Jack, Hall-Littlewood and Macdonald Polynomials
About this Title
Vadim B. Kuznetsov and Siddhartha Sahi, Editors
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and Macdonald; the original papers of Littlewood and Jack; notes on Hall's work by Macdonald; and a recently discovered unpublished manuscript by Jack (annotated by Macdonald).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.
Research mathematicians interested in algebraic combinatorics.
Table of Contents
- B. D. Sleeman – Henry Jack 1917–1978 [MR 2283247]
- Alun O. Morris – Philip Hall [MR 2284116]
- Alun O. Morris – Dudley Ernest Littlewood [MR 2284117]
- Alun O. Morris – Ian Macdonald [MR 2284118]
- Edited by Vadim B. Kuznetsov and Siddhartha Sahi – I. G. Macdonald – The algebra of partitions (This article is not available individually due to permission restrictions; to view, see the full volume PDF.) [MR 2284114]
- D. E. Littlewood – On certain symmetric functions [MR 2284120]
- Henry Jack – A class of symmetric polynomials with a parameter [MR 2284121]
- Henry Jack – A class of polynomials in search of a definition, or the symmetric group parametrized [MR 2284122]
- I. G. Macdonald – Commentary on the previous paper: “A class of polynomials in search of a definition, or the symmetric group parametrized” [in Jack, Hall-Littlewood and Macdonald polynomials, 75–106, Contemp. Math., 417, Amer. Math. Soc., Providence, RI, 2006; MR2284122] by H. Jack [MR 2284123]
- Henry Jack – First letter from Henry Jack to G. de B. Robinson [MR 2284124]
- Hasan Coskun and Robert A. Gustafson – Well-poised Macdonald functions $W_\lambda $ and Jackson coefficients $\omega _\lambda $ on $BC_n$ [MR 2284125]
- J. F. van Diejen – Asymptotics of multivariate orthogonal polynomials with hyperoctahedral symmetry [MR 2284126]
- Pavel Etingof and Alexei Oblomkov – Quantization, orbifold cohomology, and Cherednik algebras [MR 2284127]
- Bogdan Ion and Siddhartha Sahi – Triple groups and Cherednik algebras [MR 2284128]
- M. Kasatani, T. Miwa, A. N. Sergeev and A. P. Veselov – Coincident root loci and Jack and Macdonald polynomials for special values of the parameters [MR 2284129]
- Tom H. Koornwinder – Lowering and raising operators for some special orthogonal polynomials [MR 2284130]
- Vadim B. Kuznetsov and Evgeny K. Sklyanin – Factorization of symmetric polynomials [MR 2284131]
- Edwin Langmann – A method to derive explicit formulas for an elliptic generalization of the Jack polynomials [MR 2284132]
- Michel Lassalle – A short proof of generalized Jacobi-Trudi expansions for Macdonald polynomials [MR 2284133]
- Andrei Okounkov and Grigori Olshanski – Limits of $BC$-type orthogonal polynomials as the number of variables goes to infinity [MR 2284134]
- Eric M. Rains – A difference-integral representation of Koornwinder polynomials [MR 2284135]
- Michael Schlosser – Explicit computation of the $q,t$-Littlewood-Richardson coefficients [MR 2284136]
- Vyacheslav P. Spiridonov – A multiparameter summation formula for Riemann theta functions [MR 2284137]