A universal framework for Hopf algebras in combinatorics

Farid Aliniaeifard; Stephanie van Willigenburg

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arXiv · 2610.21289

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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BibTeX RIS

Farid Aliniaeifard, Stephanie van Willigenburg. 2026-10-15. A universal framework for Hopf algebras in combinatorics. https://arxiv.org/abs/2610.21289

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