Great orthogonality theorem

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, ... 

An important consequence of group theory, i.e., the so-called great orthogonality theorem, is the basis of the equation for calculating the numbers w, of the vibrations belonging to a symmetry species i ... 

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... 

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