Combinatorial Hopf algebras and generalized Dehn–Sommerville relations

Abstract

A combinatorial Hopf algebra is a graded connected Hopf algebra over a field $\Bbbk$ equipped with a character (multiplicative linear functional) $\zeta\colon{\mathcal H}\to \Bbbk$. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra ${\mathcal Q}{\mathit{Sym}}$ of quasi-symmetric functions; this explains the ubiquity of quasi-symmetric functions as generating functions in combinatorics. We illustrate this with several examples. We prove that every character decomposes uniquely as a product of an even character and an odd character. Correspondingly, every combinatorial Hopf algebra $({\mathcal H},\zeta)$ possesses two canonical Hopf subalgebras on which the character $\zeta$ is even (respectively, odd). The odd subalgebra is defined by certain canonical relations which we call the generalized Dehn–Sommerville relations. We show that, for ${\mathcal H}={\mathcal Q}{\mathit{Sym}}$, the generalized Dehn–Sommerville relations are the Bayer–Billera relations and the odd subalgebra is the peak Hopf algebra of Stembridge. We prove that ${\mathcal Q}{\mathit{Sym}}$ is the product (in the categorical sense) of its even and odd Hopf subalgebras. We also calculate the odd subalgebras of various related combinatorial Hopf algebras: the Malvenuto–Reutenauer Hopf algebra of permutations, the Loday–Ronco Hopf algebra of planar binary trees, the Hopf algebras of symmetric functions and of non-commutative symmetric functions.