Frobenius Reciprocity — Mathematics Knowledge System

Overview

Frobenius reciprocity is a fundamental adjointness relationship between induction and restriction. For a subgroup H ≤ G and representations V of G and W of H, there is a natural isomorphism Hom_G(Ind^G_H W, V) ≅ Hom_H(W, Res^G_H V). In terms of characters over ℂ, this becomes the inner product identity ⟨Ind^G_H χ_W, χ_V⟩_G = ⟨χ_W, Res^G_H χ_V⟩_H. It is the primary tool for decomposing induced representations into irreducibles.

Intuition

Frobenius reciprocity says: the number of times an irreducible V appears in Ind(W) equals the number of times W appears in Res(V). Building a big representation from a small subgroup-representation (induction) and breaking a big representation down to a subgroup (restriction) are exactly adjoint operations.

Formal Definition

Definition

Let H ≤ G be finite groups (or compact groups with Haar measure), and let W be an H-representation, V a G-representation over ℂ.

\mathrm{Hom}_G\!\bigl(\mathrm{Ind}^G_H W,\, V\bigr) \cong \mathrm{Hom}_H\!\bigl(W,\, \mathrm{Res}^G_H V\bigr)

Frobenius reciprocity: categorical adjunction

\langle \mathrm{Ind}^G_H \chi_W,\; \chi_V \rangle_G = \langle \chi_W,\; \mathrm{Res}^G_H \chi_V \rangle_H

Character-theoretic form

\langle \chi_1, \chi_2 \rangle_G = \frac{1}{|G|} \sum_{g \in G} \chi_1(g)\overline{\chi_2(g)}

Inner product of class functions on G

\bigl[\mathrm{Ind}^G_H W : V_i\bigr] = \bigl[\mathrm{Res}^G_H V_i : W\bigr]

Multiplicity form: for irreducibles V_i of G and W of H

Notation

Notation	Meaning
$\langle \cdot, \cdot \rangle_G$	Inner product of class functions on G
$[V : W]$	Multiplicity of W in V
$\mathrm{Ind}^G_H$	Induction from H to G
$\mathrm{Res}^G_H$	Restriction from G to H

Properties

Ind is left adjoint to Res

The adjunction Hom_G(Ind W, V) ≅ Hom_H(W, Res V) makes Ind the left adjoint and Res the right adjoint in the adjoint pair (Ind, Res).

Multiplicity symmetry

The multiplicity of an irreducible G-module V in Ind^G_H W equals the multiplicity of W in Res^G_H V (when W is irreducible).

Theorems

Theorem 1: Frobenius Reciprocity

For finite groups H ≤ G and a class function φ on H, the induced class function satisfies ⟨Ind^G_H φ, ψ⟩_G = ⟨φ, Res^G_H ψ⟩_H for any class function ψ on G.

Theorem 2: Nakayama Reciprocity (modular)

The same adjunction holds for projective modules in modular representation theory: Hom_G(kG ⊗_{kH} W, V) ≅ Hom_H(W, V).

Worked Examples

1
By Frobenius reciprocity, [Ind(triv_H) : triv_G]_G = [triv_H : Res(triv_G)]_H = ⟨triv_H, Res triv_G⟩_H.
$[\mathrm{Ind}^{S_3}_H(\mathrm{triv}_H) : \mathrm{triv}_{S_3}]_{S_3} = \langle \mathrm{triv}_H,\; \mathrm{Res}^{S_3}_H(\mathrm{triv}_{S_3}) \rangle_H$
2
Res^{S_3}_H(triv_{S_3}) is the trivial representation of H (restriction of the trivial representation is trivial).
$\mathrm{Res}^{S_3}_H(\mathrm{triv}_{S_3}) = \mathrm{triv}_H$
3
⟨triv_H, triv_H⟩_H = 1 (since triv_H is an irreducible H-representation).
$\langle \mathrm{triv}_H, \mathrm{triv}_H \rangle_H = 1$

✓ Answer

The trivial representation of S₃ appears exactly once in Ind^{S_3}_H(triv_H).

Practice Problems

Mediumfree response

State the character-theoretic form of Frobenius reciprocity and explain why it implies that Ind and Res are adjoint functors.

Mediumfree response

For G = S₃ and H = A₃ ≅ ℤ/3ℤ, compute Ind^{S₃}_{A₃}(ω) where ω: A₃ → ℂ× is the character (123) ↦ e^{2πi/3}. Use Frobenius reciprocity to decompose into irreducibles.

Quiz

Summary

Frobenius reciprocity: ⟨Ind^G_H χ_W, χ_V⟩_G = ⟨χ_W, Res^G_H χ_V⟩_H.
Categorically: Hom_G(Ind^G_H W, V) ≅ Hom_H(W, Res^G_H V), making Ind left adjoint to Res.
The multiplicity of an irreducible G-module V in Ind(W) equals the multiplicity of W in Res(V).
It is the main computational tool for decomposing induced representations.
The identity holds for finite groups (using sum over G) and compact groups (using Haar integral).

References

BookSerre, J.-P. — Linear Representations of Finite Groups (1977), §3.3
BookIsaacs, I.M. — Character Theory of Finite Groups (1976), Theorem 5.2