A-Species

Abstract

We introduce a generalized notion of combinatorial species called $A$-species, where $A$ is a Hopf algebra. The role played by the symmetric group, $S_n$, in the classical theory of species is now replaced with the wreath product $A\wr S_n$. We show that category of $A$-species admits a monoidal structure under the Cauchy and Hadamard product. Under certain choices of $A$, we recover the notion of Joyal's species, $\mathcal{H}$-species defined by Choquette and Bergeron, and $B_r$-species defined by Henderson. We define many bilax monoidal functors involving the category of $A$-species. The first functor we construct, $S^A$, goes from the category of species to $A$-species. This functor sends the regular representation of the symmetric group $S_n$ to the regular representation of $A\wr S_n$, and we use this functor to construct the appropriate definitions of $A$-Hopf monoids built from common Hopf monoids. When $A=\mathbb{K}C_r$, we recover the functor defined by Choquette and Bergeron. We then define $A$-Fock functors, which are bilax functors between the category of $A$-species and the category of vector spaces. We analyze the images of certain $A$-Hopf monoids under them and show that they are all isomorphic as Hopf algebras to the Hopf algebra of polynomials invariant under the hyperoctrahedral group, $\mathbb{C}\langle\langle x\rangle \rangle^{B_r}$. We end by showing a sub Hopf algebra of a quotient of the ring of $C_r$-colored set partitions functions surjects onto $\mathbb{C}\langle\langle x\rangle \rangle^{B_r}$ and has a one-sided inverse.