Clifford Theory — Mathematics Knowledge System

Overview

Clifford theory studies the relationship between representations of a group G and representations of a normal subgroup N ⊲ G. The fundamental theorem (Clifford's theorem) states that the restriction of an irreducible G-representation to N decomposes as a sum of G-conjugates of a single irreducible N-representation. Clifford theory then organises the irreducible G-representations via the inertia group of a given N-representation, reducing the classification problem to smaller pieces.

Intuition

When N is normal, G permutes the irreducible N-representations by conjugation. Clifford's theorem says: the restriction of an irreducible G-rep to N is a direct sum of the entire G-orbit of some irreducible N-rep. To lift from N to G, you look at the stabiliser (inertia group) of an N-rep and use (projective) representations of the inertia group.

Formal Definition

Definition

Let N ⊲ G be a normal subgroup and θ an irreducible representation of N. For g ∈ G, define the conjugate (gθ)(n) = θ(g⁻¹ng). The inertia group of θ is T = {g ∈ G : gθ ≅ θ}.

({}^g \theta)(n) = \theta(g^{-1}ng), \quad n \in N, \quad g \in G

Conjugate of an N-representation

T = \mathrm{Stab}_G(\theta) = \{g \in G : {}^g\theta \cong \theta\} \quad (\text{inertia group of } \theta)

Inertia group

\mathrm{Res}^G_N V \cong e \cdot \bigoplus_{x \in G/T} {}^x\theta \quad (\text{for irreducible } V \text{ of } G, \text{ where } e = \text{multiplicity})

Clifford's theorem: restriction to N

\{\text{irred. } G\text{-reps over } \theta\} \leftrightarrow \{\text{irred. } T\text{-reps extending or covering } \theta\}

Clifford correspondence

Notation

Notation	Meaning
$N \trianglelefteq G$	N is a normal subgroup of G
${}^g\theta$	Conjugate of θ by g: n ↦ θ(g⁻¹ng)
$T$	Inertia group (stabiliser) of θ in G
$\mathrm{Irr}(G\|\theta)$	Irreducible G-reps whose restriction to N contains θ

Properties

G orbits on Irr(N)

G permutes the irreducible representations of N by conjugation, and Clifford's theorem says Res^G_N V is constant on each orbit.

Degree formula

dim V = e · [G:T] · dim θ where e is the multiplicity of each conjugate of θ in Res^G_N V.

Theorems

Theorem 1: Clifford's Theorem

Let N ⊲ G, V an irreducible G-representation, and θ an irreducible constituent of Res^G_N V. Then Res^G_N V ≅ e·⊕_{x∈G/T} ˣθ, where T is the inertia group of θ and e is a positive integer. If [G:T] = [G:T], then dim V = e·[G:T]·dim θ.

Theorem 2: Clifford Correspondence

There is a bijection Irr(G|θ) ↔ Irr(T|θ), given by V ↦ (the irreducible T-constituent above θ), with inverse W ↦ Ind^G_T W. This is the Clifford correspondence.

Theorem 3: Extension case

If T = G (i.e., θ is G-stable), then every irreducible G-representation above θ is of the form θ̃ ⊗ μ, where θ̃ is an extension of θ to G and μ ∈ Irr(G/N).

Worked Examples

1
A₃ = {e, (123), (132)}, with irreducible characters: triv (all 1), ω (ζ,ζ,1 where ζ=e^{2πi/3}... wait: the three irreps of ℤ/3ℤ are χₖ: (123)↦ζᵏ for k=0,1,2).
$\mathrm{Irr}(A_3): \chi_0 = \mathrm{triv},\; \chi_1: (123) \mapsto \zeta,\; \chi_2: (123) \mapsto \zeta^2, \quad \zeta = e^{2\pi i/3}$
2
S₃ acts on Irr(A₃) by conjugation. (12)(123)(12)⁻¹ = (132), so (12) sends χ₁ ↦ χ₂ (since χ₁((132)) = ζ² = χ₂((123))). So the orbit of χ₁ is {χ₁, χ₂}, and the inertia group T of χ₁ is A₃ itself.
$(12)\cdot\chi_1 = \chi_2, \quad T = \mathrm{Stab}_{S_3}(\chi_1) = A_3$
3
The standard 2-dim irrep V of S₃: Res^{S_3}_{A_3}(V) has character values e↦2, (123)↦−1, (132)↦−1. Decompose: ⟨Res(V), χ₁⟩ = (1/3)(2+ζ̄(−1)+(ζ²)̄(−1))... Let me compute: ⟨Res V, χ₀⟩ = (1/3)(2−1−1)=0. ⟨Res V, χ₁⟩ = (1/3)(2·1+(−1)·ζ̄²+(−1)·ζ̄) = (1/3)(2+(ζ²+ζ)) = (1/3)(2−1) = 1/3... hmm. Actually χ of Res V at (123) is −1, and ⟨Res V, χ₁⟩ = (1/3)(2·1 + (−1)·ζ̄ + (−1)·ζ̄²) = (1/3)(2 − ζ̄ − ζ̄²) = (1/3)(2+1)=1 since ζ+ζ²=−1. So ⟨Res V, χ₁⟩=1 and by symmetry ⟨Res V, χ₂⟩=1. So Res(V) = χ₁ ⊕ χ₂.
$\mathrm{Res}^{S_3}_{A_3}(\mathrm{std}) = \chi_1 \oplus \chi_2$
4
By Clifford: V restricts to the full orbit {χ₁, χ₂} of χ₁, with multiplicity e=1. [S₃:T]=[S₃:A₃]=2, dim V = 1·2·1 = 2. ✓
$\dim(\mathrm{std}) = e \cdot [S_3:T] \cdot \dim \chi_1 = 1 \cdot 2 \cdot 1 = 2 \quad \checkmark$

✓ Answer

Res^{S₃}_{A₃}(std) = χ₁ ⊕ χ₂: the orbit of χ₁ under S₃ conjugation, each with multiplicity 1.

Practice Problems

Mediumfree response

For G = ℤ/4ℤ and N = ℤ/2ℤ (the unique subgroup of order 2), find the inertia group of each irreducible N-representation.

Hardproof writing

State and prove Clifford's theorem: if N ⊲ G, V is irreducible over G, and θ is an irreducible constituent of Res^G_N V, then all G-conjugates of θ appear in Res^G_N V with equal multiplicity.

Quiz

Summary

For N ⊲ G and irreducible V of G: Res^G_N V is a direct sum of an entire G-orbit of N-representations.
The inertia group T = Stab_G(θ) controls how θ ∈ Irr(N) extends to G.
Clifford correspondence: Irr(G|θ) ↔ Irr(T|θ) via induction.
When T = G (θ is G-stable): irreducibles above θ are θ̃ ⊗ μ for μ ∈ Irr(G/N).
Dimension formula: dim V = e · [G:T] · dim θ.

References

BookIsaacs, I.M. — Character Theory of Finite Groups (1976), Chapter 6
BookAlperin, J.L. — Local Representation Theory (1986), Chapter 1