Combinatorial Hopf Algebras and Generalized Dehn-sommerville Relations

A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) ζ : H → k. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra QSym 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 (H, ζ) possesses two canonical Hopf subalgebras on which the character ζ 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 H = QSym, the generalized Dehn-Sommerville relations are the Bayer-Billera relations and the odd subalgebra is the peak Hopf algebra of Stembridge. We prove that QSym 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.