A universal framework for Hopf algebras in combinatorics

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.