  We consider a Hopf algebra of simplicial complexes and provide a
cancellation-free formula for its antipode. We then obtain a family of
combinatorial Hopf algebras by defining a family of characters on this Hopf
algebra. The characters of these combinatorial Hopf algebras give rise to
symmetric functions that encode information about colorings of simplicial
complexes and their $f$-vectors. We also use characters to give a
generalization of Stanley's $(-1)$-color theorem. A $q$-analog version of this
family of characters is also studied.
