The space of complex-valued class functions of a finite group G has a natural inner product: where
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
Note that since g and h are conjugate iff they are in the same column of the character table, this implies that the columns of the character table are orthogonal.
The orthogonality relations can aid many computations including: Let
is the (finite) dimension of the irreducible representation
[1] The orthogonality relations, only valid for matrix elements of irreducible representations, are: Here
and the sum is over all elements of G. The Kronecker delta
The other two Kronecker delta's state that the row and column indices must be equal (
The great orthogonality relations immediately imply that for
permutations of three objects form a group of order 6, commonly denoted S3 (the symmetric group of degree three).
, consisting of a threefold rotation axis and three vertical mirror planes.
In the case of S3 one usually labels this representation by the Young tableau
In both cases the representation consists of the following six real matrices, each representing a single group element:[2] 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).
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.
Often one writes for the trace of a matrix in an irreducible representation with character
In this notation we can write several character formulas: which allows us to check whether or not a representation is irreducible.
(The formula means that the lines in any character table have to be orthogonal vectors.)
And which helps us to determine how often the irreducible representation
For instance, if and the order of the group is then the number of times that
has unique bi-invariant Haar measure, so that the volume of the group is 1.
is an orthonormal basis of the representation space
These orthogonality relations and the fact that all of the representations have finite dimensions are consequences of the Peter–Weyl theorem.
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).
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
ω ( α , β , γ ) = sin
while the n, ψ parametrization gives the weight
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
Since they satisfy Any physically or chemically oriented book on group theory mentions the orthogonality relations.