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Combinatorial Hopf algebras and generalized Dehn–Sommerville relations

Published online by Cambridge University Press:  13 January 2006

Marcelo Aguiar
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843, USAmaguiar@math.tamu.eduhttp://www.math.tamu.edu/˜maguiar
Nantel Bergeron
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canadabergeron@mathstat.yorku.cahttp://www.math.yorku.ca/bergeron
Frank Sottile
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843, USAsottile@math.tamu.eduhttp://www.math.tamu.edu/˜sottile
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Abstract

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A combinatorial Hopf algebra is a graded connected Hopf algebra over a field $\Bbbk$ equipped with a character (multiplicative linear functional) $\zeta\colon{\mathcal H}\to \Bbbk$. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra ${\mathcal Q}{\mathit{Sym}}$ 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 $({\mathcal H},\zeta)$ 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 ${\mathcal H}={\mathcal Q}{\mathit{Sym}}$, the generalized Dehn–Sommerville relations are the Bayer–Billera relations and the odd subalgebra is the peak Hopf algebra of Stembridge. We prove that ${\mathcal Q}{\mathit{Sym}}$ 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.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2006
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