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. Read more...
Abstract:
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. Offering a concise and self-contained exposition accessible to a wide audience, this book is a much-needed introduction to the basic concepts. Read more...
Author(s): Borodin, Alexei; Olshanski, Grigori
Series: Cambridge studies in advanced mathematics 160
Publisher: Cambridge University Press
Year: 2017
Language: English
Pages: 170
City: West Nyack
Content: Cover
Half-title page
Series page
Title page
Copyright page
Contents
Introduction
Part One Symmetric functions and Thoma's theorem
1 Preliminary Facts From Representation Theory of Finite Symmetric Groups
1.1 Exercises
1.2 Notes
2 Theory of Symmetric Functions
2.1 Exercises
2.2 Notes
3 Coherent Systems on the Young Graph
3.1 The Infinite Symmetric Group and the Young Graph
3.2 Coherent Systems
3.3 The Thoma Simplex
3.4 Integral Representation of Coherent Systems and Characters
3.5 Exercises
3.6 Notes
4 Extreme Characters and Thoma's Theorem
4.1 Thoma's Theorem. 7.4 Gibbs Measures7.5 Examples of Path Spaces for Branching Graphs
7.6 The Martin Boundary and the Vershik-Kerov Ergodic Theorem
7.7 Exercises
7.8 Notes
Part Two Unitary representations
8 Preliminaries and Gelfand Pairs
8.1 Exercises
8.2 Notes
9 Classification of General Spherical Type Representations
9.1 Notes
10 Realization of Irreducible Spherical Representations of (S(∞) × S(∞), diagS(∞))
10.1 Exercises
10.2 Notes
11 Generalized Regular Representations T[sub(z)]
11.1 Exercises
11.2 Notes
12 Disjointness of Representations T[sub(z)]
12.1 Preliminaries 12.2 Reduction to Gibbs Measures12.3 Exclusion of Degenerate Paths
12.4 Proof of Disjointness
12.5 Exercises
12.6 Notes
References
Index.