Mackey's Theorem — Mathematics Knowledge System

Overview

Mackey's theorem describes the restriction of an induced representation to a subgroup, in terms of double cosets. Specifically, if H, K ≤ G and W is an H-representation, then Res^G_K Ind^G_H(W) decomposes as a direct sum over double cosets K\G/H of induced representations. This is fundamental for studying intertwining operators between induced representations and for establishing irreducibility criteria.

Intuition

When you induce from H and then restrict to K, you must understand how the two subgroups 'see' each other inside G. Double cosets KxH partition G and measure the different ways K and H can interact. For each double coset, you get a contribution that looks like induction from the intersection K ∩ xHx⁻¹.

Formal Definition

Definition

Let G be a finite group, H and K subgroups, and W an H-representation. The Mackey decomposition formula is:

\mathrm{Res}^G_K \mathrm{Ind}^G_H W \cong \bigoplus_{x \in K \backslash G / H} \mathrm{Ind}^K_{K \cap xHx^{-1}} ({}^x W|_{K \cap xHx^{-1}})

Mackey decomposition: sum over double cosets

{}^x W: K \cap xHx^{-1} \to GL(W), \quad ({}^x W)(k) = W(x^{-1}kx) \quad \text{(conjugated representation)}

Conjugate representation by x

K \backslash G / H = \{ KxH : x \in G \} \quad \text{(set of } (K,H)\text{-double cosets)}

Double coset space

\mathrm{Hom}_G(\mathrm{Ind}^G_H W,\; \mathrm{Ind}^G_K V) \cong \bigoplus_{x \in K\backslash G/H} \mathrm{Hom}_{K\cap xHx^{-1}}({}^xW, V)

Mackey's intertwining number formula

Notation

Notation	Meaning
$K \backslash G / H$	Set of (K,H)-double cosets in G
${}^x W$	Representation of xHx⁻¹ obtained by conjugating W: k ↦ W(x⁻¹kx)
$K \cap xHx^{-1}$	Intersection subgroup for double coset representative x

Properties

Special case H = K

When K = H, Mackey's formula gives the decomposition of Res^G_H Ind^G_H(W), summing over the set of H-double cosets in G.

Number of double cosets

The number of (K,H)-double cosets in G equals the inner product of the trivial characters induced to G from K and H respectively.

Theorems

Theorem 1: Mackey's Decomposition Theorem

For finite groups K, H ≤ G and H-representation W: Res^G_K Ind^G_H(W) decomposes as a direct sum over (K,H)-double coset representatives x in G, with each summand being Ind^K_{K cap xHx^{-1}} of the twisted restriction of W.

Theorem 2: Mackey Irreducibility Criterion

Ind^G_H(W) is irreducible if and only if: (1) W is irreducible, and (2) for every g ∈ G \ H, the H∩gHg⁻¹-representations W|_{H∩gHg⁻¹} and (^g W)|_{H∩gHg⁻¹} have no common irreducible constituent.

Worked Examples

1
First find the double cosets H\S₃/H. S₃ = {e,(12),(13),(23),(123),(132)}. Double cosets: HeH = H = {e,(12)}, H(13)H = {(13),(12)(13),(13),(123)}... let us compute: (12)(13) = (132), so H(13)H = {(13),(132),(123),(23)} — but |S₃| = 6 = |H|·1 + ... Actually H(13)H consists of all h₁(13)h₂ with h₁,h₂ ∈ H: {(13), (12)(13), (13), (13)(12)} = {(13),(132),(13),(23)} = {(13),(132),(23)}. And HeH = {e,(12)}. Total: 2+... |H(13)H| = |H|²/|H∩(13)H(13)| = 4/|{e}| = 4... let me just enumerate: H = {e,(12)}, H(13)H: take (13) ∈ S₃\H: h₁(13)h₂ for h₁,h₂ ∈ {e,(12)}: e·(13)·e=(13), (12)·(13)·e=(132), e·(13)·(12)=(23)... wait (13)(12) = (132)... Actually (13)·(12): map 1→2→2, 2→2→1... let me be careful: (13)(12) means apply (12) first: 1→2, then (13): 2→2. So 1↦2. Apply (12) to 2: 2→1, then (13) to 1: 1→3. So 2↦3. Apply (12) to 3: 3→3, then (13) to 3: 3→1. So 3↦1. Result: (123). So (13)(12) = (123). And (12)(13) = (132) similarly. H(13)H = {(13), (132), (123), (23)} has 4 elements. Total: 2+4=6=|S₃|. So 2 double cosets.
$H \backslash S_3 / H = \{ HeH,\; H(13)H \}, \quad |HeH| = 2, \quad |H(13)H| = 4$
2
Mackey: Res^{S₃}_H Ind^{S₃}_H(triv) ≅ Ind^H_{H∩H}(ˢtriv) ⊕ Ind^H_{H∩(13)H(13)}(ˢtriv). For x=e: H∩H=H, ˢtriv=triv, Ind^H_H(triv) = triv. For x=(13): (13)H(13)⁻¹ = {e,(13)(12)(13)} = {e,(23)}. H∩(23)H(23) ... H∩{e,(23)} = {e}. So Ind^H_{{e}}(ˢtriv) = triv⊕sgn of H (regular rep of H = ℤ/2ℤ restricted from {e}).
$\mathrm{Res}^{S_3}_H \mathrm{Ind}^{S_3}_H(\mathrm{triv}) \cong \mathrm{triv}_H \oplus (\mathrm{triv}_H \oplus \mathrm{sgn}_H)$
3
So Res Ind(triv) ≅ 2·triv_H ⊕ sgn_H as H-representations, with total dimension 1+1+1 = 3 = [S₃:H]·1 ✓.
$\mathrm{Res}^{S_3}_H \mathrm{Ind}^{S_3}_H(\mathrm{triv}) \cong 2\,\mathrm{triv}_H \oplus \mathrm{sgn}_H$

✓ Answer

There are 2 double cosets, and Res Ind(triv) ≅ 2·triv_H ⊕ sgn_H.

Practice Problems

Mediumfree response

How many double cosets are in H\G/K when G = S₄, H = S₃ (fixing 4), K = S₃ (fixing 1)?

Hardproof writing

Use Mackey's theorem to prove that if H ∩ gHg⁻¹ = {e} for all g ∈ G \ H and W is an irreducible H-representation, then Ind^G_H(W) is irreducible.

Quiz

Summary

Mackey's theorem: Res^G_K Ind^G_H(W) ≅ ⊕_{x∈K\G/H} Ind^K_{K∩xHx⁻¹}(ˣW).
The sum is indexed by (K,H)-double cosets in G.
The conjugate representation ^xW of xHx⁻¹ is defined by (^xW)(k) = W(x⁻¹kx).
Mackey's irreducibility criterion: Ind(W) irreducible iff W irreducible and no cross-term in the inner product survives.
Special case K=H recovers a formula for ⟨Ind(W), Ind(W)⟩_G via Frobenius reciprocity.

References

BookSerre, J.-P. — Linear Representations of Finite Groups (1977), §7.3
BookIsaacs, I.M. — Character Theory of Finite Groups (1976), Chapter 5