Representations of the Infinite Symmetric Group (Cambridge Studies in Advanced Mathematics)

Borodin, Alexei/ Olshanski, Grigori

Cambridge University Press（2016/10発売）

• 製本 Hardcover:ハードカバー版／ページ数 168 p.
• 言語 ENG
• 商品コード 9781107175556
• DDC分類 512.2

Full Description

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.

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(∞) x S(∞), diagS(∞)); 11. Generalized regular representations Tz; 12. Disjointness of representations Tz; References; Index.