Frobenius Reciprocity

Frobenius Reciprocity is one of the most important theorems in representation theory, establishing an adjunction between induction and restriction functors.

TheoremFrobenius Reciprocity

Let $H \leq G$ be a subgroup, $V$ a representation of $G$ , and $W$ a representation of $H$ . Then there is a natural isomorphism: $\text{Hom}_G(\text{Ind}_H^G(W), V) \cong \text{Hom}_H(W, \text{Res}_H^G(V))$

In other words, induction from $H$ to $G$ is left adjoint to restriction from $G$ to $H$ .

This fundamental result states that giving a $G$ -homomorphism from an induced representation to $V$ is equivalent to giving an $H$ -homomorphism from $W$ to the restriction of $V$ .

ProofSketch

Direction $\Rightarrow$ : Given $\phi: \text{Ind}_H^G(W) \to V$ (a $G$ -map), define $\psi: W \to \text{Res}_H^G(V)$ by: $\psi(w) = \phi(f_w)$ where $f_w: G \to W$ is the function supported on the coset $H$ with $f_w(e) = w$ .

Direction $\Leftarrow$ : Given $\psi: W \to V$ (an $H$ -map), define $\phi: \text{Ind}_H^G(W) \to V$ by: $\phi(f) = \sum_{i=1}^n \rho_V(g_i) \psi(f(g_i))$ where $\{g_1, \ldots, g_n\}$ are coset representatives. One verifies this is well-defined and $G$ -equivariant.

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TheoremCharacter Form of Frobenius Reciprocity

For finite groups over $\mathbb{C}$ , Frobenius Reciprocity translates to characters: $\langle \chi_{\text{Ind}_H^G(W)}, \chi_V \rangle_G = \langle \chi_W, \text{Res}_H^G(\chi_V) \rangle_H$

The multiplicity of an irreducible $V$ in $\text{Ind}_H^G(W)$ equals the multiplicity of (the restriction of) $V$ in $W$ when viewed as an $H$ -representation.

ExampleApplications

Determining irreducibility: An induced representation $\text{Ind}_H^G(W)$ is irreducible if and only if:

$W$ is irreducible as an $H$ -representation
For all proper subgroups $K$ containing $H$ , the restriction $\text{Res}_K^G(\text{Ind}_H^G(W))$ does not contain any $K$ -invariant vectors except those from $H$

Computing multiplicities: To find the multiplicity of irreducible $V$ in $\text{Ind}_H^G(W)$ : $m_V = \langle \chi_{\text{Ind}_H^G(W)}, \chi_V \rangle_G = \langle \chi_W, \chi_V|_H \rangle_H$

Remark

Frobenius Reciprocity is the representation-theoretic analogue of:

Adjoint functors in category theory: $\text{Ind}_H^G \dashv \text{Res}_H^G$
Tensor-hom adjunction: $\text{Hom}(V \otimes W, U) \cong \text{Hom}(V, \text{Hom}(W, U))$
Pullback-pushforward in geometry: For a map $f: X \to Y$ , the functors $f_!$ and $f^*$ are adjoint

This adjunction is fundamental to:

Mackey theory: Understanding double coset formulas
Clifford theory: Representations of extensions
Harish-Chandra induction: Representations of reductive groups

Frobenius Reciprocity makes induction computable and connects local information (subgroup representations) to global structure (full group representations).