Combinatorial Hopf algebra of superclass functions of type $D$

TL;DR

This paper constructs a Hopf algebra structure on superclass functions for type D unipotent groups over finite fields, extending supercharacter theory to classical Lie types B and C, building on prior work for type A.

Contribution

It introduces a Hopf algebra framework for superclass functions of type D, advancing the combinatorial understanding of supercharacter theories beyond type A.

Findings

Established a Hopf algebra structure for type D superclass functions

Extended supercharacter theory insights to types B and C

Connected combinatorial structures across classical Lie types

Abstract

We provide a Hopf algebra structure on the space of superclass functions on the unipotent upper triangular group of type D over a finite field based on a supercharacter theory constructed by Andr\'e and Neto. Also, we make further comments with respect to types B and C. Type A was explores by M. Aguiar et. al (2010), thus this paper is a contribution to understand combinatorially the supercharacter theory of the other classical Lie types.