Introduction

One of the classical objects in asymptotic representation theory is the infinite symmetric group. The main objective of this course is to explore various questions and methods of the theory by examining the Plancherel measure on the set of Young diagrams, which parametrize the irreducible representations of symmetric groups. We will begin with the study of the limit shape using variational calculus techniques, followed by a discussion on the fluctuations around the limit shape. The theory of determinantal point processes provides a convenient framework for studying these fluctuations, and it also allows us to establish connections with random matrices, and also with TASEP and similar point processes.

This course serves as a continuation of the representation theory of symmetric groups course. However, we intend to review all the main concepts discussed in the previous course at the beginning of the current course. The only prerequisites are basic courses on algebra, probability and functional analysis. Additionally, we will briefly discuss the connection between this course and "From free fermions to limit shapes and beyond" by Anton Nazarov.