On the Hadamard Product of Hopf Monoids
Printed: Jun 2014
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.
species, Hopf monoid, Hadamard product, generating function, Boolean transform
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]