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# On the Hadamard Product of Hopf Monoids

Published:2013-03-08
Printed: Jun 2014
• Marcelo Aguiar,
Department of Mathematics, Texas A&M University, College Station, TX 77843
• Swapneel Mahajan,
Department of Mathematics, Indian Institute of Technology Mumbai, Powai, Mumbai 400 076, India
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## Abstract

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: species, Hopf monoid, Hadamard product, generating function, Boolean transform
 MSC Classifications: 16T30 - Connections with combinatorics 18D35 - Structured objects in a category (group objects, etc.) 20B30 - Symmetric groups 18D10 - Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 20F55 - Reflection and Coxeter groups [See also 22E40, 51F15]

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