TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Clifford correspondence

Statement

Let $G$ be finite, $N ⊴ G$ , $θ \in Irr (N)$ , and $I = I_{G} (θ)$ . Induction gives a bijection $Irr (I ∣ θ) ⟶ Irr (G ∣ θ) .$ On module isomorphism classes, the inverse takes the $θ$ -isotypical component. Conjugate normal types give the same target set, and the sets $Irr (G ∣ θ)$ , indexed by distinct $G$ -orbits in $Irr (N)$ , partition $Irr (G)$ .

Facts & Assumptions

Given: The groups, modules, characters, and hypotheses in the statement. All representations here are finite-dimensional complex left representations.

[F1]

For irreducible $V$ lying over $θ$ , the component $V_{θ}$ is irreducible over inertia and its induction is isomorphic to $V$ . (Reconstruction from the inertia component).

[F2]

An irreducible inertia module above $θ$ induces irreducibly, and its identity-coset block is the full $θ$ -component, isomorphic to the original inertia module. (Induction of an inertia constituent is irreducible).

[F3]

The normal restriction of an irreducible character is a positive common multiple of the sum of the distinct conjugates of any constituent. (Clifford restriction formula).

Proof

technique · direct

1.1F1F2given

Induction sends each element of $Irr (I ∣ θ)$ to an irreducible lying over $θ$ . Conversely, every irreducible $V$ lying over $θ$ is induced from its irreducible component $V_{θ}$ . This proves that the displayed map is defined and surjective.

2.1F2step 1.1

Taking the $θ$ -component of an induced module recovers its inducing module. A $G$ -isomorphism preserves these components, since it sends simple $N$ -submodules to simple submodules of the same type. Hence isomorphic induced modules have isomorphic inducing modules. This proves injectivity and the specified inverse.

3.1F3step 1.1step 2.1∎

The restriction formula shows that occurrence of one type is equivalent to occurrence of each conjugate, and excludes every type outside that orbit. Every irreducible $G$ -module has a nonzero finite-dimensional restriction: an $N$ -stable nonzero subspace of least possible dimension is simple, so a constituent exists. Consequently the orbit-indexed sets cover $Irr (G)$ and are pairwise disjoint. No step requires I to be proper in G or to properly contain N.

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Britta Späth, Reduction theorems for some global-local conjectures — Theorem 1.2 p.2; tom Dieck Theorem 4.2.4(1–3,5) pp.55–56 (standard reference, not scraped)