3.14. Lecture 14

3.14.1. Induced representations

Let $H\leq G$ be a subgroup. Given a a representation $(\pi,V)$ of $H$ , we will “stitch together” this with the $G$ -action on $G/H$ to create a representation of $G$ , which we call the induced representation of $(\pi,V)$ . This is often denoted $\mathrm{Ind}_{H}^{G}(\pi,V)$ or $(\pi,V)\uparrow_{H}^{G}$ .

Our definition will depend on a choice of cosets for $G/H$ : let $r_{1},r_{2},\ldots,r_{n}\in G$ be such that

G=r_{1}H\sqcup r_{2}H\sqcup\ldots\sqcup r_{n}H;

here $n=\frac{|G|}{|H|}$ , and the “ $r$ ”s stand for representatives. Note that each $g\in G$ may be written uniquely as $g=r_{i}h$ for some representative $r_{i}$ and $h\in H$ .

We use these to describe the $G$ -action on $G/H$ as follows. Define

j(\cdot,\cdot):G\times\{1,2,\ldots,n\}\rightarrow\{1,2,\ldots,n\},\qquad h_{(% \cdot,\cdot)}:G\times\{1,2,\ldots,n\}\rightarrow H

by the formula

gr_{i}=r_{j(g,i)}h_{(g,i)}

for all $g\in G$ and $i\in\{1,\ldots,n\}$ . These functions are well-defined by uniqueness of the decomposition of $G$ into $H$ -cosets.

Lemma 3.14.1.

	$\displaystyle j(g_{1}g_{2},i)=j\big{(}g_{1},j(g_{2},i)\big{)}$
	$\displaystyle h_{(g_{1}g_{2},i)}=h_{\big{(}g_{1},j(g_{2},i)\big{)}}h_{(g_{2},i)}$

for all $g_{1},g_{2}\in G$ , $i\in\{1,\ldots,n\}$ .

Proof.

From the definition of $j(\cdot,\cdot)$ and $h_{(\cdot,\cdot)}$ , we have

g_{1}g_{2}r_{i}=r_{j(g_{1}g_{2},i)}h_{(g_{1}g_{2},i)}.

On the other hand,

	$\displaystyle g_{1}g_{2}r_{i}=$	$\displaystyle g_{1}(g_{2}r_{i})=g_{1}\big{(}r_{j(g_{2},i)}h_{(g_{2},i)}\big{)}% =\big{(}g_{1}r_{j(g_{2},i)}\big{)}h_{(g_{2},i)}$
		$\displaystyle=\big{(}r_{j\big{(}g_{1},j(g_{2},i)\big{)}}h_{\big{(}g_{1},j(g_{2% },i)\big{)}}\big{)}h_{(g_{2},i)}=r_{j\big{(}g_{1},j(g_{2},i)\big{)}}\big{(}h_{% \big{(}g_{1},j(g_{2},i)\big{)}}h_{(g_{2},i)}\big{)}.$

By uniqueness of the decomposition of $G$ into $H$ -cosets, we must then have

r_{j(g_{1}g_{2},i)}=r_{j\big{(}g_{1},j(g_{2},i)\big{)}},\quad h_{(g_{1}g_{2},i% )}=h_{\big{(}g_{1},j(g_{2},i)\big{)}}h_{(g_{2},i)}.

∎

Given a representation $(\pi,V)$ of $H$ and choice of coset representatives ${\bf r}=(r_{1},r_{2},\ldots,r_{n})$ for $G/H$ , we define a vector space

V_{{\bf r}}=r_{1}V\oplus r_{2}V\oplus\ldots\oplus r_{n}V,

with vector space operations

\alpha\left(\sum_{i=1}^{n}r_{i}{\bf v}_{i}\right)+\beta\left(\sum_{i=1}^{n}r_{% i}{\bf u}_{i}\right):=\sum_{i=1}^{n}r_{i}(\alpha{\bf v}_{i}+\beta{\bf u}_{i}).

Proposition 3.14.2.

The induced representation $\mathrm{Ind}_{H}^{G}(\pi,V):=\big{(}\mathrm{Ind}_{H}^{G}\pi,V_{{\bf r}}\big{)}$ defined by extending the map

\mathrm{Ind}_{H}^{G}\pi(g)\big{(}r_{i}{\bf v}\big{)}:=r_{j(g,i)}\pi(h_{(g,i)})% {\bf v}

linearly to all of $V_{{\bf r}}$ (for each $g\in G$ ) is a representation of $G$ on $V_{{\bf r}}$ .

Proof.

We need to show that $\mathrm{Ind}_{H}^{G}\pi(g_{1}g_{2})=\mathrm{Ind}_{H}^{G}\pi(g_{1})\mathrm{Ind}% _{H}^{G}\pi(g_{2})$ . Let $r_{i}{\bf v}\in V_{{\bf r}}$ . Then

	$\displaystyle\mathrm{Ind}_{H}^{G}\pi(g_{1})\mathrm{Ind}_{H}^{G}\pi(g_{2})r_{i}% {\bf v}$	$\displaystyle=\mathrm{Ind}_{H}^{G}\pi(g_{1})r_{j(g_{2},i)}\pi(h_{(g_{2},i)}){% \bf v}$
		$\displaystyle=r_{j\big{(}g_{1},j(g_{2},i)\big{)}}\pi\big{(}h_{\big{(}g_{1},j(g% _{2},i)\big{)}}\big{)}\pi(h_{(g_{2},i)}){\bf v}$
		$\displaystyle=r_{j(g_{1}g_{2},i)}\pi(h_{(g_{1}g_{2},i)}){\bf v}$
		$\displaystyle=\mathrm{Ind}_{H}^{G}\pi(g_{1}g_{2})r_{i}{\bf v},$

where the previous lemma was used for the second-to-last equality. ∎

Remark 3.14.3.

We will later see that making a different choice of coset representatives gives rise to an isomorphic representation.

Example 3.14.4.

Let $H=\langle r\rangle<D_{n}$ . Then $D_{n}=e\langle r\rangle\sqcup s\langle r\rangle$ , and $H=C_{n}$ . Given a representation $(\psi_{j},{\mathbb{C}})$ , $\psi_{j}(r^{k})=e^{2\pi ijk/n}$ , we consider the induced representation $\mathrm{Ind}_{H}^{D_{n}}\psi_{j}$ on $W=e{\mathbb{C}}\oplus s{\mathbb{C}}$ : for $z,w\in{\mathbb{C}}$ , we have

\mathrm{Ind}_{H}^{D_{n}}\psi_{j}(s)(ez+sw)=sz+ew,

\mathrm{Ind}_{H}^{D_{n}}\psi_{j}(r)(ez+sw)=e\psi_{j}(r)z+s\psi_{j}(r^{-1})w=e^% {2\pi ij/n}ez+e^{-2\pi ij/n}sw.

The matrices of these maps with respect to the basis $e1$ , $s1$ of $W$ are then

\mathrm{Ind}_{H}^{D_{n}}\psi_{j}(s)=\left(\begin{matrix}0&1\\ 1&0\end{matrix}\right),\qquad\mathrm{Ind}_{H}^{D_{n}}\psi_{j}(r)=\left(\begin{% matrix}e^{2\pi ij/n}&0\\ 0&e^{-2\pi ij/n}\end{matrix}\right),

hence

\mathrm{Ind}_{H}^{D_{n}}(\psi_{j},{\mathbb{C}})=(\rho_{j},{\mathbb{C}}^{2}).

Note that $\operatorname{dim}\mathrm{Ind}_{H}^{G}\pi=\frac{|G|}{|H|}\operatorname{dim}\pi$ .

Recall that since $G$ acts on $G/H$ , we have the quasi-regular representation $\big{(}\lambda,C(G/H)\big{)}$ , where $C(G/H)=\{f:G/H\rightarrow{\mathbb{C}}\}$ ,

[\lambda(g)f](g_{0}H)=f(g^{-1}g_{0}H).

Proposition 3.14.5.

$\big{(}\lambda,C(G/H)\big{)}\cong\mathrm{Ind}_{H}^{G}({\mathrm{triv}},{\mathbb% {C}})$ .

Proof.

We first define $T:C(G/H)\rightarrow{\mathbb{C}}_{{\bf r}}=r_{1}{\mathbb{C}}\oplus\ldots\oplus r% _{n}{\mathbb{C}}$ as follows:

T(f):=r_{1}f(r_{1}H)+r_{2}f(r_{2}H)+\ldots+r_{n}f(r_{n}H).

It is straightforward to verify that this is an invertible linear map, so it remains to show that it is a $G$ -homomorphism. For $g\in G$ and $f\in C(G/H)$ ,

\mathrm{Ind}_{H}^{G}{\mathrm{triv}}(g)T(f)=\mathrm{Ind}_{H}^{G}{\mathrm{triv}}% (g)\left(\sum_{i}r_{i}f(r_{i}H)\right)=\sum_{i}r_{j(g,i)}f(r_{i}H).

Now reparameterising the sum with $j=j(g,i)$ , hence $i=j(g^{-1},j(g,i))$ (cf. Lemma 3.14.1), we obtain

\mathrm{Ind}_{H}^{G}{\mathrm{triv}}(g)T(f)=\sum_{j}r_{j}f(r_{(g^{-1},j)}H)=% \sum_{j}r_{j}f(g^{-1}r_{j}H)=T(\lambda(g)f).

∎

Exercise 3.14.2 generalises this type of isomorphism to any induced representation. We conclude with the following important observation about induced representations:

\mathrm{Ind}_{H}^{G}\pi(h)r_{i}{\bf v}=r_{j(h,i)}\pi(h_{(h,i)})\neq r_{i}\pi(h% ){\bf v}!

3.14.2. Exercises

Problem 81. Let $(\pi,V)$ be a representation of $H\leq G$ . Define

C_{\pi}(G,V)=\{f:G\rightarrow V\,|\,f(gh)=\pi(h^{-1})f(g)\;\forall\;g\in G,\;h% \in H\}.

(a)

Compute $\operatorname{dim}C_{\pi}(G,V)$ .
(b)

Show that $\big{(}\lambda,C_{\pi}(G,V)\big{)}$ is a $G$ -representation, where as usual $[\lambda(g)f](g^{\prime})=f(g^{-1}g^{\prime})$ for all $g,g^{\prime}\in G$ .
(c)

Given a set of representatives ${\bf r}=r_{1},\ldots,r_{n}$ , let $T_{{\bf r}}:C_{\pi}(G,V)\rightarrow V_{{\bf r}}$ be the map

$T_{{\bf r}}(f)=r_{1}f(r_{1})+r_{2}f(r_{2})+\ldots+r_{n}f(r_{n}).$

Show that $T_{{\bf r}}$ is a $G$ -isomorphism from $\big{(}\lambda,C_{\pi}(G,V)\big{)}$ to $\mathrm{Ind}_{H}^{G}(\pi,V)$ .

Remark 3.14.6.

This shows that the induced representations of $(\pi,V)$ for different choices of representatives for $G/H$ give rise to isomorphic representations!