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Tables of Contents for The Symmetric Group
Chapter/Section Title
Page #
Page Count
Preface to the 2nd Edition
v
Preface to the 1st Edition
vii
Group Representations
1
52
Fundamental Concepts
1
3
Matrix Representations
4
2
G-Modules and the Group Algebra
6
4
Reducibility
10
3
Complete Reducibility and Maschke's Theorem
13
5
G-Homomorphisms and Schur's Lemma
18
5
Commutant and Endomorphism Algebras
23
7
Group Characters
30
3
Inner Products of Characters
33
7
Decomposition of the Group Algebra
40
3
Tensor Products Again
43
2
Restricted and Induced Representations
45
3
Exercises
48
5
Representations of the Symmetric Group
53
38
Young Subgroups, Tableaux, and Tabloids
53
4
Dominance and Lexicographic Ordering
57
3
Specht Modules
60
3
The Submodule Theorem
63
3
Standard Tableaux and a Basis for Sλ
66
4
Garnir Elements
70
4
Young's Natural Representation
74
2
The Branching Rule
76
2
The Decomposition of Mμ
78
4
The Semistandard Basis for Hom(Sλ, Mμ)
82
3
Kostka Numbers and Young's Rule
85
1
Exercises
86
5
Combinatorial Algorithms
91
50
The Robinson-Schensted Algorithm
91
4
Column Insertion
95
2
Increasing and Decreasing Subsequences
97
2
The Knuth Relations
99
3
Subsequences Again
102
4
Viennot's Geometric Construction
106
6
Schutzenberger's Jeu de Taquin
112
4
Dual Equivalence
116
5
Evacuation
121
3
The Hook Formula
124
8
The Determinantal Formula
132
1
Exercises
133
8
Symmetric Functions
141
50
Introduction to Generating Functions
142
5
The Hillman-Grassl Algorithm
147
4
The Ring of Symmetric Functions
151
4
Schur Functions
155
3
The Jacobi-Trudi Determinants
158
5
Other Definitions of the Schur Function
163
4
The Characteristic Map
167
2
Knuth's Algorithm
169
5
The Littlewood-Richardson Rule
174
5
The Murnaghan-Nakayama Rule
179
6
Exercises
185
6
Applications and Generalizations
191
32
Young's Lattice and Differential Posets
191
6
Growths and Local Rules
197
7
Groups Acting on Posets
204
4
Unimodality
208
5
Chromatic Symmetric Functions
213
5
Exercises
218
5
Bibliography
223
7
Index
230