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Symmetric functions and combinatorial operators on polynomials. (English) Zbl 1039.05066

CBMS Regional Conference Series in Mathematics 99. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2871-1/pbk). xi, 268 p. (2003).
This monograph is based on the author’s lectures at the CBMS conference at North Carolina State University in 2001. It covers the application of symmetric functions to algebraic identities related to the Euclidean algorithm. It does not require extensive knowledge of symmetric functions, although some familiarity with the basic properties would be useful.
Many texts on symmetric functions view the functions as polynomials. In constrast, this monograph uses the method of \(\lambda\)-rings, in which symmetric functions are viewed as operators on the ring of polynomials. This approach is particularly natural for these applications, because the construction of the symmetric functions and all necessary properties follow from a few fundamental results.
An arbitrary polynomial can be viewed as a symmetric function in terms of its roots. In this framework, the successive remainders in the Euclidean algorithm can be expressed in terms of symmetric functions; examples include Sturm sequences and continued fractions. Divided difference operators also act on polynomials, and operations involving partial or full symmetrization can be expressed in terms of divided differences; one example is a symmetric function identity which arises in the cohomology of Grassmannians. Another viewpoint is that of orthogonal polynomials; if the “moments” are the complete symmetric functions, the resulting orthogonal polynomials are Schur functions indexed by square partitions, and other Schur functions appear in contexts such as Christoffel determinants.
A similar approach can be used to study generalizations of symmetric functions. Schubert polynomials are constructed in two ways: by generalizing Newton’s interpolation to multiple variables, and from a non-symmetric Cauchy kernel. The book concludes with a brief discussion of further generalizations to non-commutative Schur functions and Schubert polynomials.

MSC:

05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
14M15 Grassmannians, Schubert varieties, flag manifolds
20C30 Representations of finite symmetric groups
41A10 Approximation by polynomials
41A21 Padé approximation
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
05E35 Orthogonal polynomials (combinatorics) (MSC2000)
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