Combinatorial Hopf algebras and generalized DehnSommerville relations
Abstract
A combinatorial Hopf algebra is a graded connected Hopf algebra over a field $F$ equipped with a character (multiplicative linear functional) $\zeta:H\to F$. 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 combinatorial Hopf algebra $(H,\zeta)$ possesses two canonical Hopf subalgebras on which the character $\zeta$ is even (respectively, odd). The odd subalgebra is defined by certain 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 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 MalvenutoReutenauer Hopf algebra of permutations, the LodayRonco Hopf algebra of planar binary trees, the Hopf algebras of symmetric functions and of noncommutative symmetric functions.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2003
 arXiv:
 arXiv:math/0310016
 Bibcode:
 2003math.....10016A
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Quantum Algebra;
 Mathematics  Rings and Algebras;
 05A15;
 05E05;
 06A11;
 16W30;
 16W50
 EPrint:
 34 pages