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
in the mid-nineteenth century, and to the papers of F. G. Frobe-
and A. Young
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
The next stage in the development of the subject was the fundamental work of
who independently discovered a one-parameter
generalization of the Schur polynomials. Subsequent work by J. A. Green
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
had been previously studied by A. T. James
in connection with multivariate statistics.
In the 1980's,
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
the Jack parameter is the limiting
direction of approach. These polynomials were also independently discovered by K.
in connection with his investigation of the Selberg integral.
As explained above, the Macdonald symmetric polynomials are closely related
to the group
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.