On the Hadamard product of Hopf monoids

TL;DR

This paper investigates the properties of the Hadamard product of Hopf monoids in species, proving conditions for freeness and providing explicit bases, with implications for the dimension sequences of such structures.

Contribution

It establishes new conditions under which the Hadamard product of Hopf monoids is free and constructs explicit bases for these products, advancing the understanding of their algebraic structure.

Findings

Hadamard product of a connected and a free Hopf monoid is free

Explicit basis constructed for Hadamard product of two free Hopf monoids

Dimension sequence of a connected Hopf monoid has nonpositive coefficients in the reciprocal of its generating function

Abstract

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.