The (1-E)-transform in combinatorial Hopf algebras

Florent Hivert (1) , Jean-Gabriel Luque (1) , Jean-Christophe Novelli (2) , Jean-Yves Thibon (2)

1 CA - LITIS - Equipe Combinatoire et algorithmes
2 LIGM - Laboratoire d'Informatique Gaspard-Monge

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-E)-transform, where E is the exponential alphabet, whose elementary symmetric functions are e_n=1/n!. In the case of noncommutative symmetric functions, we recover Schocker's idempotents for derangement numbers [Discr. Math. 269 (2003), 239]. 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.

Domaines: Combinatoire [math.CO]

Florent Hivert, Jean-Gabriel Luque, Jean-Christophe Novelli, Jean-Yves Thibon. The (1-E)-transform in combinatorial Hopf algebras. Journal of Algebraic Combinatorics / Journal of Algebraic Combinatorics An International Journal, 2010, 33 (2), pp.277-312. ⟨10.1007/s10801-010-0245-5⟩. ⟨hal-00440924⟩