Induction and Restriction Functors — Mathematics Knowledge System

Overview

Given a subgroup H ≤ G, representation theory has two natural functors between representations of H and representations of G. The restriction functor Res^G_H takes a G-representation and views it as an H-representation by restricting the action. The induction functor Ind^G_H goes the other way, building a G-representation from an H-representation by 'extending' it to all of G. These two functors are adjoint (Frobenius reciprocity), and understanding their interaction is a central theme of the subject.

Intuition

Restriction is easy: just forget the group elements outside H and keep the action of H. Induction is the adjoint: it builds the 'largest' G-representation compatible with the given H-action. Concretely, Ind^G_H(W) consists of functions f: G → W that transform by W under H-translation on the right.

Formal Definition

Definition

Let H ≤ G be groups and k a field. For a kG-module V, the restriction is the kH-module Res^G_H(V) with the same underlying vector space but only the H-action. For a kH-module W, the induced module is

\mathrm{Res}^G_H(V) = V \text{ as a vector space, with action } h \cdot v = \rho(h)v \text{ for } h \in H

Restriction: forget the G-action outside H

\mathrm{Ind}^G_H(W) = kG \otimes_{kH} W

Induced module via tensor product over group ring

\mathrm{Ind}^G_H(W) \cong \bigoplus_{t \in G/H} t \otimes W \quad (\text{as vector spaces, choosing coset representatives } t)

Decomposition into coset-translates

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

Dimension formula

Notation

Notation	Meaning
$\mathrm{Res}^G_H$	Restriction functor from G-reps to H-reps
$\mathrm{Ind}^G_H$	Induction functor from H-reps to G-reps
$[G:H]$	Index of H in G
$kG$	Group ring of G over field k

Properties

Both functors are exact

Res^G_H is always exact. Ind^G_H = kG ⊗_{kH} − is exact when kG is a free kH-module (always for finite groups in the semisimple case).

Mackey restriction formula

Res^G_H Ind^G_K W decomposes as a direct sum over (H,K)-double cosets in G: for each representative x, the summand is Ind^H_{H cap xKx^{-1}} Res^{xKx^{-1}}_{H cap xKx^{-1}}(x.W).

Theorems

Theorem 1: Frobenius Reciprocity (adjunction)

There is a natural isomorphism \mathrm{Hom}_G(\mathrm{Ind}^G_H W, V) \cong \mathrm{Hom}_H(W, \mathrm{Res}^G_H V).

Theorem 2: Transitivity of Induction

If K ≤ H ≤ G, then \mathrm{Ind}^G_K \cong \mathrm{Ind}^G_H \circ \mathrm{Ind}^H_K (and similarly for restriction).

Theorem 3: Character of Induced Representation

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

Worked Examples

1
The sign representation of S₃ sends each permutation to its sign: e ↦ 1, (12) ↦ −1, (13) ↦ −1, (23) ↦ −1, (123) ↦ 1, (132) ↦ 1.
$\mathrm{sgn}: S_3 \to \{\pm 1\}$
2
Restricting to H = {e, (12)}: e ↦ 1, (12) ↦ −1. This is the sign representation of ℤ/2ℤ.
$\mathrm{Res}^{S_3}_H(\mathrm{sgn}) = \mathrm{sgn}_{\mathbb{Z}/2\mathbb{Z}}$

✓ Answer

The restricted representation is the non-trivial 1-dimensional representation of ℤ/2ℤ (the sign of a transposition).

Practice Problems

Mediumfree response

State the dimension formula for Ind^G_H(W) and use it to find the dimension of Ind^{S_4}_{S_3}(V) where V is the 2-dimensional standard representation of S₃.

Mediumproof writing

Prove transitivity of restriction: if K ≤ H ≤ G and V is a G-representation, then Res^H_K(Res^G_H(V)) = Res^G_K(V).

Quiz

Summary

Res^G_H restricts a G-representation to H by forgetting the G-action outside H.
Ind^G_H(W) = kG ⊗_{kH} W; as a vector space it is [G:H]·dim W dimensional.
Induction is left adjoint to restriction (Frobenius reciprocity).
The induced character formula is χ_Ind(g) = (1/|H|) Σ_{x: x⁻¹gx∈H} χ_W(x⁻¹gx).
Transitivity: Ind^G_K ≅ Ind^G_H ∘ Ind^H_K and similarly for Res.

References

BookSerre, J.-P. — Linear Representations of Finite Groups (1977), Chapter 3
BookIsaacs, I.M. — Character Theory of Finite Groups (1976), Chapter 5