Group representations

A representation of a group  is a collection of square matrices that multiply according to the multiplication table of the group.

Consider the multiplication table for the point group :

Clearly, the collections of  matrices  and  are representations of the  point group because they multiply according to the above multiplication table:

An example of a representation of the point group consisting of matrices is:

It is easy to verify that the elements of  multiply according to the multiplication table of .

These three representations of  can be summarised as follows:


Show that there are three classes for the  point group.


This is easily accomplished by inspecting the elements of and noting that group elements of the same class have the same trace.


The dimension of a representation refers to the common order of its square matrices, e.g. the dimensions of  and  are 1 and 2 respectively. A particular element of a representation is denoted by , where  refers to the representation and  refers to the corresponding symmetry operator. For example, . The matrix element of a particular element of a representation is denoted by , e.g. .

There are other representations of the  point group, e.g.  with the following elements:

If we inspect these matrices, we realise that they are of the same form, in the sense that we can group the matrix elements into submatrices that lie along the diagonals as illustrated. We call such submatrices: blocks, and the matrices containing blocks along their diagonals: block diagonal matrices.

The consequence of the submatrices in each of the  matrices being isolated by zeros is that the smaller submatrices in the multiplication of any two  matrices do not interfere with the larger submatrices. Since the collection of elements in the 1×1 blocks is the same as the collection of elements of , and the 2×2 submatrices of the bigger blocks are the same as the collection of elements of , we can conclude that the collection of the  matrices multiply according to the multiplication table of  without actually having to work out all the multiplications.

Another consequence of the way block diagonal matrices multiply with one another is that an infinite number of representations can be formed by adding elements of and  to the matrices of  to form larger matrices. For example, the addition of the elements of  to the matrices gives , with the following elements:

We call this matrix operation of adding blocks to form a larger block diagonal matrix: direct sum, which is denoted by the symbol . So, and .

With an infinite number of representations, we have to figure out which handful of representations of a group is sufficient for classifying molecules by symmetry and for analysing molecular properties. To do so, we first need to understand the difference between a reducible representation and an irreducible representation.


Next article: Reducible and irreducible representations
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