Combinatorial Hopf Algebras Research Articles

From Hurwitz numbers to Feynman diagrams: Counting rooted trees in log gravity

We show that the partition function of the logarithmic sector of critical topologically massive gravity which represents a series expansion of composition of functions, can be expressed as a sum over rooted trees. Our work brings a connection between integrable hierarchies of mathematical physics, combinatorial Hopf algebras and rooted trees, by explaining how the τ-functions of the (potential) Burgers and KP integrable hierarchies appearing in the partition function of log gravity conceal the Hopf algebra of composition of functions, known as the Faà di Bruno algebra, of the same type as the celebrated Connes-Kreimer Hopf algebra of rooted trees and Feynman diagrams. In particular, the Hurwitz numbers appearing in the partition function arise as coefficients of isomorphism classes of rooted trees. A parallel is drawn between our findings and established results in the statistical physics literature concerning certain systems with quenched disorder on trees, associated to nonlinear partial differential equations admitting traveling wave solutions. This should be of particular interest in view of a further description of the disorder observed in log gravity.

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.

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.

Hopf Structures in the Representation Theory of Direct Products

Combinatorial Hopf algebras give a linear algebraic structure to infinite families of combinatorial objects, a technique further enriched by the categorification of these structures via the representation theory of families of algebras. This paper examines a fundamental construction in group theory, the direct product, and how it can be used to build representation theoretic Hopf algebras out of towers of groups. A key special case gives us the noncommutative symmetric functions NSym, but there are many things that we can say for the general Hopf algebras, including the structure of their character groups and a formula for the antipode.

Hopf algebras of parking functions and decorated planar trees

We construct three new combinatorial Hopf algebras based on the Loday-Ronco operations on planar binary trees. The first and second algebras are defined on planar trees and labeled planar trees extending the Loday-Ronco and Malvenuto-Reutenauer Hopf algebras respectively. We show that the latter is bidendriform which implies that it is also free, cofree, and self-dual. The third algebra involves a new visualization of parking functions as decorated binary trees; it is also bidendriform, free, cofree, and self-dual, and therefore abstractly isomorphic to the algebra PQSym of Novelli and Thibon.We define partial orders on the objects indexing each of these three Hopf algebras, one of which, when restricting to (m+1)-ary trees, coarsens the m-Tamari order of Bergeron and Préville-Ratelle. We show that multiplication of dual fundamental basis elements is given by intervals in each of these orders.Finally, we use an axiomatized version of the techniques of Aguiar and Sottile on the Malvenuto-Reutenauer Hopf algebra to define a monomial basis on each of our Hopf algebras, and to show that comultiplication is cofree on the monomial elements. This in particular, implies the cofreeness of the Hopf algebra on planar trees. We also find explicit positive formulas for the multiplication on monomial basis and a cancellation-free and grouping-free formula for the antipode of monomial elements.

The "Young" and "Reverse" Dichotomy of Polynomials

A ``flip-and-reversal" involution arising in the study of quasisymmetric Schur functions provides a passage between what we term ``Young" and ``reverse" variants of bases of polynomials or quasisymmetric functions. Building on this perspective, which has found recent application in the study of q-analogues of combinatorial Hopf algebras and generalizations of dual immaculate functions, we develop and explore Young analogues of well-known bases for polynomials. We prove several combinatorial formulas for the Young analogue of the key polynomials, show that they form the generating functions for left keys, and provide a representation-theoretic interpretation of Young key polynomials as traces on certain modules. We also give combinatorial formulas for the Young analogues of Schubert polynomials, including their crystal graph structure. We moreover determine the intersections of (reverse) bases and their Young counterparts, further clarifying their relationships to one another.

Combinatorial Hopf algebras from representations of families of wreath products

We construct Hopf algebras whose elements are representations of combinatorial automorphism groups, by generalising a theorem of Zelevinsky on Hopf algebras of representations of wreath products. As an application we attach symmetric functions to representations of graph automorphism groups, generalising and refining Stanley's chromatic symmetric function.

Linear Compactness and Combinatorial Bialgebras

We present an expository overview of the monoidal structures in the category of linearly compact vector spaces. Bimonoids in this category are the natural duals of infinite-dimensional bialgebras. We classify the relations on words whose equivalence classes generate linearly compact bialgebras under shifted shuffling and deconcatenation. We also extend some of the theory of combinatorial Hopf algebras to bialgebras that are not connected or of finite graded dimension. Finally, we discuss several examples of quasi-symmetric functions, not necessarily of bounded degree, that may be constructed via terminal properties of combinatorial bialgebras.

A categorification of the Malvenuto–Reutenauer algebra via a tower of groups

There is a long tradition of categorifying combinatorial Hopf algebras by the modules of a tower of algebras (or even better via the representation theory of a tower of groups). From the point of view of combinatorics, such a categorification supplies canonical bases, inner products, and a natural avenue to prove positivity results. Recent ideas in supercharacter theory have made fashioning the representation theory of a tower of groups into a Hopf structure more tractable. This paper applies such a program to the Malvenuto–Reutenauer Hopf algebra. In particular, we design functors on the representation theory of a tower of p-groups that realize the Hopf structure of the Malvenuto–Reutenauer algebra in such a way that its well-known fundamental basis corresponds to a supercharacter basis.

Relations between infinitesimal non-commutative cumulants

Boolean, free and monotone cumulants as well as relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean cumulants were successfully used to study free infinite divisibility via the Boolean Bercovici-Pata bijection. On the other hand, in recent years the concept of infinitesimal non-commutative probability has been developed, together with the notion of infinitesimal cumulants which can be useful in the context of combinatorial questions. In this paper, we show that the known relations among free, Boolean and monotone cumulants still hold in the infinitesimal framework. Our approach is based on the use of algebra of Grassmann numbers. Formulas involving infinitesimal cumulants can be obtained by applying a formal derivation to known formulas. The relations between the various types of cumulants turn out to be captured via the shuffle algebra approach to moment-cumulant relations in non-commutative probability theory. In this formulation, (free, Boolean and monotone) cumulants are represented as elements of the Lie algebra of infinitesimal characters over a particular combinatorial Hopf algebra. The latter consists of the graded connected double tensor algebra defined over a non-commutative probability space and is neither commutative nor cocommutative. In this note it is shown how the shuffle algebra approach naturally extends to the notion of infinitesimal non-commutative probability space. The basic step consists in replacing the base field as target space of linear Hopf algebra maps by the algebra of Grassmann numbers defined over the base field. We also consider the infinitesimal analog of the Boolean Bercovici-Pata map.

Integer points enumerator of hypergraphic polytopes

For a hypergraphic polytope there is a weighted quasisymmetric function which enumerates positive integer points in its normal fan and determines its f-polynomial. This quasisymmetric function invariant of hypergraphs extends the Stanley chromatic symmetric function of simple graphs. We consider a certain combinatorial Hopf algebra of hypergraphs and show that universal morphism to quasisymmetric functions coincides with this enumerator function. We calculate the f-polynomial of uniform hypergraphic polytopes.

Lattices from graph associahedra and subalgebras of the Malvenuto–Reutenauer algebra

The Malvenuto–Reutenauer algebra is a well-studied combinatorial Hopf algebra with a basis indexed by permutations. This algebra contains a wide variety of interesting sub Hopf algebras, in particular the Hopf algebra of plane binary trees introduced by Loday and Ronco. We compare two general constructions of subalgebras of the Malvenuto–Reutenauer algebra, both of which include the Loday–Ronco algebra. The first is a construction by Reading defined in terms of lattice quotients of the weak order, and the second is a construction by Ronco in terms of graph associahedra. To make this comparison, we consider a natural partial ordering on the maximal tubings of a graph and characterize those graphs for which this poset is a lattice quotient of the weak order.

Lie groups of controlled characters of combinatorial Hopf algebras

In this article groups of “controlled” characters of a combinatorial Hopf algebra are considered from the perspective of infinite-dimensional Lie theory. A character is controlled in our sense if it satisfies certain growth bounds, e.g. exponential growth. We study these characters for combinatorial Hopf algebras. Following Loday and Ronco, a combinatorial Hopf algebra is a graded and connected Hopf algebra which is a polynomial algebra with an explicit choice of basis (usually identified with combinatorial objects such as trees, graphs, etc.). If the growth bounds and the Hopf algebra are compatible we prove that the controlled characters forminfinite-dimensional Lie groups. Further, we identify the Lie algebra and establish regularity results (in the sense of Milnor) for these Lie groups. The general construction principle exhibited here enables to treat a broad class of examples from physics, numerical analysis and control theory. Groups of controlled characters appear in renormalisation of quantum field theories, numerical analysis and control theory in the guise of groups of locally convergent power series. The results presented here generalise the construction of the (tame) Butcher group, also known as the controlled character group of the Butcher–Connes–Kreimer Hopf algebra.

Noncommutative unicellular LLT polynomials

It is known that unicellular LLT polynomials are related to the quasi-symmetric chromatic polynomials of certain graphs by the (t−1)-transform of symmetric functions. We investigate the extension of this transformation to various combinatorial Hopf algebras and prove a noncommutative version of this property.

(WEAK) INCIDENCE BIALGEBRAS OF MONOIDAL CATEGORIES

AbstractIncidence coalgebras of categories in the sense of Joni and Rota are studied, specifically cases where a monoidal product on the category turns these into (weak) bialgebras. The overlap with the theory of combinatorial Hopf algebras and that of Hopf quivers is discussed, and examples including trees, skew shapes, Milner’s bigraphs and crossed modules are considered.

Hopf algebras of m-permutations, (m + 1)-ary trees, and m-parking functions

The m-Tamari lattice of F. Bergeron is an analogue of the classical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or (m+1)-ary trees. On another hand, the Tamari order is related to the product in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is described by the m-Tamari lattices.In the same way as planar binary trees can be interpreted as sylvester classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what we call m-permutations. These objects are no longer in bijection with decreasing (m+1)-ary trees, and a finer congruence, called metasylvester, allows us to build Hopf algebras based on these decreasing trees. At the opposite, a coarser congruence, called hyposylvester, leads to Hopf algebras of graded dimensions (m+1)n−1, generalizing noncommutative symmetric functions and quasi-symmetric functions in a natural way. Finally, the algebras of packed words and parking functions also admit such m-analogues, and we present their subalgebras and quotients induced by the various congruences.

A simplicial complex splitting associativity

We introduce a simplicial object ({Dyckm}m≥0,Fi,Sj) in the category of non-symmetric algebraic operads, satisfying that Dyck0 is the operad of associative algebras and Dyck1 is J.-L. Loday’s operad of dendriform algebras. The dimensions of the operad Dyckm are given by the Fuss-Catalan numbers.Given a family of partially ordered sets P={Pn}n≥1 we show that, under certain conditions, the vector space spanned by the set of m-simpleces of P is a Dyckm algebra. This construction, applied to certain combinatorial Hopf algebras, whose associative product comes from a dendriform structure, provides examples of Dyckm algebras.

On the combinatorics of the Hopf algebra of dissection diagrams

In this article, we study the Hopf algebra $$\mathcal {H}_{\tiny \textsc {D}}$$ of dissection diagrams introduced by Dupont in his thesis, more precisely we focus on its underlying coalgebra. We use the version with a parameter x in the base field. We conjecture it is cofree if $$x=1$$ or x is not a root of unity. If $$x=-1$$, then we know there is no cofreeness. Since $$\mathcal {H}_{\tiny \textsc {D}}$$ is a free-commutative right-sided combinatorial Hopf algebra as defined by Loday and Ronco, then there exists a pre-Lie structure on the primitives of its graded dual. Furthermore $$\mathcal {H}_{\tiny \textsc {D}}^{\circledast }$$ and the enveloping algebra of its primitive elements are isomorphic. Thus, we can equip $$\mathcal {H}_{\tiny \textsc {D}}^{\circledast }$$ with a structure of Oudom–Guin. We focus on the pre-Lie structure on dissection diagrams and in particular on the pre-Lie algebra generated by the dissection diagram of degree 1. We prove that it is not free. We express a Hopf algebra morphism between the Grossman–Larson Hopf algebra and $$\mathcal {H}_{\tiny \textsc {D}}^{\circledast }$$ by using pre-Lie and Oudom–Guin structures.

Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here include ordered and unordered binary trees and ordered lists (labeled corollas). Some of the polytopes that constitute our main results are well known in other contexts. First we see the classical permutohedra, and then certain generalized permutohedra: specifically the graph associahedra of suspensions of certain simple graphs. As an aside we show that the stellohedra also appear as liftings of generalized permutohedra: graph composihedra for complete graphs. Thus our results give examples of Hopf algebras of tubings and marked tubings of graphs. We also show an alternative associative algebra structure on the graph tubings of star graphs.

Lumpings of Algebraic Markov Chains Arise from Subquotients

A function on the state space of a Markov chain is a "lumping" if observing only the function values gives a Markov chain. We give very general conditions for lumpings of a large class of algebraically-defined Markov chains, which include random walks on groups and other common constructions. We specialise these criteria to the case of descent operator chains from combinatorial Hopf algebras, and, as an example, construct a "top-to-random-with-standardisation" chain on permutations that lumps to a popular restriction-then-induction chain on partitions, using the fact that the algebra of symmetric functions is a subquotient of the Malvenuto-Reutenauer algebra.