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Journal of Algebraic Combinatorics

, Volume 33, Issue 2, pp 277–312 | Cite as

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

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

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.

Keywords

Combinatorial Hopf algebras Symmetric functions Derangements 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Florent Hivert
    • 1
  • Jean-Gabriel Luque
    • 1
  • Jean-Christophe Novelli
    • 2
  • Jean-Yves Thibon
    • 2
  1. 1.LITISUniversité de RouenSaint Étienne du RouvrayFrance
  2. 2.Institut Gaspard MongeUniversité de Marne-la-ValléeMarne-la-Vallée cedex 2France

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