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A universal framework for Hopf algebras in combinatorics

  • Farid Aliniaeifard
  • Stephanie van Willigenburg
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Abstract

A universal framework for Hopf algebras in combinatorics provides a unified understanding of combinatorial objects. To establish such a framework, we dualize the theory of combinatorial Hopf algebras developed by Aguiar, Bergeron, and Sottile, obtaining the category of rooted Hopf algebras. An object in this category is a graded connected Hopf algebra equipped with a group-like element, called a root, in its completion. These roots dualize the character theory of combinatorial Hopf algebras and provide a natural way to extend Bernstein and raising operators to other rooted Hopf algebras. The resulting constructions extend Schur, Hall--Littlewood, and Macdonald functions to rooted Hopf algebras. As an application, we develop these constructions for symmetric functions in noncommuting variables.

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Cite

BibTeXRIS
BibTeX
@misc{aliniaeifard2026universal,
  title = {{A universal framework for Hopf algebras in combinatorics}},
  author = {Farid Aliniaeifard and Stephanie van Willigenburg},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21289},
  eprint = {2610.21289},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21289},
}
RIS
TY  - UNPB
TI  - A universal framework for Hopf algebras in combinatorics
AU  - Farid Aliniaeifard
AU  - Stephanie van Willigenburg
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21289
UR  - https://arxiv.org/abs/2610.21289
AB  - A universal framework for Hopf algebras in combinatorics provides a unified understanding of combinatorial objects. To establish such a framework, we dualize the theory of combinatorial Hopf algebras developed by Aguiar, Bergeron, and Sottile, obtaining the category of rooted Hopf algebras. An object in this category is a graded connected Hopf algebra equipped with a group-like element, called a root, in its completion. These roots dualize the character theory of combinatorial Hopf algebras and provide a natural way to extend Bernstein and raising operators to other rooted Hopf algebras. The resulting constructions extend Schur, Hall--Littlewood, and Macdonald functions to rooted Hopf algebras. As an application, we develop these constructions for symmetric functions in noncommuting variables.
ER  -