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

Primary classes of compositions of numbers

The compositions, or ordered partitions, of integers, fall under certain natural classes. In this expository paper we highlight the most important classes by means of bijective proofs. Some of the proofs rely on the properties of zig-zag graphs - the graphical representations of compositions introduced by Percy A. MacMahon in his classic book Combinatory Analysis. Keywords: composition, conjugate, zig-zag graph, line graph, bit-encoding, direct detection

Canonical characters on quasi-symmetric functions and bivariate Catalan numbers

Every character on a graded connected Hopf algebra decomposes uniquely as a product of an even character and an odd character (Aguiar, Bergeron, and Sottile, math.CO/0310016). We obtain explicit formulas for the even and odd parts of the universal character on the Hopf algebra of quasi-symmetric functions. They can be described in terms of Legendre's beta function evaluated at half-integers, or in terms of bivariate Catalan numbers: $$C(m,n)=\frac{(2m)!(2n)!}{m!(m+n)!n!}.$$ Properties of characters and of quasi-symmetric functions are then used to derive several interesting identities among bivariate Catalan numbers and in particular among Catalan numbers and central binomial coefficients

Classical symmetric functions in superspace

We present the basic elements of a generalization of symmetric function theory involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal action of the symmetric group on the sets of commuting and anticommuting variables. In this work, we present the superspace extension of the classical bases, namely, the monomial symmetric functions, the elementary symmetric functions, the completely symmetric functions, and the power sums. Various basic results, such as the generating functions for the multiplicative bases, Cauchy formulas, involution operations as well as the combinatorial scalar product are also generalized.Comment: 21 pages, this supersedes the first part of math.CO/041230

Factoring Formal Maps into Reversible or Involutive Factors

An element $g$ of a group is called reversible if it is conjugate in the group to its inverse. An element is an involution if it is equal to its inverse. This paper is about factoring elements as products of reversibles in the group $\mathfrak{G}_n$ of formal maps of $(\mathbb{C}^n,0)$, i.e. formally-invertible $n$-tuples of formal power series in $n$ variables, with complex coefficients. The case $n=1$ was already understood. Each product $F$ of reversibles has linear part $L(F)$ of determinant $\pm1$. The main results are that for $n\ge2$ each map $F$ with det$(L(F))=\pm1$ is the product of $2+3c$ reversibles, and may also be factored as the product of $9+6c$ involutions, where $c$ is the smallest integer $\ge \log_2n$.Comment: 20 page

Higher order peak algebras

Using the theory of noncommutative symmetric functions, we introduce the higher order peak algebras, a sequence of graded Hopf algebras which contain the descent algebra and the usual peak algebra as initial cases (N = 1 and N = 2). We compute their Hilbert series, introduce and study several combinatorial bases, and establish various algebraic identities related to the multisection of formal power series with noncommutative coefficients.Comment: 20 pages, AMS LaTe