# Symmetric Functions and Combinatorial Operators on Polynomials

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*Alain Lascoux*

A co-publication of the AMS and CBMS

The theory of symmetric functions is an old topic in mathematics which
is used as an algebraic tool in many classical fields. With
\(\lambda\)-rings, one can regard symmetric functions as operators on
polynomials and reduce the theory to just a handful of fundamental
formulas.

One of the main goals of the book is to describe the technique of
\(\lambda\)-rings. The main applications of this technique to the theory
of symmetric functions are related to the Euclid algorithm and its occurrence
in division, continued fractions, Padé approximants, and orthogonal
polynomials.

Putting the emphasis on the symmetric group instead of symmetric functions,
one can extend the theory to non-symmetric polynomials, with Schur functions
being replaced by Schubert polynomials. In two independent chapters, the author
describes the main properties of these polynomials, following either the
approach of Newton and interpolation methods or the method of Cauchy.

The last chapter sketches a non-commutative version of symmetric functions,
using Young tableaux and the plactic monoid.

The book contains numerous exercises clarifying and extending many points of
the main text. It will make an excellent supplementary text for a graduate
course in combinatorics.

#### Reviews & Endorsements

There is a wealth of information in this book, as well an extensive bibliography and an abundance of exercises (with solutions!) for conscientious reader.

-- Australian Mathematical Society Gazette

There is much to recommend about this book ... this book is an extensive treatise on symmetric functions and their role in many classical constructions in mathematics involv~ng polynomials, by a modern master of the subject.

-- Frank Sottile for Mathematical Reviews

#### Table of Contents

# Table of Contents

## Symmetric Functions and Combinatorial Operators on Polynomials

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents iii4 free
- Preface vii8 free
- Chapter 1. Symmetric functions 114 free
- Chapter 2. Symmetric Functions as Operators and λ-Rings 3144
- Chapter 3. Euclidean Division 4760
- Chapter 4. Reciprocal Differences and Continued Fractions 6376
- 4.1. Euler's Recursions 6376
- 4.2. Continued Fraction Expression of a Formal Series 6477
- 4.3. Interpolation of a Function by a Continued Fraction 6679
- 4.4. Relation between Stieltjes and Wronski Continued Fractions 6982
- 4.5. Jacobi's Tridiagonal Matrix 7083
- 4.6. Motzkin Paths 7184
- 4.7. Dyck Paths 7285
- 4.8. Link between Enumeration of Motzkin and Dyck Paths 7386
- Exercises 7588

- Chapter 5. Division, encore 7992
- Chapter 6. Padé Approximants 87100
- Chapter 7. Symmetrizing Operators 95108
- 7.1. Divided Differences 95108
- 7.2. Compatibility with Complete Functions 97110
- 7.3. Braid Relations 97110
- 7.4. Decomposing in the Basis of Permutations 99112
- 7.5. Generating Series by Symmetrization 100113
- 7.6. Maximal Symmetrizers 101114
- 7.7. Schur Functions and Bott's Theorem 102115
- 7.8. Lagrange Interpolation 105118
- 7.9. Finite Derivation 108121
- 7.10. Calogero's Raising and Lowering Operators 110123
- Exercises 112125

- Chapter 8. Orthogonal Polynomials 117130
- 8.1. Orthogonal Polynomials as Symmetric Functions 117130
- 8.2. Reproducing Kernels 118131
- 8.3. Continued Fractions 119132
- 8.4. Higher Order Kernels 120133
- 8.5. Even Moments 123136
- 8.6. Zeros 125138
- 8.7. The Moment Generating Function 128141
- 8.8. Jacobi's Matrix and Paths 129142
- 8.9. Discrete Measures 131144
- Exercises 133146

- Chapter 9. Schubert Polynomials 141154
- 9.1. Newton Interpolation Formula 141154
- 9.2. Newton and Euclid 142155
- 9.3. Discrete Wronskian 143156
- 9.4. Schubert Polynomials 145158
- 9.5. Vanishing Properties 147160
- 9.6. Newton Interpolation in Several Variables 148161
- 9.7. Interpolation of Symmetric Functions 150163
- 9.8. Key Polynomials 152165
- Exercises 153166

- Chapter 10. The Ring of Polynomials as a Module over Symmetric Ones 157170
- 10.1. Quadratic Form on Bol 157170
- 10.2. Kernel 158171
- 10.3. Shifts 162175
- 10.4. Generating Function in the NilCoxeter Algebra 163176
- 10.5. NilPlactic Kernel 165178
- 10.6. Basis of Elementary Symmetric Functions 168181
- 10.7. Yang-Baxter Basis 170183
- 10.8. Yang-Baxter Elements as Permutations 172185
- Exercises 174187

- Chapter 11. The plactic algebra 175188
- Appendix A. Complements 185198
- Appendix B. Solutions of exercises 197210
- Bibliography 261274
- Index 267280
- Back Cover Back Cover1282