 
Summary: COMBINATORIAL HOPF ALGEBRAS
AND GENERALIZED DEHNSOMMERVILLE 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 quasisymmetric functions; this explains the ubiquity of
quasisymmetric 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 DehnSommerville relations.
We show that, for H = QSym, the generalized DehnSommerville relations are the
BayerBillera 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 MalvenutoReutenauer Hopf algebra of permuta
tions, the LodayRonco Hopf algebra of planar binary trees, the Hopf algebras of
symmetric functions and of noncommutative symmetric functions.
