Frobenius reciprocity

In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducing. It can be used to leverage knowledge about representations of a subgroup to find and classify representations of "large" groups that contain them. It is named for Ferdinand Georg Frobenius, the inventor of the representation theory of finite groups.

Statement

Character theory The theorem was originally stated in terms of character theory. Let G be a finite group with a subgroup H, let Res H G {\displaystyle \operatorname {Res} _{H}^{G}} denote the restriction of a character, or more generally, class function of G to H, and let Ind H G {\displaystyle \operatorname {Ind} _{H}^{G}} denote the induced class function of a given class function on H. For any finite group A, there is an inner product ⟨ − , − ⟩ A {\displaystyle \langle -,-\rangle _{A}} on the vector space of class functions A → C {\displaystyle A\to \mathbb {C} } (described in detail in the article Schur orthogonality relations). Now, for any class functions ψ : H → C {\displaystyle \psi :H\to \mathbb {C} } and φ : G → C {\displaystyle \varphi :G\to \mathbb {C} } , the following equality holds:

⟨ Ind H G ⁡ ψ , φ ⟩ G = ⟨ ψ , Res H G ⁡ φ ⟩ H . {\displaystyle \langle \operatorname {Ind} _{H}^{G}\psi ,\varphi \rangle _{G}=\langle \psi ,\operatorname {Res} _{H}^{G}\varphi \rangle _{H}.}

In other words, Ind H G {\displaystyle \operatorname {Ind} _{H}^{G}} and Res H G {\displaystyle \operatorname {Res} _{H}^{G}} are Hermitian adjoint.

Module theory

As explained in the section Representation theory of finite groups#Representations, modules and the convolution algebra, the theory of the representations of a group G over a field K is, in a certain sense, equivalent to the theory of modules over the group algebra K[G]. Therefore, there is a corresponding Frobenius reciprocity theorem for K[G]-modules. Let G be a group with subgroup H, let M be an H-module, and let N be a G-module. In the language of module theory, the induced module K [ G ] ⊗ K [ H ] M {\displaystyle K[G]\otimes _{K[H]}M} corresponds to the induced representation Ind H G {\displaystyle \operatorname {Ind} _{H}^{G}} , whereas the restriction of scalars K [ H ] N {\displaystyle {_{K[H]}}N} corresponds to the restriction Res H G {\displaystyle \operatorname {Res} _{H}^{G}} . Accordingly, the statement is as follows: The following sets of module homomorphisms are in bijective correspondence:

Hom K [ G ] ⁡ ( K [ G ] ⊗ K [ H ] M , N ) ≅ Hom K [ H ] ⁡ ( M , K [ H ] N ) {\displaystyle \operatorname {Hom} _{K[G]}(K[G]\otimes _{K[H]}M,N)\cong \operatorname {Hom} _{K[H]}(M,{_{K[H]}}N)} . As noted below in the section on category theory, this result applies to modules over all rings, not just modules over group algebras.

Category theory Let G be a group with a subgroup H, and let Res H G , Ind H G {\displaystyle \operatorname {Res} _{H}^{G},\operatorname {Ind} _{H}^{G}} be defined as above. For any group A and field K let Rep A K {\displaystyle {\textbf {Rep}}_{A}^{K}} denote the category of linear representations of A over K. There is a forgetful functor

Res H G : Rep G ⟶ Rep H ( V , ρ ) ⟼ Res H G ⁡ ( V , ρ ) {\displaystyle {\begin{aligned}\operatorname {Res} _{H}^{G}:{\textbf {Rep}}_{G}&\longrightarrow {\textbf {Rep}}_{H}\\(V,\rho )&\longmapsto \operatorname {Res} _{H}^{G}(V,\rho )\end{aligned}}}

This functor acts as the identity on morphisms. There is a functor going in the opposite direction:

Ind H G : Rep H ⟶ Rep G ( W , τ ) ⟼ Ind H G ⁡ ( W , τ ) {\displaystyle {\begin{aligned}\operatorname {Ind} _{H}^{G}:{\textbf {Rep}}_{H}&\longrightarrow {\textbf {Rep}}_{G}\\(W,\tau )&\longmapsto \operatorname {Ind} _{H}^{G}(W,\tau )\end{aligned}}}

These functors form an adjoint pair Ind H G ⊣ Res H G {\displaystyle \operatorname {Ind} _{H}^{G}\dashv \operatorname {Res} _{H}^{G}} . In the case of finite groups, they are actually both left- and right-adjoint to one another. This adjunction gives rise to a universal property for the induced representation (for details, see Induced representation#Properties). In the language of module theory, the corresponding adjunction is an instance of the more general relationship between restriction and extension of scalars.

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