Hopf monoids from class functions on unitriangular matrices | CU Experts

Overview

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.