Combinatorial structures which 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 which applies when the parameter is set to zero. To obtain the
second result, we introduce an operation on species which is intertwined by the
free monoid functor with the Hadamard product. As an application of the first
result, we deduce that the dimension sequence of a connected Hopf monoid
satisfies the following condition: except for the first, all coefficients of
the reciprocal of its generating function are nonpositive
