Flows on Rooted Trees and the Menous-Novelli-Thibon Idempotents

Several generating series for flows on rooted trees are introduced, as elements in the group of series associated with the Pre-Lie operad. By combinatorial arguments, one proves identities that characterise these series. One then gives a complete description of the image of these series in the group of series associated with the Dendriform operad. This allows to recover the Lie idempotents in the descent algebras recently introduced by Menous, Novelli and Thibon. Moreover, one defines new Lie idempotents and conjecture the existence of some others.

NATURAL ENDOMORPHISMS OF SHUFFLE ALGEBRAS

We show that there exist two natural endomorphism algebras for shuffle bialgebras such as Sh (X), where X is a graded set. One of these endomorphism algebras is a natural extension of the Malvenuto–Reutenauer Hopf algebra and is defined using graded permutations. The other one, the dendriform descent algebra, is a subalgebra of the first defined by mimicking the definition of the descent algebras by convolution from the graded projections in the tensor algebra. We study these algebras for their own, show that they carry bidendriform structures and establish freeness properties, study their generators, dimensions, bases, and also feature their relations to the internal structure of shuffle algebras. As an application of these ideas, we give a new proof of Chapoton's rigidity theorem for shuffle bialgebras.

A Semigroup Approach to Wreath-Product Extensions of Solomon's Descent Algebras

There is a well-known combinatorial model, based on ordered set partitions, of the semigroup of faces of the braid arrangement. We generalize this model to obtain a semigroup ${\cal F}_n^G$ associated with $G\wr S_n$, the wreath product of the symmetric group $S_n$ with an arbitrary group $G$. Techniques of Bidigare and Brown are adapted to construct an anti-homomorphism from the $S_n$-invariant subalgebra of the semigroup algebra of ${\cal F}_n^G$ into the group algebra of $G\wr S_n$. The colored descent algebras of Mantaci and Reutenauer are obtained as homomorphic images when $G$ is abelian.

Generalized Descent Algebras

If A is a subset of the set of reflections of a finite Coxeter group W, we define a sub-ℤ-module of the group algebra ℤW. We discuss cases where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra, the group algebra and, if W is of type B, the Mantaci–Reutenauer algebra.

Lie Elements and Knuth Relations

A coplactic class in the symmetric group consists of all permutations in with a given Schensted Q-symbol, and may be described in terms of local relations introduced by Knuth. Any Lie element in the group algebra of which is constant on coplactic classes is already constant on descent classes. As a consequence, the intersection of the Lie convolution algebra introduced by Patras and Reutenauer and the coplactic algebra introduced by Poirier and Reutenauer is the direct sum of all Solomon descent algebras.