Properties of Induction

Induced representations satisfy important functorial properties and compatibility with various representation-theoretic operations.

TheoremTransitivity of Induction

Let $K \leq H \leq G$ be a tower of subgroups, and let $W$ be a representation of $K$ . Then: $\text{Ind}_K^G(W) \cong \text{Ind}_H^G(\text{Ind}_K^H(W))$

Induction is transitive: we can induce in stages.

Proof idea: A function $f: G \to W$ satisfying $f(kg) = \rho_K(k)f(g)$ for $k \in K$ is equivalent to a function $F: G \to \text{Ind}_K^H(W)$ satisfying $F(hg)(x) = \rho_H(h)(F(g)(x))$ for $h \in H$ .

TheoremInduced Character Formula

For finite groups, if $\chi_W$ is the character of $W$ (an $H$ -representation), the character of $\text{Ind}_H^G(W)$ is: $\chi_{\text{Ind}_H^G(W)}(g) = \frac{1}{|H|} \sum_{\substack{x \in G \\ xgx^{-1} \in H}} \chi_W(xgx^{-1})$

Equivalently, using class functions on $H$ extended by zero: $\chi_{\text{Ind}_H^G(W)}(g) = \sum_{i=1}^n \begin{cases} \chi_W(g_i^{-1}gg_i) & \text{if } g_i^{-1}gg_i \in H \\ 0 & \text{otherwise} \end{cases}$ where $\{g_1, \ldots, g_n\}$ are coset representatives for $H$ in $G$ .

ExampleComputing Induced Characters

$S_3$ from $S_2$ : Let $H = \langle (12) \rangle \subset S_3$ with trivial representation. The coset representatives are $\{e, (13), (23)\}$ . For $g = (123)$ :

Check $(123)$ conjugated by each representative
Only $e^{-1}(123)e = (123) \notin H$ contributes 0
Result: $\chi_{\text{Ind}}((123)) = 0$

For $g = (12)$ :

$e^{-1}(12)e = (12) \in H$ : contributes $\chi_{\text{triv}}((12)) = 1$
$(13)^{-1}(12)(13) = (23) \notin H$ : contributes 0
Result: $\chi_{\text{Ind}}((12)) = 1$

The induced character is $(3, 1, 0)$ , which is the permutation representation.

TheoremCompatibility with Direct Sums and Tensor Products

Direct sums: $\text{Ind}_H^G(W_1 \oplus W_2) \cong \text{Ind}_H^G(W_1) \oplus \text{Ind}_H^G(W_2)$
Tensor products: For $V$ a $G$ -representation and $W$ an $H$ -representation: $\text{Res}_H^G(V) \otimes W \cong \text{Res}_H^G(V \otimes_G \text{Ind}_H^G(W))$ where $\otimes_G$ denotes a relative tensor product

Remark

These properties show that induction is a functor from $\text{Rep}(H)$ to $\text{Rep}(G)$ , preserving the categorical structure. This functorial perspective is crucial in modern representation theory, leading to concepts like derived functors, Mackey theory, and categorical representation theory.

The induced character formula is computationally essential: it allows us to compute characters of induced representations without explicitly constructing the representation, using only character values on the subgroup.