Symmetric Functions and Combinatorial Operators on Polynomials

Symmetric Functions and Combinatorial Operators on Polynomials

by Alain Lascoux
     
 

ISBN-10: 0821828711

ISBN-13: 9780821828717

Pub. Date: 12/31/2003

Publisher: American Mathematical Society

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

Overview

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, Pade 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.

Product Details

ISBN-13:
9780821828717
Publisher:
American Mathematical Society
Publication date:
12/31/2003
Series:
CBMS Regional Conference Series in Mathematics Ser., #99
Pages:
268
Product dimensions:
6.69(w) x 9.84(h) x (d)

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