An Introduction to Symmetric Functions and Their Combinatorics

Eric S. Egge

• Publisher: American Mathematical Society
• Year: 2019
• Print ISBN: 9781470448998
• eBook ISBN: 9781470454937
• Topic: Mathematics
• Subtopic: Counting & Numeration
• Index: Mathematics

This book is a reader-friendly introduction to the theory of symmetric functions, and it includes fundamental topics such as the monomial, elementary, homogeneous, and Schur function bases; the skew Schur functions; the Jacobi–Trudi identities; the involution $\omega$; the Hall inner product; Cauchy's formula; the RSK correspondence and how to implement it with both insertion and growth diagrams; the Pieri rules; the Murnaghan–Nakayama rule; Knuth equivalence; jeu de taquin; and the Littlewood–Richardson rule. The book also includes glimpses of recent developments and active areas of research, including Grothendieck polynomials, dual stable Grothendieck polynomials, Stanley's chromatic symmetric function, and Stanley's chromatic tree conjecture. Written in a conversational style, the book contains many motivating and illustrative examples. Whenever possible it takes a combinatorial approach, using bijections, involutions, and combinatorial ideas to prove algebraic results.The prerequisites for this book are minimal—familiarity with linear algebra, partitions, and generating functions is all one needs to get started. This makes the book accessible to a wide array of undergraduates interested in combinatorics.

Table of contents

• Cover
• Title page
• Preface
• Chapter 1. Symmetric Polynomials, the Monomial Symmetric Polynomials, and Symmetric Functions
• Chapter 2. The Elementary, Complete Homogeneous, and Power Sum Symmetric Functions
• Chapter 3. Interlude: Evaluations of Symmetric Functions
• Chapter 4. Schur Polynomials and Schur Functions
• Chapter 5. Interlude: A Rogues’ Gallery of Symmetric Functions
• Chapter 6. The Jacobi–Trudi Identities and an Involution on Λ
• Chapter 7. The Hall Inner Product
• Chapter 8. The Robinson–Schensted–Knuth Correspondence
• Chapter 9. Special Products Involving Schur Functions
• Chapter 10. The Littlewood–Richardson Rule
• Appendix A. Linear Algebra
• Appendix B. Partitions
• Appendix C. Permutations
• Bibliography
• Index
• Back cover