Tree expansions of some Lie idempotents}

TL;DR

This paper refines Catalan Lie idempotents by introducing multiple parameters, revealing their structure as sums of Lie idempotents, and connects these to noncommutative symmetric functions and tree series.

Contribution

It introduces multi-parameter refinements of Catalan Lie idempotents and links them to noncommutative symmetric functions and tree expansions.

Findings

Refined Catalan Lie idempotents with multiple parameters.

Expressed idempotents as sums of subsets of the PBW basis.

Connected idempotents to noncommutative symmetric functions and tree series.

Abstract

We prove that the Catalan Lie idempotent $D_n(a,b)$, introduced in [Menous {\it et al.}, Adv. Appl. Math. 51 (2013), 177] can be refined by introducing $n$ independent parameters $a_0,\ldots,a_{n-1}$ and that the coefficient of each monomial is itself a Lie idempotent in the descent algebra. These new idempotents are multiplicity-free sums of subsets of the Poincar\'e-Birkhoff-Witt basis of the Lie module. These results are obtained by embedding noncommutative symmetric functions into the dual noncommutative Connes-Kreimer algebra, which also allows us to interpret, and rederive in a simpler way, Chapoton's results on a two-parameter tree expanded series.