Group Theory and Symmetry, Part IV: Great or Grand, We've Got GOT

Here, we continue our treatment of symmetry and group theory by introducing a very useful mathematical tool in group theory. It has two names in common use, but thankfully they both have the same acronym: GOT.
Sep 01, 2010

In the last installment about group theory (1), I introduced character tables as the simplest representation of the behavior of the symmetry operations of a group. All well and good, but how are they used?

Reducing a Reducible Representation

Figure 1: Set of characters with C 2v symmetry.
It is not unusual in applying group theory to come across a set of characters in a particular point group that do not belong to one of the irreducible representations. One such set of characters in a system with C 2v symmetry is shown in Figure 1, where we are using "Γ" as the general symbol for a reducible representation. How do we know that this representation is reducible? Easy, we just look at the character table for C 2v (Figure 2) and note that this set of numbers is not listed in the character table. Because it's not an irreducible representation, it must be a reducible one. (NB: Any representation in any point group will either be irreducible or reducible. If it is not, then it has been constructed incorrectly.)

We need to separate this reducible representation into its irreducible components. This is the equivalent of denoting any point in three-dimensional space as so many x, so many y, and so many z. Only in this case, our "basis" is not composed of x, y, and z; it's composed of the irreducible representations of the point group. In this case, we would write the appropriate combination as

where w, x, y, and z represent the number of each irreducible representation in the reducible one, and the symbol "⊕" means "direct sum." A direct sum is the sum of the characters of each symmetry element to generate a new set of characters. Characters from different symmetry elements are never combined. Separating a reducible representation into its irreducible counterparts is called, naturally, reducing the reducible representation.

Figure 2: Character table for the C 2v point group.
We might be able to reduce it by inspection; that is, figure it out by looking at Figure 2 and maybe do a little bit of trial and error. That may work for low-magnitude reducible representations, but it's much more difficult for larger ones. What we really need is a systematic way to reduce a representation.