Representations of the infinite symmetric group / Alexei Borodin, Grigori Olshanski.

Title from publisher's bibliographic system (viewed on 02 Dec 2016).

Pt. 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
4.2.Multiplicativity
4.3.Exercises
4.4.Notes
5.A Toy Model (the Pascal Graph) and de Finetti's Theorem
5.1.Exercises
5.2.Notes
6.Asymptotics of Relative Dimension in the Young Graph
6.1.Relative Dimension and Shifted Schur Polynomials
6.2.The Algebra of Shifted Symmetric Functions
6.3.Modified Frobenius Coordinates
6.4.The Embedding Yn [→] [MARC+5D] and Asymptotic Bounds
6.5.Integral Representation of Coherent Systems: Proof
6.6.The Vershik
Kerov Theorem
6.7.Exercises
6.8.Notes
7.Boundaries and Gibbs Measures on Paths
7.1.The Category B
7.2.Projective Chains
7.3.Graded Graphs
7.4.Gibbs Measures
7.5.Examples of Path Spaces for Branching Graphs
7.6.The Martin Boundary and the Vershik
Kerov Ergodic Theorem
7.7.Exercises
7.8.Notes
pt. 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 Tz
11.1.Exercises
11.2.Notes
12.Disjointness of Representations Tz
12.1.Preliminaries
12.2.Reduction to Gibbs Measures
12.3.Exclusion of Degenerate Paths
12.4.Proof of Disjointness
12.5.Exercises
12.6.Notes

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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. 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.

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