Questions: Frobenius Reciprocity

Question 1

Using Frobenius reciprocity, how can you determine the multiplicity of an irreducible representation ρ of G in Ind_H^G(σ) without computing the induced representation explicitly?

Model answer: The multiplicity of ρ in Ind_H^G(σ) is ⟨Ind_H^G(σ), ρ⟩_G = ⟨σ, Res_H^G(ρ)⟩_H. So you restrict ρ to H (which is straightforward — just evaluate the character at elements of H) and take the inner product with σ in H's character theory.

This is enormously practical. Computing the induced representation directly requires building a [G:H]·dim(σ)-dimensional space and working out the action. Frobenius reciprocity replaces this with a computation entirely within H, which is smaller. This is the primary method for decomposing induced representations in practice.

Question 2

Frobenius reciprocity shows that induction from the trivial subgroup {e} gives which representation?

• A. The trivial representation
• B. The sign representation
• C. The regular representation
• D. An irreducible representation of maximum dimension

Answer: (C)

The trivial subgroup has only one representation: the one-dimensional trivial representation. Inducing it to G gives a [G:{e}]-dimensional = |G|-dimensional representation. This is exactly the regular representation, which acts by left multiplication on ℂ[G]. Frobenius reciprocity confirms: the multiplicity of any irreducible ρ in Ind_{e}^G(1) is ⟨1, Res_{e}^G(ρ)⟩_{e} = dim(ρ), matching the known decomposition of the regular representation.

Question 3

Frobenius reciprocity is an isomorphism of vector spaces Hom_G(Ind σ, ρ) ≅ Hom_H(σ, Res ρ). This is an example of an adjunction in category theory.

• T. True
• F. False

Answer: True

Induction is the left adjoint of restriction in the category of group representations. This means Hom_G(Ind_H^G(σ), ρ) ≅ Hom_H(σ, Res_H^G(ρ)) naturally in both σ and ρ. This categorical perspective generalizes to other contexts: compact groups (with Haar measure), Lie algebras, and algebraic groups all have analogous induction-restriction adjunctions.

Question 4

If every irreducible representation of G appears in Ind_H^G(σ) for a single representation σ of H, what does this imply about the restriction Res_H^G to H?

Model answer: By Frobenius reciprocity, every irreducible ρ of G appearing in Ind_H^G(σ) means ⟨σ, Res_H^G(ρ)⟩_H ≥ 1 for all ρ. So σ appears as a constituent of Res_H^G(ρ) for every irreducible ρ of G — the representation σ is 'seen' by every irreducible when restricted to H.

This duality between induction and restriction is Frobenius reciprocity at work. Information flows both ways: understanding how representations of G restrict to H is equivalent to understanding how representations of H induce to G. Neither direction is more fundamental — they are two sides of the same coin.