# Frobenius reciprocity

Frobenius reciprocity (Redirected from Frobenius reciprocity theorem) Ir para a navegação Ir para a pesquisa Em matemática, 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.

Conteúdo 1 Declaração 1.1 Character theory 1.2 Module theory 1.3 Category theory 2 Veja também 3 Notas 4 References Statement Character theory The theorem was originally stated in terms of character theory. Let G be a finite group with a subgroup H, deixar {nome do operador de estilo de exibição {Res} _{H}^{G}} denote the restriction of a character, or more generally, class function of G to H, e deixar {nome do operador de estilo de exibição {Ind} _{H}^{G}} denote the induced class function of a given class function on H. For any finite group A, there is an inner product {displaystyle langle -,-rangle _{UMA}} on the vector space of class functions {displaystyle Ato mathbb {C} } (described in detail in the article Schur orthogonality relations). Agora, for any class functions {estilo de exibição psi :Hto mathbb {C} } e {estilo de exibição varphi :Gto mathbb {C} } , the following equality holds: {displaystyle langle operatorname {Ind} _{H}^{G}psi ,varphi rangle _{G}=langle psi ,nome do operador {Res} _{H}^{G}varphi rangle _{H}} . Em outras palavras, {nome do operador de estilo de exibição {Ind} _{H}^{G}} e {nome do operador de estilo de exibição {Res} _{H}^{G}} are Hermitian adjoint.

show Proof of Frobenius reciprocity for class functions Module theory See also: Change of rings § Relation between the extension of scalars and the restriction of scalars 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]. Portanto, there is a corresponding Frobenius reciprocity theorem for K[G]-módulos.

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 {estilo de exibição K[G]otimes _{K[H]}M} corresponds to the induced representation {nome do operador de estilo de exibição {Ind} _{H}^{G}} , whereas the restriction of scalars {estilo de exibição {_{K[H]}}N} corresponds to the restriction {nome do operador de estilo de exibição {Res} _{H}^{G}} . De acordo, the statement is as follows: The following sets of module homomorphisms are in bijective correspondence: {nome do operador de estilo de exibição {Hom} _{K[G]}(K[G]otimes _{K[H]}M,N)cong operatorname {Hom} _{K[H]}(M,{_{K[H]}}N)} . As noted below in the section on category theory, this result applies to modules over all rings, not just modules over group algebras.