The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions

✍ Scribed by Bruce Sagan

• Publisher: Springer
• Year: 2001
• Tongue: English
• Leaves: 254
• Series: Graduate Texts in Mathematics, Vol. 203
• Edition: 2
• Category: Library

✦ Synopsis

This book brings together many of the important results in this field.

From the reviews: ""A classic gets even better....The edition has new material including the Novelli-Pak-Stoyanovskii bijective proof of the hook formula, Stanley’s proof of the sum of squares formula using differential posets, Fomin’s bijective proof of the sum of squares formula, group acting on posets and their use in proving unimodality, and chromatic symmetric functions." --ZENTRALBLATT MATH

✦ Table of Contents

Contents
Preface to the 2nd Edition
Preface to the 1st Eition
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 Sx
2.6 Garnir Elements
2.7 Young's Natural Representation
2.8 The Branching Rule
2.9 The Decomposition of M^
2.10 The Semistandard Basis for Hom(S^A,M^mu)
2.11 Kostka Numbers and Young's Rule
2.12 Exercises
3 Combinatorial Algorithms
3.1 The Robinson-Schensted Algorithm
3.2 Column Insertion
3.3 Increasing and Decreasing Subsequences
3.4 The Knuth Relations
3.5 Subsequences Again
3.6 Viennot's Geometric Construction
3.7 Schutzenberger's Jeu de Taquin
3.8 Dual Equivalence
3.9 Evacuation
3.10 The Hook Formula
3.11 The Determinantal Formula
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
5 Applications and Generalizations
5.1 Young's Lattice and Differential Posets
5.2 Growths and Local Rules
5.3 Groups Acting on Posets
5.4 Unimodality
5.5 Chromatic Symmetric Functions
5.6 Exercises
Bibliography
Index