The Hopf monoid on nonnesting supercharacters of pattern groups

TL;DR

This paper develops a new supercharacter theory for certain unipotent matrix groups, forming a Hopf monoid with combinatorial structure, and demonstrates its freeness and detailed algebraic properties.

Contribution

It introduces a novel supercharacter theory based on nonnesting set partitions and constructs a free Hopf monoid from these supercharacters.

Findings

Supercharacter tables are explicitly computed.

The Hopf monoid's product and coproduct are described combinatorially.

The Hopf monoid is proven to be free.

Abstract

We construct supercharacter theories for a collection of unipotent matrix groups and produce a Hopf monoid from the supercharacters. These supercharacter theories are coarser than those defined by Diaconis--Isaacs for algebra groups and have supercharacters and superclasses indexed by nonnesting labeled set partitions. We compute the supercharacter tables and describe the product and coproduct of the Hopf monoid combinatorially. We also show that this Hopf monoid is free.