Frobenius reciprocity

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.

As explained in the section , 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.

As noted below in the section on category theory, this result applies to modules over all rings, not just modules over group algebras.

This functor acts as the identity on morphisms. There is a functor going in the opposite direction:

In the language of module theory, the corresponding adjunction is an instance of the more general .