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Symmetric functions and orthogonal polynomials. (English) Zbl 0887.05053
University Lecture Series. 12. Providence, RI: American Mathematical Society (AMS). xv, 53 p. \$ 19.00 (1998).

In March, 1993, the author gave a series of lectures at Rutgers University on a two-parameter family of orthogonal polynomials ${P}_{\lambda }\left(q,t\right)$ that have become known as Macdonald polynomials. This booklet is an extended version of the lecture notes. The main topic discussed is the fact that Macdonald polynomials exist, are uniquely defined by orthogonality conditions with respect to a certain inner product for all root systems, and generalize many commonly known families of symmetric functions. The material contained in these notes has been well known to the specialists in the area for several years, but this booklet can serve as a self-contained introduction for anyone with some background in symmetric functions and root systems. Since 1993, there has been a huge amount of literature devoted to Macdonald polynomials in several areas of mathematics including algebraic combinatorics, representation theory and mathematical physics (which is not covered in this book). See [MR review by M. Haiman, to appear] for recent progress on the polynomiality and integrality conjectures related to Macdonald polynomials.

The first chapter contains a brief review of the theory of symmetric functions. In particular, several generalizations of Schur functions are given including zonal polynomials, Jack symmetric functions, Hall-Littlewood and Macdonald polynomials (of type $A$). All of these families of polynomials are special cases of the Macdonald polynomials and are defined by orthogonality relations pertaining to certain inner products. Uniqueness of Macdonald polynomials is easy to see from the definition, existence is much more subtle since at the time these notes were written there were no explicit formulas. Existence is proven by showing that these symmetric functions are eigenfunctions of a selfadjoint operator with distinct eigenvalues whose matrix form is upper triangular with respect to the monomial symmetric functions.

The second chapter, which contains the definition of Macdonald polynomials, is extended to all root systems generalizing the Weyl characters. Again these polynomials are defined by their leading term in the expansion with respect to orbit sums (generalization of monomial symmetric functions) and orthogonality relations with respect to a certain inner product. The proof of existence of these polynomials is similar in nature to the type $A$ case given in Chapter 1, but some complications occur for types ${D}_{n}$, ${E}_{8},{F}_{4},$ and ${G}_{2}$.

In the Postscript, the author has given a brief survey of the work of I. Cherednik who used the affine Hecke algebra to give a different proof of the existence of Macdonald polynomials in the following papers: [Invent. Math. 122, No. 1, 119-145 (1995; Zbl 0854.22021)], [Math. Res. Lett. 3, No. 3, 418-427 (1996; Zbl 0863.33016)], [Int. Math. Res. Not. 10, 483-515 (1995; Zbl 0886.05121)], [Ann. Math., II. Ser. 141, No. 1, 191-216 (1995; Zbl 0822.33008)]. This approach to orthogonal polynomials has lead to many interesting formulas, Cherednik’s generalization of the scalar product and a nonsymmetric analog.

##### MSC:
 05E05 Symmetric functions and generalizations 05E35 Orthogonal polynomials (combinatorics) (MSC2000) 05-02 Research monographs (combinatorics) 33-02 Research monographs (special functions) 33C80 Connections of hypergeometric functions with groups and algebras 33C50 Hypergeometric functions and integrals in several variables