Representations of the Infinite Symmetric Group Cambridge Studies in Advanced Mathematics Series

Representation theory of big groups is an important and quickly developing part of modern mathematics, giving rise to a variety of important applications in probability and mathematical physics. This book provides the first concise and self-contained introduction to the theory on the simplest yet very nontrivial example of the infinite symmetric group, focusing on its deep connections to probability, mathematical physics, and algebraic combinatorics. Following a discussion of the classical Thoma's theorem which describes the characters of the infinite symmetric group, the authors describe explicit constructions of an important class of representations, including both the irreducible and generalized ones. Complete with detailed proofs, as well as numerous examples and exercises which help to summarize recent developments in the field, this book will enable graduates to enhance their understanding of the topic, while also aiding lecturers and researchers in related areas.

Introduction; Part I. Symmetric Functions and Thoma's Theorem: 1. Preliminary facts from representation theory of finite symmetric groups; 2. Theory of symmetric functions; 3. Coherent systems on the Young graph; 4. Extreme characters and Thoma's Theorem; 5. A toy model (the Pascal Graph) and de Finetti's Theorem; 6. Asymptotics of relative dimension in the Young graph; 7. Boundaries and Gibbs measures on paths; Part II. Unitary Representations: 8. Preliminaries and Gelfand pairs; 9. Classification of general spherical type representations; 10. Realization of irreducible spherical representations of (S(∞) × S(∞), diagS(∞)); 11. Generalized regular representations Tz; 12. Disjointness of representations Tz; References; Index.

Alexei Borodin is a Professor of Mathematics at the Massachusetts Institute of Technology.
Grigori Olshanski is a Principal Researcher in the Section of Algebra and Number Theory at the Institute for Information Transmission Problems of the Russian Academy of Sciences, Moscow. He also holds the position of Dobrushin Professor at the National Research University Higher School of Economics, Moscow.