DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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 ramification index

Definition

Let $G$ be finite, $N ⊴ G$ , $θ \in Irr (N)$ , and $χ \in Irr (G ∣ θ)$ . The ramification index $e (χ, θ)$ is the common positive multiplicity in Clifford restriction formula. If $V$ affords $χ$ and $S$ affords $θ$ , then $e (χ, θ) = ⟨ {Res}_{N}^{G} χ, θ ⟩_{N} = \dim_{C} {Hom}_{N} (S, V) .$ The first equality is The multiplicity of an irreducible summand is a character inner product and the second is The class-function inner product $⟨ χ_{V}, χ_{W} ⟩$ equals $\dim {Hom}_{G} (W, V)$ , applied to $N$ . The formula makes the number independent of the chosen models $S, V$ and constant as $θ$ varies in its $G$ -orbit. It is defined here only for a constituent, so it is always positive.

Depends on

Used by

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

Dependency tree · two levels

10 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

Tammo tom Dieck, Representation Theory — §4.2 p.54 after Proposition 4.2.2 (standard reference, not scraped)