Descent and peak compositions yield to two remarkable and
well-studied subalgebras of the group algebra of the symmetric group $S_n$:
the descent algebra and the peak algebra. They are also sub-Hopf algebras of
the Malvenuto-Reutenauer Hopf algebra $P$ (the direct sum over $n$ of the
group algebras of $S_n$). There graded dual are commutative algebras: the
algebra QSym of Gessel's quasisymmetric functions and Stembridge's peak
algebra. They can be obtained by a nice combinatorial construction: the
standardized permutation of a word yields to a realization into words of
$P$. Letting the variables be commutative gives a morphism from $P$ to
QSym. This is the core of the theory of noncommutative symmetric functions.
In this talk, we will see how this picture can be colored to fit with the
wreath product $G\wr S_n$, where $G$ is a finite abelian
group. Coloring descent compositions yields the Mantaci-Reutenauer algebra
and coloring peak compositions yields a colored peak
subalgebra of the Mantaci-Reutenauer algebra. Coloring the Solomon's
epimorphism from the descent algebra to the ring of
symmetric functions lead us to a colored version of Gessel-Reutenauer's
formula involving the characters of the colored descent representations of
Adin-Brenti-Roichman, and Bagno-Biagioli.
(From works with Pierre Baumann and Nantel Bergeron).