Frobenius reciprocity

Frobenius reciprocity   (Redirected from Frobenius reciprocity theorem) Jump to navigation Jump to search 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.

Contents 1 Statement 1.1 Character theory 1.2 Module theory 1.3 Category theory 2 See also 3 Notes 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, let {displaystyle operatorname {Res} _{H}^{G}} denote the restriction of a character, or more generally, class function of G to H, and let {displaystyle operatorname {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 _{A}} on the vector space of class functions {displaystyle Ato mathbb {C} } (described in detail in the article Schur orthogonality relations). Now, for any class functions {displaystyle psi :Hto mathbb {C} } and {displaystyle varphi :Gto mathbb {C} } , the following equality holds: {displaystyle langle operatorname {Ind} _{H}^{G}psi ,varphi rangle _{G}=langle psi ,operatorname {Res} _{H}^{G}varphi rangle _{H}} .[1][2] In other words, {displaystyle operatorname {Ind} _{H}^{G}} and {displaystyle operatorname {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].[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. In the language of module theory, the induced module {displaystyle K[G]otimes _{K[H]}M} corresponds to the induced representation {displaystyle operatorname {Ind} _{H}^{G}} , whereas the restriction of scalars {displaystyle {_{K[H]}}N} corresponds to the restriction {displaystyle operatorname {Res} _{H}^{G}} . Accordingly, the statement is as follows: The following sets of module homomorphisms are in bijective correspondence: {displaystyle operatorname {Hom} _{K[G]}(K[G]otimes _{K[H]}M,N)cong operatorname {Hom} _{K[H]}(M,{_{K[H]}}N)} .[4][5] 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 {displaystyle operatorname {Res} _{H}^{G},operatorname {Ind} _{H}^{G}} be defined as above. For any group A and field K let {displaystyle {textbf {Rep}}_{A}^{K}} denote the category of linear representations of A over K. There is a forgetful functor {displaystyle {begin{aligned}operatorname {Res} _{H}^{G}:{textbf {Rep}}_{G}&longrightarrow {textbf {Rep}}_{H}\(V,rho )&longmapsto operatorname {Res} _{H}^{G}(V,rho )end{aligned}}} This functor acts as the identity on morphisms. There is a functor going in the opposite direction: {displaystyle {begin{aligned}operatorname {Ind} _{H}^{G}:{textbf {Rep}}_{H}&longrightarrow {textbf {Rep}}_{G}\(W,tau )&longmapsto operatorname {Ind} _{H}^{G}(W,tau )end{aligned}}} These functors form an adjoint pair {displaystyle operatorname {Ind} _{H}^{G}dashv operatorname {Res} _{H}^{G}} .[6][7] 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.

See also Mathematics portal See Restricted representation and Induced representation for definitions of the processes to which this theorem applies. See Representation theory of finite groups for a broad overview of the subject of group representations. See Selberg trace formula and the Arthur-Selberg trace formula for generalizations to discrete cofinite subgroups of certain locally compact groups. Notes ^ Serre 1977, p. 56. ^ Sengupta 2012, p. 246. ^ Specifically, there is an isomorphism of categories between K[G]-Mod and RepGK, as described on the pages Isomorphism of categories#Category of representations and Representation theory of finite groups#Representations, modules and the convolution algebra. ^ James, Gordon Douglas (1945–2001). Representations and characters of groups. Liebeck, M. W. (Martin W.) (2nd ed.). Cambridge, UK: Cambridge University Press. ISBN 9780521003926. OCLC 52220683. ^ Sengupta 2012, p. 245. ^ "Frobenius reciprocity on". Retrieved 2017-11-02. ^ "Frobenius reciprocity in nLab". Retrieved 2017-11-02. References Serre, Jean-Pierre (1926–1977). Linear representations of finite groups. New York: Springer-Verlag. ISBN 0387901906. OCLC 2202385. Sengupta, Ambar (2012). Representing finite groups : a semisimple introduction. New York. doi:10.1007/978-1-4614-1231-1_8. ISBN 9781461412304. OCLC 769756134. Weisstein, Eric. "Induced Representation". Retrieved 2017-11-02. Categories: Representation theory of finite groupsTheorems in representation theoryAdjoint functors

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