Combinatorial Hopf algebras and generalized Dehn–Sommerville relations

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 quasisymmetric 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 LodayRonco Hopf algebra of planar binary trees, the Hopf algebras of symmetric functions and of non-commutative symmetric functions.

Quasisymmetric functions and Kazhdan-Lusztig polynomials

“…The second of these is essentially in [33]. in Θ w is 2 n−|w| d , so we can conclude (1). Similarly, the coefficient of L {1,3,5,...} in Θ w is 1, so (2) follows.…”

“…The association [u, v] → F (u, v) can be viewed as a morphism of Hopf algebras, suggesting the possibility of a filtered version of the theory of combinatorial Hopf algebras [1], which predicts the existence of graded maps to the quasisymmetric functions in general combinatorial settings.…”

Colored Posets and Colored Quasisymmetric Functions

“…The product ⊔ and coproduct δ, along with the unit and counit, give P the structure of a graded connected Hopf algebra. This algebra was presented in [1,Example 2.3]. We should be clear that P is distinct from Rota's Hopf algebra of (isomorphism classes of finite) graded posets [20,11,7].…”

“…In addition to its interpretation in terms of P -partitions, the maps Γ and Λ also have simple interpretations in the context of Aguiar, Bergeron, and Sottile's theory of combinatorial Hopf algebras [1]. A combinatorial Hopf algebra is a pair (H, ϕ), where H is a graded connected Hopf algebra and ϕ is a character, i.e., an algebra homomorphism H → Q.…”

“…Examples include the Malvenuto-Reutenauer algebra of permutations, the Loday-Ronco algebra of planar binary trees, and the classic symmetric functions. The interest in Hopf algebras is both reflected in and heightened by the work of Marcelo Aguiar, Nantel Bergeron, and Frank Sottile [1], who define the category of combinatorial Hopf algebras. The algebra of quasisymmetric functions, Qsym, is the terminal object in this category.…”