Shortcut to Determine a Representation

It was quite easy to find the irreducible representation of Rz before, as the representation we worked out appeared to be an irreducible representation itself. In most cases, however, a reducible representation is found when the symmetry operations are applied to a certain basis. Now a simpler way will be shown (1) to describe the representation on a given basis without generating the matrices themselves and (2) to reduce them, if reducible, to irreducible representations. 

But how can we know the character of a matrix without writing down the whole matrix  

Returning to the diimide N-H bond length changes, let us see how the above simple rules work. The identity operation, A, leaves 

The effect of C2 has already been looked at. Its character is 0. The effect of the inversion operation is the same as that of C2, so the character will be 

Finally, operation uh leaves the two bonds unchanged, so both of them contribute +1 to the character  

Looking back at the effect of the different symmetry operations on HNNH 

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