Let
G
G
be a finite group,
f
:
G
→
C
f:G \to {\mathbf {C}}
a function, and
ρ
\rho
an irreducible representation of
G
G
. The Fourier transform is defined as
f
^
(
ρ
)
=
Σ
s
∈
G
f
(
s
)
ρ
(
s
)
\hat f(\rho ) = {\Sigma _{s \in G}}f(s)\rho (s)
. Direct computation for all irreducible representations involves order
|
G
|
2
{\left | G \right |^2}
operations. We derive fast algorithms and develop them for the symmetric group
S
n
{S_n}
. There,
(
n
!
)
2
{(n!)^2}
is reduced to
n
(
n
!
)
a
/
2
n{(n!)^{a/2}}
, where
a
a
is the constant for matrix multiplication (2.38 as of this writing). Variations of the algorithm allow efficient computation for “small” representations. A practical version of the algorithm is given on
S
n
{S_n}
. Numerical evidence is presented to show a speedup by a factor of 100 for
n
=
9
n = 9
.