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Combinatorial Hopf algebras and generalized Dehn-Sommerville relations

A combinatorial Hopf algebra is a graded connected Hopf algebra over a field $F$ equipped with a character (multiplicative linear functional) $ζ:H\to F$. 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.