Combinatorial Hopf algebras and generalized Dehn–Sommerville relations
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A combinatorial Hopf algebra is a graded connected Hopf algebra over a field doublestruck k sign equipped with a character (multiplicative linear functional) ζ: H → doublestruck k sign. 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, ζ) possesses two canonical Hopf subalgebras on which the character ζ 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. © Foundation Compositio Mathematica 2006.
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Aguiar, M., Bergeron, N., & Sottile, F.
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Character

Eulerian Poset

Generalized Dehn Sommerville Relations

Hopf Algebra

Noncommutative Symmetric Function

Quasisymmetric Function

Symmetric Function
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