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E-grāmata: Representations of the Infinite Symmetric Group

(Massachusetts Institute of Technology),
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Representations of the Infinite Symmetric GroupPalielināt
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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.

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.

Recenzijas

' 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

Papildus informācija

An introduction to the modern representation theory of big groups, exploring its connections to probability and algebraic combinatorics.
Introduction 1(12)
PART ONE SYMMETRIC FUNCTIONS AND THOMA'S THEOREM
1 Preliminary Facts From Representation Theory of Finite Symmetric Groups
13(8)
1.1 Exercises
19(1)
1.2 Notes
20(1)
2 Theory of Symmetric Functions
21(16)
2.1 Exercises
31(5)
2.2 Notes
36(1)
3 Coherent Systems on the Young Graph
37(10)
3.1 The Infinite Symmetric Group and the Young Graph
37(2)
3.2 Coherent Systems
39(2)
3.3 The Thoma Simplex
41(3)
3.4 Integral Representation of Coherent Systems and Characters
44(2)
3.5 Exercises
46(1)
3.6 Notes
46(1)
4 Extreme Characters and Thoma's Theorem
47(8)
4.1 Thoma's Theorem
47(1)
4.2 Multiplicativity
48(3)
4.3 Exercises
51(3)
4.4 Notes
54(1)
5 A Toy Model (the Pascal Graph) and de Finetti's Theorem
55(7)
5.1 Exercises
60(1)
5.2 Notes
61(1)
6 Asymptotics of Relative Dimension in the Young Graph
62(20)
6.1 Relative Dimension and Shifted Schur Polynomials
62(3)
6.2 The Algebra of Shifted Symmetric Functions
65(1)
6.3 Modified Frobenius Coordinates
66(2)
6.4 The Embedding Yn → Ω and Asymptotic Bounds
68(3)
6.5 Integral Representation of Coherent Systems: Proof
71(3)
6.6 The Vershik--Kerov Theorem
74(1)
6.7 Exercises
75(5)
6.8 Notes
80(2)
7 Boundaries and Gibbs Measures on Paths
82(19)
7.1 The Category B
82(2)
7.2 Projective Chains
84(2)
7.3 Graded Graphs
86(2)
7.4 Gibbs Measures
88(2)
7.5 Examples of Path Spaces for Branching Graphs
90(1)
7.6 The Martin Boundary and the Vershik--Kerov Ergodic Theorem
91(2)
7.7 Exercises
93(3)
7.8 Notes
96(5)
PART TWO UNITARY REPRESENTATIONS
8 Preliminaries and Gelfand Pairs
101(13)
8.1 Exercises
110(3)
8.2 Notes
113(1)
9 Classification of General Spherical Type Representations
114(7)
9.1 Notes
120(1)
10 Realization of Irreducible Spherical Representations of (S(∞) × S(∞), diagS(∞))
121(9)
10.1 Exercises
126(2)
10.2 Notes
128(2)
11 Generalized Regular Representations Tz
130(11)
11.1 Exercises
139(1)
11.2 Notes
140(1)
12 Disjointness of Representations Tz
141(9)
12.1 Preliminaries
141(2)
12.2 Reduction to Gibbs Measures
143(1)
12.3 Exclusion of Degenerate Paths
144(2)
12.4 Proof of Disjointness
146(3)
12.5 Exercises
149(1)
12.6 Notes
149(1)
References 150(8)
Index 158
Alexei Borodin is a Professor of Mathematics at the Massachusetts Institute of Technology. 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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