MA PH 464 - Group Theory in Physics: Lecture Notes

Subsection 2.2.1 Equivalent representations

We defined representations as group homomorphisms between abstract groups \(G\) and operators on a vector space \(V\text{.}\) After choosing a basis, we can think of those as invertible matrices. So if we think of representations in terms of matrices, we want to make sure that we only study “different” representations. What we mean here is that we do not want to distinguish between matrix representations that come from the same operators, but in different choices of bases.

Recall from linear algebra that if you are given a vector equation \(\mathbf{b} = A \mathbf{a}\text{,}\) and that you want to rewrite this equation in a different basis which is obtained by applying the transformation \(B\text{,}\) we get:

So, in this new basis with \(\mathbf{b}' = B \mathbf{b}\) and \(\mathbf{a}' = B \mathbf{a}\text{,}\) we get the equation

with \(A' = B A B^{-1}\text{.}\) This is what is called a similarity transformation. So we want to say that two representations are the same if they are related by a similarity transformation.

In other words, one can think of equivalent representations as mapping the group elements to the same operators on \(V\text{,}\) but in a different choice of basis.

Note that any two distinct one-dimensional representations cannot be equivalent, since complex numbers commute.

Subsection 2.2.2 Irreducible and reducible representations

Our goal is to understand and classify representations of groups in general. But not all representations are equally interesting. We want to define a notion of representations that are “minimal”, in the sense that they are somehow the building block for constructing all representations of a given group. “Minimal” representations should be representations that have no “sub-structure”. Let us first define the concept of subrepresentation.

Thus, if a representation has a subrepresentation, this means that there exists a subspace \(U \subset V\) that is closed under the action of the operators in the image of the representation. So it has some sub-structure. We want our “minimal” representations, our building block, to have no such sub-structure.

Irreducible representations are the minimal representations that we are looking for. They are the true building blocks of representation theory. In fact, a central goal of representation theory is to establish criteria to determine whether a given representation is irreducible or not, and to classify all possible irreducible representations of a given group.

By the way in the beginning of this section we said that we would focus on finite-dimensional representations, but that representations could also be defined for infinite-dimensional vector spaces. However, there is a theorem that says that:

So as far as irreducible representations of finite groups are concerned, it is sufficient to consider only finite-dimensional representations.

Subsection 2.2.3 Semisimple representations

Let us now go back to the study of general representations. There is an easy way to construct higher-dimensional representations from lower-dimensional ones. Consider the following example. Suppose that I ask you to give me a 6-dimensional representation of, say, \(S_3\text{.}\) Well, you could say, sure, I'll start with the trivial representation that assigns 1 to all permutations in \(S_3\text{,}\) and I'll assign the \(6 \times 6\) matrix:

to all permutations in \(S_3\text{.}\) That is certainly a 6-dimensional representation, but it's a rather boring one.

You could be a little fancier, and start instead with the 3-dimensional representation of \(S_3\) that we constructed in Example 2.1.4. If we let \(T(g)\) be the \(3 \times 3\) matrices associated to the group elements \(g \in S_3\) in this representation, you could construct a 6-dimensional representation by constructing the block diagonal matrices:

Those are examples of direct sums of representations.

Representations that are constructed as direct sums are certainly not minimal. More precisely, with our definition of minimality above, we can see that all direct sum representations are reducible. Indeed, consider a direct sum representation \(T \oplus S: G \to GL(V \oplus W)\text{.}\) Then the subspaces \(V \subset V \oplus W\) and \(W \subset V \oplus W\) are certainly \(T\)-invariant, as all matrices \(T(g)\) are block diagonal, and hence do not mix elements of \(V\) with elements of \(W\text{.}\) Thus the restrictions of \(T \oplus S\) to the subspaces \(V\) and \(W\) are subrepresentations, and hence \(T \oplus S\) is reducible.

So we have seen that all direct sum representations are reducible. But this raises an interesting question: is the converse true? Can all reducible representations be written as direct sums of irreducible representations? In other words, can all reducible representations be written in block diagonal form? We will see shortly that the answer is yes for finite groups, but no in general. Thus it makes sense to distinguish between representations that can or cannot be written as direct sums of irreducible representations.

With this definition, the question becomes: are all representations of a given group semisimple? (i.e. either irreducible or direct sums of irreducible representations.) We will come back to this shortly.

Subsection 2.2.4 Restriction to subgroups

Reducible representations also arise naturally when one considers a subgroup \(H\) of a group \(G\text{.}\) A representation of \(G\) clearly gives a representation of \(H\) if one keeps only the matrices corresponding to the elements of the subgroup \(H \subset G\text{.}\) However, when restricted to a subgroup \(H \subset G\text{,}\) a representation that is irreducible for \(G\) may become reducible for \(H\text{.}\) Indeed, the induced representation for \(H\) may have a subrepresentation (suppose for example that all matrices corresponding to the subgroup are block diagonal), while the original representation for \(G\) may not (suppose that the matrices corresponding to elements of \(G\) not in \(H\) are not block diagonal). A central theme in physics is to find how a given irreducible representation of \(G\) decomposes into a direct sum of irreducible representations of \(H\) upon restriction to a subgroup \(H \subset G\text{.}\)

Subsection 2.2.5 Tensor product representations

We have already seen one way of constructing higher-dimensional representations from lower-dimensional ones. Given two representations \(T: G \to GL(V)\) and \(S: G \to GL(V')\) of the same group, we have already seen how to construct a \((m+n)\)-dimensional representation on \(V \oplus V'\) by adjoining the matrices into \((m+n) \times (m+n)\) block diagonal matrices. We called this construction the direct sum, and denoted it by \(T \oplus S\text{.}\)

There is another way to construct higher-dimensional representations from lower-dimensional ones. We can also take the “product” of representations to construct a new \(m n\)-dimensional representation: this is called the tensor product representation, and we denote it by \(T \otimes S\text{.}\) What we are really doing here is define a new representation \(T \otimes S\) that is an operator on the tensor product of the underlying vector spaces. If you know what this means, great, otherwise I will simply be pedestrian here and construct the matrix representation \(T \otimes S\) explicitly.

Let us start with the simpler case where \(T\) and \(S\) are two-dimensional representations, with matrices:

Then the \(4 \times 4\) matrices \(T \otimes S(g)\) are constructed by taking the product of all matrix entries (this is not a matrix product of course):

It is easy to see how this generalizes to arbitrary \(m\) and \(n\text{.}\) The \(mn \times mn\) tensor product matrices are constructed as:

where of course one could expand out by writing down the matrices \(S(g)\) explicitly as above.

We call the corresponding representation the tensor product representation. Such tensor product representations occur frequently in physics, for instance when taking the product of wave-functions in quantum mechanics.

One important point is that even if \(T\) and \(S\) are irreducible representations of \(G\text{,}\) the tensor product \(T \otimes S\) may be reducible. In fact often in particle physics one needs to determine how the tensor product of two irreducible representations decomposes into a sum of irreducible representations.