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

  • 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.


Volume: DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)
Section: Proceedings
Published on: January 1, 2013
Imported on: November 21, 2016
Keywords: Hopf power,antipode,Eulerian idempotent,graded connected Hopf algebra,Schur indicator.,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Categories, Hopf Algebras, and Algebraic Combinatorics; Funder: National Science Foundation; Code: 1001935

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