對稱群(第2版)（簡體書）

本書是一部研究生教程，幾乎囊括了該領域的重要結果，是經典中的經典。書中包括了很多新內容。適用于數學專業的高年級本科生、研究生和相關的科研人員。11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

《對稱群(第2版)》是由世界圖書出版公司北京公司出版。

Preface to the 2nd Edition
Preface to the 1st Eition
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 λ
2.6 Garnir Elements
2.7 Young's Natural Representation
2.8 The Branching Rule
2.9 The Decomposition of Mμ
2.10 The Semistandard Basis for Hom（Sλ, Mμ）
2.11 Kostka Numbers and Young's Rule
2.12 Exercises
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 Schùtzenberger'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

where I is the appropriate identity matrix and c ∈ C is any scalar. Now C is algebraically closed, so we can take c to be an eigenvalue of T. Thus T-cI satisfies the hypothesis of Corollary 1.6.6 （with X = Y） and is not invertible by the choice of c. Our only alternative is that T-cI = O. We have proved the following result:
Corollary 1.6.8 Let X be an irreducible matrix representation of G over the complex numbers. Then the only matrices T that commute with X（g） for all g ∈ G are those of the form T = cI-i.e., scalar multiples of the identity matrix.
1.7 Commutant and Endomorphism Algebras
Corollary 1.6.8 suggests that the set of matrices that commute with those of a given representation are important. This corresponds in the module setting to the set of G-homomorphisms from a G-module to itself. We characterize these sets in this section. Extending these ideas to homomorphisms between different G-modules leads to a useful generalization of Corollary 1.6.7 （see Corollary 1.7.10）.