math> is the unique map making the following diagram commute:<ref>Thm. 2.1 from {{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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 Template:Math, then the character Template:Mvar of the induced representation is given by

<math>\psi(g) = \sum_{x\in G / H} \widehat{\chi}\left(x^{-1}gx \right),</math>

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

<math> \widehat{\chi} (k) = \begin{cases} \chi(k) & \text{if } k \in H \\ 0 & \text{otherwise}\end{cases}</math>

AnalyticEdit

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 Template:Math be a continuous unitary representation of Template:Mvar into a Hilbert space V. We can then let:

<math>\operatorname{Ind}_H^G\pi= \left\{\phi\colon G \to V \ : \ \phi(gh^{-1})=\pi(h)\phi(g)\text{ for all }h\in H,\; g\in G \text{ and } \ \phi \in L^2(G/H)\right\}.</math>

Here Template:Math means: the space G/H carries a suitable invariant measure, and since the norm of Template:Math 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, Template:Math for g,x∈G and Template:Math.

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:

<math>\operatorname{Ind}_H^G\pi= \left \{\phi \colon G \to V \ : \ \phi(gh^{-1})=\Delta_G^{-\frac{1}{2}}(h)\Delta_H^{\frac{1}{2}}(h)\pi(h)\phi(g) \text{ and } \phi\in L^2(G/H) \right \}.</math>

Here Template:Math 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:

<math>\operatorname{ind}_H^G\pi= \left\{\phi\colon G \to V \ : \ \phi(gh^{-1})=\pi(h)\phi(g) \text{ and } \phi \text{ has compact support mod } H \right\}.</math>

Note that if Template:Math is compact then Ind and ind are the same functor.

GeometricEdit

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 Template:Math. Then Template:Mvar acts on the product Template:Math as follows:

where Template:Math and Template:Math are elements of Template:Mvar and Template:Math is an element of Template:Math.

Define on Template:Math the equivalence relation

Denote the equivalence class of <math>(g,x)</math> by <math>[g,x]</math>. Note that this equivalence relation is invariant under the action of Template:Mvar; consequently, Template:Mvar acts on Template:Math . The latter is a vector bundle over the quotient space Template:Math with Template:Math as the structure group and Template:Math as the fiber. Let Template:Math be the space of sections <math>\phi : G/H \to (G \times V)/ \! \sim</math> of this vector bundle. This is the vector space underlying the induced representation <math>\operatorname{Ind}_H^G\pi : W \to \mathcal L_W</math>. The group Template:Mvar acts on a section <math>\phi : G/H \to (G \times V)/ \! \sim</math> given by <math>gH \mapsto [g,\phi_g]</math> as follows:

Systems of imprimitivityEdit

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

Lie theoryEdit

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.

NotesEdit

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