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On the Hadamard Product of Hopf Monoids

Published online by Cambridge University Press:  20 November 2018

Marcelo Aguiar  and
Swapneel Mahajan
Marcelo Aguiar
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843. e-mail: maguiar@math.tamu.edu
Swapneel Mahajan
Affiliation:
Department of Mathematics, Indian Institute of Technology Mumbai, Powai, Mumbai 400 076, India. e-mail: swapneel@math.iitb.ac.in
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Abstract

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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.

Keywords

16T3018D3520B3018D1020F55speciesHopf monoidHadamard productgenerating functionBoolean transform
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 3 , 01 June 2014 , pp. 481 - 504
Copyright
Copyright © Canadian Mathematical Society 2014

References

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