where means the complex conjugate of the value of on g. With respect to this inner product, the irreducible characters form an orthonormal basis for the space of class functions, and this yields the orthogonality relation for the rows of the character table:
For the orthogonality relation for columns is as follows:
where the sum is over all of the irreducible characters of G and the symbol denotes the order of the centralizer of .
The orthogonality relations can aid many computations including:
decomposing an unknown character as a linear combination of irreducible characters;
constructing the complete character table when only some of the irreducible characters are known;
finding the orders of the centralizers of representatives of the conjugacy classes of a group; and
where is the (finite) dimension of the irreducible representation .
The orthogonality relations, only valid for matrix elements of irreducible representations, are:
Here is the complex conjugate of and the sum is over all elements of G. The Kronecker delta is unity if the matrices are in the same irreducible representation . If and are non-equivalent it is zero. The other two Kronecker delta's state that the row and column indices must be equal ( and ) in order to obtain a non-vanishing result. This theorem is also known as the Great (or Grand) Orthogonality Theorem.
Every group has an identity representation (all group elements mapped onto the real number 1). This is an irreducible representation. The great orthogonality relations immediately imply that
for and any irreducible representation not equal to the identity representation.
Example of the permutation group on 3 objects
The 3! permutations of three objects form a group of order 6, commonly denoted by (symmetric group). This group is isomorphic to the point group, consisting of a threefold rotation axis and three vertical mirror planes. The groups have a 2-dimensional irreducible representation (l = 2). In the case of one usually labels this representation by the Young tableau and in the case of one usually writes . In both cases the representation consists of the following six real matrices, each representing a single group element:
The normalization of the (1,1) element:
In the same manner one can show the normalization of the other matrix elements: (2,2), (1,2), and (2,1). The orthogonality of the (1,1) and (2,2) elements:
Similar relations hold for the orthogonality of the elements (1,1) and (1,2), etc. One verifies easily in the example that all sums of corresponding matrix elements vanish because of the orthogonality of the given irreducible representation to the identity representation.
The generalization of the orthogonality relations from finite groups to compact groups (which include compact Lie groups such as SO(3)) is basically simple: Replace the summation over the group by an integration over the group..
Every compact group has unique bi-invariant Haar measure, so that the volume of the group is 1. Denote this measure by . Let be a complete set of irreducible representations of , and let be a matrix coefficient of the representation . The orthogonality relations can then be stated in two parts:
An example of an r = 3 parameter group is the matrix group SO(3) consisting of all 3 x 3 orthogonal matrices with unit determinant. A possible parametrization of this group is in terms of Euler angles: (see e.g., this article for the explicit form of an element of SO(3) in terms of Euler angles). The bounds are and .
Not only the recipe for the computation of the volume element depends on the chosen parameters, but also the final result, i.e., the analytic form of the weight function (measure) .
For instance, the Euler angle parametrization of SO(3) gives the weight while the n, ψ parametrization gives the weight with
It can be shown that the irreducible matrix representations of compact Lie groups are finite-dimensional and can be chosen to be unitary:
With the shorthand notation
the orthogonality relations take the form
with the volume of the group:
As an example we note that the irreducible representations of SO(3) are Wigner D-matrices , which are of dimension . Since