Combinatorial Hopf algebras and generalized Dehn–Sommerville relations

By Marcelo Aguiar, Nantel Bergeron, Frank Sottile. Compositio Mathematica, 2003

278 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f4ef84074d7ad861a2e8d25d084a25950130fe27

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Citations in the corpus

Quasisymmetric functions and Kazhdan-Lusztig polynomials (2007, background): “The association [u, v] 7→ F̃ (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.”
Modules of the 0-Hecke algebra and quasisymmetric Schur functions (2014, background): “They have also been seen to simplify many symmetric functions such as Macdonald polynomials [26, 27], Kazhdan-Lusztig polynomials [8] and identify equal ribbon Schur functions [10], as well as arising in category theory [2], topology [4, 25], and graph theory [15, 32, 51].”
Hopf algebras and dendriform structures arising from parking functions (2005, background): “With parking functions, we have other possibilities: for example, (102) j(Sn) := F11···1 is a Hopf embedding, whose dual j∗ maps PQSym∗ to QSym and therefore endows PQSym∗ with a different structure of combinatorial Hopf algebra in the sense of [1].”
A Lift of the Schur and Hall–Littlewood Bases to Non-commutative Symmetric Functions (2012, background): “For example, the composition [4, 1, 3, 1, 6, 2] is represented as”
Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras (2010, background): “Therefore, the elements κ[n] are algebraically independent in LSC, and LSC contains the free algebra C〈κ[1], κ[2], .”
Skew quasisymmetric Schur functions and noncommutative Schur functions (2010, background): “The nonsymmetric analogue QSym is the Hopf algebra of quasisymmetric functions, and since its introduction as a source of generating functions for P-partitions [24] quasisymmetric functions have been identified as generating functions for flags in graded posets [17] and matroids [12]; were shown to ”
Hopf Monoids and Generalized Permutahedra (2017, background): “This section is closely related to combinatorial Hopf algebras [1] and to [12].”
The peak algebra and the descent algebras of types B and D (2003, background): “More recent works uncovered connections between peaks and such disparate topics as the generalized Dehn-Sommerville equations [6, 10, 1] and the Schubert calculus of isotropic (cid:13)ag manifolds [11, 8, 7].”
HOPF MONOIDS IN THE CATEGORY OF SPECIES (2012, background): “e sum of the maps q(F) →q(F|S)⊗q(F|T) x1⊗···⊗xk→ ( qschS,T (F)(x i 1⊗···⊗xij )⊗(xi′ 1 ⊗···⊗xi′ k ) if Sis F-admissible, 0 otherwise. Here F = (I1,...,Ik), xi ∈q[Ii] for each i, and schS,T(F) is as in (2). In the admissible case, we have written F|S= (Ii 1,...,Iij ) and F|T = (Ii′ 1 ,...,Ii′ k ). The”
Indecomposable modules for the dual immaculate basis of quasi-symmetric functions (2013, unknown): “mon’s descent algebra in type A[MR], the algebra of quasi-symmetric functions [MR], and representation theory of the type AHecke algebra at q= 0 [KT]. It is an example of a combinatorial Hopf algebra [ABS]. While we will follow the foundational results and denitions from references such as [GKLLRT,M”