**Contemporary Mathematics**

Volume: 417;
2006;
360 pp;
Softcover

MSC: Primary 33;

Print ISBN: 978-0-8218-3683-5

Product Code: CONM/417

List Price: $105.00

Individual Member Price: $84.00

**Electronic ISBN: 978-0-8218-8096-8
Product Code: CONM/417.E**

List Price: $105.00

Individual Member Price: $84.00

# Jack, Hall-Littlewood and Macdonald Polynomials

Share this page *Edited by *
*Vadim B. Kuznetsov; Siddhartha Sahi*

The subject of symmetric functions began with the work of Jacobi, Schur, Weyl, Young and others on the Schur polynomials. In the 1950's and 60's, far-reaching generalizations of Schur polynomials were obtained by Hall and Littlewood (independently) and, in a different direction, by Jack. In the 1980's, Macdonald unified these developments by introducing a family of polynomials associated with arbitrary root systems.

The last twenty years have witnessed considerable progress in this area, revealing new and profound connections with representation theory, algebraic geometry, combinatorics, special functions, classical analysis and mathematical physics. All these fields and more are represented in this volume, which contains the proceedings of a conference on “Jack, Hall-Littlewood and Macdonald polynomials” held at ICMS, Edinburgh, during September 23–26, 2003.

In addition to new results by leading researchers, the book contains a wealth of historical material, including brief biographies of Hall, Littlewood, Jack and Macdonald; the original papers of Littlewood and Jack; notes on Hall's work by Macdonald; and a recently discovered unpublished manuscript by Jack (annotated by Macdonald). The book will be invaluable to students and researchers who wish to learn about this beautiful and exciting subject.

#### Table of Contents

# Table of Contents

## Jack, Hall-Littlewood and Macdonald Polynomials

- Contents v6 free
- Preface ix10 free
- Bibliography xv16 free
- Acknowledgments xix20 free
- Part 1. Historic Material 122 free
- Photo of Henry Jack 223
- Henry Jack 1917-1978 324
- Photo of Philip Hall 627
- Philip Hall 728
- Photo of Dudley Ernest Littlewood 1031
- Dudley Ernest Littlewood 1132
- Photo of Ian Macdonald 1637
- Ian Macdonald 1738
- The Algebra of Partitions 2344
- On Certain Symmetric Functions 4364
- A class of symmetric polynomials with a parameter 5778
- A class of polynomials in search of a definition, or the symmetric group parametrized 7596
- Commentary on the previous paper 107128
- First letter from Henry Jack to G. de B. Robinson [16.4.76] 121142

- Part 2. Research articles 125146
- Well-poised Macdonald functions Wλ and Jackson coefficients ωλ on BCn 127148
- Asymptotics of multivariate orthogonal polynomials with hyperoctahedral symmetry 157178
- Quantization, orbifold cohomology, and Cherednik algebras 171192
- Triple groups and Cherednik algebras 183204
- Coincident root loci and Jack and Macdonald polynomials for special values of the parameters 207228
- Lowering and raising operators for some special orthogonal polynomials 227248
- Factorization of symmetric polynomials 239260
- A method to derive explicit formulas for an elliptic generalization of the Jack polynomials 257278
- A short proof of generalized Jacobi-Trudi expansions for Macdonald polynomials 271292
- Limits of BC-type orthogonal polynomials as the number of variables goes to infinity 281302
- 1. Introduction 281302
- 2. Interpolation BCn polynomials and binomial formula 288309
- 3. Sufficient conditions of regularity 295316
- 4. Necessary conditions of regularity 299320
- 5. The convex set γθ 303324
- 6. Spherical functions on infinite–dimensional symmetric spaces 305326
- 7. The BCn polynomials with θ = 1 309330
- References 317338

- A difference-integral representation of Koornwinder polynomials 319340
- Explicit computation of the q, t-Littlewood-Richardson coefficients 335356
- A multiparameter summation formula for Riemann theta functions 345366

- Part 3. Vadim Kuznetsov 1963–2005 355376

#### Readership

Research mathematicians interested in algebraic combinatorics.

#### Reviews

The book will be invaluable to students and researchers who wish to learn about this beautiful and exciting subject.

-- Zentralblatt MATH