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). Now, for any class functions and , the following equality holds:
In other words, and are Hermitian adjoint.
Proof of Frobenius reciprocity for class functions
Let and be class functions.
Proof. Every class function can be written as a linear combination of irreducible characters. As is a bilinear form, we can, without loss of generality, assume and to be characters of irreducible representations of in and of in respectively. We define for all 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.
Alternative proof. 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
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]. 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 corresponds to the induced representation , whereas the restriction of scalars corresponds to the restriction . 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.
These functors form an adjoint pair . 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 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.
- 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.
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- 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". mathworld.wolfram.com. Retrieved 2017-11-02.