Characters and Character Tables

Thus far, we have defined representations and shown how they may be generated from basis functions. We have distinguished between reducible and irreducible representations and have indicated that there is an unlimited number of equivalent representations corresponding to any given two- or higher-dimensional representation. An example of a pair of equivalent, reducible, two-dimensional representations, derived in Section 13-11, is given in Table 13-17. Equivalent representations are related through unitary transformations, which are a special kind of similarity transformation (see Chapter 9), and two matrices that differ only by a similarity transformation have the same 

In practice, it is the characters of irreducihle representations that are used in most chemical applications of group theory. This means that one needs only the character table for a group, rather than the whole representation table. For the Csv group, the character table is displayed in Table 13-18. Comparison with Table 13-13 will make clear that the character is merely the sum of diagonal elements. (For one-dimensional representations, the character and the representation are identical.) Since the representation for the identity operation is always a unit matrix, the character for this operation is always the same as the dimension of the representation. Hence, the first character in a row tells us the dimension of the corresponding representation. 

TABLE 13-19 The Standard Short-Form Character Table for the C3V Group 

We can take advantage of the above mle to write our character table in abbreviated form, illustrated for the Csv group in Table 13-19. This is the standard form for character tables. A collection of such tables appears in Appendix 11. 

EXAMPLE 13-7 How many classes of symmetry operation can you count for (planar) BH3 Can you use this to select one of the possibilities from Example 13-12  

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