Schur orthogonality relations

In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. 
They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3).
Finite groups.
Intrinsic statement.
The space of complex-valued class functions of a finite group G has a natural inner product:
where formula_2 means the complex conjugate of the value of formula_3 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 formula_5, applying the same inner product to the columns of the character table yields:
where the sum is over all of the irreducible characters formula_7 of "G" and the symbol formula_8 denotes the order of the centralizer of formula_9. Note that since and 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:
Coordinates statement.
Let formula_10 be a matrix element of an irreducible matrix representation
formula_11 of a finite group formula_12 of order |"G"|, i.e. "G" has |"G"| elements. Since it can be proven that any matrix representation of any finite group is equivalent to a unitary representation, we assume formula_11 is unitary:
where formula_15 is the (finite) dimension of the irreducible representation formula_11.
The orthogonality relations, only valid for matrix elements of "irreducible" representations, are:
Here formula_18 is the complex conjugate of formula_19 and the sum is over all elements of "G".
The Kronecker delta formula_20 is unity if the matrices are in the same irreducible representation formula_21. If formula_11 and formula_23 are non-equivalent
it is zero. The other two Kronecker delta's state that 
the row and column indices must be equal (formula_24 and formula_25) 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 formula_27 and any irreducible representation formula_28 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 (symmetric group). This group is isomorphic to the point group formula_29, 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 formula_30 and in the case of formula_29
one usually writes formula_32. 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.
Direct implications.
The trace of a matrix is a sum of diagonal matrix elements,
The collection of traces is the "character" formula_37 of a representation. Often one writes for
the trace of a matrix in an irreducible representation with character formula_38
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 formula_11 is contained within the reducible representation formula_43 with character formula_44.
For instance, if
and the order of the group is
then the number of times that formula_47 is contained within the given
"reducible" representation formula_43 is
"See Character theory for more about group characters." 
Compact Groups.
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 formula_50 has unique bi-invariant Haar measure, so that the volume of the group is 1. Denote this measure by formula_51. Let formula_52 be a complete set of irreducible representations of formula_50, and let formula_54 be a matrix coefficient of the representation formula_55. The orthogonality relations can then be stated in two parts:
1) If formula_56 then
2) If formula_58 is an orthonormal basis of the representation space formula_55 then
where formula_61 is the dimension of formula_55. These orthogonality relations and the fact that all of the representations have finite dimensions are consequences of the Peter–Weyl theorem.
An Example SO(3).
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: formula_63 (see e.g., this article for the explicit form of an element of SO(3) in terms of Euler angles). The bounds are formula_64 and formula_65.
Not only the recipe for the computation of the volume element formula_66 depends on the chosen parameters, but also the final result, i.e. the analytic form of the weight function (measure) formula_67.
For instance, the Euler angle parametrization of SO(3) gives the weight formula_68 while the n, ψ parametrization gives the weight formula_69
with formula_70
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 formula_75, which are of dimension formula_76. Since
they satisfy
References.
Any physically or chemically oriented book on group theory mentions the orthogonality relations. The following more advanced books give the proofs:
The following books give more mathematically inclined treatments: