| Summary: | 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 coe cients. Such formulas can provide clues to deep connections
with other elds. 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 coe cients 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 coe cients which hold even in this generality. In the simplest case, the Fourier coe cients
of Eisenstein series are described in terms of symmetric functions known as Schur polynomials, so
that is where our story begins.
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