Incidence Hopf Algebra Research Articles

Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulants

We consider the group (G,⁎) of unitized multiplicative functions in the incidence algebra of non-crossing partitions, where “⁎” denotes the convolution operation. We introduce a larger group (G˜,⁎) of unitized functions from the same incidence algebra, which satisfy a weaker semi-multiplicativity condition. The natural action of G˜ on sequences of multilinear functionals of a non-commutative probability space captures the combinatorics of transitions between moments and some brands of cumulants that are studied in the non-commutative probability literature. We use the framework of G˜ in order to explain why the multiplication of free random variables can be very nicely described in terms of Boolean cumulants and more generally in terms of t-Boolean cumulants, a one-parameter interpolation between free and Boolean cumulants arising from work of Bożejko and Wysoczanski.It is known that the group G can be naturally identified as the group of characters of the Hopf algebra Sym of symmetric functions. We show that G˜ can also be identified as group of characters of a Hopf algebra T, which is an incidence Hopf algebra in the sense of Schmitt. Moreover, the inclusion G⊆G˜ turns out to be the dual of a natural bialgebra homomorphism from T onto Sym.

The antipode of the noncrossing partition lattice

We prove the cancellation-free formula for the antipode of the noncrossing partition lattice in the reduced incidence Hopf algebra of posets due to Einziger. The proof is based on a map from chains in the noncrossing partition lattice to noncrossing hypertrees and expressing the alternating sum over these fibers as an Euler characteristic.

Antipodes and involutions

If H is a connected, graded Hopf algebra, then Takeuchi's formula can be used to compute its antipode. However, there is usually massive cancellation in the result. We show how sign-reversing involutions can sometimes be used to obtain cancellation-free formulas. We apply this idea to nine different examples. We rederive known formulas for the antipodes in the Hopf algebra of polynomials, the shuffle Hopf algebra, the Hopf algebra of quasisymmetric functions in both the monomial and fundamental bases, the Hopf algebra of multi-quasisymmetric functions in the fundamental basis, and the incidence Hopf algebra of graphs. We also find cancellation-free expressions for particular values of the antipode in the immaculate basis for the noncommutative symmetric functions as well as the Malvenuto–Reutenauer and Poirier–Reutenauer Hopf algebras, some of which are the first of their kind. We include various conjectures and suggestions for future research.

Semi-pointed partition posets and species

We define semi-pointed partition posets, which are a generalization of partition posets, and show that they are Cohen---Macaulay. We then use multichains to compute the dimension and the character for the action of the symmetric groups on their homology. We finally study the associated incidence Hopf algebra, which is similar to the Faa di Bruno Hopf algebra.

Semi-pointed partition posets

We present here a family of posets which generalizes both partition and pointed partition posets. After a short description of these new posets, we show that they are Cohen-Macaulay, compute their Moebius numbers and their characteristic polynomials. The characteristic polynomials are obtained using a combinatorial interpretation of the incidence Hopf algebra associated to these posets. Nous introduisons ici une famille de posets qui généralise à la fois les poset de partitions et les posets de partitions pointées. Après une description rapide de ces nouveaux posets, nous montrons qu’ils sont Cohen-Macaulay et nous calculons leurs nombres de Moebius et leurs polynômes caractéristiques. Ces derniers sont obtenus grâce à une interprétation combinatoire de l’algèbre de Hopf d’incidence associée à ces posets.

Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras

Bell polynomials appear in several combinatorial constructions throughout mathematics. Perhaps most naturally in the combinatorics of set partitions, but also when studying compositions of diffeomorphisms on vector spaces and manifolds, and in the study of cumulants and moments in probability theory. We construct commutative and noncommutative Bell polynomials and explain how they give rise to Faà di Bruno Hopf algebras. We use the language of incidence Hopf algebras, and along the way provide a new description of antipodes in noncommutative incidence Hopf algebras, involving quasideterminants. We also discuss Möbius inversion in certain Hopf algebras built from Bell polynomials.

An antipode formula for the natural Hopf algebra of a set operad

A symmetric set operad is a monoid in the category of combinatorial species with respect to the operation of substitution. From a symmetric set operad, we give here a simple construction of a commutative and non-co-commutative Hopf algebra, that we call the natural Hopf algebra of the operad. We obtain a combinatorial formula for its antipode in terms of Schröder trees, generalizing the Haiman–Schmitt formula for the Faá di Bruno Hopf algebra. From there we derive antipode formulas for specific operads. The classical Lagrange inversion formula is obtained in this way from the set operad of pointed sets. We also derive antipodes formulas for the natural Hopf algebra corresponding to the operads of connected graphs, the NAP operad, and for its generalization, the set operad of trees enriched with a monoid. When the set operad is left cancellative, we can construct a family of posets. The natural Hopf algebra is then obtained as a reduced incidence Hopf algebra, by taking a suitable equivalence relation over the intervals of that family of posets. We also present a simple combinatorial construction of an epimorphism from the natural Hopf algebra corresponding to the NAP operad, to the Connes and Kreimer Hopf algebra. For non-symmetric operads a similar construction leads to the world of non-commutative Hopf algebras. We recover from our general formula the Novelli–Thibon combinatorial form of the antipode for the non-commutative Hopf algebra of formal diffeomorphisms.

The Incidence Hopf Algebra of Graphs

The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite simple graphs and whose Hopf product and coproduct admit simple combinatorial descriptions. We give a new formula for the antipode in the graph algebra in terms of acyclic orientations; our formula contains many fewer terms than Schmitt's more general formula for the antipode in an incidence Hopf algebra. Applications include several formulas (some old and some new) for evaluations of the Tutte polynomial. L'algèbre de graphes est une algèbre d'incidence de Hopf commutative, cocommutative, graduée, et connexe, dont les éléments de base correspondent à des graphes finis simples et dont le produit et coproduit de Hopf admettent une description combinatoire simple. Nous présentons une nouvelle formule de l'antipode dans l'algèbre de graphes utilisant les orientations acycliques; notre formule contient beaucoup moins de termes que la formule générale de Schmitt pour l'antipode dans une algèbre d'incidence de Hopf. Les applications incluent plusieurs formules (connues et inconnues) pour les évaluations du polynôme de Tutte.

The Incidence Hopf Algebra of Graphs

The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite graphs, and whose Hopf product and coproduct admit simple combinatorial descriptions. We give a new formula for the antipode in the graph algebra in terms of acyclic orientations; our formula contains many fewer terms than Takeuchi's and Schmitt's more general formulas for the antipode in an incidence Hopf algebra. Applications include several formulas (some old and some new) for evaluations of the Tutte polynomial.

Relating Two Hopf Algebras Built from An Operad

Starting from an operad, one can build a family of posets. From this family of posets, one can define an incidence Hopf algebra. By another construction, one can also build a group directly from the operad. We then consider its Hopf algebra of functions. We prove that there exists a surjective morphism from the latter Hopf algebra to the former one. This is illustrated by the case of an operad built on rooted trees, the $\NAP$ operad, where the incidence Hopf algebra is identified with the Connes-Kreimer Hopf algebra of rooted trees.

Incidence categories

Given a family F of posets closed under disjoint unions and the operation of taking convex subposets, we construct a category C F called the incidence category of F . This category is “nearly abelian” in the sense that all morphisms have kernels/cokernels, and possesses a symmetric monoidal structure akin to direct sum. The Ringel–Hall algebra of C F is isomorphic to the incidence Hopf algebra of the collection P ( F ) of order ideals of posets in F . This construction generalizes the categories introduced by K. Kremnizer and the author, in the case when F is the collection of posets coming from rooted forests or Feynman graphs.

Pointed and multi-pointed partitions of type A and B

The aim of this paper is to define and study pointed and multi-pointed partition posets of type A and B (in the classification of Coxeter groups). We compute their characteristic polynomials, incidence Hopf algebras and homology groups. As a corollary, we show that some operads are Koszul over \(\mathbb{Z}\).

Zimmermann type cancellation in the free Faà di Bruno algebra

Haiman and Schmitt showed that one can use the antipode S F of the colored Faà di Bruno Hopf algebra F to compute the (compositional) inverse of a multivariable formal power series. It is shown that the antipode S H of an algebraically free analogue H of F may be used to invert non-commutative power series. Whereas F is the incidence Hopf algebra of the colored partitions of finite colored sets, H is the incidence Hopf algebra of the colored interval partitions of finite totally ordered colored sets. Haiman and Schmitt showed that the monomials in the geometric series for S F are labeled by trees. By contrast, the non-commuting monomials of S H are labeled by colored planar trees. The order of the factors in each summand is determined by the breadth first ordering on the vertices of the planar tree. Finally there is a parallel to Haiman and Schmitt's reduced tree formula for the antipode, in which one uses reduced planar trees and the depth first ordering on the vertices. The reduced planar tree formula is proved by recursion, and again by an unusual cancellation technique. The one variable case of H has also been considered by Brouder, Frabetti, and Krattenthaler, who point out its relation to Foissy's free analogue of the Connes–Kreimer Hopf algebra.

Economic analysis of a transesophageal echocardiography-guided approach to cardioversion of patients with atrial fibrillation: The ACUTE economic data at eight weeks

A symmetric set operad is a monoid in the category of combinatorial species with respect to the operation of substitution. From a symmetric set operad, we give here a simple construction of a commutative and non-co-commutative Hopf algebra, that we call the natural Hopf algebra of the operad. We obtain a combinatorial formula for its antipode in terms of Schröder trees, generalizing the Haiman–Schmitt formula for the Faá di Bruno Hopf algebra. From there we derive antipode formulas for specific operads. The classical Lagrange inversion formula is obtained in this way from the set operad of pointed sets. We also derive antipodes formulas for the natural Hopf algebra corresponding to the operads of connected graphs, the NAP operad, and for its generalization, the set operad of trees enriched with a monoid. When the set operad is left cancellative, we can construct a family of posets. The natural Hopf algebra is then obtained as a reduced incidence Hopf algebra, by taking a suitable equivalence relation over the intervals of that family of posets. We also present a simple combinatorial construction of an epimorphism from the natural Hopf algebra corresponding to the NAP operad, to the Connes and Kreimer Hopf algebra. For non-symmetric operads a similar construction leads to the world of non-commutative Hopf algebras. We recover from our general formula the Novelli–Thibon combinatorial form of the antipode for the non-commutative Hopf algebra of formal diffeomorphisms.

Galois Connections for Incidence Hopf Algebras of Partially Ordered Sets

An important well-known result of Rota describes the relationship between the Möbius functions of two posets related by a Galois connection. We present an analogous result relating the antipodes of the corresponding incidence Hopf algebras, from which the classical formula can be deduced. To motivate the derivation of this more general result, we first observe that a simple conceptual proof of Rota's classical formula can be obtained by interpreting it in terms of bimodules over the incidence algebras. Bimodules correct the apparent lack of functoriality of incidence algebras with respect to monotone maps. The theory of incidence Hopf algebras is reviewed from scratch and centered around the notion of cartesian posets. Also, the universal multiplicative function on a poset is constructed and an analog for antipodes of the classical Möbius inversion formula is presented.

Hopf Algebras and Edge-Labeled Posets

Given a finite graded poset with labeled Hasse diagram, we construct a quasi-symmetric generating function for chains whose labels have fixed descents. This is a common generalization of a generating function for the flag f-vector defined by Ehrenborg and of a symmetric function associated with certain edge-labeled posets that arose in the theory of Schubert polynomials. We show that this construction gives a Hopf morphism from an incidence Hopf algebra of edge-labeled posets to the Hopf algebra of quasi-symmetric functions.

On Posets and Hopf Algebras

We generalize the notion of the rank-generating function of a graded poset. Namely, by enumerating different chains in a poset, we can assign a quasi-symmetric function to the poset. This map is a Hopf algebra homomorphism between the reduced incidence Hopf algebra of posets and the Hopf algebra of quasi-symmetric functions. This work implies that the zeta polynomial of a poset may be viewed in terms Hopf algebras. In the last sections of the paper we generalize the reduced incidence Hopf algebra of posets to the Hopf algebra of hierarchical simplicial complexes.

Incidence Hopf algebras

We present several results about incidence Hopf algebras of families of partially ordered sets, including a characterization of their algebra structure, a combinatorial technique for finding generating sets of primitive elements in the cocommutative case, and a determinant formula for the antipode which holds for a class including the Faá di Bruno Hopf algebra. We introduce a variety of examples of incidence Hopf algebras arising from families of graphs, matroids and distributive lattices, many of which generalize well-known Hopf algebras.

Ultimate chromatic polynomials

We outline an approach to enumeration problems which relies on the algebra of free abelian groups, giving as our main application a generalisation of the chromatic polynomial of a simple graph G. Our polynomial lies in the free abelian group generated by the poset K( G) of contractions of G, and reduces to the classical case after a simple substitution. Its main properties may be stated in terms of an evaluation homomorphism, one for each positive integer t, which when applied to the polynomial yields an explicit list of the colourings of G with t colours. Considering posets larger than K( G), and enriching the algebra accordingly, we extend the whole construction to incorporate the corresponding incidence Hopf algebras. The enriched polynomial reduces by a similar substitution to the umbral chromatic polynomial of G.