Mackey Irreducibility Criterion

Mackey's irreducibility criterion provides a practical test for determining when an induced representation is irreducible, essential for classification problems.

TheoremMackey's Irreducibility Criterion

Let $H \leq G$ be a subgroup and $W$ an irreducible representation of $H$ . Then $\text{Ind}_H^G(W)$ is irreducible if and only if:

Intertwining condition: For all $g \in G \setminus H$ , the representations $W$ and ${}^gW$ are disjoint (non-isomorphic) as representations of $H \cap gHg^{-1}$ where ${}^gW$ denotes conjugation: ${}^gW(h) = W(g^{-1}hg)$
$W$ is $H$ -irreducible: The representation $W$ is irreducible as an $H$ -representation

Equivalently, $\text{Res}_H^G(\text{Ind}_H^G(W))$ contains $W$ with multiplicity 1, and all other irreducible summands are non-isomorphic to $W$ .

Proof idea: Use Frobenius Reciprocity and Mackey's decomposition formula. The condition ensures that $\text{End}_G(\text{Ind}_H^G(W)) \cong \mathbb{C}$ , so by Schur's Lemma, the representation is irreducible.

ExampleSymmetric Group $S_3$

Let $H = \langle (12) \rangle$ (subgroup of order 2) and $W = \mathbb{C}_{\text{sgn}}$ (sign representation of $H$ , so $(12)$ acts as $-1$ ).

Check conjugates: For $g = (23) \notin H$ :

$gHg^{-1} = \langle (13) \rangle$
${}^gW$ on $H \cap gHg^{-1} = \{e\}$ is just the trivial representation
$W$ restricted to $\{e\}$ is also trivial

Since $H \cap gHg^{-1} = \{e\}$ , all representations of it are isomorphic (trivial), so the intertwining condition fails. Thus $\text{Ind}_H^G(W)$ is not irreducible.

Indeed, computing: $\text{Ind}_H^{S_3}(\mathbb{C}_{\text{sgn}}) \cong \mathbb{C}_{\text{sgn}} \oplus V_{\text{std}}$ where $V_{\text{std}}$ is the 2-dimensional standard representation.

ExampleWhen the Criterion Succeeds

Dihedral group: For $D_n = \langle r, s : r^n = s^2 = 1, srs = r^{-1} \rangle$ , let $H = \langle s \rangle$ with a non-trivial character $\chi(s) = -1$ .

For many $g \in D_n \setminus H$ , we have $gHg^{-1} \neq H$ and the conjugated representations are non-isomorphic on the intersection. Mackey's criterion often confirms that $\text{Ind}_H^{D_n}(\chi)$ is an irreducible 2-dimensional representation.

TheoremCriterion for Simply Transitive Actions

If $G$ acts simply transitively on the cosets $G/H$ (i.e., $H$ is maximal and has no non-trivial intersection with its conjugates), then $\text{Ind}_H^G(W)$ is irreducible for any irreducible $W$ of $H$ .

This is because the intertwining condition is automatically satisfied when conjugate subgroups intersect trivially.

Remark

Mackey's criterion is fundamental for:

Constructing irreducibles: Many irreducible representations arise as inductions from maximal subgroups
Representation theory of wreath products: Irreducibles are often induced from subgroups
Automorphic representations: Generalization to p-adic and adelic groups
Geometric representation theory: Irreducibility of perverse sheaves

The criterion transforms the abstract problem of checking irreducibility into a concrete computation involving conjugacy and restriction.