Orthogonality for characters

Section 2.8 Orthogonality for characters

Objectives

You should be able to:

Recall and prove the orthogonality relations for characters.
Calculate the number of irreducible representations of a finite group.
Constrain the dimensions of the irreducible representations of a finite group.
Constrain the character table of a group using orthogonality relations for characters.

Characters are very useful to, indeed, characterize representations. This becomes clear when we state the orthogonality relations in terms of characters. We know that equivalent representations have the same character. We will see that inequivalent representations have different characters, in fact orthogonal.

Subsection 2.8.1 First orthogonality theorem

Recall that the character of a representation \(T\) is a complex-valued function \(\chi^{(T)}: G \to \mathbb{C}\text{.}\) We can think of it as a vector \(\chi^{(T)}\) in a \(|G|\)-dimensional complex vector space. The standard inner product is then given by

\begin{equation*} \langle \chi^{(T)}, \chi^{(S)} \rangle = \sum_{g \in G} \left(\chi^{(T)}\right)^*(g) \chi^{(S)}(g). \end{equation*}

In fact, we can do better. We know that the character is a class function, that is, all group elements in a conjugacy class have the same character. If \(c\) is the number of conjugacy classes in the group \(G\text{,}\) then we can write

\begin{equation*} \langle \chi^{(T)}, \chi^{(S)} \rangle = \sum_{i=1}^c n_i \left(\chi^{(T)}_i\right)^* \chi^{(S)}_i, \end{equation*}

where \(n_i\) is the number of elements in the \(i\)'th conjugacy class, and we use \(\chi^{(T)}_i\) to denote the character of these elements. What this means is that we can think of the characters as vectors in a \(c\)-dimensional complex vector space. Note that we have reduced the dimension of the vector space from \(|G|\) to \(c\text{,}\) the number of conjugacy classes in \(G\text{.}\)

What we will now see is that the simple characters (characters of irreducible representations) are in fact orthogonal vectors. In particular, inequivalent irreducible representations must have different (in fact orthogonal) characters.

Theorem 2.8.1. Orthogonality of characters.

The simple characters of a finite group \(G\) (the characters of its irreducible unitary representations) are orthogonal:

\begin{equation*} \sum_{g \in G} \left(\chi^{(T)}\right)^*(g) \chi^{(S)}(g) = \sum_{i=1}^c n_i \left(\chi^{(T)}_i\right)^* \chi^{(S)}_i = |G| \delta_{T S}, \end{equation*}

where \(\delta_{T S}\) is zero if \(T\) and \(S\) are inequivalent, and one if they are equivalent. Here \(n_i\) denotes the number of elements in the \(i\)'th conjugacy class.

Proof.

This is a direct consequence of the great orthogonality theorem Theorem 2.6.1. Using the notation \(\delta_{TS}\text{,}\) we can state Theorem 2.6.1 as

\begin{equation*} \sum_{g \in G} T^\dagger(g)_{ij} S(g)_{kl} = \frac{|G|}{d} \delta_{il} \delta_{j k} \delta_{TS}. \end{equation*}

We look at the special case with \(i=j\text{,}\) \(k=l\text{,}\) and we sum over \(i\) and \(k\text{.}\) We get:

\begin{equation*} \sum_{i,k} \sum_{g \in G}T^\dagger(g)_{ii} S(g)_{kk} = \sum_{i,k} \frac{|G|}{d} \delta_{ik} \delta_{ik} \delta_{TS}. \end{equation*}

The left-hand-side is simply \(\sum_{g \in G}\left(\chi^{(T)}\right)^*(g) \chi^{(S)}(g)\text{.}\) For the right-hand-side, we notice that

\begin{equation*} \sum_{i,k}\delta_{ik} \delta_{ik} = \sum_{i,k}\delta_{ik} = \sum_{i=1}^d 1 = d. \end{equation*}

Therefore

\begin{equation*} \sum_{g \in G} \left(\chi^{(T)}\right)^*(g) \chi^{(S)}(g) = |G| \delta_{TS}. \end{equation*}

We have shown that the characters are orthogonal vectors in the vector space of class functions, which has dimensions \(c\text{,}\) the number of conjugacy classes. It thus follows immediately that:

Corollary 2.8.2. Upper bound on the number of irreps.

The number \(\rho\) of inequivalent irreducible representations of a finite group is less or equal than the number \(c\) of conjugacy classes.

Subsection 2.8.2 Second orthogonality theorem

So far we looked at the character \(\chi^{(T)}: G \to \mathbb{C}\) of an irreducible representation \(T\) as a vector in the \(c\)-dimensional space of class functions. Now we want to look at a different type of orthogonality. Let \(\rho\) be the number of irreducible representations of a finite group \(G\text{,}\) which we know is finite and bounded by \(c\text{,}\) and label the irreducible representations as \(T^{(\alpha)}\text{.}\) We denote their characters by \(\chi^{(\alpha)}\text{.}\) Now, for each conjugacy class \(i \in \{1, \ldots, c\}\text{,}\) we can think of the \(\chi^{(\alpha)}_i\) as being the components of a vector in the \(\rho\)-dimensional vector space of irreducible representations. We can also show that the characters are orthogonal from this point of view.

Theorem 2.8.3. Orthogonality of characters II.

Let \(\chi^{(\alpha)}_i\) be the character of the irreducible representation \(T^{(\alpha)}\) of a finite group \(G\text{,}\) in the \(i\)'th conjugacy class. Then:

\begin{equation*} \sum_{\alpha=1}^\rho \left(\chi_i^{(\alpha)} \right)^* \chi_j^{(\alpha)} = \frac{|G|}{n_i} \delta_{i j}. \end{equation*}

That is, the characters are orthogonal in the \(\rho\)-dimensional vector space of irreducible representations. Here \(n_i\) denotes the number of elements in the \(i\)'th conjugacy class.

We will skip the proof of this orthogonality theorem.

Subsection 2.8.3 Some important consequences

A direct consequence of this theorem is that it completely fixes the number of irreducible representations of a finite group \(G\text{:}\)

Theorem 2.8.4. The number of irreps of a finite group.

The number \(\rho\) of inequivalent irreducible representations of a finite group is equal to the number \(c\) of conjugacy classes of the group.

Proof.

We have already seen that \(\rho \leq c\text{.}\) Since the characters are orthogonal in the vector space of irreducible representations, we also get that \(c \leq \rho\text{.}\) Thus \(\rho = c\text{.}\)

Remark 2.8.5.

It follows from this statement that simple characters form an orthogonal basis on the space of class functions. Thus any class function can be written as a linear combination of simple characters.

Using orthogonality of characters we can also determine the possible dimensions of irreducible representations. Perhaps surprisingly, not only is the number of irreducible representations fixed, but their dimensions is also highly constrained. In other words, for a group of a certain size, its irreducible representations cannot be too large.

Theorem 2.8.6. Dimensions of irreducible representations.

Let \(d_\alpha\) be the dimension of the irreducible representation \(T^{(\alpha)}\) of a finite group \(G\text{.}\) Then

\begin{equation*} \sum_{\alpha=1}^c d_\alpha^2 = |G|, \end{equation*}

where the sum is over all inequivalent irreducible representations of \(G\text{.}\)

Proof.

This follows from the second orthogonality relation Theorem 2.8.3. Set \(i=j=1\text{,}\) where the first conjugacy class is identified with the conjugacy class which contains only the identity element. We get:

\begin{equation*} \sum_{\alpha=1}^c (\chi_1^{(\alpha)})^* \chi_1^{(\alpha)} = \sum_{\alpha=1}^c d_\alpha^2 = |G|. \end{equation*}

Using this result, we can prove the converse of Theorem 2.5.3, which stated that all irreducible representations of finite abelian groups are one-dimensional.

Theorem 2.8.7. Irreducible representations of abelian groups.

A finite group is abelian if and only if all its irreducible representations are one-dimensional.

Proof.

We already know that the irreducible representations of a finite abelian group are one-dimensional. We need to show that if a finite group only has one-dimensional irreducible representations, then it is abelian.

Suppose that all irreducible representations are one-dimensional. Then from Theorem 2.8.6 we get that

\begin{equation*} \sum_{\alpha=1}^c 1 = |G|, \end{equation*}

which means that the number of inequivalent irreducible representations \(c\text{,}\) which is equal to the number of conjugacy classes in \(G\text{,}\) must be equal to the order of the group \(|G|\text{.}\) In other words, all conjugacy classes of \(G\) must contain only one element, which implies that the group is abelian.

Subsection 2.8.4 Character table

The character table of a finite group is a common way of summarizing the characters of the irreducible representations of a finite group \(G\text{.}\) It is very useful in physics, for instance to determine the direct sum decomposition of reducible representations using the decomposition theorem (see Theorem 2.9.2). In fact some textbook on group theory for physics compile character tables for various groups.

The rows of a character table are labeled by the inequivalent irreducible representations of \(G\text{,}\) while the columns are labeled by the conjugacy classes of \(G\text{.}\) Since the number of inequivalent irreducible representations is equal to the number of conjugacy classes, the character table is always square. Each entry in the table gives the character of the given irreducible representation in the corresponding conjugacy class. Orthogonality relations between characters are very useful to fill in the character table of a given group, even without knowing all details of its irreducible representations. We will see how this goes in the example of \(S_3\) shortly (see Section 2.10).