Last update: 20 September 2012
In this section we use Theorem 2.2 to give expansions of the nonsymmetric Macdonald polynomials (Theorem 3.1) and the symmetric Macdonald polynomials (Theorem 3.4).
Let be the double affine Hecke algebra (defined in (2.31)) and let be the subalgebra of generated by and The polynomial representation of is
The monomials form a –basis of is the basis of nonsymmetric Macdonald polynomials
with the minimal length element in the coset Note that for since for
If is the set of dominant integral coweights (analogous to defined in (3.8)), and is a reduced word, then
since is the number of hyperplanes parallel to which are between and 1. If then for some and so, for all
More generally, if is the minimal length element of the coset then
where, in the last line, the action of on is as in (2.15). Thus the are eigenvectors for the action of the on the polynomial representation
Retain the notation of (2.36-2.37) so that if is a reduced word then denotes the set of alcove walks of type beginning at For define the weight and the final direction of by
In other words, is the "hexagon where ends". For define
If are as defined in (2.35) then, by (3.4),
with By (2.30) and the definition of in (3.4), the constant is a monomial in the symbols To simplify the notation for these constants write so that
Let and let be the minimal length element in the coset Fix a reduced word for and let be as defined in (2.35). With notations as in (3.5-3.7) the nonsymmetric Macdonald polynomial
where the sum is over the set of alcove walks of type beginning at 1.
applying the formula for in Theorem 2.2 to gives the formula in the statement.
From the expansion of in Theorem 3.1, the nonsymmetric MAcdonald polynomial has top term where is the minimal length representative of the coset This term is the term corresponding to the unique alcove walk in with no folds.
If is a reduced word for the minimal length element of the coset then is a walk from 1 to which stays completely in the dominant chamber. This has the effect that the roots are all of the form with (positive coroots) and The height of a coroot is
In the case that all the parameters are equal the values which appear in Theorem 3.1,
The set of dominant integral weights is
Recall the notation for from (3.6). For the symmetric Macdonald polynomial (see [14, Remarks after (6.8)]) is
so that for and has top term with coefficient 1. The symmetric Macdonald polynomials are –symmetric polynomials in which are eigenvectors for the action of –symmetric polynomials in the
Let and let be a reduced word for the minimal length element in the coset Let be as defined in (2.35) and let
be the set of alcove walks of type beginning at an element Then the symmetric Macdonald polynomial
where is the initial alcove of the path
which is computed by the same method as in Theorem 2.2 and Theorem 3.1.
The Hall-Littlewood polynomials or Macdonald spherical functions are and the Schur functions or Weyl characters are In the first case the formula in Theorem 3.4 reduces to the formula for the Macdonald spherical functions in terms of positively folded alcove walks as given in [17, Thm. 1.1] (see also [16, Thm. 4.2(a)]). In the case the formula in Theorem 3.4 reduces to the formula for the Weyl characters in terms of maximal dimensional positively folded alcove walks (the Littelmann path model) as given in [2, Cor. 1 p. 62], or tha –chain formulation of ,.
When is not a regular weight, the formula for in Theorem 3.4 has an alternate
formulation as a sum over paths whose initial alcove is in the minimal coset
To see this, suppose for some
Further, let be the minimal length element in the coset
for some so
This page is taken from a paper entitled A combinatorial formula for Macdonald Polynomials by Arun Ram and Martha Yip.