Character theory

The theorem was originally stated in terms of character theory. Let G be a finite group with a subgroup H, let ${\displaystyle \operatorname {Res} _{H}^{G}}$ denote the restriction of a character, or more generally, class function of G to H, and let ${\displaystyle \operatorname {Ind} _{H}^{G}}$ denote the induced class function of a given class function on H. For any finite group A, there is an inner product ${\displaystyle \langle -,-\rangle _{A}}$ on the vector space of class functions ${\displaystyle A\to \mathbb {C} }$ (described in detail in the article Schur orthogonality relations). Now, for any class functions ${\displaystyle \psi :H\to \mathbb {C} }$ and ${\displaystyle \varphi :G\to \mathbb {C} }$, the following equality holds:^[1][2]

$${\displaystyle \langle \operatorname {Ind} _{H}^{G}\psi ,\varphi \rangle _{G}=\langle \psi ,\operatorname {Res} _{H}^{G}\varphi \rangle _{H}.}$$

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

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Proof of Frobenius reciprocity for class functions

Let ${\displaystyle \psi :H\to \mathbb {C} }$ and ${\displaystyle \varphi :G\to \mathbb {C} }$ be class functions. Proof. Every class function can be written as a linear combination of irreducible characters. As ${\displaystyle \langle \cdot ,\cdot \rangle }$ is a bilinear form, we can, without loss of generality, assume ${\displaystyle \psi }$ and ${\displaystyle \varphi }$ to be characters of irreducible representations of ${\displaystyle H}$ in ${\displaystyle W}$ and of ${\displaystyle G}$ in ${\displaystyle V,}$ respectively. We define ${\displaystyle \psi (s)=0}$ for all ${\displaystyle s\in G\setminus H.}$ Then we have $${\displaystyle {\begin{aligned}\langle {\text{Ind}}(\psi ),\varphi \rangle _{G}&={\frac {1}{|G|}}\sum _{t\in G}{\text{Ind}}(\psi )(t)\varphi (t^{-1})\\&={\frac {1}{|G|}}\sum _{t\in G}{\frac {1}{|H|}}\sum _{s\in G \atop s^{-1}ts\in H}\psi (s^{-1}ts)\varphi (t^{-1})\\&={\frac {1}{|G|}}{\frac {1}{|H|}}\sum _{t\in G}\sum _{s\in G}\psi (s^{-1}ts)\varphi ((s^{-1}ts)^{-1})\\&={\frac {1}{|G|}}{\frac {1}{|H|}}\sum _{t\in G}\sum _{s\in G}\psi (t)\varphi (t^{-1})\\&={\frac {1}{|H|}}\sum _{t\in G}\psi (t)\varphi (t^{-1})\\&={\frac {1}{|H|}}\sum _{t\in H}\psi (t)\varphi (t^{-1})\\&={\frac {1}{|H|}}\sum _{t\in H}\psi (t){\text{Res}}(\varphi )(t^{-1})\\&=\langle \psi ,{\text{Res}}(\varphi )\rangle _{H}\end{aligned}}}$$ In the course of this sequence of equations we used only the definition of induction on class functions and the properties of characters. ${\displaystyle \Box }$ Alternative proof. In terms of the group algebra, i.e. by the alternative description of the induced representation, the Frobenius reciprocity is a special case of a general equation for a change of rings: ${\displaystyle {\text{Hom}}_{\mathbb {C} [H]}(W,U)={\text{Hom}}_{\mathbb {C} [G]}(\mathbb {C} [G]\otimes _{\mathbb {C} [H]}W,U).}$ This equation is by definition equivalent to [how?] ${\displaystyle \langle W,{\text{Res}}(U)\rangle _{H}=\langle W,U\rangle _{H}=\langle {\text{Ind}}(W),U\rangle _{G}.}$ As this bilinear form tallies the bilinear form on the corresponding characters, the theorem follows without calculation. ${\displaystyle \Box }$

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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].^[3] 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 ${\displaystyle K[G]\otimes _{K[H]}M}$ corresponds to the induced representation ${\displaystyle \operatorname {Ind} _{H}^{G}}$, whereas the restriction of scalars ${\displaystyle {_{K[H]}}N}$ corresponds to the restriction ${\displaystyle \operatorname {Res} _{H}^{G}}$. Accordingly, the statement is as follows: The following sets of module homomorphisms are in bijective correspondence:

$${\displaystyle \operatorname {Hom} _{K[G]}(K[G]\otimes _{K[H]}M,N)\cong \operatorname {Hom} _{K[H]}(M,{_{K[H]}}N)}$$

.^[4][5]

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

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

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

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

These functors form an adjoint pair ${\displaystyle \operatorname {Ind} _{H}^{G}\dashv \operatorname {Res} _{H}^{G}}$.^[6] 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.