Frobenius reciprocity


In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducting. 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 be a finite group with a subgroup, let denote the restriction of a character, or more generally, class function of to, and let denote the induced class function of a given class function on. For any finite group, there is an inner product on the vector space of class functions . Now, for any class functions and, the following equality holds:
In other words, and are Hermitian adjoint.
Let and be class functions.
Proof. Every class function can be written as a linear combination of irreducible characters. As is a bilinear form, we can, without loss of generality, assume and to be characters of irreducible representations of in and of in respectively.
We define for all Then we have
In the course of this sequence of equations we used only the definition of induction on class functions and the [|properties of characters].
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:
This equation is by definition equivalent to
As this bilinear form tallies the bilinear form on the corresponding characters, the theorem follows without calculation.

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 over a field is, in a certain sense, equivalent to the theory of modules over the group algebra . Therefore, there is a corresponding Frobenius reciprocity theorem for -modules.
Let be a group with subgroup, let be an -module, and let be a -module. In the language of module theory, the induced module corresponds to the induced representation, whereas the restriction of scalars corresponds to the restriction. Accordingly, the statement is as follows: The following sets of module homomorphisms are in bijective correspondence:
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 be a group with a subgroup, and let be defined as above. For any group and field let denote the category of linear representations of over. There is a forgetful functor
This functor acts as the identity on morphisms. There is a functor going in the opposite direction:
These functors form an adjoint pair. 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.
In the language of module theory, the corresponding adjunction is an instance of the more general relationship between restriction and extension of scalars.