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

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

1.1

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.

F1F2given

2.1

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.

F2step 1.1

3.1

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.

F3step 1.1step 2.1∎

Depends on

Used by

Ramification indices account for the inertia quotient Corollary
Boundary normal subgroups in Clifford theory Example
Clifford correspondence for A3 in S3 Example
Gallagher correspondence for an extendible type Theorem

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)