Representations of the Infinite Symmetric Group (Cambridge Studies in Advanced Mathematics, Series Number 160)

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... 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. Alexei Borodin, Massachusetts Institute Of Technology And Institute For Information Transmission Problems Of The Russian Academy Of Sciences, Grigori Olshanski, Institute For Information Transmission Problems Of The Russian Academy Of Sciences, And National Research University Higher School Of Economics, Moscow. Includes Bibliographical References And Index.