The symmetric group algebra 
(SGA )
in which 
stands for any ring
of coefficients is defined as the algebra
generated by the elementary transpositions 
.
Thus it has a natural linear basis consisting of all
permutations in 
.
We give special elements, which are called
Jucis- Murphy elements:
Jucis(
, sum on all transpositions
exchanging i and k, 
.
The algebra generated by the Jucis-Murphy elements is a maximal commutative
sub-algebra of the symmetric group algebra. The symmetric functions in these
elements span the center of the group algebra 
.
Using these elements, one can build central idempotents corresponding to standard
tableaux, which are mutually orthogonal
(StdTab2Idemp ).
Carre(perm) and
Nabla(perm) ) are other distinguished
linear bases of 
. One can obtain idempotents from them:
Carre[LastPerm(n)] is the sum of all permutations in 
and
Nabla[LastPerm(n)] is the alternated sum of all permutations in
.
More generally, corresponding to a given shape, e.g. 
, one will have the
idempotent Carre([2,4,1,3,5]) 
Nabla([2,5,1,4,3])
that one gets from reading the following special tableaux and contretableaux:
