3 Induced representations 3 Induced representations 3.2 Frobenius reciprocity

3.1 Definition

Suppose that $H$ is a subgroup of $G$ . Given a representation of $G$ , we may construct a representation of $H$ by restriction. What about the other direction — given a representation of $H$ , is there a natural way to construct a representation of $G$ ? The answer is yes, and this is the induced representation.

More precisely, we start with a representation $(\sigma,W)$ of $H$ and want to construct a representation $(\rho,V)$ of $G$ which contains $W$ as an $H$ -subrepresentation. Suppose we have such a thing. Then, for each $g\in G$ ,

gW=\{\rho(g)w:w\in W\}

is a subspace of $W$ (not necessarily an $H$ -subrepresentation!). We should have that, if $g=g^{\prime}h$ for some $h\in H$ , then

gW=g^{\prime}hW=g^{\prime}W

so that $g W$ only depends on the left coset $g H$ . The representation is induced if there are “no more relations”. Formally:

Definition 3.1.

Let $G$ be a finite group and let $H$ be a subgroup. Let $V$ be a representation of $H$ and let $W$ be a representation of $G$ . We say that $W$ is induced from $V$ if:

1.

There is an $H$ -subrepresentation $V_{0}$ of $W$ that is isomorphic to $V$ ; and
2.

If $g_{1},\ldots,g_{r}$ are a set of left coset representatives for $H$ in $G$ , then

$W=g_{1}V_{0}\oplus\ldots\oplus g_{r}V_{0}.$

The action of $G$ is determined by the direct sum decomposition: if $g\in G$ and $g_{i}v\in g_{i}V_{0}$ then we may write $gg_{i}=g_{j}h$ for some $j$ and some $h\in H$ , and then

g(g_{i}v)=(gg_{i})v=(g_{j}h)v=g_{j}(hv)\in g_{j}V_{0}

and $h v$ is known since we have that $V_{0}$ is isomorphic to $V$ as an $H$ -representation.

One can show that induced representations always exist (we didn’t do this in class). Here is one construction: Let

W=\{\text{$f:G\to V$ such that $f(gh)=hf(g)$ for all $h\in H,g\in g$}\}

with $G$ -action $(gf)(x)=f(g^{-1}x)$ for all $g,x\in G$ , $f\in W$ . For any $f\in W$ we can define its support

\operatorname{supp}(f)=\{g\in G:f(g)\neq 0\}

which will be a union of left cosets of $H$ . We can then take $V_{0}=\{f:\operatorname{supp}(f)\subset H\}$ ; then $g_{i}V_{0}=\{f:\operatorname{supp}(f)\subset g_{i}H\}$ and $W=\bigoplus_{i=1}^{r}g_{i}V_{0}$ .

Lemma 3.2.

If $V$ is a representation of $H$ then any two representation of $G$ induced from $V$ are isomorphic.

Proof.

We defer this proof until after the discussion of Frobenius reciprocity below. ∎

The significance of the lemma is that we can talk about ‘the’ induced representation, since it is unique up to isomorphism.

Example 3.3.

Let $(\sigma,\mathbb{C}^{2})$ be the two-dimensional representation of $G=D_{n}$ such that $\sigma(r)=\begin{pmatrix}\omega&0\\ 0&\omega^{-1}\end{pmatrix}$ and $\sigma(s)=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$ — irreducible if $\omega\neq\pm 1$ . Let $(\chi,\mathbb{C})$ be the one-dimensional representation of the subgroup $H=\left\langle r\right\rangle\cong C_{n}$ with $\chi(r)=\omega$ .

The cosets of $C_{n}$ in $D_{n}$ are $C_{n},sC_{n}$ . If $V_{0}=\left\langle e_{1}\right\rangle\subset\mathbb{C}^{2}$ then $V_{0}$ is a $C_{n}$ -subrepresentation isomorphic to $\chi$ , and $sV_{0}=\left\langle e_{2}\right\rangle$ . We clearly have $\mathbb{C}^{2}=V_{0}\oplus sV_{0}$ , so this shows

\operatorname{Ind}_{C_{n}}^{D_{n}}\chi\cong\sigma.

Example 3.4.

Suppose that $V=\mathbb{C}$ is the trivial representation of $H$ . Then the induced representation $\operatorname{Ind}_{H}^{G}(\mathbb{C})$ coincides with the permutation representation associated to the left action of $G$ on $G/H$ . Indeed, $\operatorname{Ind}_{H}^{G}(\mathbb{C})$ contains a vector $v$ fixed by $H$ (that is, a copy of the trivial representation of $H$ ), and if $g_{1}H,\ldots,g_{r}H$ are the left cosets of $H$ in $G$ then

g_{1}v,\ldots,g_{r}v

are a basis for $\operatorname{Ind}_{H}^{G}(\mathbb{C})$ . Moreover, if $g(g_{i}H)=g_{j}H$ then $gg_{i}=g_{j}h$ for some $h\in H$ and so

gg_{i}v=g_{j}hv=g_{j}v.

This shows that $G$ acts on these basis vectors ‘in the same way’ as it acts on the elements $G/H$ , which is what we have to prove.

Proposition 3.5.

Inducing multiplies the dimension by the index of the subgroup:

\dim\operatorname{Ind}_{H}^{G}V=[G:H]\dim V.

Proof.

If $g_{1}H,\ldots,g_{r}H$ are the cosets of $H$ in $G$ then

\operatorname{Ind}_{H}^{G}V=g_{1}V_{0}\oplus\ldots\oplus g_{r}V_{0}

for some copy $V_{0}$ of $V$ in $G$ . The dimension formula follows. ∎