An Introduction to Symmetric Functions and Their Combinatorics

Contents Preface 1. Symmetric Polynomials, the Monomial Symmetric Polynomials, and Symmetric Functions 1.1. Symmetric Polynomials 1.2. The Monomial Symmetric Polynomials 1.3. Symmetric Functions 1.4. Problems 1.5. Notes 2. The Elementary, Complete Homogeneous, and Power Sum Symmetric Functions 2.1. The Elementary Symmetric Functions 2.2. The Complete Homogeneous Symmetric Functions 2.3. The Power Sum Symmetric Functions 2.4. Problems 3. Interlude: Evaluations of Symmetric Functions 3.1. Symmetric Function Identities 3.2. Binomial Coefficients 3.3. Stirling Numbers of the First and Second Kinds 3.4. q-Binomial Coefficients 3.5. Problems 3.6. Notes 4. Schur Polynomials and Schur Functions 4.1. Schur Functions and Semistandard Tableaux 4.2. Schur Polynomials as Ratios of Determinants 4.3. Problems 4.4. Notes 5. Interlude: A Rogues’ Gallery of Symmetric Functions 5.1. Skew Schur Functions 5.2. Stable Grothendieck Polynomials 5.3. Dual Stable Grothendieck Polynomials 5.4. The Chromatic Symmetric Function 5.5. Problems 5.6. Notes 6. The Jacobi–Trudi Identities and an Involution on 6.1. The First Jacobi–Trudi Identity 6.2. The Second Jacobi–Trudi Identity 6.3. The Involution 6.4. Problems 6.5. Notes 7. The Hall Inner Product 7.1. Inner Products on 7.2. The Hall Inner Product and Cauchy’s Formula 7.3. The Hall Inner Product on the Power Sum Symmetric Functions 7.4. Problems 7.5. Notes 8. The Robinson–Schensted–Knuth Correspondence 8.1. RSK Insertion: Constructing 8.2. Constructing 8.3. Implementing RSK with Growth Diagrams 8.4. Problems 8.5. Notes 9. Special Products Involving Schur Functions 9.1. The Pieri Rules 9.2. The Murnaghan–Nakayama Rule 9.3. Problems 10. The Littlewood–Richardson Rule 10.1. Products of Tableaux 10.2. Knuth Equivalence 10.3. The Relationship Between P and word 10.4. The Littlewood–Richardson Rule 10.5. Problems 10.6. Notes Appendix A. Linear Algebra A.1. Fields and Vector Spaces A.2. Bases and Linear Transformations A.3. Inner Products and Dual Bases A.4. Problems Appendix B. Partitions B.1. Partitions and a Generating Function B.2. Problems Appendix C. Permutations C.1. Permutations as Bijections C.2. Determinants and Permutations C.3. Problems Bibliography Index