Induced Representations — Mathematics Knowledge System

Overview

Given a subgroup H ≤ G and a representation W of H, the induced representation Ind_H^G(W) is a representation of G constructed by 'extending' W to all of G by acting on cosets. Induction is the adjoint operation to restriction, a duality captured by Frobenius reciprocity. Induced representations are the primary tool for constructing all irreducible representations of a group from those of its subgroups, and they are central to the representation theory of symmetric groups, p-adic groups, and automorphic forms.

Intuition

If H acts on a space W, then G acts on the space of functions f: G → W satisfying f(hg) = ρ_W(h)f(g) for all h ∈ H. This gives a G-representation whose dimension is [G:H]·dim(W). Induction 'copies' the H-action across all cosets of H.

Formal Definition

Definition

Let H ≤ G be a subgroup (G and H finite), and let (ρ, W) be a representation of H over k. The induced representation is defined as follows.

\mathrm{Ind}_H^G(W) = kG \otimes_{kH} W = \bigoplus_{g \in G/H} g \otimes W

Induced representation as tensor product over kH

\dim \mathrm{Ind}_H^G(W) = [G:H] \cdot \dim W

Dimension formula

\chi_{\mathrm{Ind}_H^G W}(g) = \frac{1}{|H|}\sum_{\substack{x \in G \\ x^{-1}gx \in H}} \chi_W(x^{-1}gx)

Character of induced representation

\langle \chi_{\mathrm{Ind}_H^G W},\, \chi_V \rangle_G = \langle \chi_W,\, \chi_{\mathrm{Res}_H^G V} \rangle_H

Frobenius reciprocity

Notation

Notation	Meaning
$\mathrm{Ind}_H^G(W)$	Representation of G induced from W on H
$\mathrm{Res}_H^G(V)$	Restriction of V (a G-rep) to H
$[G:H]$	Index of H in G

Properties

Induction from the trivial subgroup gives the regular representation

\mathrm{Ind}_{\{e\}}^G(k) \cong kG \quad (\text{regular representation})

Induction and restriction are adjoint functors

\mathrm{Ind}_H^G : \mathrm{Rep}(H) \rightleftharpoons \mathrm{Rep}(G) : \mathrm{Res}_H^G

Character formula

\chi_{\mathrm{Ind}_H^G W}(g) = \frac{1}{|H|}\sum_{x \in G,\, x^{-1}gx \in H} \chi_W(x^{-1}gx)

Theorems

Theorem 1: Frobenius Reciprocity

\mathrm{Hom}_G(\mathrm{Ind}_H^G W,\, V) \cong \mathrm{Hom}_H(W,\, \mathrm{Res}_H^G V) \quad \text{i.e.,} \quad \langle \mathrm{Ind}_H^G \chi_W, \chi_V \rangle_G = \langle \chi_W, \mathrm{Res}_H^G \chi_V \rangle_H

Theorem 2: Mackey's Irreducibility Criterion

\mathrm{Ind}_H^G W \text{ is irreducible} \iff W \text{ is irreducible and for all } s \notin H,\; \mathrm{Res}_{H \cap sHs^{-1}}^H W \text{ and } \mathrm{Res}_{H \cap sHs^{-1}}^{sHs^{-1}} {}^s W \text{ have no common component}

Theorem 3: Transitivity of induction

\mathrm{Ind}_K^G \cong \mathrm{Ind}_H^G \circ \mathrm{Ind}_K^H \quad \text{for } K \leq H \leq G

Worked Examples

1
Take H = ⟨(12)⟩ ≅ ℤ/2ℤ ≤ S₃, and W = sign representation of H (1-dim, (12) acts by -1).
$H = \langle (12) \rangle,\; W = \mathrm{sgn}_H$
2
The induced representation has dimension [S₃:H]·1 = 3.
$\dim \mathrm{Ind}_H^{S_3} W = 3$
3
Use the character formula to compute χ_{Ind}. The result is the permutation representation on cosets minus the trivial representation — decomposing, this gives sgn_{S₃} ⊕ std.
$\mathrm{Ind}_H^{S_3}(\mathrm{sgn}_H) \cong \mathrm{sgn}_{S_3} \oplus V_{\mathrm{std}}$

✓ Answer

The induced representation is 3-dimensional and decomposes as sgn ⊕ std.

Practice Problems

Hardfree response

Compute the character of Ind_{A₃}^{S₃}(ρ) where ρ is the non-trivial 1-dimensional representation of A₃ ≅ ℤ/3ℤ.

Hardproof writing

Prove that every irreducible representation of G appears as a summand of an induced representation from some subgroup.

Common Mistakes

Common Mistake

Thinking Ind_H^G and Res_H^G are inverse to each other

They are adjoint, not inverse. Res takes G-representations to H-representations; Ind goes the other way. Their composition is not the identity — it is described by Mackey's decomposition theorem.

Common Mistake

Assuming induced representations are always irreducible

Induced representations are almost never irreducible (unless [G:H] = 1 and W is irreducible). Mackey's criterion gives the conditions for irreducibility.

Quiz

Frobenius reciprocity states:

Ind and Res are inverse functors⟨Ind_H^G χ_W, χ_V⟩_G = ⟨χ_W, Res_H^G χ_V⟩_HEvery induced representation is irreducibledim Ind_H^G W = dim G

The dimension of Ind_H^G(W) is:

|G|·dim W[G:H]·dim Wdim W|H|·dim W

Historical Background

Frobenius introduced induced representations in 1898 to construct representations of a group from representations of subgroups. He proved the reciprocity theorem (Frobenius reciprocity) in the same paper. Mackey's theory (1949–1952) generalised this to locally compact groups and provided the criterion (Mackey's criterion) for when an induced representation is irreducible. The representation theory of the symmetric group, worked out by Young and Frobenius, relies crucially on induction from Young subgroups.

1898
Frobenius introduces induced representations and proves Frobenius reciprocity
Georg Frobenius
1949
Mackey's subgroup theorem and irreducibility criterion for induced representations
George Mackey
1955
Induced representations central to Harish-Chandra's work on real Lie groups
Harish-Chandra

Summary

Ind_H^G(W) is a G-representation of dimension [G:H]·dim(W) built from cosets of H.
Its character is χ_{Ind}(g) = (1/|H|) Σ_{x: x⁻¹gx ∈ H} χ_W(x⁻¹gx).
Frobenius reciprocity: Ind and Res are adjoint; multiplicities are preserved across the adjunction.
Mackey decomposition: Res_K^G Ind_H^G W = ⊕_s Ind_{K∩sHs⁻¹}^K ({}^s W).
Every irreducible representation of G appears in the induction from {e} (the regular representation).

References

BookSerre, J.-P. — Linear Representations of Finite Groups (1977), Chapter 3 (Frobenius reciprocity) and §7 (Mackey)
BookFulton, W. & Harris, J. — Representation Theory: A First Course (1991), Lecture 3
WebsiteWikipedia — Induced representation
WebsitenLab — induced representation