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

1.1F1F2given

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.

2.1step 1.1algebra∎

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.

Depends on

Used by

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)