Hopf monoids from class functions on unitriangular matrices. (English) Zbl 1276.05127

Aguiar, Marcelo; Bergeron, Nantel; Thiem, Nathaniel

Algebra Number Theory 7, No. 7, 1743-1779 (2013).

Summary: 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.

MSC:

05E15	Combinatorial aspects of groups and algebras (MSC2010)
05E10	Combinatorial aspects of representation theory
05E05	Symmetric functions and generalizations
16T05	Hopf algebras and their applications
16T30	Connections of Hopf algebras with combinatorics
18D35	Structured objects in a category (MSC2010)
20C33	Representations of finite groups of Lie type

Keywords:

unitriangular matrix; class function; superclass function; Hopf monoid; Hopf algebra

Software:

OEIS