The $(1-\mathbb{E})$-transform in combinatorial Hopf algebras

  • Authors:

  • Florent Hivert;Jean-Gabriel Luque;Jean-Christophe Novelli;Jean-Yves Thibon

  • Affiliations:

  • LITIS, Université de Rouen, Saint Étienne du Rouvray, France 76801;LITIS, Université de Rouen, Saint Étienne du Rouvray, France 76801;Institut Gaspard Monge, Université de Marne-la-Vallée, Marne-la-Vallée cedex 2, France 77454;Institut Gaspard Monge, Université de Marne-la-Vallée, Marne-la-Vallée cedex 2, France 77454

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 2011

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Abstract

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 $(1-\mathbb{E})$ -transform, where $\mathbb{E}$ is the "exponential alphabet," whose elementary symmetric functions are $e_{n}=\frac{1}{n!}$ . 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.