Combinatorial Hopf Algebras of Simplicial Complexes

Abstract

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.

To appear in SIAM J. of Discrete Math. This version has new material, including a proof of the antipode formula. The even and odd subalgebras section is corrected. An extended abstract version of this article appeared in the Proceedings of FPSAC'15

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