Preface

1. Historical perspective

The subject of symmetric functions arose initially in connection with the represen-

tation theory of the symmetric group, however it has since found wide applicability.

In the last twenty years or so, there have far-reaching new developments in the sub-

ject, as well as a general broadening of the areas of applicability, especially within

combinatorics, classical analysis and mathematical physics.

The subject has a particularly distinguished history going back to the work of

C. G. Jacobi

[Ja]

in the mid-nineteenth century, and to the papers of F. G. Frobe-

nius

[F],

I.

Schur

[S], H.

Weyl

[W],

M.A. MacMahon

[M],

and A. Young

[Y]

in

the early twentieth century. These papers singled out a certain family of symmet-

ric polynomials, now called Schur functions, which played a significant role in the

representation theory of the symmetric group Bn as well as the complex general

linear group GLn(IC). This dual role of the Schur functions is often referred to as

"Schur-Weyl duality".

The next stage in the development of the subject was the fundamental work of

P. Hall

[H]

and

D.

E. Littlewood

[L]

who independently discovered a one-parameter

generalization of the Schur polynomials. Subsequent work by J. A. Green

[G]

and

I.

G. Macdonald

[Ml]

showed that these polynomials, now called the Hall-

Littlewood polynomials, play a crucial role in the representation theory of G Ln

over finite and p-adic fields.

In the late 1960's, Henry Jack [Jl, J2] discovered a totally different one-

parameter generalization of Schur functions. These polynomials, now called Jack

polynomials, include as a special case the zonal polynomials, which are related to

the group GLn(F) with F

=~,and

had been previously studied by A. T. James

[J]

in connection with multivariate statistics.

In the 1980's,

I.

G. Macdonald unified these developments by introducing a two-

parameter family of symmetric polynomials, now called Macdonald polynomials.

The Hall-Littlewood polynomials are a special case of Macdonald polynomials, and

arise by specializing one of the parameters to 0. The Jack polynomials too arise as a

limiting case when both parameters approach

1-

the Jack parameter is the limiting

direction of approach. These polynomials were also independently discovered by K.

Kadell,

[Kad]

in connection with his investigation of the Selberg integral.

As explained above, the Macdonald symmetric polynomials are closely related

to the group

G

Ln and hence to root systems of type A. In subsequent work Macdon-

ald constructed analogous polynomials associated to arbitrary root system. These

polynomials arise as the 'discrete spectrum' of a class of q-difference operators.

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