Representations of the infinite symmetric group:
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 ve...
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| Hauptverfasser: | , |
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| Format: | Elektronisch E-Book |
| Sprache: | English |
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Cambridge
Cambridge University Press
2017
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| Schriftenreihe: | Cambridge studies in advanced mathematics
160 |
| Schlagworte: | |
| Online-Zugang: | BSB01 FHN01 UBR01 URL des Erstveröffentlichers Inhaltsverzeichnis |
| Zusammenfassung: | 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 |
| Beschreibung: | 1 online resource (vii, 160 Seiten) |
| ISBN: | 9781316798577 |
| DOI: | 10.1017/CBO9781316798577 |
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Datensatz im Suchindex
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| adam_text | Titel: Representations of the infinite symmetric group
Autor: Borodin, Alexei
Jahr: 2017
Representations of the Infinite Symmetric Group ALEXEJ/ B ORODIN Massachusetts Inštitúte 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 Kl Cambridge UNIVERSITY PRESS
Contents Introduction page 1 PART ONE SYMMETRIC FUNCTIONS AND THOMA’S THEOREM 1 Preliminary Facts From Representation Theory of Finite Symmetric Groups 13 1.1 Exercises 19 1.2 Notes 20 2 Theory of Symmetric Functions 21 2.1 Exercises 31 2.2 Notes 36 3 Coherent Systems on the Young Graph 37 3.1 The Infinite Symmetric Group and the Young Graph 37 3.2 Coherent Systems 39 3.3 The Thoma Simplex 41 3.4 Integral Representation of Coherent Systems and Characters 44 3.5 Exercises 46 3.6 Notes 46 4 Extreme Characters and Thoma’s Theorem 47 4.1 Thoma’s Theorem 47 4.2 Multiplicativity 48 4.3 Exercises 51 4.4 Notes 54 5 A Toy Model (the Pascal Graph) and de Finetti’s Theorem 55 5.1 Exercises 60 5.2 Notes 61 v
VI Contents 6 Asymptotics of Relative Dimension in the Young Graph 62 6.1 Relative Dimension and Shifted Schur Polynomials 62 6.2 The Algebra of Shifted Symmetric Functions 65 6.3 Modified Frobenius Coordinates 66 6.4 The Embedding Y„ — ? Q and Asymptotic Bounds 68 6.5 Integral Representation of Coherent Systems: Proof 71 6.6 The Vershik-Kerov Theorem 74 6.7 Exercises 75 6.8 Notes 80 7 Boundaries and Gibbs Measures on Paths 82 7.1 The Category SS 82 7.2 Projective Chains 84 7.3 Graded Graphs 86 7.4 Gibbs Measures 88 7.5 Examples of Path Spaces for Branching Graphs 90 7.6 The Martin Boundary and the Vershik-Kerov Ergodic Theorem 91 7.7 Exercises 93 7.8 Notes 96 PART TWO UNITARY REPRESENTATIONS 8 Preliminaries and Gelfand Pairs 101 8.1 Exercises 110 8.2 Notes 113 9 Classification of General Spherical Type Representations 114 9.1 Notes 120 10 Realization of Irreducible Spherical Representations of (5(oo) x 5(oo), diag5(oo)) 121 10.1 Exercises 126 10.2 Notes 128 11 Generalized Regular Representations T z 130 11.1 Exercises 139 11.2 Notes 140 12 Disjointness of Representations T z 141 12.1 Preliminaries 141 12.2 Reduction to Gibbs Measures 143 12.3 Exclusion of Degenerate Paths 144
Contents vii 12.4 Proof of Disjointness 146 12.5 Exercises 149 12.6 Notes 149 References 150 Index 158
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| isbn | 9781316798577 |
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| spelling | Borodin, Alexei 1975- (DE-588)133797724 aut Representations of the infinite symmetric group Alexei Borodin, Grigori Olshanski Cambridge Cambridge University Press 2017 1 online resource (vii, 160 Seiten) txt rdacontent c rdamedia cr rdacarrier Cambridge studies in advanced mathematics 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 Hopf algebras Algebraic topology Representations of groups Symmetry groups Hopf-Algebra (DE-588)4160646-2 gnd rswk-swf Symmetrische Gruppe (DE-588)4184204-2 gnd rswk-swf Hopf-Algebra (DE-588)4160646-2 s Symmetrische Gruppe (DE-588)4184204-2 s DE-604 Olšanskij, Grigori I. aut Erscheint auch als Druck-Ausgabe 978-1-107-17555-6 https://doi.org/10.1017/CBO9781316798577 Verlag URL des Erstveröffentlichers Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029405606&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Borodin, Alexei 1975- Olšanskij, Grigori I. Representations of the infinite symmetric group Hopf algebras Algebraic topology Representations of groups Symmetry groups Hopf-Algebra (DE-588)4160646-2 gnd Symmetrische Gruppe (DE-588)4184204-2 gnd |
| subject_GND | (DE-588)4160646-2 (DE-588)4184204-2 |
| title | Representations of the infinite symmetric group |
| title_auth | Representations of the infinite symmetric group |
| title_exact_search | Representations of the infinite symmetric group |
| title_full | Representations of the infinite symmetric group Alexei Borodin, Grigori Olshanski |
| title_fullStr | Representations of the infinite symmetric group Alexei Borodin, Grigori Olshanski |
| title_full_unstemmed | Representations of the infinite symmetric group Alexei Borodin, Grigori Olshanski |
| title_short | Representations of the infinite symmetric group |
| title_sort | representations of the infinite symmetric group |
| topic | Hopf algebras Algebraic topology Representations of groups Symmetry groups Hopf-Algebra (DE-588)4160646-2 gnd Symmetrische Gruppe (DE-588)4184204-2 gnd |
| topic_facet | Hopf algebras Algebraic topology Representations of groups Symmetry groups Hopf-Algebra Symmetrische Gruppe |
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