Pattern groups and a poset based Hopf monoid

TL;DR

This paper generalizes the Hopf algebra structure related to supercharacter theory from algebra groups to pattern groups, introducing a new basis and computing key structure constants.

Contribution

It extends the supercharacter-based Hopf monoid framework to pattern groups and introduces a new canonical basis with explicit structure constant calculations.

Findings

Introduces a third canonical basis for the Hopf monoid

Computes coproducts and antipodes for the new bases

Generalizes the supercharacter theory to a broader class of groups

Abstract

The supercharacter theory of algebra groups gave us a representation theoretic realization of the Hopf algebra of symmetric functions in noncommuting variables. The underlying representation theoretic framework comes equipped with two canonical bases, one of which was completely new in terms of symmetric functions. This paper simultaneously generalizes this Hopf structure by considering a larger class of groups while also restricting the representation theory to a more combinatorially tractable one. Using the normal lattice supercharacter theory of pattern groups, we not only gain a third canonical basis, but also are able to compute numerous structure constants in the corresponding Hopf monoid, including coproducts and antipodes for the new bases.