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

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Normal restriction has one orbit of constituents

Statement

Let $G$ be finite, $N ⊴ G$ , and $V$ an irreducible complex $G$ -module. If $θ$ is a constituent of $V ∣_{N}$ , the constituents of $V ∣_{N}$ are exactly the $G$ -orbit of $θ$ under left conjugation, and each has the same positive integer multiplicity.

Facts & Assumptions

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

[F1]

Translation carries $V_{ψ}$ onto $V_{^{g} ψ}$ , and the restriction is the direct sum of its isotypical components. (Translation permutes normal isotypical components).

Proof

technique · direct

1.1

The sum of the components indexed by the orbit of $θ$ is nonzero because $V_{θ} \neq 0$ . Translation permutes these components, so their sum is $G$ -stable. Irreducibility makes it all of $V$ , leaving no other types.

F1given

2.1

The linear isomorphism $v \mapsto g v$ gives $\dim V_{^{g} θ} = \dim V_{θ}$ . Conjugate simple modules have the same dimension, since twisting changes only the action. Dividing component dimensions by this common simple dimension proves equality of multiplicities, each positive because its component is nonzero. This also covers a one-element orbit.

F1step 1.1algebra∎

Depends on

Translation permutes normal isotypical components

Used by

Induction of an inertia constituent is irreducible Lemma
Reconstruction from the inertia component Lemma
Clifford restriction formula Theorem
Gallagher correspondence for an extendible type Theorem

Dependency tree · two levels

4 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, Proposition 4.2.2 and following formula; Späth Theorem 1.1 (standard reference, not scraped)