1,527 research outputs found

### 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

### 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

### 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

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