Properties of representations

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Section 2.2 Properties of representations

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Objectives

You should be able to:

Recall the definition of equivalent representations.
Recall the definition of reducible and irreducible representations.
Construct the restriction of a representation of a group $G$ to a given subgroup $H \subset G\text{.}$
Construct the tensor product of two given matrix representations.

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Subsection 2.2.1 Equivalent representations

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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.

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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:

$\begin{equation*} B \mathbf{b} = B A \mathbf{a} = B A B^{-1} B\mathbf{a}. \end{equation*}$

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So, in this new basis with

$\mathbf{b}' = B \mathbf{b}$ and

$\mathbf{a}' = B \mathbf{a}\text{,}$ we get the equation

$\begin{equation*} \mathbf{b}' = A' \mathbf{a}', \end{equation*}$

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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.

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Definition 2.2.1. Equivalent representations.

We say that two representations $T: G \to GL(V)$ and $T' : G \to GL(V)$ are equivalent if, given a choice of basis for $V\text{,}$ all matrices $T(g)$ and $T'(g)$ are related by a similarity transformation:

$\begin{equation*} T'(g) = B T(g) B^{-1}, \end{equation*}$

for some invertible matrix $B\text{.}$

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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.

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

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Subsection 2.2.2 Irreducible and reducible representations

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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.

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Definition 2.2.2. Subrepresentation.

Let $T: G \to GL(V)$ be a representation of a finite group $G\text{.}$ A vector subspace $U \subset V$ is said to be $T$ -invariant if $T(g) u \in U$ for all $u \in U\text{.}$ The restriction of $T$ to the subspace $U$ is called a subrepresentation.

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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.

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Definition 2.2.3. Reducible and irreducible representations.

If a representation $T: G \to GL(V)$ has a non-trivial subrepresentation, then it is said to be reducible. It is otherwise irreducible.

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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.

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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:

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Theorem 2.2.4. Irreducible representations of a finite group.

All irreducible representations of a finite group are finite-dimensional.

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

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Subsection 2.2.3 Semisimple representations

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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:

$\begin{equation*} \begin{pmatrix} 1 \amp 0 \amp 0 \amp 0 \amp 0 \amp 0\\ 0 \amp 1 \amp 0 \amp 0 \amp 0 \amp 0\\ 0 \amp 0 \amp 1 \amp 0 \amp 0 \amp 0\\ 0 \amp 0 \amp 0 \amp 1 \amp 0 \amp 0\\ 0 \amp 0 \amp 0 \amp 0 \amp 1 \amp 0\\ 0 \amp 0 \amp 0 \amp 0 \amp 0 \amp 1 \end{pmatrix} \end{equation*}$

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to all permutations in

$S_3\text{.}$ That is certainly a 6-dimensional representation, but it's a rather boring one.

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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:

$\begin{equation*} \begin{pmatrix} T(g) \amp 0 \\ 0 \amp T(g) \end{pmatrix} \end{equation*}$

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Those are examples of direct sums of representations.

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Definition 2.2.5. Direct sum of representations.

Suppose that $T: G \to GL(V)$ and $S: G \to GL(W)$ are $m$ - and $n$ -dimensional representations of a group $G\text{.}$ We construct the $(m+n)$ -dimensional direct sum representation $T \oplus S: G \to GL(V \oplus W)$ by forming block diagonal matrices, with the $m$ -dimensional matrices $T(g)$ in the upper left block and the $n$ -dimensional matrices $S(g)$ in the bottom right block:

$\begin{equation*} T(g) \oplus S(g) = \begin{pmatrix} T(g) \amp 0 \\ 0 \amp S(g) \end{pmatrix}. \end{equation*}$

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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.

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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.

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Definition 2.2.6. Semisimple representations.

A representation is semisimple (also called completely reducible) if it is a direct sum of irreducible representations.

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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.

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Remark 2.2.7.

A consequence of the direct sum construction is that for any group, there exists an infinite number of reducible representations, which we can construct as direct sums. This is why, from a classification viewpoint, it is much more interesting to focus on irreducible representations.

However, in physics one often encounters representations that are reducible (in fact semisimple). So it is also interesting to develop methods to find the decomposition of a semisimple representation as a direct sum of irreducible representations. For instance, in the case of a quantum mechanical Hamiltonian $H$ with a symmetry group $G\text{,}$ we know that solutions will transform according to some semisimple representation $T$ of $G\text{.}$ In other words, we can decompose the space of solutions as a direct sum of invariant subspaces according to the decomposition of the semisimple representation as a direct sum of irreducible representations. As we will see later on, all states in a given invariant subspace must be eigenstates of the Hamiltonian with the same energy eigenvalue. Thus, if we are interested in the energy levels of our system, what we first want to do is decompose the space of states into invariant subspaces, that is, decompose the representation $T$ into irreducible representations of the symmetry group $G\text{.}$

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Subsection 2.2.4 Restriction to subgroups

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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

$H \subset G\text{.}$

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Example 2.2.8. Restriction to $\mathbb{Z}_2 \subset S_3$ .

As an example of this, let us look back at the two-dimensional representation $T: S_3 \to GL(V)$ that we constructed in Example 2.1.6. We reproduce it here for convenience:

\begin{align*} T(e) =\amp \begin{pmatrix} 1 \amp 0 \\ 0 \amp 1 \end{pmatrix}, \qquad T(a) = \frac{1}{2} \begin{pmatrix} 1 \amp - \sqrt{3} \\ - \sqrt{3} \amp -1 \end{pmatrix}, \qquad T(b) = \frac{1}{2} \begin{pmatrix} 1 \amp \sqrt{3} \\ \sqrt{3}\amp -1 \end{pmatrix},\\ T(c) =\amp \begin{pmatrix} -1 \amp 0 \\ 0 \amp 1 \end{pmatrix}, \qquad T(d) = \frac{1}{2} \begin{pmatrix} -1 \amp -\sqrt{3} \\ \sqrt{3} \amp -1 \end{pmatrix}, \qquad T(f) = \frac{1}{2} \begin{pmatrix} -1 \amp \sqrt{3} \\ - \sqrt{3} \amp -1 \end{pmatrix}. \end{align*}

One can show that this representation is irreducible. Let us now consider the order two subgroup $H= \{e,c \} \subset S_3\text{.}$ The restriction of $T$ to the subgroup $H \simeq \mathbb{Z}_2$ gives the two-dimensional representation:

\begin{equation*} T(e) = \begin{pmatrix} 1 \amp 0 \\ 0 \amp 1 \end{pmatrix}, \qquad T(c) = \begin{pmatrix} -1 \amp 0 \\ 0 \amp 1 \end{pmatrix}, \end{equation*}

which is now reducible (in fact semisimple), as a representation of $H\text{,}$ since it is in block diagonal form. It is the direct sum of the trivial representation of $\mathbb{Z}_2$ and its non-trivial one-dimensional representation.

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Subsection 2.2.5 Tensor product representations

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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

$T \oplus S\text{.}$

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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.

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Let us start with the simpler case where

$T$ and

$S$ are two-dimensional representations, with matrices:

$\begin{equation*} T(g) = \begin{pmatrix} T_{11}(g) \amp T_{12}(g) \\ T_{21}(g) \amp T_{22}(g) \end{pmatrix}, \qquad S(g) = \begin{pmatrix} S_{11}(g) \amp S_{12}(g) \\ S_{21}(g) \amp S_{22}(g) \end{pmatrix}. \end{equation*}$

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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):

$\begin{align*} T \otimes S(g) =\amp \begin{pmatrix} T_{11}(g) S(g) \amp T_{12}(g) S(g) \\ T_{21}(g) S(g) \amp T_{22}(g) S(g) \end{pmatrix}\\ =\amp \begin{pmatrix} T_{11}(g) S_{11}(g) \amp T_{11}(g) S_{12}(g) \amp T_{12}(g) S_{11}(g) \amp T_{12}(g) S_{12}(g) \\ T_{11}(g) S_{21}(g) \amp T_{11}(g) S_{22}(g) \amp T_{12}(g) S_{21}(g) \amp T_{12}(g) S_{22}(g) \\ T_{21}(g) S_{11}(g) \amp T_{21}(g) S_{12}(g) \amp T_{22}(g) S_{11}(g) \amp T_{22}(g) S_{12}(g) \\ T_{21}(g) S_{21}(g) \amp T_{21}(g) S_{22}(g) \amp T_{22}(g) S_{21}(g) \amp T_{22}(g) S_{22}(g) \end{pmatrix}. \end{align*}$

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It is easy to see how this generalizes to arbitrary

$m$ and

$n\text{.}$ The

$mn \times mn$ tensor product matrices are constructed as:

$\begin{equation*} T \otimes S(g) = \begin{pmatrix} T_{11}(g) S(g)\amp \ldots\amp T_{1n}(g) S(g) \\ \vdots \amp \ddots \amp \vdots \\ T_{n1}(g) S(g) \amp \ldots \amp T_{n n}(g)S(g) \end{pmatrix}, \end{equation*}$

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where of course one could expand out by writing down the matrices

$S(g)$ explicitly as above.

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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.

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Checkpoint 2.2.9.

Check that the tensor product representation constructed above is indeed a representation, i.e. that $T \otimes S : G \to GL(mn, \mathbb{C})$ is a group homomorphism.

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One important point is that even if