· 2012· Preprint
Combinatorial Hopf algebras and generalized DehnSommerville relations
- 217citations
- 2012year
Short summary
Researchers introduce the concept of a combinatorial Hopf algebra, a structure with a character mapping to a field, and demonstrate that the algebra of quasi-symmetric functions is the universal example.
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Key points
- Introduced the concept of a combinatorial Hopf algebra with a character.
- Demonstrated that the algebra of quasi-symmetric functions (QSym) is the terminal object in this category.
- Proved that any character uniquely decomposes into an even and an odd component.
- Identified generalized Dehn-Sommerville relations defining the odd subalgebra.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract. A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) ζ: H → k. 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, 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 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 authors' abstract, as published at the source., 2012 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics