Hyperoctahedral species

TL;DR

This paper introduces H-species, a new algebraic framework for type B combinatorial structures involving signed permutations, and explores their algebraic properties and connections to Hopf monoids.

Contribution

It defines H-species analogous to classical species but incorporating signed permutations, and demonstrates their algebraic structures including examples of Hopf monoids.

Findings

H-species generalize classical species to type B structures

The paper constructs Hopf monoids within the H-species framework

The functorial approach links classical set compositions to the Hopf algebra DQSym.

Abstract

We introduce a new definition for the species of type B, or H-species, analog to the classical species (of type A), but on which we consider the action of the groups Bn of signed permutations. We are interested in algebraic structure on these H-species and give examples of Hopf monoids. The natural way to get a graded vector space from a species, given in this paper in terms of functors, will allow us to deepen our understanding of these species. In particular, the image of the classical species of set compositions under a given functor is isomorphic to the combinatorial Hopf algebra DQSym.