Induced Representations, Restriction and Frobenius Reciprocity

The subgroup boundary

Let H be a subgroup of a finite group G. The inclusion H≤G creates a natural question: how should representations move between H and G?

• Restriction: start with a G-representation and forget how elements outside H act.
• Induction: start with an H-representation and build a G-representation that contains copies of the H-space indexed by cosets.

Restriction is easy to define and hard to interpret. Induction is harder to define and often easier to use than it first appears.

Large symmetry → restrict → inspect locally. Small symmetry → induce → propagate globally.

Restriction

Suppose ρ:G→GL(V) is a representation. The restricted representation Res_H^G V is the same vector space V with the action limited to H:

Res_H^G ρ : H → GL(V), h ↦ ρ(h).

No matrices change. No vectors change. Only the acting group becomes smaller.

Yet decomposition can change dramatically. An irreducible G-representation may become reducible when viewed only as an H-representation. This is not a contradiction. Irreducibility means there is no proper subspace invariant under all of G. If we demand invariance under fewer transformations, new invariant subspaces may appear.

Example: restricting the standard representation of S₃

The two-dimensional standard representation of S₃ is irreducible over C. Let H=A₃≅C₃. When we restrict the standard representation to C₃, the 120° rotation has two complex eigenvalues ω and ω², where ω=e^{2πi/3}.

Since C₃ is abelian, all irreducible complex representations are one-dimensional. Therefore the restricted two-dimensional representation splits as

Res_{C₃}^{S₃}(standard) ≅ χ_ω ⊕ χ_{ω²}.

A representation that was indivisible under the full S₃ symmetry separates into two one-dimensional modes when only rotational symmetry remains.

Branching rules

A branching rule records how irreducible representations of G decompose after restriction to H. For a chain of groups

G₀ ≤ G₁ ≤ G₂ ≤ ···,

branching rules can organise representations stage by stage. In symmetric groups, for example, restricting an irreducible representation of S_n to S_{n−1} has an elegant combinatorial description in terms of removing a box from a Young diagram.

Branching is powerful because a large representation problem can be replaced by a controlled descent through smaller groups. Instead of solving the whole system at once, we inspect how complexity changes at each subgroup boundary.

Why induction is needed

Restriction moves downward effortlessly. But suppose we know a representation W of H and want a representation of G. We cannot simply declare that elements outside H act arbitrarily. Their action must be compatible with the whole group law.

Induction solves this by allowing G to move copies of W across cosets of H. The induced space is large enough that the whole group can act naturally.

At a conceptual level:

one H-space × one copy for each left coset of H in G → a G-space.

Dimension of an induced representation

If W is a finite-dimensional representation of H, then

dim Ind_H^G W = [G:H] · dim W.

The index [G:H] counts the number of left cosets of H in G. This formula is an essential verification test. If your constructed induced representation has the wrong dimension, something in the coset bookkeeping is wrong.

Induction by functions

One standard construction defines Ind_H^G W as the space of functions f:G→W satisfying an H-equivariance condition. Depending on conventions, a common form is

f(hg)=h·f(g).

The group G then acts by translating the argument. The equivariance condition means the values of f are determined by their values on coset representatives. That is why the dimension grows by [G:H].

This functional construction is especially useful because it generalises naturally to topological groups, compact groups and Lie groups, where induced representations become central to harmonic analysis.

Induction by tensor products over group algebras

There is another compact algebraic construction. Regard W as a module over the group algebra F[H]. Then define

Ind_H^G W = F[G] ⊗_{F[H]} W.

This formula packages the coset construction into module theory. F[G] carries both left G-action and right H-action. Tensoring over F[H] identifies the H-action built into F[G] with the H-action already present on W.

The two constructions—equivariant functions and tensor products—describe the same induced representation in equivalent ways. The best choice depends on the problem.

The easiest induced representation: a coset permutation representation

Take the trivial one-dimensional representation 1_H of H. Then Ind_H^G 1_H is naturally the permutation representation of G on the left coset space G/H.

This is one of the most important examples in the subject. It connects subgroup structure, permutation actions and induced representations.

The character of this permutation representation at g counts the cosets fixed by g. Therefore subgroup geometry becomes character data.

Example: induce the trivial representation from A₃ to S₃

A₃ has index 2 in S₃. Therefore Ind_{A₃}^{S₃} 1 has dimension 2. Since A₃ is normal, S₃ acts on the two cosets, giving the two-dimensional permutation representation on a two-point set.

That permutation representation decomposes as

1_{S₃} ⊕ sign.

This example is small, but it previews the general principle: inducing a simple subgroup representation can produce several irreducible representations of the larger group.

Induced characters

Characters make induction computational. Let ψ be a character of H. The induced character Ind_H^G ψ has a formula that sums contributions from group elements that conjugate g into H.

One common form is

(Ind_H^G ψ)(g) = (1/|H|) Σ_{x∈G, x⁻¹gx∈H} ψ(x⁻¹gx).

The formula looks complicated until we interpret it. If no conjugate of g lies inside H, the induced character vanishes at g. If conjugates do meet H, the value records how the subgroup character is seen across all compatible cosets.

For ψ=1_H, this becomes the fixed-coset count for the permutation action on G/H.

Frobenius reciprocity

Frobenius reciprocity is the central bridge between restriction and induction. Let W be an H-representation and V a G-representation. Then there is a natural correspondence

Hom_G(Ind_H^G W, V) ≅ Hom_H(W, Res_H^G V).

At the character level, for finite groups over C, this becomes

⟨Ind_H^G ψ, χ⟩_G = ⟨ψ, Res_H^G χ⟩_H.

This equality is much more than a neat identity. It converts a multiplicity calculation in G into one in H. If χ is irreducible, the left side asks how many times V occurs inside the induced representation. The right side asks how many times W occurs inside the restriction of V.

Multiplicity upstairs = multiplicity downstairs.

Worked Frobenius reciprocity example with S₃

Let H=A₃≅C₃, and let ψ be a nontrivial one-dimensional character of C₃. We want to know which irreducible representations of S₃ occur in Ind_H^{S₃} ψ.

The irreducible representations of S₃ are trivial, sign and standard.

• The trivial representation restricts to the trivial representation of A₃, so ψ occurs zero times.
• The sign representation also restricts trivially to A₃, because every element of A₃ is even. Again ψ occurs zero times.
• The standard representation restricts to ψ⊕ψ̄, so ψ occurs once.

By Frobenius reciprocity, the standard representation occurs once inside Ind_H^{S₃} ψ. The induced representation has dimension [S₃:A₃]·1=2. Since the standard representation also has dimension 2, we conclude

Ind_{A₃}^{S₃} ψ ≅ standard.

We have identified the induced representation without constructing its matrices.

Why reciprocity is an adjunction

Frobenius reciprocity says induction is left adjoint to restriction. In category language, induction is not merely an inverse-like operation. It is the operation naturally paired with restriction through homomorphism spaces.

This distinction matters. Restricting and then inducing usually does not return the original representation. Inducing and then restricting usually does not return the original subgroup representation. Instead, the two operations satisfy a universal relationship about symmetry-respecting maps.

This is a recurring mathematical theme: adjoint operations need not undo each other, but they translate problems between two environments in the most structured possible way.

Restriction after induction

If W is an H-representation, what happens when we induce it to G and then restrict back to H?

The answer is usually larger than W. Other cosets contribute copies twisted by conjugation, and intersections of H with conjugate subgroups appear. Mackey’s decomposition theorem organises this structure using double cosets H\G/H.

Even without the full theorem, the conceptual lesson is valuable:

moving up and back down remembers how the subgroup sits inside the larger group.

Double cosets

A double coset HgH is the set {h₁gh₂ : h₁,h₂∈H}. Double cosets classify the relative positions of a subgroup and its conjugates inside G.

They appear naturally when restriction and induction are composed because the induced representation is built from cosets, while restricting asks how H acts on those cosets. The resulting orbit structure is governed by double cosets.

This is a good example of representation theory revealing that apparently technical combinatorics is actually measuring subgroup geometry.

Normal subgroups simplify induction

If H is normal in G, then conjugation preserves H. The quotient G/H acts on cosets cleanly, and induced characters can be easier to analyse.

Normality also connects induction and restriction with quotient representations. Representations whose kernel contains H factor through G/H. By comparing these quotient-type representations with representations induced from H, one can separate information coming from the subgroup from information coming from the quotient.

Transitivity of induction

If K≤H≤G, then induction can be performed in stages:

Ind_H^G(Ind_K^H W) ≅ Ind_K^G W.

This is useful computationally and conceptually. A representation can be lifted through a subgroup chain without changing the final induced representation up to equivalence.

Restriction is also transitive:

Res_K^H(Res_H^G V)=Res_K^G V.

These identities let subgroup towers become representation-theoretic pipelines.

Induction and the regular representation

The regular representation of G can be obtained by inducing the trivial representation from the identity subgroup:

Ind_{\{e\}}^G 1 ≅ F[G].

The index is |G|, so the induced dimension is |G|. This connects induction with one of representation theory’s canonical constructions.

Since every irreducible complex representation appears in the regular representation with multiplicity equal to its dimension, the regular representation becomes a universal induced object containing all irreducible symmetry types.

Permutation representations from subgroups

Every subgroup H≤G gives a transitive permutation action of G on G/H. Conversely, every transitive G-set is isomorphic to G/H for the stabiliser H of a point.

Therefore transitive permutation representations are induced trivial representations. This is a major bridge between group actions and representation theory:

stabiliser subgroup ↔ coset action ↔ induced trivial representation.

Orbit counting through reciprocity

Let G act on a finite set X. The permutation representation C[X] decomposes into irreducibles. The multiplicity of the trivial representation equals the dimension of the fixed-vector space, which equals the number of G-orbits on X.

If X=G/H is transitive, there is exactly one orbit, so the trivial representation occurs once in Ind_H^G 1. Frobenius reciprocity predicts the same result:

⟨Ind_H^G 1, 1_G⟩_G = ⟨1_H, Res_H^G 1_G⟩_H = 1.

Induction in symmetric groups

Symmetric groups provide one of the richest settings for induction and restriction. The inclusion S_{n−1}≤S_n produces branching rules described by Young diagrams. Restriction corresponds to removing a box. Induction corresponds to adding a box.

This makes subgroup transport combinatorial. Irreducible representations are labelled by partitions, and movement between groups becomes movement through a graph of partitions. The resulting branching graph is not merely a picture; it encodes multiplicities of restrictions and inductions.

Induction in harmonic analysis

For locally compact groups and Lie groups, induced representations become a foundational construction. Many important infinite-dimensional representations are built from representations of subgroups, especially stabilisers, Borel subgroups or parabolic subgroups.

The finite-group version is therefore not an isolated chapter. It is the discrete model of a much broader strategy: construct global symmetry from local or subgroup data.

Induction in physics

In symmetry-based physics, a state may be classified first by a stabiliser subgroup of a momentum, direction or configuration. Induced-representation methods then construct representations of a larger spacetime or symmetry group from those stabiliser representations.

The physical interpretation requires additional structure, but the mathematical architecture is recognisable: choose a subgroup that fixes a reference object, classify the subgroup action, then induce to the full symmetry group.

A computational workflow

• Identify H≤G and compute the index [G:H].
• Describe the H-representation W and its character ψ.
• Use the dimension formula to predict dim Ind_H^G W.
• Compute the induced character, or avoid doing so directly if reciprocity gives the desired multiplicities more cheaply.
• List irreducible G-characters χ_i.
• Compute ⟨ψ,Res_H^G χ_i⟩_H.
• Those values are the multiplicities of χ_i in Ind_H^G ψ.
• Check that the dimensions reconstruct the induced dimension.

This workflow is often much cheaper than building induced matrices coset by coset.

Common mistakes

• Treating induction as extension: an H-representation need not extend to G on the same vector space. Induction typically enlarges the space.
• Forgetting the index: induced dimension is multiplied by [G:H].
• Assuming irreducibility survives restriction: it often does not.
• Assuming induction preserves irreducibility: induced representations may decompose.
• Ignoring conjugation: induced character values depend on whether conjugates of a group element enter H.
• Confusing reciprocity with inverse operations: induction and restriction are adjoint, not mutually inverse.
• Skipping dimension checks: a multiplicity decomposition must reconstruct the induced dimension.

Practice set

1. Dimension

H has index 5 in G and W is a three-dimensional H-representation. Find dim Ind_H^G W.

Answer: 15.

2. Restriction

Does restriction change the underlying vector space?

Answer: no. It changes only the acting group from G to H.

3. Induced trivial representation

What familiar representation is Ind_H^G 1_H?

Answer: the permutation representation on G/H.

4. Reciprocity

If an irreducible G-representation V restricts to H with W appearing twice, how often does V appear in Ind_H^G W?

Answer: twice, by Frobenius reciprocity.

5. Transitivity

If K≤H≤G, how does inducing from K to G compare with inducing first to H and then to G?

Answer: the two constructions are equivalent: Ind_H^G Ind_K^H W ≅ Ind_K^G W.

What Frobenius reciprocity teaches beyond representation theory

There is a broader mathematical lesson here. A difficult question can often be moved to a different level where it becomes cheaper to answer. Frobenius reciprocity does this with multiplicities. Instead of decomposing a large induced representation directly, we restrict a candidate irreducible and solve a smaller subgroup problem.

The theorem is powerful because it does not throw information away. It translates the question across an adjunction where the relevant homomorphism spaces match exactly.

Good representation theory moves the question to the level where symmetry is easiest to see, then carries the answer back without breaking equivariance.

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