The Symmetric Group

Title
Contents
Preface
List of Symbols
1. Group Representations
1.1 Fundamental Concepts
1.2 Matrix Representations
1.3 G-modules and the Group Algebra
1.4 Reducibility
1.5 Complete Reducibility and Maschke's Theorem
1.6 G-homomorphisms and Schur’s Lemma
1.7 Commutant and Endomorphism Algebras
1.8 Group Characters
1.9 Inner Products of Characters
1.10 Decomposition of the Group Algebra
1.11 Tensor Products Again
1.12 Restricted and Induced Representations
1.13 Exercises
2. Representations of the Symmetric Group
2.1 Young Subgroups, Tableaux, and Tabloids
2.2 Dominance and Lexicographic Ordering
2.3 Specht Modules
2.4 The Submodule Theorem
2.5 Standard Tableaux and a Basis for S^lambda
2.6 Garnir Elements
2.7 Young’s Natural Representation
2.8 The Branching Rule
2.9 The Decomposition of M^mu
2.10 The Semistandard Basis for Hom(S^lambda, M^mu)
2.11 Kostka Numbers and Young’s Rule
2.12 Exercises
3. Combinatorial Algorithms
3.1 The Hook Formula
3.2 The Determinantal Formula
3.3 The Robinson-Schensted Algorithm
3.4 Column Insertion
3.5 Increasing and Decreasing Subsequences
3.6 The Knuth Relations
3.7 Subsequences Again
3.8 Viennot’s Geometric Construction
3.9 Schützenberger’s Jeu de Taquin
3.10 Dual Equivalence
3.11 Evacuation
3.12 Exercises
4. Symmetric Functions
4.1 Introduction to Generating Functions
4.2 The Hillman-Grassl Algorithm
4.3 The Ring of Symmetric Functions
4.4 Schur Functions
4.5 The Jacobi-Trudi Determinants
4.6 Other Definitions of the Schur Function
4.7 The Characteristic Map
4.8 Knuth’s Algorithm
4.9 The Littlewood-Richardson Rule
4.10 The Murnaghan-Nakayama Rule
4.11 Exercises
Bibliography
Index