An introduction to the half-infinite wedge

TL;DR

This paper introduces the half-infinite wedge representation of the infinite symmetric group, explaining its tools and applications in deriving limit shapes of distributions on partitions, along with variational methods for computation.

Contribution

It provides an accessible exposition of the half-infinite wedge representation and demonstrates its use in analyzing limit shapes of partition distributions.

Findings

Derived limit shapes for various distributions on partitions

Explained the application of the half-infinite wedge representation

Reviewed variational methods for computing limit shapes

Abstract

After a quick review of the representation theory of the symmetric group, we give an exposition of the tools brought about by the so-called half-infinite wedge representation of the infinite symmetric group. We show how these can be applied to find the limit shapes of several distributions on partitions. We also briefly review the variational methods available to compute these limit shapes.