Duality between the process of restricting and inducting in representation theory
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 G be a finite group with a subgroup H, let formula_1 denote the restriction of a character, or more generally, class function of G to H, and let formula_2 denote the induced class function of a given class function on H. For any finite group A, there is an inner product formula_3 on the vector space of class functions formula_4 (described in detail in the article Schur orthogonality relations). Now, for any class functions formula_5 and formula_6, the following equality holds:
formula_7
In other words, formula_2 and formula_1 are Hermitian adjoint.
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 G over a field K is, in a certain sense, equivalent to the theory of modules over the group algebra K[G]. 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. In the language of module theory, the induced module formula_10 corresponds to the induced representation formula_2, whereas the restriction of scalars formula_12 corresponds to the restriction formula_1. Accordingly, the statement is as follows: The following sets of module homomorphisms are in bijective correspondence:
formula_14.
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 G be a group with a subgroup H, and let formula_15 be defined as above. For any group A and field K let formula_16 denote the category of linear representations of A over K. There is a forgetful functor
formula_17
This functor acts as the identity on morphisms. There is a functor going in the opposite direction:
formula_18
These functors form an adjoint pair formula_19. 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.
Notes.
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References.
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