2 Character theory 2.1 Characters 2.3 Example: S₄

2.2 Orthogonality of characters

Definition 2.16.

If $\chi$ and $\psi$ are two class functions on $G$ , then their inner product is

\left\langle\chi,\psi\right\rangle=\frac{1}{|G|}\sum_{g\in G}\overline{\chi(g)% }\psi(g).

You could define this for any old functions on $G$ , but we will only use it for class functions. Note that this is a Hermitian inner product on the space of class functions of $G$ , whose dimension is the number of conjugacy classes of $G$ .

Over the next few section, we will prove the following theorem-

Theorem 2.17.

Let $G$ be a finite group with conjugacy classes $\mathcal{C}_{1},\ldots,\mathcal{C}_{r}$ . If $\chi$ is a class function, write $\chi(\mathcal{C}_{i})=\chi(g)$ for any $g\in\mathcal{C}_{i}$ .

1.

The irreducible characters are orthonormal with respect to the inner product defined above. This means that, if $\chi$ and $\psi$ are irreducible characters, then

$\frac{1}{|G|}\sum_{g\in G}\overline{\chi(g)}\psi(g)=\begin{cases}1&\text{if $% \chi=\psi$}\\ 0&\text{otherwise.}\end{cases}$

We can also see this inner product as the standard inner product of the rows of the character table with the entries weighted by $\sqrt{\frac{|\mathcal{C}_{i}|}{|G|}}$ :

$\left\langle\chi,\psi\right\rangle=\frac{1}{|G|}\sum_{i=1}^{r}|\mathcal{C}_{i}% |\overline{\chi(\mathcal{C}_{i})}\psi(\mathcal{C}_{i}).$
2.

The number of irreducible representations is equal to the number of conjugacy classes. In other words, the character table is square.
3.

The columns of the character table are orthonormal with respect to the weighted inner product from part₍₁₎. That is, if $\mathcal{C}_{i}$ , $\mathcal{C}_{j}$ are distinct conjugacy classes, then

$\sum_{\chi}\overline{\chi(\mathcal{C}_{i})}\chi(\mathcal{C}_{j})=0$

while if they are the same conjugacy class then

$\sum_{\chi}\overline{\chi(\mathcal{C}_{i})}\chi(\mathcal{C}_{i})=\frac{|G|}{|% \mathcal{C}_{i}|}.$

To complement this, we have:

Theorem 2.18.

Two irreducible representations of $G$ are isomorphic if and only if they have the same character.

In other words, each row of the character table corresponds to exactly one isomorphism class of irreducible representation.

Theorem 2.19.

1.

A representation is determined up to isomorphism by its character.
2.

If $V$ is a representation with character $\chi$ , then $V$ is irreducible if and only if $\left\langle\chi,\chi\right\rangle=1$ .

Proof.

Let the irreducible representations be $V_{1},\ldots,V_{r}$ and let $\chi_{1},\ldots,\chi_{r}$ be their characters (distinct, by Theorem 2.18 above). By Maschke’s Theorem, any representation $V$ with character $\chi$ can be written as $\bigoplus V_{i}^{a_{i}}$ for some integers $a_{i}\geq 0$ . Its character is then $\bigoplus a_{i}\chi_{i}$ . We then have, for each $i$ ,

\left\langle\chi,\chi_{i}\right\rangle=\left\langle\sum a_{i}\chi_{i},\chi_{i}% \right\rangle=a_{i}\left\langle\chi_{i},\chi_{i}\right\rangle=a_{i}

since $\left\langle\chi_{j},\chi_{i}\right\rangle=0$ for all $j\neq i$ . Thus the integers $a_{i}$ are determined by $c h i$ , the character of $V$ . But the integers $a_{i}$ determine $V$ up to isomorphism so we are done.

For the second part, note that

\left\langle\chi,\chi\right\rangle=\sum a_{i}^{2}.

Since the $a_{i}$ are nonnegative integers, this holds if and only if exactly one of the $a_{i}$ is equal to one and the rest are zero. This means that $V$ is irreducible. ∎