Frobenius reciprocity

This fact is related to: linear representation theory
View other facts related to linear representation theory | View terms related to linear representation theory

Statement

Plain statement

Let $H\leq G$ 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 $H\leq G$.

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