Great orthogonality theorem
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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, 
, are given as 
, 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:
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, 
, 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
where 
is the character of any reducible or irreducible representation having the j-th irreducible representation nj times. Hence we can write several character formulas:
which allows us to check the whether or not a representation is irreducible and, if so, which representation it is and
which helps determine the number of irreducible representations, from nj, that are contained within the reducible representation 
with character .
For instance, if
and
then
or
And lastly,
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