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Representations of the Infinite Symmetric Group

$70.99

Part of Cambridge Studies in Advanced Mathematics

  • Authors:
  • Alexei Borodin, Massachusetts Institute of Technology
  • Grigori Olshanski, Institute for Information Transmission Problems, Russian Academy of Sciences
  • Date Published: October 2016
  • availability: Available
  • format: Hardback
  • isbn: 9781107175556

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About the Authors
  • 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.

    • Explores the connections of representation theory to seemingly distant areas of mathematics, including probability and algebraic combinatorics
    • The only book offering an accessible and up-to-date introduction to this material
    • Includes numerous exercises and examples which help to summarize recent developments
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    Reviews & endorsements

    '… the aim of this book is to provide a detailed introduction to the representation theory of S(∞) in such a way that would be accessible to graduate and advanced undergraduate students. At the end of each section of the book, there are exercises and notes which are helpful for students who choose the book for the course.' Mohammad-Reza Darafsheh, Zentralblatt MATH

    '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

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    Product details

    • Date Published: October 2016
    • format: Hardback
    • isbn: 9781107175556
    • length: 168 pages
    • dimensions: 235 x 157 x 15 mm
    • weight: 0.37kg
    • contains: 2 b/w illus. 80 exercises
    • availability: Available
  • Table of Contents

    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.

  • Authors

    Alexei Borodin, Massachusetts Institute of Technology
    Alexei Borodin is a Professor of Mathematics at the Massachusetts Institute of Technology.

    Grigori Olshanski, Institute for Information Transmission Problems, Russian Academy of Sciences
    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.

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