Linear Molecules Groups of Infinite Order

The reducible representation in these groups can be assigned in the normal way. We consider the effect of an example operation on each member of the basis to assign a character for each class of operations in the group. To proceed with the reduction into the set of standard irreducible representations we return to the basic idea from which the reduction formula was derived in Section 5.5. 

Equation (5.15) states that, within each class, the sum of the characters from the set of irreducible representations which make up a given reducible representation F sum to the character obtained for that F, /r( C). This sum must work for every class so, once a particular alternative combination of irreducible representations is shown to be inconsistent with the Xr( C) in any class, we need not consider that mix again. 

Now the standard character table is shown in Table 5.21, and so we have to look for combinations of irreducible representations that correspond to F. The first restriction is that the irreducible representations must give only two objects, because we have used 

Under the principal axis, the 11- or A-type representations give cosine functions of the angle of rotation. For our reducible representation, Xr(Co )= 2, and so fl- or A-type representations do not fit at all. 

The correct combination must be two XI representations. At this point, any of these would be allowed because they each have character 1 under 2Coo in Table 5.21. However, under the ooo-v class the 11 (both gerade and ungerade cases) have -1. This means that the inclusion of these in our linear combination will lead to a total character of less than the required 2. So the set of irreducible representations we seek can only contain X1+ types. 

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