Matrix elements
Finite groups
For any finite group and irreps and of dimensions , Schur orthogonality relations consist of "row-orthogonality" [eq. ( Ham89, p.3.148 )] (orthogonality of matrix–entries across group elements):
and "column-orthogonality" [eq. ( Ham89, p.3.178 )] (completeness/orthogonality across irreps):
See also eqs. (2.19), (2.20) of [FH91]. The last equation is nothing more than
For finite groups we can always choose our representation to be unitary [Ham89, sec.3.11]. The consequence of it is
So othogonality relations become
and
Eq. (6) can be used to define Fourier tranform on (see Exercise 3.32 in [FH91])
where
Symmetric group
All irreps of symmetric group are not only unitary, but real, what implies
Schur orthogonality relations for are then
where
Irreps of are indexed by , so and we have
Compact groups
Unitary group
Schur orthogonality relations for are
where . The general form of this these relations for arbitrary compact groups doesn't differ much and is due to Peter and Weyl. Schur proved such a formula for finite groups and now everything of that type is called by his name. See eqs. (3.1) and (3.17) of [IZ80], as well as eqs. (2), (3) on p. 44 and eqs. (4), (5) on p. 46 of [Vil68].
Haar measure above is of course assumed to be properly normalized
Fourier transform on reads as
where
Characters
Finite groups
"Raw-orthogonality" for characters
is easily derived from (5) by summing over and and using the definition of character:
"Column-orthogonality" for the characters
where denotes the order of the centralizer of , is a bit more involved in proof.
Symmetric group
For the symmetric group with the irreducible characters coincide on the elements of the same conjugacy class (cycle-type) and thus are denoted by . Size of the centralizer of any permutation of cycle type is denoted as . The order of conjugacy class is then . All these allows us to write "column-orthogonality" relations for the symmetric group
from (23), as well as "row-orthogonality" ones
The latter follows from (20) by the following observation
Compact groups
Unitary group
"Raw-orthogonality" again can be easily derived from the corresponding relation for matrix elements and reads as
provided we interpret as the character of the irreducible polynomial –representation of highest weight (i.e. a partition with ). As are class functions (constant on conjugacy classes of — symmetric functions of eigenvalues), Weyl integration formula helps us to rewrite this expression in the form
where is the Vandermonde determinant. This observation can help one motivate the standard inner product on the maximal torus on the space of symmetric functions
"Column-orthogonality" relation is
with as before. Again, the derivation of this statement seems to be a bit more involved.