Combinatorial Hopf Algebras of Simplicial Complexes

TL;DR

This paper develops a Hopf algebra framework for simplicial complexes, providing formulas for antipodes, characters, and connections to symmetric functions, colorings, and generalizations of Stanley's theorem.

Contribution

It introduces a new Hopf algebra structure for simplicial complexes and explores its combinatorial and algebraic properties, including antipode formulas and character-based symmetric functions.

Findings

Cancellation-free antipode formula derived

Characters encode coloring and f-vector information

Generalization of Stanley's (-1)-color theorem

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.