Combinatorial Hopf algebras and generalized Dehn-Sommerville relations
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A combinatorial Hopf algebra is a graded connected Hopf algebra over a field double-struck k sign equipped with a character (multiplicative linear functional) : H double-struck k sign. 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, ) possesses two canonical Hopf subalgebras on which the character 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 Stem-bridge. 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. Foundation Compositio Mathematica 2006.