Summary: COMBINATORIAL HOPF ALGEBRAS
AND GENERALIZED DEHN-SOMMERVILLE RELATIONS
MARCELO AGUIAR, NANTEL BERGERON, AND FRANK SOTTILE
Abstract. A combinatorial Hopf algebra is a graded connected Hopf algebra over
a field k equipped with a character (multiplicative linear functional) : 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 com-
binatorial Hopf algebra (H, ) possesses two canonical Hopf subalgebras on which
the character is even (respectively, odd). The odd subalgebra is defined by cer-
tain 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 permuta-
tions, the Loday-Ronco Hopf algebra of planar binary trees, the Hopf algebras of
symmetric functions and of non-commutative symmetric functions.