Let
G
be a nite group and let
W
be a nitely generated
R
G
-
module with a positive de nite bilinear form (
;
)
W
. Assume that
G
permutes
transitively a generating set
X
of
W
and that (
;
)
W
is constant on each
orbital of
G
on
X
. We show a new method for computing the dimensions of
the irreducible constituents of
W
. Further, we apply that method to Majorana
representations of the symmetric groups proving that the symmetric group
S
n
has a Majorana representation, in which every permutation of type (2
;
2) of
S
n
corresponds to a Majorana axis, if and only if
n≤
12