The Great Orthogonality Theorem

The following set of characters is for a linear combination of irreducible representations for a system having symmetry (say, ammonia)  

Use equation 13.6 to determine what linear combination of Aj, A2, and E symmetry species is being represented. 

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

We need the characters for Aj, A2, and E from the character table of the C y point group. For the number of times A appears in the linear combination, 

Therefore, this is a sum of two Aj, one A2, and two E symmetry species. This is how the great orthogonality theorem is applied to reduce character sets into their unique set of irreducible representations. 

---

This equation (16) is known as the great orthogonality theorem for the irreducible representations of a group and occupies a central position in the theory of group representations. 

Each irreducible representation of a group consists of a set of square matrices of order lt. The set of matrix elements with the same index, grouped together, one from each matrix in the set, constitutes a vector in -dimensional space. The great orthogonality theorem (16) states that all these vectors are mutually orthogonal and that each of them is normalized so that the square of its length is equal to g/li. This interpretation becomes more obvious when (16) is unpacked into separate expressions ... 

From eqn (7-2.1) (the Great Orthogonality Theorem) we can obtain for the non-equivalent irreducible representations T and Tv ... 

Now we ask the parallel question—what is the new choice of basis functions for the function space (the one which produced rred) which will produce matrices in their fully reduced form Once again we are looking at the opposite side of the coin whose two faces are a similarity transformation and a change of basis functions. To answer the question we have posed, we will invoke the Great Orthogonality Theorem and carry out a certain amount of straightforward algebra. 

Proof of the Great Orthogonality Theorem This theorem states that... 

The great orthogonality theorem may then be stated as follows ... 

The most powerful theorem in group theory, for our purposes, is the great orthogonality theorem (GOT) which states that for irreps D and D, of respective dimensions na and n, ... 

Very often we can decompose a sum of numbers such as those in 4.13 by inspection by using the character table. If not, then there is an equation that comes from a result called the great orthogonality theorem which does this for us (equation 4.15),... 

In his book Molecular Symmetry and Group Theory, Robert Carter introduces a tabular method for applying the great orthogonality theorem, which is illustrated... 

TABLE 8.10 Tabular method for reducing representations using the great orthogonality theorem. 

Example 8-19. Use the great orthogonality theorem and tabular method to reduce the Tdof representation for methane in the Tj point group. 

Example 8-21. Use a set of three orthogonal miniature vectors on each atom in the CO2 molecule to generate the representation in the subgroup of Then use the great orthogonality theorem and the correlation table in Table 8.12 to determine the IRRs that comprise the Tjof representation for the CO2 molecule. 

The remainder of the table below uses the tabular method of application of the great orthogonality theorem to reduce Tdof into its irreducible components. 

Use the character tables in Appendix B and the great orthogonality theorem to determine each of the following direct products ... 

---