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CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-30
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Frobenius reciprocity for complex characters

Statement

Let G be a finite group, let H≤G, let χ be a complex character of H, and let ψ be a complex character of G. Then

⟨Ind⁡HGχ,ψ⟩G=⟨χ,Res⁡HGψ⟩H.

Facts & Assumptions

Given: A finite group G, a subgroup H≤G, a finite-dimensional complex representation W of H with character χ, and a finite-dimensional complex representation V of G with character ψ.

[F1]

The inner product of two complex characters equals the dimension of the intertwiner space between the corresponding representations (The class-function inner product ⟨χV,χW⟩ equals dim⁡Hom⁡G(W,V)).

[F2]

Induction is left adjoint to restriction: Hom⁡G(Ind⁡HGW,V)≅Hom⁡H(W,Res⁡HGV) (Induction is left adjoint to restriction for finite-group modules over a commutative ring).

[F3]

The character Ind⁡HGχ is the character of Ind⁡HGW (The induced character Ind⁡HGχ of a complex character).

Proof

technique · direct
1.1

By [F3] and then [F1], ⟨Ind⁡HGχ,ψ⟩G=dim⁡Hom⁡G(Ind⁡HGW,V).

F1F3given
1.2

By [F2], this dimension equals dim⁡Hom⁡H(W,Res⁡HGV); applying [F1] again on H gives dim⁡Hom⁡H(W,Res⁡HGV)=⟨χ,Res⁡HGψ⟩H.

F1F2given
2.1

The expressions in steps 1.1 and 1.2 are equal, which is exactly the Frobenius reciprocity identity.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

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Sources