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Great orthogonality theorem

From Wikipedia, the free encyclopedia

The great orthogonality theorem (GOT) defines properties of irreducible matrix representations of symmetry groups and is the underlying groundwork for group theory.

[edit] Theory

Take a symmetry operator constructed in standard basis relation or standard order. The matrix elements within this symmetry transformation operator, \hat R, are given as \Gamma_i (\hat R)_{mn}, where m and n are the row and column indices. Since the basis functions are orthogonal to each other, the Great orthogonality theorem holds for finite groups and is written as follows:

\sum_{\hat R}^N \overline{\Gamma_i(\hat R)_{mn}} \, \Gamma_j(\hat R)_{m'n'} = \delta_{ij}\delta_{mm'}\delta_{nn'} \frac{h}{l_i}

where h is the order (number of elements in the group) and li is the dimension of the i-th representation. The δ's are 1 only if the matrices, \Gamma (\hat R), are from the same irreducible representation and also in the same row and column (m = m' and n = n').

[edit] Direct implications

The trace, or character, of these matrices can be written as a summation of diagonal matrix elements

\Chi_i (\hat R) = \sum_{m=1}^N \Gamma_i(\hat R)_{mm'}

where \Chi (\hat R) is the character of any reducible or irreducible representation having the j-th irreducible representation nj times. Hence we can write several character formulas:

\sum_{\hat R}^N \overline{\Chi_i(\hat R)} \, \Chi_j(\hat R) = \delta_{ij} h

which allows us to check the whether or not a representation is irreducible and, if so, which representation it is and

\sum_{\hat R}^N \overline{\Chi_j(\hat R)} \, \Chi(\hat R) = n_jh

which helps determine the number of irreducible representations, from nj, that are contained within the reducible representation \Gamma_j \, with character \Chi (\hat R).

For instance, if

n_j h = 96 \,

and

h = 24 \,

then

n_j = 4 = 2^2 \,

or

n_j = 1^2 1^2 1^2 1^2. \,

And lastly,

\sum_{\hat R}^N {\begin{vmatrix}{\Chi_i(\hat R)} \end{vmatrix}}^2 = h \sum_{j}^N n_j^2

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