3 Induced representations 3.1 Definition 3.3 Characters

3.2 Frobenius reciprocity

Theorem 3.6.

(Frobenius reciprocity) Let $H\subset G$ be finite groups, let $V$ be a representation of $H$ , and let $W$ be a representation of $G$ induced from $V$ . Then, for any representation $U$ of $G$ , there is an isomorphism of vector spaces

\operatorname{Hom}_{G}(W,U)\xrightarrow{\sim}\operatorname{Hom}_{H}(V,U).

Proof.

We use the decomposition of $W$ as

\bigoplus_{g_{i}H\in G/H}g_{i}V_{0}.

Since $V_{0}$ is isomorphic to $V$ as an $H$ -representation, and the right hand side only depends on $V$ up to isomorphism, we may as well assume that $V$ is actually a subrepresentation of $\operatorname{Ind}_{H}^{G}V$ .

If $\tilde{\phi}:W\to U$ is a $G$ -homomorphism, then the restriction $\phi$ of $\tilde{\phi}$ to $V$ is an element of $\operatorname{Hom}_{H}(V,U)$ . We show that the map $\tilde{\phi}\mapsto\phi$ is a (linear) bijection, which will prove the theorem.

Given $\phi:V\rightarrow U$ an $H$ -homomorphism, we must show that there is a unique $G$ -homomorphism $\tilde{\phi}:W\rightarrow U$ such that $\tilde{\phi}|_{V}=\phi$ .

Let $g_{1}H,\ldots,g_{r}H$ be the left cosets as usual. We must have

\tilde{\phi}(g_{i}v)=g_{i}\phi(v)

for all $i=1,\ldots,r$ and $v\in V$ , which shows the uniqueness of $\tilde{\phi}$ . To show existence, define $\psi:W\to U$ by $\psi(g_{i}v)=g_{i}\psi(v)$ for all $i=1,\ldots r$ and $v\in V$ ; since $W=\bigoplus_{i=1}^{r}g_{i}V$ this defined a unique linear map. Since we may take $g_{1}=e$ , we see that $\psi$ extends $\phi$ . To show it $\psi$ is a $G$ -homomorphism it is enough to show that $\psi(gg_{i}v)=g\psi(g_{i}v)$ for all $v\in V$ , $i=1,\ldots,r$ , and $g\in G$ . Then $gg_{i}=g_{j}h$ for some $j\in\{1,\ldots,r\}$ and $h\in H$ and we have

	$\displaystyle\psi(gg_{i}v)$	$\displaystyle=\psi(g_{j}hw)$
		$\displaystyle=g_{j}\phi(hw)$
		$\displaystyle=g_{j}h\phi(w)$
as $\phi$ is an $H$ -homomorphism
		$\displaystyle=gg_{i}\phi(w)$
		$\displaystyle=g\psi(g_{i}w).$

This shows the existence of $\tilde{\phi}=\psi$ , and we are done. ∎

Corollary 3.7.

Any two representations induced from isomorphic representations of $H$ are isomorphic.

Proof.

If $W,W^{\prime}$ are both induced from $V$ , and $V_{0},V_{0}^{\prime}$ are their respective $H$ -subrepresentations isomorphic to $V$ , then let $\phi:V_{0}\to V_{0}^{\prime}$ be an $H$ -isomorphism. Then the construction in the previous proof provides a $G$ -homomorphism $\tilde{\phi}:W\to W^{\prime}$ such that $\tilde{\phi}(v)=\phi(v)$ if $v\in V_{0}$ . As $\tilde{\phi}$ then defines an isomorphism of vector spaces $g_{i}V_{0}\to g_{i}V_{0}^{\prime}$ for each left coset representative $g_{i}H$ and each of $W$ and $W^{\prime}$ is the direct sum of these, $\tilde{\phi}$ is an isomorphism. ∎

Corollary 3.8.

Let $(\rho,V)$ be a representation of $H$ with character $\chi$ , and let $\psi$ be any class function on $G$ . Then

\left\langle\operatorname{Ind}_{H}^{G}\chi,\psi\right\rangle_{G}=\left\langle% \chi,\operatorname{Res}^{G}_{H}\psi\right\rangle_{H}.

Proof.

This is true when $\psi$ is the character of a representation of $G$ by Frobenius reciprocity. In general, we can write $\chi$ as a linear combination of characters of representations of $G$ , and deduce the result by linearity. ∎

If $X$ and $Y$ are vector spaces with inner products, then the adjoint of a linear map $T:X\rightarrow Y$ is a linear map $T^{*}:Y\rightarrow X$ such that

\left\langle Tx,y\right\rangle=\left\langle x,T^{*}y\right\rangle

for all $x\in X$ , $y\in Y$ . Thus we say that induction is adjoint to restriction.

Exercise 3.9.

If $W_{1},W_{2}$ are representations of $H$ , then

\operatorname{Ind}_{H}^{G}(W_{1}\oplus W_{2})\cong\operatorname{Ind}_{H}^{G}W_% {1}\oplus\operatorname{Ind}_{H}^{G}W_{2}.

Exercise 3.10.

If $H\subset K\subset G$ are subgroups, and $W$ is a representation of $H$ , then

\operatorname{Ind}_{H}^{G}W\cong\operatorname{Ind}_{K}^{G}(\operatorname{Ind}_% {H}^{K}W).

Exercise 3.11.

If $V$ is a representation of $G$ and $H$ is a subgroup, then

\operatorname{Ind}_{H}^{G}\operatorname{Res}_{H}^{G}V\cong V\otimes% \operatorname{Ind}_{H}^{G}\mathbb{1}.

3.2.1 Example: S₃ to S₄

Table 3 below shows the irreducible characters $\psi_{i}$ of $S_{4}$ and $\chi_{i}$ of $S_{3}$ .

\begin{array}[]{l|rrrrr}&e&(12)&(12)(34)&(123)&(1234)\\ \hline\cr\psi_{0}&1&1&1&1&1\\ \psi_{1}&1&-1&1&1&-1\\ \psi_{2}&2&0&2&-1&-1\\ \psi_{3}&3&1&-1&0&-1\\ \psi_{4}&3&-1&-1&0&1\\ \hline\cr\hline\cr\chi_{0}&1&1&NA&1&NA\\ \chi_{1}&1&-1&NA&1&NA\\ \chi_{2}&2&0&NA&-1&NA\\ \end{array}

Table 3: Characters of

S_{3}

and

S_{4}

We regard $S_{3}$ as the subgroup of $S_{4}$ of elements that fix 4, and use Frobenius reciprocity to compute $\operatorname{Ind}_{S_{3}}^{S_{4}}\chi_{2}$ .

For each $i$ , Frobenius reciprocity implies that

\left\langle\operatorname{Ind}_{S_{3}}^{S_{4}}\chi_{2},\psi_{i}\right\rangle_{% S_{4}}=\left\langle\chi_{2},\psi_{i}|_{S_{3}}\right\rangle_{S_{3}}.

The right hand side is easily seen to be zero for $i=0,1$ and one for $i=2,3,4$ . We therefore have

\operatorname{Ind}_{S_{3}}^{S_{4}}\chi_{2}=\psi_{2}+\psi_{3}+\psi_{4}.

3.2 Frobenius reciprocity

Theorem 3.6.

Proof.

Corollary 3.7.

Proof.

Corollary 3.8.

Proof.

Exercise 3.9.

Exercise 3.10.

Exercise 3.11.

3.2.1 Example: S3 to S4

3.2.1 Example: S₃ to S₄