Contemporary Mathematics 417. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3683-8/pbk). xix, 360 p. $ 79.00 (2006).

The articles of this volume will be reviewed individually.
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and Macdonald; the original papers of Littlewood and Jack; notes on Hall’s work by Macdonald; and a recently discovered unpublished manuscript by Jack (annotated by Macdonald).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. Indexed articles: {\it Sleeman, B.D.}, Henry Jack (1917--1978), 3-5, 2 [

Zbl 1135.01333] {\it Morris, Alun O.}, Philip Hall, 7-9, 4 [

Zbl 1135.01328] {\it Morris, Alun O.}, Dudley Ernest Littlewood, 11-15, 10 [

Zbl 1135.01326] {\it Morris, Alun O.}, Ian MacDonald, 17-22, 16 [

Zbl 1135.01327] {\it MacDonald, I.G.}, The algebra of partitions (reprint), 23-41 [

Zbl 1134.05102] {\it Littlewood, D.E.}, On certain symmetric functions (reprint), 43-56 [

Zbl 1133.05100] {\it Jack, Henry}, A class of symmetric polynomials with a parameter. Reprint, 57-74 [

Zbl 1188.05155] {\it Jack, Henry}, A class of polynomials in search of a definition, or the symmetric group parametrized, 75-106 [

Zbl 1143.33007] {\it Macdonald, I.G.}, Commentary on the previous paper, 107-120 [

Zbl 1188.05156] {\it Coskun, Hasan; Gustafson, Robert A.}, Well-poised Macdonald functions $W_{\lambda}$ and Jackson coefficients $\omega_{\lambda}$ on $BC_n$, 127-155 [

Zbl 1154.05059] {\it van Diejen, J.F.}, Asymptotics of multivariate orthogonal polynomials with hyperoctahedral symmetry, 157-169 [

Zbl 1132.33334] {\it Etingof, Pavel; Oblomkov, Alexei}, Quantization, orbifold cohomology, and Cherednik algebras., 171-182 [

Zbl 1137.16014] {\it Ion, Bogdan; Sahi, Siddhartha}, Triple groups and Cherednik algebras., 183-206 [

Zbl 1158.20002] {\it Kasatani, M.; Miwa, T.; Sergeev, A.N.; Veselov, A.P.}, Coincident root loci and Jack and Macdonald polynomials for special values of the parameters, 207-225 [

Zbl 1188.33019] {\it Koornwinder, Tom H.}, Lowering and raising operators for some special orthogonal polynomials, 227-238 [

Zbl 1132.33315] {\it Kuznetsov, Vadim B.; Sklyanin, Evgeny K.}, Factorization of symmetric polynomials, 239-256 [

Zbl 1132.33318] {\it Langmann, Edwin}, A method to derive explicit formulas for an elliptic generalization of the Jack polynomials, 257-270 [

Zbl 1136.81028] {\it Lassalle, Michel}, A short proof of generalized Jacobi-Trudi expansions for Macdonald polynomials, 271-280 [

Zbl 1133.05099] {\it Okounkov, Andrei; Olshanski, Grigori}, Limits of $BC$-type orthogonal polynomials as the number of variables goes to infinity, 281-318 [

Zbl 1151.33008] {\it Rains, Eric M.}, A difference-integral representation of Koornwinder polynomials, 319-333 [

Zbl 1132.33332] {\it Schlosser, Michael}, Explicit computation of the $q,t$-Littlewood-Richardson coefficients, 335-343 [

Zbl 1132.33333] {\it Spiridonov, Vyacheslav P.}, A multiparameter summation formula for Riemann theta functions, 345-353 [

Zbl 1132.33340] {\it Sklyanin, Evgeny; Sleeman, Brian D.}, Vadim Borisovich Kuznetsov, 1963--2005, 357-360, 356 [

Zbl 1188.01023]