Structure of the Malvenuto-Reutenauer Hopf algebra of permutations (Extended Abstract)

We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration and show that it decomposes as a crossed product over the Hopf algebra of quasi-symmetric functions. We also describe the structure constants of the multiplication as a certain number of facets of the permutahedron. Our results reveal a close relationship between the structure of this Hopf algebra and the weak order on the symmetric groups.Comment: 12 pages, 2 .eps figures. (minor revisions) Extended abstract for Formal Power Series and Algebraic Combinatorics, Melbourne, July 200

On the Hadamard product of Hopf monoids

Combinatorial structures which compose and decompose give rise to Hopf monoids in Joyal's category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The first one states that if one factor is connected and the other is free as a monoid, their Hadamard product is free (and connected). The second provides an explicit basis for the Hadamard product when both factors are free. The first main result is obtained by showing the existence of a one-parameter deformation of the comonoid structure and appealing to a rigidity result of Loday and Ronco which applies when the parameter is set to zero. To obtain the second result, we introduce an operation on species which is intertwined by the free monoid functor with the Hadamard product. As an application of the first result, we deduce that the dimension sequence of a connected Hopf monoid satisfies the following condition: except for the first, all coefficients of the reciprocal of its generating function are nonpositive

Structure of the Loday-Ronco Hopf algebra of trees

Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of non-commutative symmetric functions in the Malvenuto-Reutenauer Hopf algebra of permutations factors through their Hopf algebra of trees, and these maps correspond to natural maps from the weak order on the symmetric group to the Tamari order on planar binary trees to the boolean algebra. We further study the structure of this Hopf algebra of trees using a new basis for it. We describe the product, coproduct, and antipode in terms of this basis and use these results to elucidate its Hopf-algebraic structure. We also obtain a transparent proof of its isomorphism with the non-commutative Connes-Kreimer Hopf algebra of Foissy, and show that this algebra is related to non-commutative symmetric functions as the (commutative) Connes-Kreimer Hopf algebra is related to symmetric functions.Comment: 32 pages, many .eps pictures in color. Minor revision

The characteristic polynomial of the Adams operators on graded connected Hopf algebras

The Adams operators $\Psi_n$ on a Hopf algebra $H$ are the convolution powers of the identity of $H$. We study the Adams operators when $H$ is graded connected. They are also called Hopf powers or Sweedler powers. The main result is a complete description of the characteristic polynomial (both eigenvalues and their multiplicities) for the action of the operator $\Psi_n$ on each homogeneous component of $H$. The eigenvalues are powers of $n$. The multiplicities are independent of $n$, and in fact only depend on the dimension sequence of $H$. These results apply in particular to the antipode of $H$ (the case $n=-1$). We obtain closed forms for the generating function of the sequence of traces of the Adams operators. In the case of the antipode, the generating function bears a particularly simple relationship to the one for the dimension sequence. In case H is cofree, we give an alternative description for the characteristic polynomial and the trace of the antipode in terms of certain palindromic words. We discuss parallel results that hold for Hopf monoids in species and $q$-Hopf algebras.Comment: 36 pages; two appendice

Combinatorics of the free Baxter algebra

We study the free (associative, non-commutative) Baxter algebra on one generator. The first explicit description of this object is due to Ebrahimi-Fard and Guo. We provide an alternative description in terms of a certain class of trees, which form a linear basis for this algebra. We use this to treat other related cases, particularly that in which the Baxter map is required to be quasi-idempotent, in a unified manner. Each case corresponds to a different class of trees. Our main focus is on the underlying combinatorics. In several cases, we provide bijections between our various classes of trees and more familiar combinatorial objects including certain Schroeder paths and Motzkin paths. We calculate the dimensions of the homogeneous components of these algebras (with respect to a bidegree related to the number of nodes and the number of angles in the trees) and the corresponding generating series. An important feature is that the combinatorics is captured by the idempotent case; the others are obtained from this case by various binomial transforms. We also relate free Baxter algebras to Loday's dendriform trialgebras and dialgebras. We show that the free dendriform trialgebra (respectively, dialgebra) on one generator embeds in the free Baxter algebra with a quasi-idempotent map (respectively, with a quasi-idempotent map and an idempotent generator). This refines results of Ebrahimi-Fard and Guo.Comment: Fixed errata about grading in the idempotent cas