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.
The theorem was originally stated in terms of character theory.
Let G be a finite group with a subgroup H, let
denote the restriction of a character, or more generally, class function of G to H, and let
denote the induced class function of a given class function on H. For any finite group A, there is an inner product
on the vector space of class functions
(described in detail in the article Schur orthogonality relations).
, the following equality holds:[1][2] In other words,
Every class function can be written as a linear combination of irreducible characters.
is a bilinear form, we can, without loss of generality, assume
to be characters of irreducible representations of
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.
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 [how?]
As this bilinear form tallies the bilinear form on the corresponding characters, the theorem follows without calculation.
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.
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.
denote the category of linear representations of A over K. There is a forgetful functor This functor acts as the identity on morphisms.
[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.