 
Summary: HOPF MONOIDS FROM CLASS FUNCTIONS ON
UNITRIANGULAR MATRICES
MARCELO AGUIAR, NANTEL BERGERON, AND NATHANIEL THIEM
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.
Introduction
A Hopf monoid (in Joyal's category of species) is an algebraic structure akin to that
of a Hopf algebra. Combinatorial structures which compose and decompose give rise
to Hopf monoids. These objects are the subject of [4, Part II]. The few basic notions
and examples needed for our purposes are reviewed in Section 1, including the Hopf
monoids of linear orders, set partitions, and simple graphs, and the Hadamard product
