2.8. Lecture 8

2.8.1. Irreducible characters and the character table

Definition 2.8.1.

A character is called irreducible if the representation corresponding to it is irreducible.

Recall from Proposition 2.7.9 (ii) that the character of a representation is constant on conjugacy classes. Observe also that by Proposition 2.7.9 (iii) and Theorem 1.5.12, every character may be written as an finite sum of irreducible characters. Thus, all the characters are completely determined by the values of the irreducible characters on each conjugacy class of $G$ .

We often collect this data in a character table. This has columns labelled by the conjugacy classes of $G$ , and rows labelled by the irreducible representations. The entries are the values of the characters of the irreducible representations on elements of the conjugacy class.

Example 2.8.2.

Here is the character table of $S_{3}$ . We label each column by a representative element of the conjugacy class. It is also common, as here, to write the number of elements in the conjugacy class in the second row.

\begin{array}[h]{c|rrr}\text{class:}&e&(12)&(123)\\ \text{size:}&1&3&2\\ \hline\cr{\mathrm{triv}}&1&1&1\\ \mathrm{sgn}&1&-1&1\\ \pi&2&0&-1\end{array}

Example 2.8.3.

Let $G=C_{n}$ and $\omega=e^{2\pi i/n}$ . Then we can write down the character table of $C_{n}$ ; we will do this for $n=5$ for concreteness. Since $G$ is Abelian, all conjugacy classes are singletons so we will omit the second row.

\begin{array}[h]{c|rrrrrrr}\text{Class}&e&g&g^{2}&g^{3}&g^{4}\\ \hline\cr{\mathrm{triv}}&1&1&1&1&1\\ \chi&1&\omega&\omega^{2}&\omega^{3}&\omega^{4}\\ \chi^{2}&1&\omega^{2}&\omega^{4}&\omega&\omega^{3}\\ \chi^{3}&1&\omega^{3}&\omega&\omega^{4}&\omega^{2}\\ \chi^{4}&1&\omega^{4}&\omega^{3}&\omega^{2}&\omega\end{array}

Remark 2.8.4.

It is standard practice to put the identity conjugacy class and the trivial representation in the the first column and row respectively.

From these two examples, we conjecture:

Theorem 2.8.5.

The character table is square, i.e. the number of (isomorphism classes) of irreducible representations of a finite group is equal to the number of conjugacy classes.

This will follow from a stronger theorem, which is the main result of this chapter.

2.8.2. Class functions

Definition 2.8.6.

A class function is a function $G\rightarrow{\mathbb{C}}$ that is constant on conjugacy classes. The set of all ( ${\mathbb{C}}$ -valued) class functions on a group $G$ is denoted $CF(G)$ .

Observe that $CF(G)$ is a vector space. It has dimension equal to the number of conjugacy classes of $G$ (since the indicator functions of the conjugacy classes form a basis of $CF(G)$ ), and a natural inner product

\langle f_{1},f_{2}\rangle_{G}:=\frac{1}{|G|}\sum_{g\in G}f_{1}(g)\overline{f_% {2}(g)}=\frac{1}{|G|}\sum_{\mathrm{conj.\;classes}\;{\mathcal{C}}\subseteq G}|% {\mathcal{C}}|f_{1}({\mathcal{C}})\overline{f_{2}({\mathcal{C}})}

for all $f_{1},f_{2}\in CF(G)$ .

Theorem 2.8.7.

The set $\{\chi_{\rho}\}_{\rho\in\mathrm{Irr}(G)}$ is an orthonormal basis of $CF(G)$ with respect to the inner product $\langle\cdot,\cdot\rangle_{G}$ .

We will prove this over the course of the next two lectures. First we give a corollary to it.

Corollary 2.8.8.

Let ${\mathcal{C}}$ be a conjugacy class of $G$ . We denote the column of the character table corresponding to ${\mathcal{C}}$ by $[{\mathcal{C}}]$ , when considered as an element of ${\mathbb{C}}^{|\mathrm{Irr}(G)|}$ . We have

[{\mathcal{C}}]\cdot[{\mathcal{D}}]=\begin{cases}0&\quad\mathrm{if}\;{\mathcal% {C}}\neq{\mathcal{D}},\\ \frac{|G|}{|{\mathcal{C}}|}&\quad\mathrm{if}\;{\mathcal{C}}={\mathcal{D}}.\end% {cases}

Proof.

Let $\mathbbold{1}_{{\mathcal{C}}}$ denote the indicator function of ${\mathcal{C}}$ , i.e. $\mathbbold{1}_{{\mathcal{C}}}(g)=1$ if $g\in{\mathcal{C}}$ , and zero otherwise. Since conjugacy classes are disjoint, we have

\langle\mathbbold{1}_{{\mathcal{C}}},\mathbbold{1}_{{\mathcal{D}}}\rangle_{G}=% \frac{1}{|G|}\sum_{g\in G}\mathbbold{1}_{{\mathcal{C}}}(g)\overline{\mathbbold% {1}_{{\mathcal{D}}}(g)}=\begin{cases}0&\quad\mathrm{if}\;{\mathcal{C}}\neq{% \mathcal{D}}\\ \frac{|{\mathcal{C}}|}{|G|}&\quad\mathrm{if}\;{\mathcal{C}}={\mathcal{D}}.\end% {cases}

By Theorem 2.8.7, the irreducible characters form an orthonormal basis of $CF(G)$ , hence

\mathbbold{1}_{{\mathcal{C}}}(g)=\sum_{\rho\in\mathrm{Irr}(G)}\langle% \mathbbold{1}_{{\mathcal{C}}},\chi_{\rho}\rangle_{G}\,\chi_{\rho}(g),\quad% \mathbbold{1}_{{\mathcal{D}}}(g)=\sum_{\sigma\in\mathrm{Irr}(G)}\langle% \mathbbold{1}_{{\mathcal{D}}},\chi_{\sigma}\rangle_{G}\,\chi_{\sigma}(g).

Substituting in these two expressions into the previous equality gives

\frac{1}{|G|}\sum_{g\in G}\sum_{\rho,\sigma\in\mathrm{Irr}(G)}\langle% \mathbbold{1}_{{\mathcal{C}}},\chi_{\rho}\rangle_{G}\,\chi_{\rho}(g)\overline{% \langle\mathbbold{1}_{{\mathcal{D}}},\chi_{\sigma}\rangle_{G}\,\chi_{\sigma}(g% )}=\begin{cases}0&\quad\mathrm{if}\;{\mathcal{C}}\neq{\mathcal{D}}\\ \frac{|{\mathcal{C}}|}{|G|}&\quad\mathrm{if}\;{\mathcal{C}}={\mathcal{D}}.\end% {cases}

By swapping the order of summation,

	$\displaystyle\frac{1}{\|G\|}$	$\displaystyle\sum_{g\in G}\sum_{\rho,\sigma\in\mathrm{Irr}(G)}\langle% \mathbbold{1}_{{\mathcal{C}}},\chi_{\rho}\rangle_{G}\overline{\langle% \mathbbold{1}_{{\mathcal{D}}},\chi_{\sigma}\rangle_{G}\,\chi_{\sigma}(g)}$
		$\displaystyle=\sum_{\rho,\sigma\in\mathrm{Irr}(G)}\langle\mathbbold{1}_{{% \mathcal{C}}},\chi_{\rho}\rangle_{G}\overline{\langle\mathbbold{1}_{{\mathcal{% D}}},\chi_{\sigma}\rangle_{G}}\frac{1}{\|G\|}\sum_{g\in G}\chi_{\rho}(g)% \overline{\chi_{\sigma}(g)}.$

Again using Theorem 2.8.7, we have $\sum_{g\in G}\chi_{\rho}(g)\overline{\chi_{\sigma}(g)}=0$ unless $\sigma=\rho$ , in which case it equals one. This gives

\sum_{\rho\in\mathrm{Irr}(G)}\langle\mathbbold{1}_{{\mathcal{C}}},\chi_{\rho}% \rangle_{G}\overline{\langle\mathbbold{1}_{{\mathcal{D}}},\chi_{\rho}\rangle_{% G}}=\begin{cases}0&\quad\mathrm{if}\;{\mathcal{C}}\neq{\mathcal{D}}\\ \frac{|{\mathcal{C}}|}{|G|}&\quad\mathrm{if}\;{\mathcal{C}}={\mathcal{D}},\end% {cases}

and the claim then follows from the identity $\langle\mathbbold{1}_{{\mathcal{C}}},\chi_{\rho}\rangle_{G}=\frac{|{\mathcal{C% }}|}{|G|}\chi_{\rho}({\mathcal{C}})$ and noting that

[{\mathcal{C}}]\cdot[{\mathcal{D}}]=\sum_{\rho\in\mathrm{Irr}(G)}\chi_{\rho}({% \mathcal{C}})\overline{\chi_{\rho}({\mathcal{D}})}.

. ∎

2.8.3. Exercises

Problem 47. Verify that Theorem 2.8.7 holds for the character tables of $S_{3}$ , $C_{5}$ , and $D_{4}$ .

Problem 48. Compute the character tables of the following groups (You shouldn’t need to use Theorem 2.8.7 for this):

(a)

$C_{9}$
(b)

$C_{3}\times C_{3}$
(c)

$D_{5}$
(d)

$Q_{8}$
(e)

$D_{n}$

(You will need to find the conjugacy classes and irreducible representations for each of these groups. See the exercises from previous lectures! )

Problem 49. Write the character table of a group $G$ as a matrix $T$ . Use Theorem 2.8.7 to show that

|\det(T)|=\sqrt{\prod_{\mathrm{con.\,classes}\,{\mathbb{C}}\subseteq G}\frac{|% G|}{|{\mathcal{C}}|}}.