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Section 2.4 Semisimplicity

In Definition 2.2.3 and Definition 2.2.6 we introduced reducible and semisimple representations. We then asked whether all representations of a given group are semisimple. That is, whether all representations are either irreducible or can be written as direct sums of irreducible representations. With our detour study of unitary representations, we are now in a position to answer this question for finite groups.

We said earlier that unitary representations are quite nice, and satisfy properties that are not shared by all representations. An example of this is the following important theorem.

Let \(T:G \to GL(V)\) be a unitary representation. If \(T\) is irreducible, then we are done. So let us assume that \(T\) is reducible. Then there exists a \(T\)-invariant subspace \(U \subset V\text{.}\) Let \(W\) be the orthogonal complement of \(U\) in \(V\text{.}\) Then \(V = U \oplus W\text{.}\) We now show that \(W\) is also \(T\)-invariant, and hence the representation \(T\) must leave both \(U\) and its orthogonal complement \(W\) invariant. This means that it must decompose in block diagonal form, i.e. it is a direct sum of representations.

To prove this, we formulate the \(T\)-invariance condition using projection operators. Recall a few facts from linear algebra. A projection \(P\) on a vector space \(V\) is an operator \(P: V \to V\) such that \(P^2 = P\text{.}\) Let \(U \subset V\) be the range of \(P\text{,}\) and \(W \subset V\) be its kernel. Then one can think of \(P\) as projecting vectors onto the subspace \(U\text{.}\) Further, it follows that \(V = U \oplus W\text{,}\) and that \(Q = I - P \) is also a projection, but now onto the subspace \(W \subset V\text{.}\) If we have a notion of inner product on \(V\text{,}\) we can define an orthogonal projection \(P\text{:}\) a projection for which the range \(U\) and the kernel \(W\) are orthogonal. A projection is orthogonal if and only if it is given by a Hermitian matrix.

Let us then introduce an orthogonal (thus Hermitian) projection matrix onto the subspace \(U \subset V\text{,}\) that is \(P:V \to U\text{,}\) with \(P^2 = P\) and \(P = P^\dagger\text{.}\) Then the statement that \(U\) is \(T\)-invariant can be written as the condition that

\begin{equation*} P T(g) P = T(g) P \end{equation*}

for all \(g \in G\text{.}\) This is because the projection operator \(P\) acts as the identity operator on the subspace \(U \subset V\text{,}\) and \(P v \in U\) for all \(v \in V\text{,}\) so the condition that \(T(g) u \in U\) for all \(u \in U\) will be satisfied if and only if \(P T(g) P v = T(g) P v\) for all \(v \in V\text{.}\)

Take the Hermitian conjugate on both sides. We get the condition \(P T^\dagger(g) P = P T^\dagger (g)\text{.}\) We know that the matrices \(T(g)\) are unitary, and hence \(T^\dagger(g) = T^{-1}(g) = T(g^{-1})\text{.}\) Thus the condition becomes \(P T(g^{-1}) P = P T(g^{-1})\text{,}\) but since this must be true for all \(g \in G\text{,}\) we can write it simply as

\begin{equation*} P T(g) P = P T(g) \end{equation*}

for all \(g \in G\text{.}\) Reorganizing, we get:

\begin{equation*} P T(g) P - P T(g) - T(g) P + T(g) = - T(g) P + T(g), \end{equation*}

that is,

\begin{equation*} (1-P)T(g)(1-P) = T(g) (1-P). \end{equation*}

But the projection matrix \(1-P\) projects on the orthogonal complement \(W\) of the subspace \(U \subset V\text{.}\) Since \(V = U \oplus W\text{,}\) this means that \(T\) must decompose in block diagonal form, i.e. it is a direct sum of representations.

We continue this process inductively. Since the representation is finite-dimensional, the process must stop at some point, and we end up with \(T\) being a direct sum of irreducible representations, that is, a semisimple representation.

Now we can use this key result to study whether representations of finite groups can be written as direct sums of irreducible representations. Indeed, in Theorem 2.3.8 we showed that all finite-dimensional representations of finite groups are equivalent to unitary representations. Therefore Theorem 2.4.1 implies the following corollary:

Thus as far as finite groups are concerned, irreducible representations really are the true building blocks, since all finite-dimensional representations can be constructed as direct sums of irreducible representations.

Remark 2.4.3.

The key result here is Theorem 2.4.1, which applies to finite-dimensional unitary representations in general, regardless of whether the group is finite or infinite. Moreover, for the case of compact groups, such as \(SU(n)\) and \(SO(n)\text{,}\) the unitarity theorem Theorem 2.3.8 also holds. Thus one can generalize Corollary 2.4.2 to compact groups: all finite-dimensional representations of compact groups are either irreducible or equivalent to direct sums of irreducible representations.

However, just like the unitarity theorem, this is not true in general for infinite groups. For instance, consider the two-dimensional representation of \((\mathbb{R},+)\) given by

\begin{equation*} \begin{pmatrix}1 \amp 0 \\ u \amp 1 \end{pmatrix}, \qquad u \in \mathbb{R} \end{equation*}

This representation is not a direct sum, hence it is not semisimple, but it is reducible. Indeed, the subspace spanned by \(\begin{pmatrix}0\\1 \end{pmatrix}\) is invariant, hence this two-dimensional representation has a one-dimensional subrepresentation, even if it is not block diagonal and cannot be brought into block diagonal form by a similarity transformation.

The outcome of the last two sections is that, as long as finite (or compact) groups are concerned, we only have to deal with direct sums of irreducible representations (up to equivalence). Phew!

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