Representations of the Infinite Symmetric Group

An introduction to the modern representation theory of big groups, exploring its connections to probability and algebraic combinatorics.

Alexei Borodin (Author), Grigori Olshanski (Author)

9781107175556, Cambridge University Press

Hardback, published 27 October 2016

168 pages, 2 b/w illus. 80 exercises
23.5 x 15.7 x 1.5 cm, 0.38 kg

'This book by A. Borodin and G. Olshanski is devoted to the representation theory of the infinite symmetric group, which is the inductive limit of the finite symmetric groups and is in a sense the simplest example of an infinite-dimensional group. … This book is the first work on the subject in the format of a conventional book, making the representation theory accessible to graduate students and undergraduates with a solid mathematical background. The book is very well written, with clean and clear exposition, and has a nice collection of exercises to help the engaged reader absorb the material. It does not assume a lot of background material, just some familiarity with the representation theory of finite groups, basic probability theory and certain results from functional analysis. … Among the many useful features of the book are its comprehensive list of references and notes after every section that direct the reader to the relevant literature to further explore the topics discussed.' Sevak Mkrtchyan, Mathematical Reviews

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.

Subject Areas: Mathematical physics [PHU], Algebra [PBF]