Frobenius reciprocity

This fact is related to: linear representation theory
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Statement

Plain statement

Let $H \le G$ be finite groups, $k$ be a field whose characteristic does not divide the order of $G$, and $f_1,f_2$ be class functions on $H$ and $G$ respectively taking values in $k$. In other words, $f_1:H \to k$ and $f_2:G \to k$ are functions with the property that $f_1$ is constant on conjugacy classes in $H$ and $f_2$ is constant on conjugacy classes in $G$.

Then, the inner product of $\operatorname{Ind}(f_1)$ with $f_2$ (in $G$) equals the inner product of $f_1$ and $\operatorname{Res}(f_2)$ (in $H$).

In symbols:

$\langle \operatorname{Ind}(f_1)_H^G,f_2 \rangle_G = \langle f_1, \operatorname{Res}(f_2)_H^G\rangle_H$

This also applies in particular to the case when $f_1,f_2$ are the characters of linear representations.

Category-theoretic statement

Let $H \le G$.

Consider the category of representations of $G$ (viz $G$-modules) and the category of representations of $H$, with a homomorphism of objects within a category, a vector space homomorphism that commutes with the $G$-action. Then, induction defines a functor from the category of $H$-modules to the category of $G$-modules, and restriction defines a functor from the category of $G$-modules to the category of $H$-modules. Frobenius reciprocity says that these two functors are adjoint functors.

Applications

Applications to relation between representation theory of group and subgroup

• Maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
• Maximum degree of irreducible representation of subgroup is less than or equal to maximum degree of irreducible representation of whole group