Antipode and Primitive elements in the Hopf Monoid of Super Characters

TL;DR

This paper explores the Hopf monoid structure of supercharacters in unipotent upper-triangular groups, providing explicit formulas for the antipode and primitive elements, advancing algebraic understanding of these structures.

Contribution

It introduces new cancelation-free formulas for the antipode on supercharacter bases and describes the primitives of the Hopf monoid, enhancing algebraic tools for supercharacter theory.

Findings

Cancelation-free antipode formulas for class functions and power sum functions

Partial results for antipode on supercharacter basis

Explicit description of primitives in the Hopf monoid

Abstract

From a recent paper, we recall the Hopf monoid structure on the supercharacters of the unipotent uppertriangular groups over a finite field. We give cancelation free formula for the antipode applied to the bases of class functions and power sum functions, giving new cancelation free formulae for the standard Hopf algebra of supercharacters and symmetric functions in noncommuting variables. We also give partial results for the antipode on the supercharacter basis, and explicitly describe the primitives of this Hopf monoid.