Hopf monoids from class functions on unitriangular matrices

TL;DR

This paper constructs Hopf monoids from class functions on unitriangular matrices, linking algebraic structures with combinatorial models and deriving estimates on conjugacy classes.

Contribution

It introduces a new Hopf monoid structure on class functions of unitriangular matrices and provides a combinatorial model using species theory.

Findings

Hopf monoids are built from class functions on unitriangular matrices.

The structure is a free monoid in species with a canonical Hopf structure.

Estimates on the number of conjugacy classes of unitriangular matrices are derived.

Abstract

We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal's category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatorial model for the Hopf monoid of superclass functions, in terms of the Hadamard product of the Hopf monoids of linear orders and of set partitions. This implies a recent result relating the Hopf algebra of superclass functions on unitriangular matrices to symmetric functions in noncommuting variables. We determine the algebraic structure of the Hopf monoid: it is a free monoid in species, with the canonical Hopf structure. As an application, we derive certain estimates on the number of conjugacy classes of unitriangular matrices.