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Symmetric Functions and Combinatorial Operators on Polynomials
Alain Lascoux, Institut Gaspard Monge, Université de Marne-la-Vallée, France
A co-publication of the AMS and CBMS.
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CBMS Regional Conference Series in Mathematics
2003; 268 pp; softcover
Number: 99
ISBN-10: 0-8218-2871-1
ISBN-13: 978-0-8218-2871-7
List Price: US$61
Member Price: US$48.80
All Individuals: US$48.80
Order Code: CBMS/99
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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.

Readership

Graduate students and research mathematicians interested in combinatorics.

Reviews

"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

  • Symmetric functions
  • Symmetric functions as operators and \(\lambda\)-rings
  • Euclidean division
  • Reciprocal differences and continued fractions
  • Division, encore
  • Padé approximants
  • Symmetrizing operators
  • Orthogonal polynomials
  • Schubert polynomials
  • The ring of polynomials as a module over symmetric ones
  • The plactic algebra
  • Complements
  • Solutions of exercises
  • Bibliography
  • Index
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