Eisenstein Series, Crystals and Ice

Automorphic forms are generalizations of periodic functions; they are functions on a group that are invariant under a discrete subgroup. A natural way to arrange this invariance is by averaging. Eisenstein series are an important class of functions obtained in this way. It is possible to give explicit formulas for their Fourier coefficients. Such formulas can provide clues to deep connections with other fields. As an example, Langlands’ study of Eisenstein series inspired his far-reaching conjectures that dictate the role of automorphic forms in modern number theory. In this article, we present two new explicit formulas for the Fourier coefficients of (certain) Eisenstein series, each given in terms of a combinatorial model: crystal graphs and square ice. Crystal graphs encode important data associated to Lie group representations while ice models arise in the study of statistical mechanics. Both will be described from scratch in subsequent sections. We were led to these surprising combinatorial connections by studying Eisenstein series not just on a group, but more generally on a family of covers of the group. We will present formulas for their Fourier coefficients which hold even in this generality. In the simplest case, the Fourier coefficients of Eisenstein series are described in terms of symmetric functions known as Schur polynomials, so that is where our story begins.