Symmetric Functions and Combinatorial Operators on Polynomials
About this Title
Alain Lascoux, Institut Gaspard Monge, Université de Marne-la-Vallée, Marne-la-Vallée, France
Publication: CBMS Regional Conference Series in Mathematics
Publication Year: 2003; Volume 99
ISBNs: 978-0-8218-2871-7 (print); 978-1-4704-2459-6 (online)
MathSciNet review: MR2017492
MSC: Primary 05E05; Secondary 14N15, 42C05
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 and the diagonalization of a kernel generalizing the resultant.
The last chapter sketches a non-commutative version of symmetric functions, with the help of Young tableaux and the plactic monoid.
The book also contains numerous exercises clarifying and extending many points of the main text.
Graduate students and research mathematicians interested in combinatorics.
Table of Contents
- Chapter 1. Symmetric functions
- Chapter 2. Symmetric functions as operators and $\lambda$-rings
- Chapter 3. Euclidean division
- Chapter 4. Reciprocal differences and continued fractions
- Chapter 5. Division, encore
- Chapter 6. Padé approximants
- Chapter 7. Symmetrizing operators
- Chapter 8. Orthogonal polynomials
- Chapter 9. Schubert polynomials
- Chapter 10. The ring of polynomials as a module over symmetric ones
- Chapter 11. The plactic algebra
- Appendix A. Complements
- Appendix B. Solutions of exercises