Multiplication Tables for Character Representations

To be valid representations, the sets of numbers we have produced for the AivA2,B, and B2 representations must each act in the same way as the symmetry operations themselves. For example, we found in Chapter 2 that C2 Tv = ad for the C2-, point group of H2O. In the Aj representation, both C2 and a have character 1 and so the character product is 1, which is also the character of ad- This must hold for the whole multiplication table between the symmetry operators. For Ai, the table is quite trivial, since it contains only Is. However, the other representations must also give consistent multiplication tables. 

The C2V group multiplication table was constructed from the group operations in Problem 2.1 and the result is reproduced in Table 4.4a. The same table using the Bi representation characters is given in Table 4.4b. In this representation version, the operation symbols for the row and column headings are replaced by the corresponding Bi characters. The result of a product of operations then becomes a simple numerical multiplication of 

Problem 4.4 Write out multiplication tables for the Aj and B2 representations in C2V and confirm that they are also consistent with Table 4.4a. 

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