Relating two Hopf algebras built from an operad

Starting from an operad, one can build a family of posets. From this family of posets, one can define an incidence Hopf algebra. By another construction, one can also build a group directly from the operad. We then consider its Hopf algebra of functions. We prove that there exists a surjective morphism from the latter Hopf algebra to the former one. This is illustrated by the case of an operad built on rooted trees, the \NAP operad, where the incidence Hopf algebra is identified with the Connes-Kreimer Hopf algebra of rooted trees.Comment: 21 pages, use graphics, 12 figures Version 2 : references added, minor changes. This version has not been corrected after submission. The final and corrected version will appear in IMRN and can be obtained from the author

Character formulas for the operad of two compatible brackets and for the bihamiltonian operad

We compute dimensions of the components for the operad of two compatible brackets and for the bihamiltonian operad. We also obtain character formulas for the representations of the symmetric groups and the $SL_2$ group in these spaces.Comment: 24 pages, accepted by Functional Analysis and its Applications, a few typos correcte

Some Combinatorial Operators in Language Theory

Multitildes are regular operators that were introduced by Caron et al. in order to increase the number of Glushkov automata. In this paper, we study the family of the multitilde operators from an algebraic point of view using the notion of operad. This leads to a combinatorial description of already known results as well as new results on compositions, actions and enumerations.Comment: 21 page