Reducibility and decomposition

Section 2.9 Reducibility and decomposition

Objectives

You should be able to:

Use the decomposition theorem to calculate the direct sum decomposition of a reducible representation.
Use the character of a representation to determine whether it is reducible or not.

In the previous section we used characters to characterize inequivalent irreducible representations of finite groups. We found that the number of inequivalent irreducible representations of \(G\) is equal to the number of conjugacy classes in \(G\text{,}\) and that the dimensions of the irreducible representations are constrained by the order \(G\text{.}\) We saw that the orthogonality relations for characters can be neatly encoded in terms of a character table.

In this section we show how characters can also be used to study whether a given representation is reducible or not: the so-called “reducibility criterion”. In the case of a reducible (semisimple) representation, we also study how characters can be used to find the decomposition of the representation as a direct sum of irreducible representations: the so-called “decomposition theorem”.

Subsection 2.9.1 A criterion for reducibility

Using orthogonality of characters we can find a simple criterion to determine whether a representation is reducible or not.

Let \(T: G \to GL(V)\) be a semisimple representation. It can be decomposed as a sum of irreducible representations

\begin{equation*} T = \bigoplus_{\alpha=1}^c m_\alpha T^{(\alpha)}, \end{equation*}

where \(m_\alpha\) is an integer that denotes the number of times that \(T^{(\alpha)}\) appears in the decomposition. Taking the trace, we get a similar relation for the characters:

\begin{equation*} \chi^{(T)}(g) = \sum_{\alpha=1}^c m_\alpha \chi^{(\alpha)}(g). \end{equation*}

Therefore, the character of a semisimple representation is a linear combination of simple characters with non-negative coefficients. (Any class function can be written as a linear combination of simple characters, but what is special here is that the coefficients are non-negative.)

A semisimple representation will then be irreducible if and only if only one of the coefficients \(m_\alpha\) is non-zero and equal to one. Our goal is to find a simple criterion in terms of the characters of a representation to determine when this is the case, without first having to calculate the explicit direct sum decomposition of the representation.

Theorem 2.9.1. Criterion for reducibility.

A representation \(T\) of a finite group \(G\) is irreducible if and only if

\begin{equation*} \sum_{i=1}^c n_i |\chi_i^{(T)}|^2 = |G|, \end{equation*}

where \(n_i\) is the number of elements in the \(i\)'th conjugacy class. It it is reducible, then

\begin{equation*} \sum_{i=1}^c n_i |\chi_i^{(T)}|^2 > |G|. \end{equation*}

Proof.

As explained above, the characters of a semisimple representation are a linear combination of simple characters with non-negative coefficients:

\begin{equation*} \chi^{(T)}_i = \sum_{\alpha=1}^k m_\alpha \chi^{(\alpha)}_i. \end{equation*}

The representation is irreducible if and only if only one of the coefficients \(m_\alpha\) is non-zero and equal to one.

Taking the complex conjugate we also have

\begin{equation*} (\chi^{(T)}_i)^* = \sum_{\alpha=1}^k m_\alpha (\chi^{(\alpha)}_i)^*. \end{equation*}

We take the product of these two equations, multiply by \(n_i\) (the number of elements in the \(i\)'th conjugacy class), and sum over \(i\text{.}\) We get:

\begin{align*} \sum_{i=1}^c n_i \chi^{(T)}_i (\chi^{(T)}_i)^* =\amp\sum_{\alpha=1}^k \sum_{\beta=1}^k m_\beta m_\alpha \left( \sum_{i=1}^c n_i \chi^{(\alpha)}_i (\chi^{(\beta)}_i)^* \right)\\ =\amp |G| \sum_{\alpha=1}^k m_\alpha^2 , \end{align*}

where we used the first orthogonality theorem Theorem 2.8.1. In particular, \(T\) is irreducible if and only if all \(m_\alpha\) are zero except one that is equal to one, so we get that it is irreducible if and only if

\begin{equation*} \sum_{i=1}^c n_i |\chi^{(T)}_i |^2 = |G|. \end{equation*}

It thus follows that it is reducible if and only if

\begin{equation*} \sum_{i=1}^c n_i |\chi_i^{(T)}|^2 > |G|. \end{equation*}

Subsection 2.9.2 The decomposition theorem

We now introduce the decomposition theorem, which gives us a way of determining the irreducible representations that appear in the direct sum decomposition of a semisimple representation. The decomposition theorem uses orthogonality of characters to determine the coefficients \(m_\alpha\) in the decomposition.

Theorem 2.9.2. The decomposition theorem.

Given a semisimple representation \(T\) of a finite group \(G\) with characters \(\chi_i^{(T)}\text{,}\) where \(i\) indexes the conjugacy classes of \(G\text{,}\) the coefficients \(m_\alpha\) in the direct sum decomposition

\begin{equation*} T = \bigoplus_{\alpha=1}^c m_\alpha T^{(\alpha)} \end{equation*}

are given by

\begin{equation*} m_\alpha = \frac{1}{|G|} \sum_{i=1}^c n_i \chi_i^{(T)} (\chi_i^{(\alpha)})^*, \end{equation*}

where \(n_i\) is the number of elements in the \(i\)'th conjugacy class, and \(\chi_i^{(\alpha)}\) are the characters of the irreducible representation \(T^{(\alpha)}\text{.}\)

Proof.

We know that

\begin{equation*} \chi^{(T)}_i = \sum_{\beta=1}^c m_\beta \chi^{(\beta)}_i. \end{equation*}

We multiply by \(n_i (\chi_i^{(\alpha)})^*\) and sum over \(i\text{.}\) We get:

\begin{align*} \sum_{i=1}^c n_i \chi^{(T)}_i (\chi_i^{(\alpha)})^* =\amp \sum_{\beta=1}^c m_\beta \left(\sum_{i=1}^c n_i \chi^{(\beta)}_i (\chi_i^{(\alpha)})^* \right)\\ =\amp |G| m_\alpha, \end{align*}

where we used the first orthogonality theorem Theorem 2.8.1. Therefore

\begin{equation*} n_\alpha = \frac{1}{|G|} \sum_{i=1}^c n_i \chi^{(T)}_i (\chi_i^{(\alpha)})^*. \end{equation*}

The decomposition theorem is useful to determine the direct sum decomposition of reducible representations. If one knows the characters of all irreducible representations of a given group, then calculating the coefficients \(m_\alpha\) of the decomposition of a reducible representation becomes a simple algebraic calculation.