MA PH 464 - Group Theory in Physics: Lecture Notes

We start with the most boring example. For any group \(G\text{,}\) consider the mapping \(G \to GL(V)\text{,}\) with \(V = \mathbb{C}\text{,}\) given by sending \(g \mapsto 1\) for all \(g \in G\text{.}\) This is certainly a group homomorphism, but it is a rather boring one. It exists for any group \(G\text{.}\) It is called the identity representation (or trivial representation). For any non-trivial \(G\text{,}\) it is a dimension one, unfaithful, representation.

Let us look at a more interesting example. Consider the symmetric group \(S_n\text{,}\) which can be understood as the group of permutations of \(n\) objects. Let us focus on \(S_3\) for simplicity. We can think of the three objects as vectors \(v_1 = \begin{pmatrix}1 \\ 0 \\ 0 \end{pmatrix}\text{,}\) \(v_2 = \begin{pmatrix}0 \\ 1 \\ 0 \end{pmatrix}\) and \(v_3 = \begin{pmatrix}0 \\ 0 \\ 1 \end{pmatrix}\text{.}\) Then a permutation can be represented as a \(3 \times 3\) matrix acting on the vector space spanned by these vectors. That is, we can define a representation \(T: S_3 \to GL(V)\) with \(V\) the three-dimensional vector space spanned by the vectors \(\{v_1, v_2, v_3 \}\text{.}\) For instance, the cyclic permutation \(\pi = (1 2 3)\) can be represented by the matrix:

since by matrix multiplication \(T(\pi) v_1 = v_2\text{,}\) \(T(\pi) v_2 = v_3\) and \(T(\pi) v_3 = v_1\text{.}\) We could do the same thing for the other permutations in \(S_3\text{:}\) this gives a three-dimensional representation of \(S_3\text{.}\) In this way, for \(S_n\) we would construct an \(n\)-dimensional representation. This is a faithful representation, and is sometimes called the permutation representation of \(S_n\).

Using the previous example we can easily construct a faithful representation for all finite groups \(G\text{.}\) We know from Cayley's theorem that every finite group \(G\) of order \(n\) is isomorphic to a subgroup of \(S_n\text{.}\) Using the construction of the \(n\)-dimensional representation of \(S_n\) above, this gives us for free an \(n\)-dimensional representation of \(G\text{,}\) by keeping only the subset of matrices corresponding to the subgroup isomorphic to \(G\text{.}\) This is called the regular representation of \(G\text{;}\) its dimension is the order of \(G\text{.}\)

For example, one can think of the group of order three \(\mathbb{Z}_3\) as a subgroup of \(S_3\text{.}\) It is easy to see that \(\mathbb{Z}_3\) is isomorphic to the subgroup of \(S_3\) given by the cyclic permutations \(\{ e, (123), (132) \}\text{.}\) Thus, if we denote the elements of \(\mathbb{Z}_3\) by \(\{e, \omega, \omega^2\}\text{,}\) we can write a three-dimensional representation as

You can check that these matrices satisfy the multiplication rules in \(\mathbb{Z}_3\text{.}\)

Going back to \(S_n\text{,}\) we can construct other representations of different dimensions. For instance, there is a one-dimensional representation \(T: S_n \to GL(V)\) with \(V = \mathbb{C}\) given by assigning \(1\) if \(g \in S_n\) is even, and \(-1\) if it is odd. We have already seen that this is a group homomorphism. This is an unfaithful representation.

As a further example, one can construct a two-dimensional representation for \(S_3\text{.}\) The idea is to think of symmetries of an equilateral triangle. The group of symmetries is the dihedral group of order 6, that is \(D_3\text{,}\) which turns out to be isomorphic to \(S_3\text{,}\) as you can check. One can then write two-dimensional matrices that represent the symmetries of the triangle in the plane:

The quaternion group \(Q\) is defined by the set \(Q = \{1,-1,i,-i,j,-j,k,-k \}\) with multiplication rules \(i^2 = j^2 = k^2 = -1\text{,}\) \(ij = -ji = k\text{,}\) \(jk = -kj = i\text{,}\) and \(ki = -ik = j\text{.}\) One can show that the following \(2 \times 2\) complex-valued matrices (\(i\) here is the imaginary number) form a faithful representation of \(Q\text{:}\)

In Example 1.1.9, we defined the cyclic group \(\mathbb{Z}_n\) as the set of \(n\)'th roots of unity under multiplication. This is in fact a one-dimensional representation of the abstract group \(\mathbb{Z}_n\text{.}\) Indeed, writing the group elements as \(a_k = e^{2 \pi i k/ n}\) for \(k \in \{0, 1, \ldots, n-1 \}\) is in fact a faithful one-dimensional representation of \(\mathbb{Z}_n\text{.}\) But there are other one-dimensional representations. One could write \(a_k = e^{2 \pi i k \ell/n}\) for some fixed \(\ell \in \{0,1,\ldots,n-1\}\text{;}\) this gives in total \(n\) different one-dimensional representations for \(\mathbb{Z}_n\text{.}\) We recover the original one for \(\ell =1\text{.}\)

Are these all faithful? Certainly not. Setting \(\ell=0\) gives the trivial representation \(a_k = 1\) for all \(k \in \{0,1,\ldots, n-1 \}\text{,}\) which is clearly not faithful. In general, the representation will be faithful if \(\ell\) and \(n\) are coprime. Can you check that?

For instance, for \(\mathbb{Z}_3\text{,}\) if we denote the third root of unity \(\omega = e^{2 \pi i /3}\text{,}\) we get three one-dimensional representations, namely \(\{1, 1, 1\}, \{1, \omega, \omega^2 \}\) and \(\{1, \omega^2, \omega \}\text{.}\) The last two are faithful.

For \(\mathbb{Z}_4\text{,}\) if we denote the fourth root of unity \(\alpha = e^{2 \pi i /4}\text{,}\) we get four one-dimensional representations, namely \(\{1,1,1,1\}, \{1,\alpha, \alpha^2, \alpha^3 \} = \{1, i, -1, -i\}, \{1, \alpha^2, 1, \alpha^2\} = \{1, -1, 1, -1\}\) and \(\{1, \alpha^3, \alpha^2, \alpha \} = \{1, -i, -1, i\}\text{.}\) The first and third ones are not faithful, while the second and fourth ones are.

Consider now the group of real numbers \(\mathbb{R}\) under addition. To build a representation for this group, we need to rewrite addition as matrix multiplication. How can we do that?

One simple way is to use the exponential map. For any \(u \in \mathbb{R}\text{,}\) we define the one-dimensional matrix \(D(u) = e^u\text{.}\) Then \(D(u+v) = D(u) D(v)\text{,}\) hence it is a group homomorphism. In fact, it is an isomorphism, so it gives a faithful one-dimensional representation.

We can also build a two-dimensional representation easily. Consider the matrices

Then \(D(u+v) = D(u) D(v)\text{,}\) so it is a group homomorphism. Again, it is one-to-one, so it gives a two-dimensional faithful representation.