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2013 Hopf monoids from class functions on unitriangular matrices
Marcelo Aguiar, Nantel Bergeron, Nathaniel Thiem
Algebra Number Theory 7(7): 1743-1779 (2013). DOI: 10.2140/ant.2013.7.1743


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.


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Marcelo Aguiar. Nantel Bergeron. Nathaniel Thiem. "Hopf monoids from class functions on unitriangular matrices." Algebra Number Theory 7 (7) 1743 - 1779, 2013.


Received: 10 April 2012; Revised: 8 August 2012; Accepted: 7 October 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1276.05127
MathSciNet: MR3117506
Digital Object Identifier: 10.2140/ant.2013.7.1743

Primary: 05E10
Secondary: 05E05 , 05E15 , 16T05 , 16T30 , 18D35 , 20C33

Keywords: class function , Hopf algebra , Hopf monoid , superclass function , unitriangular matrix

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 7 • 2013
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