2 Character theory 2 Character theory 2.2 Orthogonality of characters

2.1 Characters

Throughout this section, $G$ is a finite group and $V$ is a finite dimensional complex vector space.

Definition 2.1.

Let $(\rho,V)$ be a finite-dimensional complex representation of $G$ . The character of $(\rho,V)$ is the function $\chi_{\rho}:G\rightarrow\mathbb{C}$ defined by

\chi_{\rho}(g)=\operatorname{tr}(\rho(g)).

We might also write $\chi_{V}$ instead of $\chi_{\rho}$ .

Remark 2.2.

It is not totally obvious that the process ‘Start with an element $\rho(g)\in\operatorname{GL}(V)$ , choose a basis for $V$ , write $\rho(g)$ as a matrix using that basis, and take the trace’ gives a number that is independent of the choice of basis. One way to see this is to use the change of basis formula and the fact that $\operatorname{tr}PMP^{-1}=\operatorname{tr}M$ .

The same argument shows:

Lemma 2.3.

Isomorphic representations have the same character.

It seems that we throw away a lot of information when we pass to the character; the remarkable thing is that, in fact, the character completely determines the representation. Moreover, there is a lot of structure to the characters and it is often possible to find all of the characters of a group, even when it is not clear how to construct the representations!

Example 2.4.

If $\chi:G\rightarrow\mathbb{C}^{\times}$ is a one-dimensional representation, then it is its own character (the trace of a scalar is just itself).

Example 2.5.

Let $\rho$ be the irreducible two-dimensional representation of $S_{3}$ and let $\chi$ be its character. Then

\chi(e)=2\qquad\chi((12))=\chi((23))=\chi((31))=0\qquad\chi((123))=\chi((132))% =-1.

Lemma 2.6.

If $(\rho,V)$ is a representation of $G$ , then $\chi_{\rho}(e)=\dim\rho$ .

Proof.

Exercise. ∎

We call $\dim\rho$ the degree (or the dimension) of the character $\chi_{\rho}$ .

Lemma 2.7.

If $V$ and $W$ are representations with characters $\chi$ and $\psi$ , then $V\oplus W$ has character $\chi+\psi$ .

Proof.

Use block matrices. ∎

Lemma 2.8.

If $\chi$ is the character of a representation $V$ of $G$ , then

\chi(g^{-1})=\overline{\chi(g)}.

Proof.

As $g$ has finite order, say $m$ , we can find a basis such that $\rho(g)$ is diagonal with eigenvalues $\lambda_{1},\ldots,\lambda_{n}$ , and the $\lambda_{i}$ are $m$th roots of unity. Then $\rho(g)^{-1}$ is diagonal with eigenvalues $\lambda^{-1}_{1},\ldots,\lambda^{-1}_{n}$ , which are $\overline{\lambda}_{1},\ldots,\overline{\lambda}_{n}$ , giving the result. ∎

Remark 2.9.

It follows that, if $g$ is conjugate to $g^{-1}$ , then $\chi(g)$ is real for every character $\chi$ .

Lemma 2.10.

Let $\chi$ be the character of a representation $\rho$ of $G$ . If $g$ and $h$ are in the same conjugacy class of $G$ , then

\chi(g)=\chi(h).

Proof.

Let $h=xgx^{-1}$ for some $x\in G$ . Then

\chi(h)=\operatorname{tr}\rho(xgx^{-1})=\operatorname{tr}\left(\rho(x)\rho(g)% \rho(x)^{-1}\right)=\operatorname{tr}\rho(g)=\chi(g)

since conjugate matrices have the same trace. ∎

Definition 2.11.

A class function is a function $G\rightarrow\mathbb{C}$ that is constant on conjugacy classes.

The previous lemma then says that the character of a representation is a class function. We often organise the information into a character table. This has columns labeled by the conjugacy classes of $G$ , and rows labeled by the irreducible representations. The entries are the values of the characters of the irreducible representations on elements of the conjugacy class.

We usually write $\mathbbm{1}$ for the character of the trivial representation.

Example 2.12.

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\mathbf{1}&1&1&1\\ \epsilon&1&-1&1\\ \rho&2&0&-1\end{array}

Example 2.13.

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

\begin{array}[h]{c|rrrrrrr}\text{Class}&e&g&g^{2}&g^{3}&g^{4}\\ \hline\cr\mathbbm{1}&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}

Exercise 2.14.

Let $G$ act on a finite set $X$ , and let $\rho$ be the permutation representation. Then its character $\chi$ is given by

\chi(g)=|\mathrm{Fix}(g)|,

the number of fixed points of $g$ .

Exercise 2.15.

What is the character of the regular representation?