1.3. Lecture 3

One of the main goals of representation theory is to classify all the irreducible representations of a group $G$ (or more correctly, the isomorphism classes of irreducible representations). This entails producing a list of representations of $G$ and showing that

(i)

any irreducible representation of $G$ is isomorphic to some entry of the list, and
(ii)

there are no redundancies in the list, i.e all entries are non-isomorphic.

1.3.1. Example: dihedral groups

We list the elements of the dihedral group $D_{n}$ as

\{r^{k},sr^{k}\,|\,k=0,\ldots,n-1\}.

We aim to show that Table 1.1 gives the complete list of irreducible representations of $D_{n}$ , for $n$ odd. We leave the case of $n$ even as an exercise (there are two more 1-dimensional representations in this case).

Table 1.1. Representations of

D_{n}

	Dimension	$\rho(r)$	$\rho(s)$
$(\mathrm{Id},{\mathbb{C}})$	$1$	$1$	$1$
$(\epsilon,{\mathbb{C}})$	$1$	$1$	$-1$
$\underset{1\leq k<n/2}{\rho_{k}}$	$2$	$\left(\begin{smallmatrix}e^{\frac{2\pi ik}{n}}&0\\ 0&e^{\frac{-2\pi ik}{n}}\end{smallmatrix}\right)$	$\left(\begin{smallmatrix}0&1\\ 1&0\end{smallmatrix}\right)$

We start by noting the following:

Proposition 1.3.1.

Let $(\pi,V)$ be an irreducible representation of a finite group $G$ . Then $\operatorname{dim}\pi\leq|G|$ ; in particular, $V$ is finite-dimensional.

Proof.

Let ${\bf v}\in V$ be a non-zero vector. Consider the subspace $U=\mathrm{span}\{\pi(g){\bf v}\,|\,g\in G\}\subseteq V$ . Since $U$ is spanned by the vectors $\pi(g){\bf v}$ as $g$ runs over all elements of $G$ , we have $1\leq\operatorname{dim}U\leq|G|$ . We claim that $(\pi,U)$ is a subrepresentation of $(\pi,V)$ . Given ${\bf u}\in U$ , we may write it as a linear combination of the spanning elements, i.e.

{\bf u}=\sum_{h\in G}z_{h}\pi(h){\bf v},

where each $z_{h}\in{\mathbb{C}}$ . Given any $g\in G$ , we then have

\pi(g){\bf u}=\pi(g)\left(\sum_{h\in G}z_{h}\pi(h){\bf v}\right)=\sum_{h\in G}% z_{h}\pi(g)\pi(h){\bf v}=\sum_{h\in G}z_{h}\pi(gh){\bf v}=\sum_{h\in G}z_{g^{-% 1}h}\pi(h){\bf v}.

The vector $\sum_{h\in G}z_{g^{-1}h}\pi(h){\bf v}$ is a linear combination of vectors in $U$ , and so is also in $U$ . The subspace $U$ is therefore $G$ -invariant, and hence a subrepresentation. Since $(\pi,V)$ is irreducible and $U\neq 0$ , $V=U$ . ∎

Theorem 1.3.2.

Let $n$ be odd. Then Table 1.1 is a complete list of non-isomorphic irreducible representations of $D_{n}$ .

Proof.

Firstly, we observe that the matrices in the table satisfy the group relations for $D_{n}$ , and so do in fact define representations of $D_{n}$ .

Now let $(\pi,V)$ to be an irreducible (complex) representation of $D_{n}$ . Since $V$ is finite-dimensional, by Proposition 1.3.1, $\pi(r)$ has an eigenvector ${\bf v}\in V$ with eigenvalue $\lambda$ . Note that ${\bf v}=\pi(e){\bf v}=\pi(r^{n}){\bf v}=\pi(r)^{n}{\bf v}=\lambda^{n}{\bf v}$ ; $\lambda$ is therefore an $n$ -th root of unity.

Consider the vector ${\bf w}=\pi(s){\bf v}$ . The key calculation is:

\pi(r){\bf w}=\pi(r)\pi(s){\bf v}\\ =\pi(rs){\bf v}=\pi(sr^{-1}){\bf v}=\pi(s)\pi(r)^{-1}{\bf v}=\pi(s)(\lambda^{-% 1}{\bf v})=\lambda^{-1}{\bf w}.

We also have $\pi(s){\bf w}=\pi(s)^{2}{\bf v}={\bf v}$ and so $\mathrm{span}\{{\bf v},{\bf w}\}$ is a subrepresentation of $V$ . As $V$ is irreducible, we see that $V=\mathrm{span}\{{\bf v},{\bf w}\}$ .

Case 1:

Suppose that $\lambda\neq\lambda^{-1}$ . Then ${\bf v}$ and ${\bf w}$ are eigenvectors of $\pi(r)$ with distinct eigenvalues, and so are linearly independent. Thus $\operatorname{dim}V=2$ . In the basis ${\bf v},{\bf w}$ , the representation is

	$\displaystyle\pi(r)$	$\displaystyle=\left(\begin{matrix}\lambda&0\\ 0&\lambda^{-1}\end{matrix}\right)$
	$\displaystyle\pi(s)$	$\displaystyle=\left(\begin{matrix}0&1\\ 1&0\end{matrix}\right).$

If $\lambda=e^{2\pi ik/n}$ for some $1\leq k<n/2$ , then we get the representation $\rho_{k}$ . Otherwise, $\lambda=e^{-2\pi ik/n}$ for some $1\leq k<n/2$ and we instead take the basis ${\bf w},{\bf v}$ to get $\rho_{k}$ again.

Case 2:

Suppose that $\lambda=\lambda^{-1}$ . Then $\lambda=1$ as $n$ is odd. Since

$\pi(r)({\bf v}+{\bf w})=\pi(s)({\bf v}+{\bf w})={\bf v}+{\bf w},$

we see that ${\bf v}+{\bf w}$ spans a subrepresentation of $V$ . If ${\bf v}+{\bf w}\neq 0$ , then $V={\mathbb{C}}({\bf v}+{\bf w})$ is the trivial representation. Otherwise, $\pi(s){\bf v}={\bf w}=-{\bf v}$ , and we get the representation $\epsilon$ .

It remains to show that the representations in the table are non-isomorphic; this is left as Exercise 1.3.2 below.

∎

1.3.2. Exercises

Problem 25.

(a)

Show that if $(\pi,V)$ and $(\rho,W)$ are two isomorphic finite-dimensional representations of a group $G$ , then $\pi(g)$ and $\rho(g)$ have the same eigenvalues for all $g\in G$ .
(b)

Complete the proof of Theorem 1.3.2.

Problem 26. Find all the irreducible representations of $D_{n}$ for $n$ even.

Problem 27. Show that the irreducible representations of $S_{3}$ consist of (a) The 1-dimensional trivial representation (b) the sign representation $(\mathrm{sgn},{\mathbb{C}})$ , where $\mathrm{sgn}(\sigma)\in\{\pm 1\}$ is the sign of the permutation $\sigma$ , and (c) the representation $(\pi,W_{0})$ , where $\pi$ is the usual permutation representation of $S_{3}$ on ${\mathbb{C}}^{3}$ , and $W_{0}=\{(z_{1},z_{2},z_{3})\in{\mathbb{C}}^{3}\,|\,z_{1}+z_{2}+z_{3}=0\}$ . Hint: use the fact that we have a complete classification of the irreducible representations of $D_{n}$ .

Problem 28. Classify the irreducible representations of $Q_{8}$ . Hint: adapt the strategy we used for $D_{n}$ .