Convolution Powers of the Identity

Abstract

We study convolution powers $\mathtt{id}^{\ast n}$ of the identity of graded connected Hopf algebras $H$. (The antipode corresponds to $n=-1$.) The chief result is a complete description of the characteristic polynomial - both eigenvalues and multiplicity - for the action of the operator $\mathtt{id}^{\ast n}$ on each homogeneous component $H_m$. The multiplicities are independent of $n$. This follows from considering the action of the (higher) Eulerian idempotents on a certain Lie algebra $\mathfrak{g}$ associated to $H$. In case $H$ is cofree, we give an alternative (explicit and combinatorial) description in terms of palindromic words in free generators of $\mathfrak{g}$. We obtain identities involving partitions and compositions by specializing $H$ to some familiar combinatorial Hopf algebras.

Keywords

• hopf power
• antipode
• eulerian idempotent
• graded connected hopf algebra
• schur indicator.
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]