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 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.

Keywords

• combinatorial hopf algebra
• quasi-symmetric functions
• simplicial complex
• colorings
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]