On the Hopf algebra of multi-complexes

Abstract

We introduce a general class of combinatorial objects, which we call \emph{multi-complexes}, which simultaneously generalizes graphs, multigraphs, hypergraphs and simplicial and delta complexes. We introduce a natural algebra of multi-complexes which is defined as the algebra which has a formal basis $\mathcal{C}$ of all isomorphism types of multi-complexes, and multiplication is to take the disjoint union. This is a Hopf algebra with an operation encoding the dissasembly information for such objects, and extends the Hopf algebra of graphs. In our main result, we explicitly describe here the structure of this Hopf algebra of multi-complexes $H$. We find an explicit basis $\mathcal{B}$ of the space of primitives, which is of combinatorial relevance: it is such that each multi-complex is a polynomial with non-negative integer coefficients of the elements of $\mathcal{B}$, and each $b\in\mathcal{B}$ is a polynomial with integer coefficients in $\mathcal{C}$. Using this, we find the cancellation and grouping free formula for the antipode. The coefficients appearing in all these polynomials are, up to signs, numbers counting multiplicities of sub-multi-complexes in a multi-complex. We also explicitly illustrate how our results specialize to the graph Hopf algebra, and observe how they specialize to results in all of the above mentioned particular cases. We also investigate applications of these results to the graph reconstruction conjectures, and rederive some results in the literature on these questions.