Group Theory and Symmetry, Part III: Representations and Character Tables

In the previous installments of this column, we introduced symmetry elements and point groups. Now we'll start getting more into the mathematical aspects of group theory; aspects that ultimately become useful in spectroscopy.
Apr 01, 2010

David W. Ball
In the first installment of this series (1), I introduced the five fundamental symmetry elements that we consider for physical objects. In the second installment (2), I discussed why these symmetry elements form mathematical groups. Here, we will extend the group concept in terms that can be used more mathematically — ultimately, a form that can be used in spectroscopy.

Matrix Representations — Again

In the last column, I made the point that a symmetry operation can be represented by matrix multiplication. This is a point worth making again, so here is a quick review. Consider the point (3, 4) in two-dimensional space, as shown in Figure 1a. If a reflection plane of symmetry σ were to cut the plane of the page coincident with the y axis, that point would be reflected to the point (-3, 4). This is equivalent to performing the following matrix multiplication:

Figure 1
(You may want to review the procedures for performing matrix multiplication.) The 2 x 2 matrix in the equation above is a matrix representation of σ, the reflection symmetry element.

Figure 1b shows that this concept is not limited to points, but to shapes as well: if every point on the shape in Figure 1b, represented by (x, y), were operated on by the 2 x 2 matrix element representing σ, it would transfer every point to another point on the shape. That is, the expression

holds for each and every point on the shape. Therefore, the shape has a reflection symmetry element. Furthermore, if this equation were not true for every point on the shape, the shape would not have the symmetry element σ.

As was mentioned in the last installment, matrix multiplication is extendable to three dimensions and appropriate to all five symmetry elements.