Hopf Monoids Research Articles

Hopf Monoids of Ordered Simplicial Complexes

Abstract We study Hopf classes: families of pure ordered simplicial complexes that give rise to Hopf monoids under join and deletion/contraction. The prototypical Hopf class is the family of ordered matroids. The idea of a Hopf class leads to a systematic study of simplicial complexes related to matroids, including shifted complexes and broken-circuit complexes. We compute the Hopf antipodes in two cases: facet-initial complexes (which generalize shifted complexes) and unbounded ordered matroids. The latter calculation uses the topological method of Aguiar and Ardila, complicated by certain auxiliary simplicial complexes that we call Scrope complexes, whose Euler characteristics control the coefficients of the antipode. The resulting antipode formula is multiplicity-free and cancellation-free.

Semi-abelian condition for color Hopf algebras

Recently, in [22], it was shown that the category of cocommutative Hopf algebras over an arbitrary field k is semi-abelian. We extend this result to the category of cocommutative color Hopf algebras, i.e. of cocommutative Hopf monoids in the symmetric monoidal category of G-graded vector spaces with G an abelian group, given an arbitrary skew-symmetric bicharacter on G, when G is finitely generated and the characteristic of k is different from 2 (not needed if G is finite of odd cardinality). We also prove that this category is action representable and locally algebraically cartesian closed, then algebraically coherent. In particular, these results hold for the category of cocommutative super Hopf algebras by taking G=Z2. Furthermore, we prove that, under the same assumptions on G and k, the abelian category of abelian objects in the category of cocommutative color Hopf algebras is given by those cocommutative color Hopf algebras which are also commutative.

Hopf Monoids and Generalized Permutahedra

Generalized permutahedra are polytopes that arise in combinatorics, algebraic geometry, representation theory, topology, and optimization. They possess a rich combinatorial structure. Out of this structure we build a Hopf monoid in the category of species.Species provide a unifying framework for organizing families of combinatorial objects. Many species carry a Hopf monoid structure and are related to generalized permutahedra by means of morphisms of Hopf monoids. This includes the species of graphs, matroids, posets, set partitions, linear graphs, hypergraphs, simplicial complexes, and building sets, among others. We employ this algebraic structure to define and study polynomial invariants of the various combinatorial structures.We pay special attention to the antipode of each Hopf monoid. This map is central to the structure of a Hopf monoid, and it interacts well with its characters and polynomial invariants. It also carries information on the values of the invariants on negative integers. For our Hopf monoid of generalized permutahedra, we show that the antipode maps each polytope to the alternating sum of its faces. This fact has numerous combinatorial consequences.We highlight some main applications:We obtain uniform proofs of numerous old and new results about the Hopf algebraic and combinatorial structures of these families. In particular, we give optimal formulas for the antipode of graphs, posets, matroids, hypergraphs, and building sets. They are optimal in the sense that they provide explicit descriptions for the integers entering in the expansion of the antipode, after all coefficients have been collected and all cancellations have been taken into account.We show that reciprocity theorems of Stanley and Billera–Jia–Reiner (BJR) on chromatic polynomials of graphs, order polynomials of posets, and BJR-polynomials of matroids are instances of one such result for generalized permutahedra.We explain why the formulas for the multiplicative and compositional inverses of power series are governed by the face structure of permutahedra and associahedra, respectively, providing an answer to a question of Loday.We answer a question of Humpert and Martin on certain invariants of graphs and another of Rota on a certain class of submodular functions.We hope our work serves as a quick introduction to the theory of Hopf monoids in species, particularly to the reader interested in combinatorial applications. It may be supplemented with Marcelo Aguiar and Swapneel Mahajan’s 2010 and 2013 works, which provide longer accounts with a more algebraic focus.

A [formula omitted]-compatible Hopf algebra of unitriangular class functions

This paper constructs a novel Hopf algebra cf(UT•) on the class functions of the unipotent upper triangular groups UTn(Fq) over a finite field. This construction is representation theoretic in nature and uses the machinery of Hopf monoids in the category of vector species. In contrast with a similar known construction, this Hopf algebra has the property that induction to the finite general linear group induces a homomorphism to Zelevinsky's Hopf algebra of GLn(Fq) class functions. Furthermore, cf(UT•) contains a Hopf subalgebra which is isomorphic to a known combinatorial Hopf algebra, previously used to prove a conjecture about chromatic quasisymmetric functions. Some additional Hopf algebraic properties are also established.

Functors for Long dimodules and Yetter–Drinfeld modules in a weak setting

In this paper, for two weak Hopf monoids H and B with invertible antipode, we define a functor between the category of left-left H\otimes B -Yetter–Drinfeld modules and the category of H - B -Long dimodules. We also show that, if moreover H is quasitriangular and B is coquasitriangular, this functor is a retraction of the well-known injective functor between left-left H - B -Long dimodules and left-left H\otimes B -Yetter–Drinfeld modules.

Hopf monoids, permutohedral cones, and generalized retarded functions

The commutative Hopf monoid of set compositions is a fundamental Hopf monoid internal to vector species, having undecorated bosonic Fock space the combinatorial Hopf algebra of quasisymmetric functions. We construct a geometric realization of this Hopf monoid over the adjoint of the (essentialized) braid hyperplane arrangement, which identifies the monomial basis with signed characteristic functions of the interiors of permutohedral tangent cones. We show that the indecomposable quotient Lie coalgebra is obtained by restricting functions to chambers of the adjoint arrangement, i.e., by quotienting out the higher codimensions. The resulting functions are characterized by the Steinmann relations of axiomatic quantum field theory, demonstrating an equivalence between the Steinmann relations, tangent cones to (generalized) permutohedra, and having algebraic structure internal to species. Our results give a new interpretation of a construction appearing in the mathematically rigorous formulation of renormalization by Epstein–Glaser, called causal perturbation theory. In particular, we show that operator products of time-ordered products correspond to the H-basis of the cocommutative Hopf monoid of set compositions, and generalized retarded products correspond to a spanning set of its primitive part Lie algebra.

On the Signature of a Path in an Operator Algebra

We introduce a class of operators associated with the signature of a smooth path $X$ with values in a $C^{\star}$ algebra $\mathcal{A}$. These operators serve as the basis of Taylor expansions of solutions to controlled differential equations of interest in noncommutative probability. They are defined by fully contracting iterated integrals of $X$, seen as tensors, with the product of $\mathcal{A}$. Were it considered that partial contractions should be included, we explain how these operators yield a trajectory on a group of representations of a combinatorial Hopf monoid. To clarify the role of partial contractions, we build an alternative group-valued trajectory whose increments embody full-contractions operators alone. We obtain therefore a notion of signature, which seems more appropriate for noncommutative probability.

Coloring complexes and combinatorial Hopf monoids

We generalize the notion of a coloring complex of a graph to linearized combinatorial Hopf monoids. We determine when a linearized combinatorial Hopf monoid has such a construction, and discover some inequalities that are satisfied by the quasisymmetric function invariants associated to the combinatorial Hopf monoid. We show that the collection of all such coloring complexes forms a linearized combinatorial Hopf monoid, which is the terminal object in the category of combinatorial Hopf monoids with convex characters. We also study several examples of combinatorial Hopf monoids.

Valuations and the Hopf Monoid of Generalized Permutahedra

AbstractThe goal of this paper is to show that valuation theory and Hopf theory are compatible on the class of generalized permutahedra. We prove that the Hopf structure $\textbf {GP}^+$ on these polyhedra descends, modulo the inclusion-exclusion relations, to an indicator Hopf monoid $\mathbb {I}(\textbf {GP}^+)$ of generalized permutahedra that is isomorphic to the Hopf monoid of weighted ordered set partitions. This quotient Hopf monoid $\mathbb {I}(\textbf {GP}^+)$ is cofree. It is the terminal object in the category of Hopf monoids with polynomial characters; this partially explains the ubiquity of generalized permutahedra in the theory of Hopf monoids. This Hopf theoretic framework offers a simple, unified explanation for many new and old valuations on generalized permutahedra and their subfamilies. Examples include, for matroids: the Chern–Schwartz–MacPherson cycles, Eur’s volume polynomial, the Kazhdan–Lusztig polynomial, the motivic zeta function, and the Derksen–Fink invariant; for posets: the order polynomial, Poincaré polynomial, and poset Tutte polynomial; for generalized permutahedra: the universal Tutte character and the corresponding class in the Chow ring of the permutahedral variety. We obtain several algebraic and combinatorial corollaries; for example, the existence of the valuative character group of $\textbf {GP}^+$ and the indecomposability of a nestohedron into smaller nestohedra.

The polytope algebra of generalized permutahedra

The polytope subalgebra of deformations of a zonotope can be endowed with the structure of a module over the Tits algebra of the corresponding hyperplane arrangement. We explore this construction and find relations between statistics on (signed) permutations and the module structure in the case of (type B) generalized permutahedra. In type B, the module structure surprisingly reveals that any family of generators (via signed Minkowski sums) for generalized permutahedra of type B will contain at least $2^{d-1}$ full-dimensional polytopes. We find a generating family of simplices attaining this minimum. Finally, we prove that the relations defining the polytope algebra are compatible with the Hopf monoid structure of generalized permutahedra.

Mapping class group actions from Hopf monoids and ribbon graphs

We show that any pivotal Hopf monoid H in a symmetric monoidal category \mathcal{C} gives rise to actions of mapping class groups of oriented surfaces of genus g\geq 1 with n\geq 1 boundary components. These mapping class group actions are given by group homomorphisms into the group of automorphisms of certain Yetter–Drinfeld modules over H . They are associated with edge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. We give a concrete description of these mapping class group actions in terms of generating Dehn twists and defining relations. For the case where \mathcal{C} is finitely complete and cocomplete, we also obtain actions of mapping class groups of closed surfaces by imposing invariance and coinvariance under the Yetter–Drinfeld module structure.

Long Dimodules and Quasitriangular Weak Hopf Monoids

In this paper, we prove that for any pair of weak Hopf monoids H and B in a symmetric monoidal category where every idempotent morphism splits, the category of H-B-Long dimodules HBLong is monoidal. Moreover, if H is quasitriangular and B coquasitriangular, we also prove that HBLong is braided. As a consequence of this result, we obtain that if H is triangular and B cotriangular, HBLong is an example of a symmetric monoidal category.

Integrals Along Bimonoid Homomorphisms

We introduce a notion of an integral along a bimonoid homomorphism as a simultaneous generalization of the integral and cointegral of bimonoids. The purpose of this paper is to characterize an existence of a specific integral, called a normalized generator integral, along a bimonoid homomorphism in terms of the kernel and cokernel of the homomorphism. We introduce a notion of a volume on an abelian category as a generalization of the dimension of vector spaces and the order of abelian groups. In applications, we show that there exists a nontrivial volume partially defined on a category of bicommutative Hopf monoids. The volume yields a notion of Fredholm homomorphisms between bicommutative Hopf monoids, which gives an analogue of the Fredholm index theory. This paper gives a technical preliminary of our subsequent paper about a construction of TQFT’s.

Heap and ternary self-distributive cohomology

Heaps are algebraic structures endowed with para-associative ternary operations, bijectively exemplified by groups via the operation They are also ternary self-distributive and have a diagrammatic interpretation in terms of framed links. Motivated by these properties, we define para-associative and heap cohomology theories and also a ternary self-distributive cohomology theory with abelian heap coefficients. We show that one of the heap cohomologies is related to group cohomology via a long exact sequence. Moreover, we construct maps between second cohomology groups of normalized group cohomology and heap cohomology, and show that the latter injects into the ternary self-distributive second cohomology group. We proceed to study heap objects in symmetric monoidal categories providing a characterization of pointed heaps as involutory Hopf monoids in the given category. Finally, we prove that heap objects are also “categorically” self-distributive in an appropriate sense.

The kernel of chromatic quasisymmetric functions on graphs and hypergraphic polytopes

The chromatic symmetric function on graphs is a celebrated graph invariant. Analogous chromatic maps can be defined on other objects, as presented by Aguiar, Bergeron and Sottile. The problem of identifying the kernel of some of these maps was addressed by Féray, for the Gessel quasisymmetric function on posets.On graphs, we show that the modular relations and isomorphism relations span the kernel of the chromatic symmetric function. This helps us to construct a new invariant on graphs, which may be helpful in the context of the tree conjecture. We also address the kernel problem in the Hopf algebra of generalized permutahedra, introduced by Aguiar and Ardila. We present a solution to the kernel problem on the Hopf algebra spanned by hypergraphic polytopes, which is a subfamily of generalized permutahedra that contains a number of polytope families.Finally, we consider the non-commutative analogues of these quasisymmetric invariants, and establish that the word quasisymmetric functions, also called non-commutative quasisymmetric functions, form the terminal object in the category of combinatorial Hopf monoids. As a corollary, we show that there is no combinatorial Hopf monoid morphism between the combinatorial Hopf monoid of posets and that of hypergraphic polytopes.

Quasisymmetric functions from combinatorial Hopf monoids and Ehrhart Theory

We investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes. This class of complexes, called forbidden composition complexes, also forms a Hopf monoid, thus demonstrating a link between Hopf algebras, Ehrhart theory, and commutative algebra. We also study various specializations of quasisymmetric functions.

The Hopf Monoid of Hypergraphs and its Sub-Monoids: Basic Invariant and Reciprocity Theorem

In arXiv:1709.07504 Aguiar and Ardila give a Hopf monoid structure on hypergraphs as well as a general construction of polynomial invariants on Hopf monoids. Using these results, we define in this paper a new polynomial invariant on hypergraphs. We give a combinatorial interpretation of this invariant on negative integers which leads to a reciprocity theorem on hypergraphs. Finally, we use this invariant to recover well-known invariants on other combinatorial objects (graphs, simplicial complexes, building sets, etc) as well as the associated reciprocity theorems.

The Hopf monoid of orbit polytopes

Many families of combinatorial objects have a Hopf monoid structure. Aguiar and Ardila introduced the Hopf monoid of generalized permutahedra and showed that it contains various other notable combinatorial families as Hopf submonoids, including graphs, posets, and matroids. We introduce the Hopf monoid of orbit polytopes, which is generated by the generalized permutahedra that are invariant under the action of the symmetric group. We show that modulo normal equivalence, these polytopes are in bijection with integer compositions. We interpret the Hopf structure through this lens, and we show that applying the first Fock functor to this Hopf monoid gives a Hopf algebra of compositions. We describe the character group of the Hopf monoid of orbit polytopes in terms of noncommutative symmetric functions, and we give a combinatorial interpretation of the basic character and its polynomial invariant.

Pattern groups and a poset based Hopf monoid

The supercharacter theory of algebra groups gave us a representation theoretic realization of the Hopf algebra of symmetric functions in noncommuting variables. The underlying representation theoretic framework comes equipped with two canonical bases, one of which was completely new in terms of symmetric functions. This paper simultaneously generalizes this Hopf structure by considering a larger class of groups while also restricting the representation theory to a more combinatorially tractable one. Using the normal lattice supercharacter theory of pattern groups, we not only gain a third canonical basis, but also are able to compute numerous structure constants in the corresponding Hopf monoid, including coproducts and antipodes for the new bases.

The antipode of linearized Hopf monoids

The antipode of linearized Hopf monoids