Orthogonality theorem

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... 

Equation (30) can be derived from the general orthogonality theorem, Eq. (44). 

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 ... 

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 ... 

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, ... 

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... 

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... 

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