Hopf monoids from class functionson unitriangular matrices

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 of Hopf monoids. The main goal of this paper is to construct a Hopf monoid out of the groups of unitriangular matrices with entries in a finite field, and to do this in a transparent manner. The structure exists on the collection of function spaces on these groups, and also on the collections of class function and superclass function spaces. It is induced by two simple operations on this collection of groups: the passage from a matrix to its principal minors gives rise to the product, and direct sum of matrices gives rise to the coproduct. Class functions are defined for arbitrary groups. An abstract notion and theory of superclass functions (and supercharacters) for arbitrary groups exists [12]. While a given group may admit several such theories, there is a canonical choice of superclasses for a special class of groups known as algebra groups. These notions are briefly discussed Date: October 16, 2012. 2010 Mathematics Subject Classification. 05E05; 05E10; 05E15; 16T05; 16T30; 18D35; 20C33.