CorollaryStatement: 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.

Ramification indices account for the inertia quotient

Statement

Let $G$ be finite, $N ⊴ G$ , and $θ \in Irr (N)$ . List the distinct characters in $Irr (G ∣ θ)$ as $χ_{1}, \dots, χ_{r}$ , and set $e_{j} = e (χ_{j}, θ)$ . Then ${Ind}_{N}^{G} θ = \sum_{j = 1}^{r} e_{j} χ_{j}, \sum_{j = 1}^{r} e_{j}^{2} = [I_{G} (θ) : N] .$

Facts & Assumptions

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

[F1]

Finite-dimensional complex representations of a finite group are completely reducible. (If $char k ∤ ∣ G ∣$ , every finite-dimensional representation of $G$ is completely reducible).

[F2]

Character induction and restriction satisfy Frobenius reciprocity for finite groups. (Frobenius reciprocity for complex characters).

[F3]

The multiplicity of an irreducible in a complex representation is the character inner product. (The multiplicity of an irreducible summand is a character inner product).

[F4]

For a character lying over $θ$ , its ramification index is the multiplicity of $θ$ in its normal restriction. (Clifford ramification index).

[F5]

The self-inner-product of ${Ind}_{N}^{G} θ$ equals $[I_{G} (θ) : N]$ . (Normal subgroup induction criterion).

Proof

technique · direct

1.1

Decompose the nonzero induced module into simple $G$ -modules by complete reducibility. The coefficient of an irreducible character $ψ$ is $⟨ {Ind}_{N}^{G} θ, ψ ⟩_{G} = ⟨ θ, {Res}_{N}^{G} ψ ⟩_{N}$ . The latter is the nonnegative integer multiplicity of $θ$ : conjugate symmetry of the inner product does not change that real integer. It is zero exactly outside the lying-over set and equals $e_{j}$ for $ψ = χ_{j}$ . This proves the first identity and also that the finite list is nonempty.

F1F2F3F4given

2.1

Applying the multiplicity formula to each simple module itself gives $⟨ χ_{j}, χ_{k} ⟩_{G} = δ_{j k}$ . Taking the norm of the finite sum in step 1.1 therefore gives $\sum_{j} e_{j}^{2}$ . The normal-induction norm formula identifies this with $[I_{G} (θ) : N]$ . If the index is one there is exactly one term with e=1; the formula also includes N=1 and N=G.

F3F5step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Tammo tom Dieck, Representation Theory — Theorem 4.2.4(4), equation (4.7), pp.55–56 (standard reference, not scraped)