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.

Statement

Character theory

The theorem was originally stated in terms of character theory. Let G be a finite group with a subgroup H, let $\operatorname {Res} _{H}^{G}$ denote the restriction of a character, or more generally, class function of G to H, and let $\operatorname {Ind} _{H}^{G}$ denote the induced class function of a given class function on H. For any finite group A, there is an inner product $\langle -,-\rangle _{A}$ on the vector space of class functions $A\to \mathbb {C}$ (described in detail in the article Schur orthogonality relations). Now, for any class functions $\psi :H\to \mathbb {C}$ and $\varphi :G\to \mathbb {C}$ , the following equality holds:^[1]^[2]

\langle \operatorname {Ind} _{H}^{G}\psi ,\varphi \rangle _{G}=\langle \psi ,\operatorname {Res} _{H}^{G}\varphi \rangle _{H}.

In other words, $\operatorname {Ind} _{H}^{G}$ and $\operatorname {Res} _{H}^{G}$ are Hermitian adjoint.

Proof of Frobenius reciprocity for class functions
Let $\psi :H\to \mathbb {C}$ and $\varphi :G\to \mathbb {C}$ be class functions. Proof. Every class function can be written as a linear combination of irreducible characters. As $\langle \cdot ,\cdot \rangle$ is a bilinear form, we can, without loss of generality, assume $\psi$ and $\varphi$ to be characters of irreducible representations of $H$ in $W$ and of $G$ in $V,$ respectively. We define $\psi (s)=0$ for all $s\in G\setminus H.$ Then we have ${\begin{aligned}\langle {\text{Ind}}(\psi ),\varphi \rangle _{G}&={\frac {1}{\|G\|}}\sum _{t\in G}{\text{Ind}}(\psi )(t)\varphi (t^{-1})\\&={\frac {1}{\|G\|}}\sum _{t\in G}{\frac {1}{\|H\|}}\sum _{s\in G \atop s^{-1}ts\in H}\psi (s^{-1}ts)\varphi (t^{-1})\\&={\frac {1}{\|G\|}}{\frac {1}{\|H\|}}\sum _{t\in G}\sum _{s\in G}\psi (s^{-1}ts)\varphi ((s^{-1}ts)^{-1})\\&={\frac {1}{\|G\|}}{\frac {1}{\|H\|}}\sum _{t\in G}\sum _{s\in G}\psi (t)\varphi (t^{-1})\\&={\frac {1}{\|H\|}}\sum _{t\in G}\psi (t)\varphi (t^{-1})\\&={\frac {1}{\|H\|}}\sum _{t\in H}\psi (t)\varphi (t^{-1})\\&={\frac {1}{\|H\|}}\sum _{t\in H}\psi (t){\text{Res}}(\varphi )(t^{-1})\\&=\langle \psi ,{\text{Res}}(\varphi )\rangle _{H}\end{aligned}}$ In the course of this sequence of equations we used only the definition of induction on class functions and the properties of characters. $\Box$ 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: ${\text{Hom}}_{\mathbb {C} [H]}(W,U)={\text{Hom}}_{\mathbb {C} [G]}(\mathbb {C} [G]\otimes _{\mathbb {C} [H]}W,U).$ This equation is by definition equivalent to [how?] $\langle W,{\text{Res}}(U)\rangle _{H}=\langle W,U\rangle _{H}=\langle {\text{Ind}}(W),U\rangle _{G}.$ As this bilinear form tallies the bilinear form on the corresponding characters, the theorem follows without calculation. $\Box$

Proof of Frobenius reciprocity for class functions

Let $\psi :H\to \mathbb {C}$ and $\varphi :G\to \mathbb {C}$ be class functions.

Proof. Every class function can be written as a linear combination of irreducible characters. As $\langle \cdot ,\cdot \rangle$ is a bilinear form, we can, without loss of generality, assume $\psi$ and $\varphi$ to be characters of irreducible representations of $H$ in $W$ and of $G$ in $V,$ respectively. We define $\psi (s)=0$ for all $s\in G\setminus H.$ Then we have

{\begin{aligned}\langle {\text{Ind}}(\psi ),\varphi \rangle _{G}&={\frac {1}{|G|}}\sum _{t\in G}{\text{Ind}}(\psi )(t)\varphi (t^{-1})\\&={\frac {1}{|G|}}\sum _{t\in G}{\frac {1}{|H|}}\sum _{s\in G \atop s^{-1}ts\in H}\psi (s^{-1}ts)\varphi (t^{-1})\\&={\frac {1}{|G|}}{\frac {1}{|H|}}\sum _{t\in G}\sum _{s\in G}\psi (s^{-1}ts)\varphi ((s^{-1}ts)^{-1})\\&={\frac {1}{|G|}}{\frac {1}{|H|}}\sum _{t\in G}\sum _{s\in G}\psi (t)\varphi (t^{-1})\\&={\frac {1}{|H|}}\sum _{t\in G}\psi (t)\varphi (t^{-1})\\&={\frac {1}{|H|}}\sum _{t\in H}\psi (t)\varphi (t^{-1})\\&={\frac {1}{|H|}}\sum _{t\in H}\psi (t){\text{Res}}(\varphi )(t^{-1})\\&=\langle \psi ,{\text{Res}}(\varphi )\rangle _{H}\end{aligned}}

In the course of this sequence of equations we used only the definition of induction on class functions and the properties of characters. $\Box$

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:

{\text{Hom}}_{\mathbb {C} [H]}(W,U)={\text{Hom}}_{\mathbb {C} [G]}(\mathbb {C} [G]\otimes _{\mathbb {C} [H]}W,U).

This equation is by definition equivalent to [how?]

\langle W,{\text{Res}}(U)\rangle _{H}=\langle W,U\rangle _{H}=\langle {\text{Ind}}(W),U\rangle _{G}.

As this bilinear form tallies the bilinear form on the corresponding characters, the theorem follows without calculation. $\Box$

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].^[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 $K[G]\otimes _{K[H]}M$ corresponds to the induced representation $\operatorname {Ind} _{H}^{G}$ , whereas the restriction of scalars ${_{K[H]}}N$ corresponds to the restriction $\operatorname {Res} _{H}^{G}$ . Accordingly, the statement is as follows: The following sets of module homomorphisms are in bijective correspondence:

\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 $\operatorname {Res} _{H}^{G},\operatorname {Ind} _{H}^{G}$ be defined as above. For any group A and field K let ${\textbf {Rep}}_{A}^{K}$ denote the category of linear representations of A over K. There is a forgetful functor

{\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:

{\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 $\operatorname {Ind} _{H}^{G}\dashv \operatorname {Res} _{H}^{G}$ .^[6] 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

^ Serre 1977, p. 56.
^ Sengupta 2012, p. 246.
^ Specifically, there is an isomorphism of categories between K[G]-Mod and Rep_G^K, 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 in nLab". ncatlab.org. Retrieved 2017-11-02.

References

Serre, Jean-Pierre (1977). Linear representations of finite groups. New York: Springer-Verlag. ISBN 0387901906. OCLC 2202385.
Sengupta, Ambar (2012). "Induced Representations". Representing finite groups : a semisimple introduction. New York. pp. 235–248. doi:10.1007/978-1-4614-1231-1_8. ISBN 9781461412304. OCLC 769756134.{{cite book}}: CS1 maint: location missing publisher (link)
Weisstein, Eric. "Induced Representation". mathworld.wolfram.com. Retrieved 2017-11-02.