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

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Clifford restriction formula

Statement

Let $G$ be finite, $N ⊴ G$ , $θ \in Irr (N)$ , and $χ \in Irr (G ∣ θ)$ . Put $I = I_{G} (θ)$ . There is a positive integer $e$ such that ${Res}_{N}^{G} χ = e \sum_{g I \in G / I}^{g} θ, χ (1) = e [G : I] θ (1) .$ Here $G / I$ indexes left cosets, one for each distinct conjugate. In particular, the entire restriction is isotypical precisely when $I = G$ ; it need not be isotypical in general.

Facts & Assumptions

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

[F1]

An irreducible complex module restricts to one orbit of normal types with equal positive multiplicities. (Normal restriction has one orbit of constituents).

[F2]

For finite-dimensional complex representations of a finite group, the character of a direct sum is the sum of the characters. (Characters add on direct sums, multiply on tensor products, and conjugate on duals).

Proof

technique · direct

1.1

Let $V$ afford $χ$ . Its restriction is a direct sum with one common positive multiplicity $e$ for the orbit of $θ$ . The map $g I \mapsto^{g} θ$ is well defined and bijective: two conjugates agree exactly when the corresponding elements differ on the right by an element of the stabilizer. Additivity of trace now gives the restriction formula.

F1F2given

2.1

At the identity every conjugate takes value $θ (1)$ , giving the degree formula. Since $e > 0$ , precisely $[G : I]$ types occur. Thus an isotypical restriction forces $[G : I] = 1$ , hence $I = G$ ; conversely $I = G$ makes the sum a single type. The calculation includes index one and both N=1 and N=G.

step 1.1algebra∎

Depends on

Used by

Clifford ramification index Definition
Clifford correspondence 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 1.1 p.2; tom Dieck §4.2 p.54 (standard reference, not scraped)