Geodesic
The $(1-\mathbb{E})$-transform in combinatorial Hopf algebras
Journal of Algebraic Combinatorics, Tome 33 (2011) no. 2, pp. 277-312.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We extend to several combinatorial Hopf algebras the endomorphism of symmetric functions sending the first power-sum to zero and leaving the other ones invariant. As a "transformation of alphabets", this is the $(1 -\mathbb E)$ (1-mathbbE)-transform, where $\mathbb E$ mathbbE is the "exponential alphabet," whose elementary symmetric functions are $e _{ n}=\frac1 n$! e_n=frac1n!. In the case of noncommutative symmetric functions, we recover Schocker's idempotents for derangement numbers (Schocker, Discrete Math. 269:239-248, 2003). From these idempotents, we construct subalgebras of the descent algebras analogous to the peak algebras and study their representation theory. The case of WQSym leads to similar subalgebras of the Solomon-Tits algebras. In FQSym, the study of the transformation boils down to a simple solution of the Tsetlin library in the uniform case.
Keywords: keywords combinatorial Hopf algebras, symmetric functions, derangements 
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     title = {The $(1-\mathbb{E})$-transform in combinatorial {Hopf} algebras},
     journal = {Journal of Algebraic Combinatorics},
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     number = {2},
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     url = {http://geodesic.mathdoc.fr/item/JAC_2011__33_2_a1/}
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Hivert, Florent; Luque, Jean-Gabriel; Novelli, Jean-Christophe; Thibon, Jean-Yves. The $(1-\mathbb{E})$-transform in combinatorial Hopf algebras. Journal of Algebraic Combinatorics, Tome 33 (2011) no. 2, pp. 277-312. http://geodesic.mathdoc.fr/item/JAC_2011__33_2_a1/