Reduction in Symmetry and The Splitting of Energy Levels

To tackle this problem, we first need to know how a given representation F is reduced to its irreducible representations T) in other words, to determine the coefficients a, in the equation F= Sat Fi. Although this is a key problem in group theory, here we only explain how to perform this reduction without entering into formal details, which can easily be found in specialized textbooks. 

It can be shown that the number of times a, that the irreducible representation /] appears in F is given by 

Let us consider the character table of the C4 group and the representation F displayed below (see Table 7.3). F is, reducible, because its character does not coincide with any one of the irreducible representations of the Cav group. According to Equation (7.4) and the character table (Table 7.3), we can write 

Consequently, the other irreducible representations, A2, Si, S2, and E, also appear only once. Therefore, the reduction equation can be written as follows  

The same result can sometimes be easily inferred from direct inspection of the character table. In fact, we can directly observe from Table 7.3 that X + x + X + + X = X - Because there is only one possible de- 

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