Marcelo Aguiar ; Aaron Lauve

Convolution Powers of the Identity
dmtcs:2365 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2013,
DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)

https://doi.org/10.46298/dmtcs.2365
Convolution Powers of the IdentityArticle
Authors: Marcelo Aguiar ^{1}; Aaron Lauve ^{2}
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Marcelo Aguiar;Aaron Lauve
1 Department of Mathematics [Austin]
2 Department of Mathematics and Statistics [Chicago]
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.