Group orthogonality theorem

For two irreducible representations a and f, the following group orthogonality theorem is satisfied ... 

The most important consequence of the group orthogonality theorem is the equation... 

Equation (C.7) can be obtained from the group orthogonality theorem after setting m = n and m = n, and then summing up over m and m ... 

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 ... 

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... 

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

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 ... 

The decomposition of the characters of the integrand into those of the irreducible representations is difficult to do by inspection, but when accomplished it is seen to contain A. Therefore the transition to Ti would become allowed. It is easier to use the formula below which is obtained from what is referred to as the little orthogonality theorem" of group theory. (See the Justification in Section 15.5 of the 5th edition of this text.) The coefficient of Ai in the integrand is given as... 

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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