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MIT Combinatorics Seminar

Coloring Descents and Peaks


Christophe Hohlweg (University of Toronto)

Wednesday, May 17, 2006   4:30 PM;   Building 2 Room 105; 

ABSTRACT

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).