Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

TL;DR

This paper establishes an isomorphism between supercharacters used for Fourier analysis on unipotent upper-triangular matrices and the ring of symmetric functions in noncommuting variables, linking two Hopf algebra structures.

Contribution

It reveals a deep connection between supercharacters and noncommutative symmetric functions, enabling transfer of results and enriching the study of combinatorial Hopf algebras.

Findings

Supercharacters form a Hopf algebra structure.

The ring of symmetric functions in noncommuting variables is a Hopf algebra.

The two Hopf algebras are isomorphic, allowing cross-application of theories.

Abstract

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.