# Frobenius reciprocity

This fact is related to: linear representation theory

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## Contents

## Statement

### Plain statement

Let be finite groups, be a field whose characteristic does not divide the order of , and be class functions on and respectively taking values in . In other words, and are functions with the property that is constant on conjugacy classes in and is constant on conjugacy classes in .

Then, the inner product of with (in ) equals the inner product of and (in ).

In symbols:

This also applies in particular to the case when are the characters of linear representations.

### Category-theoretic statement

Let .

Consider the category of representations of (viz -modules) and the category of representations of , with a homomorphism of objects within a category, a vector space homomorphism that commutes with the -action. Then, induction defines a functor from the category of -modules to the category of -modules, and restriction defines a functor from the category of -modules to the category of -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