**Student Mathematical Library**

Volume: 91;
2019;
342 pp;
Softcover

MSC: Primary 05;

**Print ISBN: 978-1-4704-4899-8
Product Code: STML/91**

List Price: $55.00

Individual Price: $44.00

**Electronic ISBN: 978-1-4704-5493-7
Product Code: STML/91.E**

List Price: $55.00

Individual Price: $44.00

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#### Supplemental Materials

# An Introduction to Symmetric Functions and Their Combinatorics

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*Eric S. Egge*

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.

#### Readership

Undergraduate and graduate students interested in algebra and combinatorics.

#### Reviews & Endorsements

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 ww; 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.

-- Anthony Mendesm, Cal Poly San Luis Obispo, MAA Reviews

#### Table of Contents

# Table of Contents

## An Introduction to Symmetric Functions and Their Combinatorics

- Cover Cover11
- Title page iii4
- Preface ix10
- Chapter 1. Symmetric Polynomials, the Monomial Symmetric Polynomials, and Symmetric Functions 116
- Chapter 2. The Elementary, Complete Homogeneous, and Power Sum Symmetric Functions 2338
- Chapter 3. Interlude: Evaluations of Symmetric Functions 5368
- Chapter 4. Schur Polynomials and Schur Functions 7590
- Chapter 5. Interlude: A Rogues’ Gallery of Symmetric Functions 119134
- Chapter 6. The Jacobi–Trudi Identities and an Involution on Λ 157172
- Chapter 7. The Hall Inner Product 191206
- Chapter 8. The Robinson–Schensted–Knuth Correspondence 209224
- Chapter 9. Special Products Involving Schur Functions 247262
- Chapter 10. The Littlewood–Richardson Rule 271286
- Appendix A. Linear Algebra 309324
- Appendix B. Partitions 323338
- Appendix C. Permutations 327342
- Bibliography 337352
- Index 341356
- Back cover Back cover1359