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no. 5
Tree expansions of some Lie idempotents
Menous, Frédéric1; Novelli, Jean-Christophe2; Thibon, Jean-Yves2
1 Laboratoire de Mathématiques Bât. 425 Université Paris-Sud 91405 Orsay Cedex France
2 Laboratoire d’Informatique Gaspard-Monge, Université Gustave Eiffel, CNRS, ENPC, ESIEE-Paris 5 Boulevard Descartes Champs-sur-Marne 77454 Marne-la-Vallée cedex 2 France
Algebraic Combinatorics, Volume 7 (2024) no. 5, pp. 1261-1282.

We prove that the Catalan Lie idempotent D n (a,b), introduced in [Menous et al., Adv. Appl. Math. 51 (2013), 177] can be refined by introducing n independent parameters a 0 ,...,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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.373
Classification: 16T30, 05E05, 17D25
Keywords: Noncommutative symmetric functions, Lie idempotents, Free Lie algebra, Dendriform algebras, PreLie algebras

Menous, Frédéric 1; Novelli, Jean-Christophe 2; Thibon, Jean-Yves 2

1 Laboratoire de Mathématiques Bât. 425 Université Paris-Sud 91405 Orsay Cedex France
2 Laboratoire d’Informatique Gaspard-Monge, Université Gustave Eiffel, CNRS, ENPC, ESIEE-Paris 5 Boulevard Descartes Champs-sur-Marne 77454 Marne-la-Vallée cedex 2 France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Menous, Frédéric; Novelli, Jean-Christophe; Thibon, Jean-Yves. Tree expansions of some Lie idempotents. Algebraic Combinatorics, Volume 7 (2024) no. 5, pp. 1261-1282. doi : 10.5802/alco.373. https://alco.centre-mersenne.org/articles/10.5802/alco.373/

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