Kung Young Tableaux in Combinatorics, Invariant Theory, and Algebra

An Anthology of Recent Work
1. Auflage 2014
ISBN: 978-1-4832-7202-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 342 Seiten, Web PDF

Young Tableaux in Combinatorics, Invariant Theory, and Algebra: An Anthology of Recent Work is an anthology of papers on Young tableaux and their applications in combinatorics, invariant theory, and algebra. Topics covered include reverse plane partitions and tableau hook numbers; some partitions associated with a partially ordered set; frames and Baxter sequences; and Young diagrams and ideals of Pfaffians. Comprised of 16 chapters, this book begins by describing a probabilistic proof of a formula for the number f? of standard Young tableaux of a given shape f?. The reader is then introduced to the generating function of R. P. Stanley for reverse plane partitions on a tableau shape; an analog of Schensted's algorithm relating permutations and triples consisting of two shifted Young tableaux and a set; and a variational problem for random Young tableaux. Subsequent chapters deal with certain aspects of Schensted's construction and the derivation of the Littlewood-Richardson rule for the multiplication of Schur functions using purely combinatorial methods; monotonicity and unimodality of the pattern inventory; and skew-symmetric invariant theory. This volume will be helpful to students and practitioners of algebra.

Autoren/Hrsg.

Kung, Joseph P. S.

Inhaltsverzeichnis

1;Cover;1
2;Young Tableaux in Combinatorics, Invariant Theory, and Algebra: An Anthology of Recent Work;3
3;Copyright Page;4
4;Table of Contents;5
5;Contributors;7
6;Introduction;9
6.1;COMBINATORICS;10
6.2;INVARIANT THEORY;13
6.3;ALGEBRA;15
6.4;BIBLIOGRAPHY;18
7;Section 1 : Combinatorics;25
7.1;Chapter 1. A Probabilistic Proof of a Formula for the Number of Young Tableaux of a Given Shape;27
7.1.1;1. INTRODUCTION;27
7.1.2;2. PROOF OF THE FORMULA;28
7.1.3;3. FURTHER REMARKS;30
7.1.4;REFERENCES;31
7.2;Chapter 2. Reverse Plane Partitions and Tableau Hook Numbers;33
7.2.1;1. ORDERING OF TABLEAU NODES;33
7.2.2;2. REVERSE PLANE PARTITION;34
7.2.3;3. HOOK NUMBER MULTIPLICITIES FOR P;34
7.2.4;4. THE ZIGZAG PATH AND THE DERIVED rpp;35
7.2.5;5. RETURN PATHS;36
7.2.6;6. THE BIJECTION µ;36
7.2.7;7. AN EXAMPLE;37
7.2.8;ACKNOWLEDGMENT;38
7.2.9;REFERENCES;38
7.3;Chapter 3. An Analog of Schensted's Algorithm for Shifted Young Tableaux;39
7.3.1;1. DEFINITIONS AND MOTIVATION;39
7.3.2;2. ALGORITHMIC PROOF OF THE THEOREM;40
7.3.3;3. PROPERTIES AND CHARACTERIZATIONS;44
7.3.4;4. PROBLEMS AND CONJECTURE;46
7.3.5;ACKNOWLEDGMENT;47
7.3.6;REFERENCES;47
7.4;Chapter 4. An Extension of Schensted's Theorem;49
7.4.1;1. INTRODUCTION;49
7.4.2;2. SCHENSTED'S ALGORITHM;51
7.4.3;3. EXTENSIONS OF SCHENSTED'S THEOREM;53
7.4.4;4. CONSTRUCTION OF SUBSEQUENCES AND PARTITIONS;56
7.4.5;REFERENCES;60
7.5;Chapter 5. Some Partitions Associated with a Partially Ordered Set;61
7.5.1;1. INTRODUCTION;61
7.5.2;2. PROOFS OF THE MAIN RESULTS;65
7.5.3;3. PERFECT GRAPH THEOREMS;69
7.5.4;REFERENCES;70
7.6;Chapter 6. A Variational Problem for Random Young Tableaux;73
7.6.1;1. INTRODUCTION;74
7.6.2;2. THE VARIATIONAL PROBLEM;78
7.6.3;3. MINIMIZATION UNDER CONSTRAINTS;85
7.6.4;REFERENCES;88
7.7;Chapter 7. On Schensted's Construction and the Multiplication of Schur Functions;91
7.7.1;1. INTRODUCTION;91
7.7.2;2. DEFINITIONS;92
7.7.3;3. SCHENSTED'S CONSTRUCTION;93
7.7.4;4. FURTHER LEMMAS ON SCHENSTED'S CONSTRUCTION;96
7.7.5;5.
AN EXTENSION OF SCHENSTED'S CONSTRUCTION;102
7.7.6;6. A Q-SYMBOL FOR THE EXTENSION OF SCHENSTED'S CONSTRUCTION;104
7.7.7;7. FURTHER RESULTS ON THE
Q-SYMBOL OF F(.). F(µ);110
7.7.8;8. THE MULTIPLICATION OF SCHUR FUNCTIONS;112
7.7.9;REFERENCES;114
7.8;Chapter 8. Frames, Young Tableaux, and Baxter Sequences;117
7.8.1;1. INTRODUCTION;117
7.8.2;2. FRAMES AND NUMBERINGS;118
7.8.3;3. THE INVENTORY OF A FRAME;120
7.8.4;4. BAXTER SEQUENCES;121
7.8.5;5. MAIN RESULTS;124
7.8.6;6. SCHUR FUNCTIONS;127
7.8.7;7. CONJUGATE FRAMES;128
7.8.8;8. OTHER NUMBERINGS OF FRAMES;130
7.8.9;REFERENCES;130
7.9;Chapter 9. Monotonicity and Unimodality of the Pattern Inventory;133
7.9.1;INTRODUCTION;133
7.9.2;I. TABLEAUX AND KOSTKA NUMBERS;133
7.9.3;II. MONOTONICITY OF THE KOSTKA NUMBERS;135
7.9.4;III. MONOTONICITY OF THE POLYA PATTERNS;137
7.9.5;ACKNOWLEDGMENT;139
7.9.6;REFERENCES;139
8;Section 2: Invariant Theory;141
8.1;Chapter 10. Invariant Theory, Young Bitableaux, and Combinatorics;143
8.1.1;1. INTRODUCTION;143
8.1.2;2. YOUNG TABLEAUX;146
8.1.3;3. THE STRAIGHTENING FORMULA;146
8.1.4;4. THE BASIS THEOREM;152
8.1.5;5. INVARIANT THEORY;159
8.1.6;REFERENCES;172
8.2;Chapter 11. Skew-Symmetric Invariant
Theory;173
8.2.1;1. INTRODUCTION;173
8.2.2;2. THE SKEW-STRAIGHTENING FORMULA;174
8.2.3;3. SKEW INVARIANTS;176
8.2.4;REFERENCES;178
8.3;Chapter 12. A Characteristic Free Approach to Invariant Theory;179
8.3.1;0. INTRODUCTION;179
8.3.2;1. THE STRAIGHTENING FORMULA;181
8.3.3;2. ABSOLUTE INVARIANTS;185
8.3.4;3. THE FIRST FUNDAMENTAL THEOREM;187
8.3.5;4. THE SYMMETRIC GROUP;190
8.3.6;5. THE ORTHOGONAL GROUP;192
8.3.7;6. THE SYMPLECTIC GROUP;198
8.3.8;7. THE BRAUER–WEYL
ALGEBRA;201
8.3.9;REFERENCES;203
9;Section 3 : Algebra;205
9.1;Chapter 13. Letter Place Algebras and a Characteristic-Free Approach to the Representation Theory of the General Linear and Symmetric Groups,
I;207
9.1.1;1. INTRODUCTION;207
9.1.2;2. NOTATIONS AND PRELIMINARIES;209
9.1.3;3. LP-ALGEBRAS UNDER A REPRESENTATION THEORETICAL POINT OF VIEW;212
9.1.4;4. A STRAIGHTENING ALGORITHM FOR SYMMETRIZED BIDETERMINANTS;218
9.1.5;5. SPECHT AND WEYL MODULES;225
9.1.6;6. A CHARACTERISTIC-FREE CONSTRUCTION OF THE IRREDUCIBLE MODULES FOR GENERAL LINEAR AND SYMMETRIC GROUPS;229
9.1.7;ACKNOWLEDGMENTS;236
9.1.8;REFERENCES;236
9.2;Chapter 14. Letter Place Algebras and a Characteristic-Free Approach to the Representation Theory of the General Linear and Symmetric Groups, II;238
9.2.1;7. ON THE CONSTRUCTION OF SPECHT AND W E Y L SERIES: GENERAL REMARKS;238
9.2.2;8. SHUFFLE PRODUCTS;243
9.2.3;9. THE BRANCHING THEOREM FOR SPECHT AND WEYL MODULES;246
9.2.4;10. A SPECHT SERIES FOR
R;248
9.2.5;11.
A SPECHT SERIES FOR LITTLEWOOD–RICHARDSON PRODUCTS;252
9.2.6;REFERENCES;263
9.3;Chapter 15. Young Diagrams and Ideals of Pfaffians;265
9.3.1;INTRODUCTION;265
9.3.2;1. PRELIMINARIES;266
9.3.3;2. REPRESENTATION THEORY
FOR;269
9.3.4;3. THE CLASSIFICATION OF
G-INVARIANT IDEALS;272
9.3.5;4. PRODUCT OF PFAFFIAN IDEALS;275
9.3.6;5. ORDER OF VANISHING ON PFAFFIAN VARIETIES;277
9.3.7;6. INTEGRALLY CLOSED G-IDEALS;278
9.3.8;REFERENCES;285
9.4;Chapter 16. On the Variety of Complexes;287
9.4.1;INTRODUCTION;287
9.4.2;1. YOUNG TABLEAUX AND THE VARIETY OF COMPLEXES;289
9.4.3;2. SOME PROPERTIES OF THE VARIETIES OF COMPLEXES;300
9.4.4;ACKNOWLEDGMENTS;307
9.4.5;REFERENCES;307
9.5;Chapter 17. Syzygies des variétés déterminantales;309
9.5.1;1. REPRÉSENTATIONS DES GROUPES LINÉAIRES;311
9.5.2;2. LES VARIÉTÉS DE SCHUBERT ONT DES SINGULARITÉS RATIONNELLES;322
9.5.3;3. SYZYGIES DES VARIÉTÉS DE SCHUBERT;326
9.5.4;4. PROPRIÉTÉS DES SYZYGIES;335
9.5.5;5. ADDENDUM: THÉORÈME DE BOTT;338
9.5.6;REFERENCES;343