Combinatorial Hopf algebras of simplicial complexes. (Q2820857)

A Hopf algebra \(\mathcal A\) based on simplicial complexes is here defined; its product is given by disjoint union and its coproduct by extraction of vertices. A cancellation-free formula is given for its antipode. A family of characters makes it a combinatorial Hopf algebra, in the sense defined by Aguiar and Bergeron, for any integer \(s\); this gives morphisms from \(\mathcal A\) into the Hopf algebra of quasi-symmetric functions -- in fact, here, of symmetric functions. These morphisms encode informations about colorings of simplicial complexes. Specializations of these symmetric functions give a generalization of Stanley's \((-1)\)-color theorem. A \(q\)-deformation of these characters is also studied.