Frobenius Reciprocity

Core Idea

Frobenius reciprocity states that for a subgroup H ≤ G, induction and restriction are adjoint functors: ⟨Ind_H^G(σ), ρ⟩_G = ⟨σ, Res_H^G(ρ)⟩_H for any representation σ of H and ρ of G. Equivalently, Hom_G(Ind_H^G(σ), ρ) ≅ Hom_H(σ, Res_H^G(ρ)). This fundamental adjunction links the representation theories of a group and its subgroups, providing a powerful tool for computing multiplicities without explicitly constructing induced representations.

Explainer

Frobenius reciprocity connects two fundamental operations in representation theory: induction (building representations of G from those of a subgroup H) and restriction (obtaining representations of H by forgetting part of the G-structure). The theorem states a precise numerical relationship: the multiplicity of an irreducible G-representation ρ in the induced representation Ind_H^G(σ) equals the multiplicity of an irreducible H-representation σ in the restricted representation Res_H^G(ρ). In inner product notation: ⟨Ind_H^G(σ), ρ⟩_G = ⟨σ, Res_H^G(ρ)⟩_H.

The proof at the character level is a direct computation. The left side is (1/|G|) Σ_{g∈G} χ_{Ind}(g) conjugate(χ_ρ(g)). Substituting the induction formula for χ_{Ind} and rearranging the sums (using the fact that summing x⁻¹gx over all x ∈ G and all g covers all elements of H with the correct multiplicity) yields (1/|H|) Σ_{h∈H} χ_σ(h) conjugate(χ_ρ(h)), which is exactly ⟨σ, Res_H^G(ρ)⟩_H.

The practical power of Frobenius reciprocity is that it replaces a computation in G (decomposing a potentially large induced representation) with a computation in H (decomposing a restriction). Since H is smaller, this is usually far easier. For example, to find the irreducible constituents of a representation induced from a cyclic subgroup, you only need to know how the irreducibles of G look when restricted to that cyclic subgroup — and representations of cyclic groups are completely understood (they are all one-dimensional over ℂ).

In modern algebra, Frobenius reciprocity is recognized as a special case of an adjunction between functors. The induction functor Ind_H^G is left adjoint to the restriction functor Res_H^G. This categorical perspective has been enormously fruitful: it generalizes to compact groups (where induction uses integration against Haar measure), Lie algebras (parabolic induction), and algebraic geometry (push-forward and pull-back of sheaves). The same structural relationship appears throughout mathematics, with Frobenius's original group-theoretic version as the prototype.