Recently, starting from the quasi-determinants of Gelfand and Retakh [,] a noncommutative theory of symmetric functions has been developed [], in such a way that most of the classical applications can be lifted to the noncommutative case. In particular, the character theory of the symmetric group has a natural noncommutative analogue, in which the rôle of the character ring is played by Solomon's descent algebra [].
The algebra of formal
noncommutative symmetric functions is the free associative
algebra 
generated by
an infinite sequence of noncommuting indeterminates 
, called
the elementary functions. It is convenient to set 
.
Let t be another indeterminate, commuting with the 
. One
introduces the generating series
The 
are called complete symmetric functions, and the 
are
the power sums of the second kind. The power sums of the first kind,
denoted by 
are the coefficients of the formal series
defined by the equation
The algebra 
is graded by the weight function w defined by
, and its homogeneous component of weight n is denoted
by 
. If 
is a sequence of noncommutative symmetric functions
such that 
, we set for any composition 
Then, 
, 
, 
and 
are homogeneous bases of 
(ToL ,
ToS ,
ToPh ,
ToPs ).
The set of all compositions
(ListCompo )
of a given integer n is equipped with
the reverse refinement order
(ListCompoFiner ,
IsFiner ),
denoted 
. For example,
the compositions J of 5 such that 
are 
,
, 
and 
. The basis 
of ribbon Schur functions,
originally defined in terms of quasi-determinants in [], can
also be defined by either of the two equivalent equations
being the length of the composition I
(ToR ).
The commutative image of a noncommutative symmetric function
is given by the algebra morphism 
.
Then, 
, 
, 
,
and 
is sent to an ordinary ribbon Schur function, which is
denoted by 
. Ribbon Schur functions have been defined by
McMahon ( see [], t. 1, p. 200), and are denoted in his book by 
.
Let 
be a permutation with descent set
.
The descent composition
(Perm2Desc )
is the composition 
of n defined by
, where 
and 
. We
also set 
, and conversely, the subset
A of 
associated to a composition I of n
is denoted by 
.
The sum in the group algebra
of all permutations with descent composition I is denoted by 
. The
with |I|=n form a basis of a subalgebra 
, called the
descent algebra of 
[]. We denote by 
the same algebra, with scalars extended to 
. There is
an isomorphism of graded vector spaces
such that
for any composition I.
The direct sum 
can be given an algebra structure, by extending the
natural product of its components 
by setting xy=0 for
and 
with 
. The
internal product
(Internal ),
denoted 
, on 
is defined by requiring that 
be an anti-isomorphism. That is, we set
One also defines on 
a coproduct 
by any of the following
equivalent conditions:
The fundamental property for computing with the internal product is the following splitting formula:
In the commutative case, this formula can be regarded as a particular case of the Mackey tensor product theorem.
With this coproduct, the involution 
,
and the natural unit and counit, 
is a Hopf algebra,
whose graded dual is Gessel's algebra of quasisymmetric functions
[].
It has been shown by Klyachko [] that for a primitive
n-th root of unity 
, the element
is a Lie idempotent. In terms of noncommutative symmetric functions,
this element is the specialization at 
of the
symmetric function 
, where 
is given by the
generating series
The symmetric functions of the virtual alphabet
(SpDirect )
are defined by the equation
where A is just a label.
The inverse transformation is given by the function SpInverse.
For example, 
corresponds in the
descent algebra to the q-bracket operator.
As an application, one can show that the element
is a Lie idempotent [,], interpolating between several known ones.
As shown in [], this leads to a q-deformation of the Eulerian idempotents.
