Symmetric functions, Schubert polynomials, and degeneracy loci /

"This text serves as an introduction to the combinatories of symmetric functions, more precisely to Schur and Schubert polynomials. Simultaneously, it studies the geometry of Grassmannians, flag varieties and especially their Schubert varieties, and examines the profound connections that unite...

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Bibliographic Details
Main Author:	Manivel, Laurent
Format:	Book
Language:	English French
Published:	Providence, RI : American Mathematical Society, 2001 Providence, R.I. : [Paris] : American Mathematical Society ; Société Mathématique de France, [2001]
Series:	SMF/AMS texts and monographs, v. 6 Collection SMF Cours spécialisés ; no 3. Collection SMF Cours spécialisés no 3. Collection SMF Cours spécialisés ; no 3. SMF/AMS texts and monographs ; v. 6 SMF/AMS texts and monographs ; v. 6 SMF/AMS texts and monographs v. 6 SMF/AMS texts and monographs v. 6
Subjects:	Homology theory Schubert varieties Symmetric functions Fonctions symétriques Homologie Schubert, Variétés de Homologietheorie Schubert-Polynom Symmetrische Funktion Combinatieleer Polynomen Symmetrie
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Table of Contents:

Ch. 1 The Ring of Symmetric Functions
1.1. Ordinary Functions
1.2. Schur Functions
1.3. The Knuth Correspondence
1.4. Some Applications to Symmetric Functions
1.5. The Littlewood-Richardson Rule
1.6. The Characters of the Symmetric Group
1.7. Kostka-Foulkes Polynomials
1.8. How the Symmetric Group Acts on Tableaux
Ch. 2. Schubert Polynomials
2.1. Permutations and the Bruhat Order
2.2. Some Classes of Permutations
2.3. Schubert Polynomials
2.4. Some Properties of Schubert Polynomials
2.5. Simple Schubert Polynomials
2.6. Flagged Schur Functions
2.7. Multiplication of Schur Polynomials
2.8. Enumeration of Reduced Words
Ch. 3. Schubert Varieties
3.1. Grassmannians
3.2. Schubert Varieties of Grassmannians
3.3. Standard Monomials
3.4. Singularities of Schubert Varieties
3.5. Characteristic Classes and Degeneracy Loci
3.6. Flag Varieties
3.7. Singularities of Schubert Varieties, reprise
3.8. Degeneracy Loci and Schubert Polynomials. A. A Brief Introduction to Singular Homology
A.1. Singular Homology
A.2. Singular Cohomology
A.3. The Fundamental Class and Poincare Duality
A.4. Intersection of Algebraic Subvarieties.
Ch. 1 The Ring of Symmetric Functions. 1.1. Ordinary Functions. 1.2. Schur Functions. 1.3. The Knuth Correspondence. 1.4. Some Applications to Symmetric Functions. 1.5. The Littlewood-Richardson Rule. 1.6. The Characters of the Symmetric Group. 1.7. Kostka-Foulkes Polynomials. 1.8. How the Symmetric Group Acts on Tableaux
Ch. 2. Schubert Polynomials. 2.1. Permutations and the Bruhat Order. 2.2. Some Classes of Permutations. 2.3. Schubert Polynomials. 2.4. Some Properties of Schubert Polynomials. 2.5. Simple Schubert Polynomials. 2.6. Flagged Schur Functions. 2.7. Multiplication of Schur Polynomials. 2.8. Enumeration of Reduced Words
Ch. 3. Schubert Varieties. 3.1. Grassmannians. 3.2. Schubert Varieties of Grassmannians. 3.3. Standard Monomials. 3.4. Singularities of Schubert Varieties. 3.5. Characteristic Classes and Degeneracy Loci. 3.6. Flag Varieties. 3.7. Singularities of Schubert Varieties, reprise.
Chapter 1 The Ring of Symmetric Functions 7
1.1. Ordinary Functions 7
1.2. Schur Functions 9
1.3. The Knuth Correspondence 15
1.4. Some Applications to Symmetric Functions 18
1.5. The Littlewood-Richardson Rule 24
1.6. The Characters of the Symmetric Group 34
1.7. Kostka-Foulkes Polynomials 46
1.8. How the Symmetric Group Acts on Tableaux 52
Chapter 2. Schubert Polynomials 57
2.1. Permutations and the Bruhat Order 57
2.2. Some Classes of Permutations 64
2.3. Schubert Polynomials 69
2.4. Some Properties of Schubert Polynomials 75
2.5. Simple Schubert Polynomials 78
2.6. Flagged Schur Functions 85
2.7. Multiplication of Schur Polynomials 92
2.8. Enumeration of Reduced Words 97
Chapter 3. Schubert Varieties 101
3.1. Grassmannians 101
3.2. Schubert Varieties of Grassmannians 104
3.3. Standard Monomials 111
3.4. Singularities of Schubert Varieties 115
3.5. Characteristic Classes and Degeneracy Loci 121
3.6. Flag Varieties 132
3.7. Singularities of Schubert Varieties, reprise 142
3.8. Degeneracy Loci and Schubert Polynomials 147
A Brief Introduction to Singular Homology 153
A.1. Singular Homology 153
A.2. Singular Cohomology 155
A.3. The Fundamental Class and Poincare Duality 156
A.4. Intersection of Algebraic Subvarieties 158.
The Ring of Symmetric Functions
Ordinary Functions
Schur Functions
The Knuth Correspondence
Some Applications to Symmetric Functions
The Littlewood-Richardson Rule
The Characters of the Symmetric Group
Kostka-Foulkes Polynomials
How the Symmetric Group Acts on Tableaux
Schubert Polynomials
Permutations and the Bruhat Order
Some Classes of Permutations
Schubert Polynomials
Some Properties of Schubert Polynomials
Simple Schubert Polynomials
Flagged Schur Functions
Multiplication of Schur Polynomials
Enumeration of Reduced Words
Schubert Varieties
Grassmannians
Schubert Varieties of Grassmannians
Standard Monomials
Singularities of Schubert Varieties
Characteristic Classes and Degeneracy Loci
Flag Varieties
Singularities of Schubert Varieties, reprise
Degeneracy Loci and Schubert Polynomials
A Brief Introduction to Singular Homology
Singular Homology
Singular Cohomology
The Fundamental Class and Poincare Duality
Intersection of Algebraic Subvarieties
3.8 Degeneracy Loci and Schubert Polynomials. A. A Brief Introduction to Singular Homology. A.1. Singular Homology. A.2. Singular Cohomology. A.3. The Fundamental Class and Poincare Duality. A.4. Intersection of Algebraic Subvarieties.

Symmetric functions, Schubert polynomials, and degeneracy loci /

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