Tables and Properties of Irreducible Representations

The sum of the squares of the dimensions of all irreducible representations in a group is equal to the order of the group. The dimension of an irreducible representation is simply the dimension of any of its matrices, which is the number of rows or columns of the matrix. Since the identity operation always leaves the molecules unchanged, its representation is a unit matrix. The character of a unit matrix is equal to the number of rows or columns of that matrix, as is demonstrated below  

From this it follows that the character under E is always the dimension of the given irreducible representation. The onedimensional representations are nondegenerate, and the two- or higher-dimensional representations are degenerate. The meaning of degeneracy will be discussed in Chapter 6. 

The sum of the squares of the absolute values of characters of any irreducible representation in a group is equal to the order of the group. 

The sum of the products of the corresponding characters (or one character with the conjugate of another in the case of imaginary characters) of any two different irreducible representations of the same group is zero. 

The characters of all matrices belonging to operations in the same class are identical in a given irreducible representation. 

We do not know yet, however, whether this representation is reducible or irreducible. To answer this question, first we have to know the characters of the irreducible representations of the C2h point group. 

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