Combinatorial Hopf algebras and generalized Dehn-Sommerville 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 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 $(H,\zeta)$ 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 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.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- October 2003
- DOI:
- 10.48550/arXiv.math/0310016
- arXiv:
- arXiv:math/0310016
- Bibcode:
- 2003math.....10016A
- Keywords:
-
- Mathematics - Combinatorics;
- Mathematics - Quantum Algebra;
- Mathematics - Rings and Algebras;
- 05A15;
- 05E05;
- 06A11;
- 16W30;
- 16W50
- E-Print:
- 34 pages