The Representation Theory of the Symmetric Group

Gordon James, Adalbert Kerber9780201135152, 0-201-13515-9

The Representation Theory of the Symmetric Group provides an account of both the ordinary and modular representation theory of the symmetric groups. The range of applications of this theory is vast, varying from theoretical physics, through combinatories to the study of polynomial identity algebras; and new uses are still being found.

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Table of contents :
The Representation Theory of the Symmetric Group……Page 1
Contents……Page 3
Editor’s Statement……Page 6
Foreword……Page 7
Alfred Young……Page 9
Introduction……Page 10
Preface……Page 14
List of Symbols……Page 16
1.1 Symmetric and Alternating Groups……Page 23
1.2 The Conjugacy Classes of Symmetric and Alternating Groups……Page 30
1.3 Young Subgroups of Sn and Their Double Cosets……Page 37
1.4 The Diagram Lattice……Page 43
1.5 Young Subgroups as Horizontal and Vertical Groups of Young Tableaux……Page 51
Exercises……Page 55
2.1 The Ordinary Irreducible Representations of Sn……Page 56
2.2 The Permutation Characters Induced by Young Subgroups……Page 60
2.3 The Ordinary Irreducible Characters as Z-linear Combinations of Permutation Characters……Page 67
2.4 A Recursion Formula for the Irreducible Characters……Page 80
2.5 Ordinary Irreducible Representations and Characters of An……Page 87
2.6 Sn is Characterized by its Character Table……Page 94
2.7 Cores and Quotients of Partitions……Page 97
2.8 Young’s Rule and the Littlewood-Richardson Rule……Page 109
2.9 Inner Tensor Products……Page 117
Exercises……Page 122
3.1 A Decomposition of the Group Algebra QSn into Minimal Left Ideals……Page 123
3.2 The Seminormal Basis of QSn……Page 131
3.3 The Representing Matrices……Page 137
3.4 The Orthogonal and the Natural Form of [α]……Page 148
Exercises……Page 153
4.1 Wreath Products……Page 154
4.2 The Conjugacy Classes of G wr Sn……Page 160
4.3 Representations of Wreath Products over Algebraically Closed Fields……Page 168
4.4 Special Cases and Properties of Representations of Wreath Products……Page 177
Exercises……Page 183
5. Applicationis to Combinatorics and Representation Theory……Page 184
5.1 The Pólya Theory of Enumeration……Page 185
5.2 Symmetrization of Representations……Page 206
5.3 Permutrization of Representations……Page 224
5.4 Plethysms of Representations……Page 240
5.5 Multiply Transitive Groups……Page 249
Exercises……Page 259
6.1 The p-block Structure of the Ordinary Irreducibles of Sn and An; Generalized Decomposition Numbers……Page 262
6.2 The Dimensions of a p-block; u-numbers; Defect Groups……Page 276
6.3 Techniques for Finding Decomposition Matrices……Page 287
Exercises……Page 314
7.1 Specht Modules……Page 316
7.2 The Standard Basis of the Specht Module……Page 323
7.3 On the Role of Hook Lengths……Page 328
Exercises……Page 340
8. Representations of General Linear Groups……Page 341
8.1 Weyl Modules……Page 342
8.2 The Hyperalgebra……Page 349
8.3 Irreducible GL(m,F)-modules over F……Page 356
8.4 Further Connections between Specht and Weyl Modules……Page 363
Exercises……Page 368
I.A Character Tables……Page 370
I.B Class Multiplication Coefficients……Page 378
I.C Representing Matrices……Page 390
I.D Decompositions of Symmetrizations and Permutrizations……Page 402
I.E Decomposition Numbers……Page 435
I.F Irreducible Brauer Characters……Page 452
I.G Littlewood-Richardson Coefficients……Page 458
I.H Character Tables of Wreath Products of Symmetric Groups……Page 464
I.I Decompositions of Inner Tensor Powers……Page 473
II.A Books and Lecture Notes……Page 481
II.B Comments on the Chapters……Page 482
II.D References……Page 490
Index……Page 529