Tree expansions of some Lie idempotents

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. T...

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Published in:arXiv.org 2023-07
Main Authors: Menous, Frédéric, Novelli, Jean-Christophe, Jean-Yves Thibon
Format: Article
Language:English
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Mathematical analysis
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Summary: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 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.
ISSN:2331-8422