Let 
be a ring of coefficients and 
. The usual Hecke
algebra of type A
(HEQA ),
denoted 
is a deformation of the symmetric
group algebra. With 
as set of generators, its
presentation is
As the braid relations appear in the presentation, the product
is independent of the reduced
decomposition 
of the
permutation w. It is well known that 
is a
linear basis of the free module 
. Hence, for n<m, we get a
natural embedding 
, and define
The typical element of 
then is the sum
.
There is a unique scalar product
(HeqaScal )
on 
satisfying
It will be called invariant
and, if 
is a field and q not a root
of unity, it makes 
into a symmetric algebra.
Remarkable self-adjoint elements are the Jucis-Murphy ones
(HeqaJucis ). They are
the q-deformation of the elements previously defined for the symmetric
group. The k-th (
) Jucis-Murphy element is equal to
In case 
is semi-simple, they generate a (self-adjoint) maximal
commutative subalgebra. Minimal idempotents are polynomials in these elements
(polynôme gallois
computed by the HeqaGall function )
indexed by tableaux.
We can construct a complete family of irreducible representations
(HeqaRepT and
HeqaRepCT )
as orbits
of q-deformation of the Vandermonde determinant. For an integral vector
, let 
, 
, 
, 
,
. The q-Vandermonde of index I is by definition
The deformation of the usual Specht modules is then provided by the
orbit of 
under an action on polynomials which is the quotient
of the faithful representation of 
as divided difference operators
[] that is, for h<k,
When I is increasing (resp. decreasing) a natural basis
of the Specht module is indexed by contretableaux (resp. tableaux).
In the semi-simple case, inverses in 
can be computed by means of
the minimal polynomials
(HeqaMinT and
HeqaMinCT ).
For 
, I an integral vector
(a shape) and x an indeterminate, let 
denote the minimal
polynomial for the matrix of e in a representation of shape I. The
minimal polynomial Q of e in 
is then
which can be written 
. If 
, we get
(these elements being computed by the use of the
HeqaInv function ).