The orthogonality theorem

Alternatively, we can regard T as reduced into T1 and T2. A representation of G is reducible if it can be transformed by a similarity transformation into an equivalent representation, each matrix of which has the same block-diagonal form. Then, each of the smaller representations T1, T2 is also a representation of G. A representation that cannot be reduced any further is called an irreducible representation (IR). 

Example 4.2-1 Show that the matrix representation found for C3v consists of the totally symmetric representation and a 2-D representation (T3). 

Many of the properties of IRs that are used in applications of group theory in chemistry and physics follow from one fundamental theorem called the orthogonality theorem (OT). If F, F are two irreducible unitary representations of G which are inequivalent if i -/ j and identical if i = j, then 

Note that Tl(T) q means the element common to the pt row and qth column of the MR for the group element T in the ith IR, complex conjugated. The sum is over all the elements of the group. If the matrix elements T (T)pq, TJ(T)rs are corresponding elements, that is from the same row p r and the same column q = s, and from the same IR, i = j, then the sum is unity, but otherwise it is zero. The proof of the OT is quite lengthy, and it is therefore given in Appendix A1.5. Here we verify eq. (1) for some particular cases. 

So we may interpret eq. (1) as a statement about the orthogonality of vectors in a g-dimensional vector space, where the components of the vectors are chosen from the elements of the /,-dimensional matrix representations r ( 2 ), F T), i.e. from the / th row and qth column of the /th IR, and from the rth row and sth column of the /th IR. If these are corresponding elements (p = r, q = s) from the same representation (i=j), then the theorem states that a vector whose components are T (T)pq, T 6 G, is of length fgJT,. But if the components are not corresponding elements of matrices from the same representation, then these vectors are orthogonal. The maximum number of mutually orthogonal vectors in ag-dimensional space is g. Now p may be chosen in /, ways (p = 1,. ..,/,) and similarly q may be chosen in /, ways ( 7 = 1,. .., /,) so that Y T)pq may be chosen in if from the /th IR and in if from all IRs. Therefore, 

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The orthogonality theorem can also be extended to cover a somewhat more general form of the eigenvalue equation. For the sake of convenience, we present in detail the case of a single variable, although the treatment can be generalized to any number of variables. Suppose that instead of the eigenvalue equation (3.5), we have for a hermitian operator 4 of one variable... 

A matrix of order l has l2 elements. Each irreducible representation T, must therefore contribute If -dimensional vectors. The orthogonality theorem requires that the total set of Y f vectors must be mutually orthogonal. Since there can be no more than g orthogonal vectors in -dimensional space, the sum Y i cannot exceed g. For a complete set (19) is implied. Since the character of an identity matrix always equals the order of the representation it further follows that... 

For every integral value ol m. there is an irreducible representation of 0(2), given by (23). The orthogonality theorem for characters in this case becomes... 

If we specify in the previous item that rj = /, the orthogonality theorem results ... 

The usefulness of the characters - 7 (R) of a representation j stems largely from the orthogonality theorem of Section 4.4, which for a finite group of order g, is that... 

The orthogonality theorem The inequivalent irreducible unitary matrix representations of a group G satisfy the orthogonality relations... 

In more complicated cases, inspection is not a practical way to proceed. For these we use the orthogonality theorem, which we must simply state without proof. This theorem can be written in the special form... 

Using the orthogonality theorem, Eq. (23.35), for the second sum on the right side, we obtain... 

Step III. The orthogonality theorem is finally proved making use of the above theorem. 

Proof of Step III. The preceding results now allow a proof of the orthogonality theorem to be set up immediately. Consider the dy by dy matrix A defined by... 

But the orthogonality theorem in Appendix XI shows that the sum enclosed in parentheses in (4) A anishes unless and unless... 

In order to derive character tables for more complicated groups, it is convenient to develop some general relations Avhich must be satisfied by the characters of any group. From the orthogonality theorem [Eq. (1), Ipp. XIJ one such set of relations may be obtained directly. In the Orthogonality theorem... 

The principal purpose of this appendix is to prove the orthogonality theorem which was used repeatedly in Chap, (i, namely,... 

The reducible representation (2.106) will be reduced to the irreducible representations of the molecular group, using the molecular formula of detached decomposition from the orthogonally theorem (2.102)... 

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