Tree expansions of some Lie idempotents

2023/07/06 by Menous, Frédéric, Novelli, Jean-Christophe, Thibon, Jean-Yves
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2307.03001

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 a₀,…,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 Poincaré-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.

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