Formal group laws and hypergraph colorings

Abstract

This thesis demonstrates a connection between formal group laws and chromatic symmetric functions of hypergraphs, two seemingly unrelated topics in the theory of symmetric functions. A formal group law is a symmetric function of the form $f(f^{-1}(x_1) + f^{-1}(x_2) + \cdots)$ for a formal power series $f(x) \in \mathbb{Q}[[x]]$. A \emph{hypergraph} $H$ is a generalized graph where edges can contain more than two vertices, and a coloring $\chi$ of $H$ is \emph{proper} if no edge of $H$ is monochromatic under $\chi$. The \emph{chromatic symmetric function} $X_H$ of hypergraph $H$ on a vertex set $V$ is a sum of monomials corresponding to proper colorings of $H$. Our main result gives a combinatorial interpretation for certain formal group laws. In particular, we show that if $f(x)$ is a generating function for a type of combinatorial object with a certain recursive structure then the associated formal group law is a sum of chromatic symmetric functions. For example, this occurs when $f(x)$ is the generating function for trees, permutations, lattice paths or graphs. We develop a unifying framework by defining \emph{contractible species}, classes of combinatorial objects having generating functions with this property. This will also allow us to give combinatorial interpretations to products of polynomials in some well-known families of polynomials, such as the Bell, Laguerre and Hermite polynomials. Finally, we observe that many of the hypergraphs arising from formal group laws are special hypergraphs called \emph{hypertrees}, and we show that the chromatic symmetric functions of hypertrees are positive in Gessel's fundamental quasisymmetric functions when they have prime-sized edges.