Vol. 7, No. 7, 2013

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ISSN: 1944-7833 (e-only)
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Hopf monoids from class functions on unitriangular matrices

Marcelo Aguiar, Nantel Bergeron and Nathaniel Thiem

Vol. 7 (2013), No. 7, 1743–1779
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.

Keywords
unitriangular matrix, class function, superclass function, Hopf monoid, Hopf algebra
Mathematical Subject Classification 2010
Primary: 05E10
Secondary: 05E05, 05E15, 16T05, 16T30, 18D35, 20C33
Milestones
Received: 10 April 2012
Revised: 8 August 2012
Accepted: 7 October 2012
Published: 12 October 2013
Authors
Marcelo Aguiar
Department of Mathematics
Texas A&M University
College Station, TX
77843
United States
http://www.math.tamu.edu/~maguiar
Nantel Bergeron
Department of Mathematics and Statistics
York University
Toronto, ON
M3J 1P3
Canada
http://www.math.yorku.ca/bergeron
Nathaniel Thiem
Department of Mathematics
University of Colorado
Boulder, CO
80309
United States
http://math.colorado.edu/~thiemn