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.

$P$-Partitions and a Multi-Parameter Klyachko Idempotent

Because they play a role in our understanding of the symmetric group algebra, Lie idempotents have received considerable attention. The Klyachko idempotent has attracted interest from combinatorialists, partly because its definition involves the major index of permutations. For the symmetric group $S_n$, we look at the symmetric group algebra with coefficients from the field of rational functions in $n$ variables $q_1, \ldots, q_n$. In this setting, we can define an $n$-parameter generalization of the Klyachko idempotent, and we show it is a Lie idempotent in the appropriate sense. Somewhat surprisingly, our proof that it is a Lie element emerges from Stanley's theory of $P$-partitions.

Noncommutative Symmetric Functions II: Transformations of Alphabets

Noncommutative analogues of classical operations on symmetric functions are investigated, and applied to the description of idempotents and nilpotents in descent algebras. It is shown that any sequence of Lie idempotents (one in each descent algebra) gives rise to a complete set of indecomposable orthogonal idempotents of each descent algebra, and various deformations of the classical sequences of Lie idempotents are obtained. In particular, we obtain several q-analogues of the Eulerian idempotents and of the Garsia-Reutenauer idempotents.