The Macdonald polynomials P_\lambda are a two-parameter family of
orthogonal polynomials indexed by a positive weight λ of a
root system, introduced by
Ian G. Macdonald (1987). They generalize several other families of orthogonal polynomials, such as
Jack polynomials and
Hall–Littlewood polynomials. They are known to have deep relationships with
affine Hecke algebras and
Hilbert schemes, which were used to prove several conjectures made by Macdonald about them. introduced a new basis for the space of
symmetric functions, which specializes to many of the well-known bases for the symmetric functions, by suitable substitutions for the parameters
q and
t. In fact, we can obtain in this manner the
Schur functions, the Hall–Littlewood symmetric functions, the Jack symmetric functions, the
zonal symmetric functions, the
zonal spherical functions, and the elementary and monomial symmetric functions. The so-called
q,
t-
Kostka polynomials are the coefficients of a resulting
transition matrix. Macdonald conjectured that they are polynomials in
q and
t, with non-negative
integer coefficients. It was
Adriano Garsia's idea to construct an appropriate
module in order to prove positivity (as was done in his previous joint work with
Procesi on Schur positivity of
Kostka–Foulkes polynomials). In an attempt to prove Macdonald's conjecture, introduced the bi-graded module H_\mu of
diagonal harmonics and conjectured that the (modified) Macdonald polynomials are the Frobenius image of the character generating function of
Hμ, under the diagonal action of the
symmetric group. The proof of Macdonald's conjecture was then reduced to the
n! conjecture; i.e., to prove that the dimension of
Hμ is
n!. In 2001, Haiman proved that the dimension is indeed
n! (see [4]). This breakthrough led to the discovery of many hidden connections and new aspects of
symmetric group representation theory, as well as combinatorial objects (e.g., insertion tableaux, Haglund's inversion numbers, and the role of parking functions in
representation theory). ==References==