Young Tableaux in Combinatorics, Invariant Theory, and Algebra (eBook)

Cover 1
Young Tableaux in Combinatorics, Invariant Theory, and Algebra: An Anthology of Recent Work 3
Copyright Page 4
Table of Contents 5
Contributors 7
Introduction 9
COMBINATORICS 10
INVARIANT THEORY 13
ALGEBRA 15
BIBLIOGRAPHY 18
Section 1 : Combinatorics 25
Chapter 1. A Probabilistic Proof of a Formula for the Number of Young Tableaux of a Given Shape 27
1. INTRODUCTION 27
2. PROOF OF THE FORMULA 28
3. FURTHER REMARKS 30
REFERENCES 31
Chapter 2. Reverse Plane Partitions and Tableau Hook Numbers 33
1. ORDERING OF TABLEAU NODES 33
2. REVERSE PLANE PARTITION 34
3. HOOK NUMBER MULTIPLICITIES FOR P 34
4. THE ZIGZAG PATH AND THE DERIVED rpp 35
5. RETURN PATHS 36
6. THE BIJECTION µ 36
7. AN EXAMPLE 37
ACKNOWLEDGMENT 38
REFERENCES 38
Chapter 3. An Analog of Schensted's Algorithm for Shifted Young Tableaux 39
1. DEFINITIONS AND MOTIVATION 39
2. ALGORITHMIC PROOF OF THE THEOREM 40
3. PROPERTIES AND CHARACTERIZATIONS 44
4. PROBLEMS AND CONJECTURE 46
ACKNOWLEDGMENT 47
REFERENCES 47
Chapter 4. An Extension of Schensted's Theorem 49
1. INTRODUCTION 49
2. SCHENSTED'S ALGORITHM 51
3. EXTENSIONS OF SCHENSTED'S THEOREM 53
4. CONSTRUCTION OF SUBSEQUENCES AND PARTITIONS 56
REFERENCES 60
Chapter 5. Some Partitions Associated with a Partially Ordered Set 61
1. INTRODUCTION 61
2. PROOFS OF THE MAIN RESULTS 65
3. PERFECT GRAPH THEOREMS 69
REFERENCES 70
Chapter 6. A Variational Problem for Random Young Tableaux 73
1. INTRODUCTION 74
2. THE VARIATIONAL PROBLEM 78
3. MINIMIZATION UNDER CONSTRAINTS 85
REFERENCES 88
Chapter 7. On Schensted's Construction and the Multiplication of Schur Functions 91
1. INTRODUCTION 91
2. DEFINITIONS 92
3. SCHENSTED'S CONSTRUCTION 93
4. FURTHER LEMMAS ON SCHENSTED'S CONSTRUCTION 96
5. 
102 
6. A Q-SYMBOL FOR THE EXTENSION OF SCHENSTED'S CONSTRUCTION 104
7. FURTHER RESULTS ON THE 
110 
8. THE MULTIPLICATION OF SCHUR FUNCTIONS 112
REFERENCES 114
Chapter 8. Frames, Young Tableaux, and Baxter Sequences 117
1. INTRODUCTION 117
2. FRAMES AND NUMBERINGS 118
3. THE INVENTORY OF A FRAME 120
4. BAXTER SEQUENCES 121
5. MAIN RESULTS 124
6. SCHUR FUNCTIONS 127
7. CONJUGATE FRAMES 128
8. OTHER NUMBERINGS OF FRAMES 130
REFERENCES 130
Chapter 9. Monotonicity and Unimodality of the Pattern Inventory 133
INTRODUCTION 133
I. TABLEAUX AND KOSTKA NUMBERS 133
II. MONOTONICITY OF THE KOSTKA NUMBERS 135
III. MONOTONICITY OF THE POLYA PATTERNS 137
ACKNOWLEDGMENT 139
REFERENCES 139
Section 2: Invariant Theory 141
Chapter 10. Invariant Theory, Young Bitableaux, and Combinatorics 143
1. INTRODUCTION 143
2. YOUNG TABLEAUX 146
3. THE STRAIGHTENING FORMULA 146
4. THE BASIS THEOREM 152
5. INVARIANT THEORY 159
REFERENCES 172
Chapter 11. Skew-Symmetric Invariant 
173 
1. INTRODUCTION 173
2. THE SKEW-STRAIGHTENING FORMULA 174
3. SKEW INVARIANTS 176
REFERENCES 178
Chapter 12. A Characteristic Free Approach to Invariant Theory 179
0. INTRODUCTION 179
1. THE STRAIGHTENING FORMULA 181
2. ABSOLUTE INVARIANTS 185
3. THE FIRST FUNDAMENTAL THEOREM 187
4. THE SYMMETRIC GROUP 190
5. THE ORTHOGONAL GROUP 192
6. THE SYMPLECTIC GROUP 198
7. THE BRAUER–WEYL 
201 
REFERENCES 203
Section 3 : Algebra 205
Chapter 13. Letter Place Algebras and a Characteristic-Free Approach to the Representation Theory of the General Linear and Symmetric Groups, 
207 
1. INTRODUCTION 207
2. NOTATIONS AND PRELIMINARIES 209
3. LP-ALGEBRAS UNDER A REPRESENTATION THEORETICAL POINT OF VIEW 212
4. A STRAIGHTENING ALGORITHM FOR SYMMETRIZED BIDETERMINANTS 218
5. SPECHT AND WEYL MODULES 225
6. A CHARACTERISTIC-FREE CONSTRUCTION OF THE IRREDUCIBLE MODULES FOR GENERAL LINEAR AND SYMMETRIC GROUPS 229
ACKNOWLEDGMENTS 236
REFERENCES 236
Chapter 14. Letter Place Algebras and a Characteristic-Free Approach to the Representation Theory of the General Linear and Symmetric Groups, II 238
7. ON THE CONSTRUCTION OF SPECHT AND W E Y L SERIES: GENERAL REMARKS 238
8. SHUFFLE PRODUCTS 243
9. THE BRANCHING THEOREM FOR SPECHT AND WEYL MODULES 246
10. A SPECHT SERIES FOR 
248 
11. 
252 
REFERENCES 263
Chapter 15. Young Diagrams and Ideals of Pfaffians 265
INTRODUCTION 265
1. PRELIMINARIES 266
2. REPRESENTATION THEORY 
269 
3. THE CLASSIFICATION OF 
272 
4. PRODUCT OF PFAFFIAN IDEALS 275
5. ORDER OF VANISHING ON PFAFFIAN VARIETIES 277
6. INTEGRALLY CLOSED G-IDEALS 278
REFERENCES 285
Chapter 16. On the Variety of Complexes 287
INTRODUCTION 287
1. YOUNG TABLEAUX AND THE VARIETY OF COMPLEXES 289
2. SOME PROPERTIES OF THE VARIETIES OF COMPLEXES 300
ACKNOWLEDGMENTS 307
REFERENCES 307
Chapter 17. Syzygies des variétés déterminantales 309
1. REPRÉSENTATIONS DES GROUPES LINÉAIRES 311
2. LES VARIÉTÉS DE SCHUBERT ONT DES SINGULARITÉS RATIONNELLES 322
3. SYZYGIES DES VARIÉTÉS DE SCHUBERT 326
4. PROPRIÉTÉS DES SYZYGIES 335
5. ADDENDUM: THÉORÈME DE BOTT 338
REFERENCES 343