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
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
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
(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.)
Figure 1: Set of characters with C 2v symmetry.
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
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.
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.
Figure 2: Character table for the C 2v point group.