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On the Hadamard Product of Hopf Monoids

DSpace at IIT Bombay

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Title On the Hadamard Product of Hopf Monoids
 
Creator AGUIAR, M
MAHAJAN, S
 
Subject species
Hopf monoid
Hadamard product
generating function
Boolean transform
NONCOMMUTING VARIABLES
SYMMETRIC FUNCTIONS
ALGEBRAS
 
Description Combinatorial structures that compose and decompose give rise to Hopf monoids in Joyal's category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The first one states that if one factor is connected and the other is free as a monoid, their Hadamard product is free (and connected). The second provides an explicit basis for the Hadamard product when both factors are free. The first main result is obtained by showing the existence of a one-parameter deformation of the comonoid structure and appealing to a rigidity result of Loday and Ronco that applies when the parameter is set to zero. To obtain the second result, we introduce an operation on species that is intertwined by the free monoid functor with the Hadamard product. As an application of the first result, we deduce that the Boolean transform of the dimension sequence of a connected Hopf monoid is nonnegative.
 
Publisher CANADIAN MATHEMATICAL SOC
 
Date 2014-12-28T12:23:54Z
2014-12-28T12:23:54Z
2014
 
Type Article
 
Identifier CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 66(3)481-504
0008-414X
1496-4279
http://dx.doi.org/10.4153/CJM-2013-005-x
http://dspace.library.iitb.ac.in/jspui/handle/100/16506
 
Language English