Induced representation

In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup $H$ to a representation of the (whole) group $G$ itself. 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.

Constructions

Algebraic

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 $g 1, ..., g n$ be a full set of representatives in $G$ of the left cosets in $G / H$ . The induced representation $Ind G H π$ can be thought of as acting on the following space:

W=\bigoplus _{i=1}^{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 $v \in V$ . For each g in $G$ and each g_i there is an h_i in $H$ and j(i) in {1, ..., n} such that $g g i = g j(i) h i$ . (This is just another way of saying that $g 1, ..., g n$ is a full set of representatives.) Via the induced representation $G$ acts on $W$ 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 using the tensor product: any K-linear representation π of the group H can be viewed as a module V over the group ring K[H]. We can then define

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

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

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 $H$ 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 $G$ 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 $ρ$ . The theorem is useful (in the typical case of non-modular representations, anyway - say with $K = C$ ) for computing the decomposition of the induced representation: we can do calculations on the side of $H$ , which is the 'small' group.

A more precise statement, also valid for infinite groups, is given by the universal property of the induced representation, which can be stated as follows. If $(\sigma ,V)$ is a representation of H and $(\operatorname {Ind} (\sigma ),{\hat {V}})$ is the representation of G induced by $\sigma$ , then there exists a $H$ -equivariant linear map $j:V\to {\hat {V}}$ with the following property: given any representation $(ρ, W)$ of $G$ and $H$ -equivariant linear map $f:V\to W$ , there is a unique $G$ -equivariant linear map ${\hat {f}}:{\hat {V}}\to W$ with ${\hat {f}}j=f$ .

This shows that the functor Res (from the category KG-Mod of $K$ -linear representations of $G$ , with $G$ -equivariant linear maps as morphisms, to the category KH-Mod) and the functor Ind (from KH-Mod to KG-Mod) are adjoint functors. More precisely, Ind is the left adjoint to Res. But in the finite group case, it is also a right adjoint, so (Res, Ind) is a Frobenius pair.

The reciprocity formula can sometimes be generalized to more general topological groups; for example, the Selberg trace formula and the Arthur-Selberg trace formula are generalizations of Frobenius reciprocity to discrete cofinite subgroups of certain locally compact groups.

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

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

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

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

Analytic

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:

\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\}.

Here $φ \in L 2 (G / H)$ means: the group 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, $(g . φ)(x)= φ (g -1 x)$ for g,x∈G and $φ \in Ind G H π$ .

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:

\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\}.

Here $Δ G, Δ H$ 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:

\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\}.

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

Geometric

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 $G \times V$ as follows:

g'.(g,x)=(g'g,x)

where $g$ and $g'$ are elements of $G$ and $x$ is an element of $V$ .

Define on $G \times V$ the equivalence relation

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

Note that this equivalence relation is invariant under the action of $G$ . Consequently, $G$ acts on $(G \times V)/~$ . 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 of this vector bundle. This is the vector space underlying the induced representation $Ind G H π$ . The group $G$ acts on $W$ as follows:

(g.\phi )({\overline {k}})=g.\left(\phi (g^{-1}{\overline {k}})\right)\ {\text{ for }}g\in G,\ \phi \in W,\ {\overline {k}}\in G/H.

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.

References

Alperin, J.L.; Rowen B. Bell (1995). Groups and Representations. Springer-Verlag. pp. 164–177. ISBN 0-387-94526-1.
Folland, G.B. (1995). A Course in Abstract Harmonic Analysis. CRC Press. pp. 151–200. ISBN 0-8493-8490-7.

↑ Brown, Cohomology of Groups, III.5

Kaniuth, E.; Taylor, K. (2013). Induced Representations of Locally Compact Groups. Cambridge University Press.
Mackey, G.W. (1951), "On induced representations of groups", American Journal of Mathematics 73 (3): 576–592, doi:10.2307/2372309
Mackey, G.W. (1952), "Induced representations of locally compact groups I", Annals of Mathematics 55 (1): 101–139, doi:10.2307/1969423
Mackey, G.W. (1953), "Induced representations of locally compact groups II : the Frobenius reciprocity theorem", Annals of Mathematics 58 (2): 193–220, doi:10.2307/1969786

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