Macdonald polynomials

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 20 September 2012

Macdonald polynomials

In this section we use Theorem 2.2 to give expansions of the nonsymmetric Macdonald polynomials Eμ (Theorem 3.1) and the symmetric Macdonald polynomials Pμ (Theorem 3.4).

Let H be the double affine Hecke algebra (defined in (2.31)) and let H be the subalgebra of H generated by T0,,Tn and Ω. The polynomial representation of H is

[X]= IndHH (1)=–span { qkXμ1 k12 ,μ𝔥* } (3.1)

with

Ti1= ti1/21 andg1=1, forgΩ. (3.2)

The monomials Xμ1,μ 𝔥*, form a [q±1/e]–basis of [X] is the basis of nonsymmetric Macdonald polynomials

{Eμμ𝔥*}, whereEμ= τXμm1 (3.3)

with Xμm the minimal length element in the coset XμW0. Note that τw1=0 for wW0 since τi1=0 for i=1,2,,n.

If 𝔥+ is the set of dominant integral coweights (analogous to (𝔥*)+ defined in (3.8)), λ𝔥+ and Yλ=si1si is a reduced word, then

Yλ 1=Ti1 Ti1= ti112 ti121= q 12 ( ci1++ ci ) 1 q 12 αR+ cαλ,α 1= qλ,ρc 1,

since λ,α is the number of hyperplanes parallel to 𝔥α which are between Yλ and 1. If λ𝔥 then λ=μ-ν for some μ,ν𝔥+ and so, for all λ𝔥,

Yλ1= qλ,ρc 1,whereρc= 12αR+ cαα. (3.4)

More generally, if Xμm is the minimal length element of the coset XμW0 then

YλEμ = YλτXμm 1=τXμm Y m-1X-μ λ 1=τXμm Y m-1 ( λ+ λ,μd ) 1 = τXμm Ym-1λ q-λ,μ 1= q m-1λ,ρc -λ,μ τXμm1= q λ,mρc-μ Eμ = q λ,X-μ m.ρc Eμ,

where, in the last line, the action of W on 𝔥* is as in (2.15). Thus the Eμ are eigenvectors for the action of the Yλ on the polynomial representation [X].

Retain the notation of (2.36-2.37) so that if s=si1 si is a reduced word then (v,w) denotes the set of alcove walks of type w=(i1,,i) beginning at v. For p(v,w) define the weight wt(p) and the final direction φ(p) of p by

Xend(p)= Xwt(p) Tφ(p), withwt(p) 𝔥*and φ(p)W0. (3.5)

In other words, wt(p) is the "hexagon where p ends". For wW define

tw1/2= ti11/2 ti1/2, ifw=si1 si is a reduced word. (3.6)

If βk si sik+1 αik are as defined in (2.35) then, by (3.4),

Y-βk 1= Y -(-γ+jd) 1=qj qγ,ρc 1,ifβk =-γ+jd

with γR, j. By (2.30) and the definition of ρc in (3.4), the constant qγ,ρc is a monomial in the symbols ti1/2. To simplify the notation for these constants write qj qγ,ρc= q-βk,ρc so that

Y-βk1= q-βk,ρc 1. (3.7)

Let μ𝔥* and let w=Xμm be the minimal length element in the coset XμW0. Fix a reduced word w= ss1 ss for w and let β,. β1 be as defined in (2.35). With notations as in (3.5-3.7) the nonsymmetric Macdonald polynomial

Eμ= p(u) Xwt(p) tφ(p)12 ( kf+(p) tβk-12 (1-tβk) 1- q-βk,ρc ) ( kf-(p) tβk-12 (1-tβk) q-βk,ρc 1- q-βk,ρc ) ,

where the sum is over the set (μ)= (1,w) of alcove walks of type i1,,i beginning at 1.

Proof.

Since Eμ= τXμm1,

Xend(p)1 =Xwt(p) Tφ(p)1= Xwt(p) tφ(p)12 1andYλ 1= qλ,ρc 1,

applying the formula for τXμm in Theorem 2.2 to 1 gives the formula in the statement.

From the expansion of Eμ in Theorem 3.1, the nonsymmetric MAcdonald polynomial Eμ has top term tm1/2Xμ, where Xμm is the minimal length representative of the coset XμW0. This term is the term corresponding to the unique alcove walk in (μ) with no folds.

If w=Xμm= si1 si is a reduced word for the minimal length element of the coset XμW0 then w-1= si si1 is a walk from 1 to w-1 which stays completely in the dominant chamber. This has the effect that the roots β,, β1 are all of the form -γ+jd with γ(R)+ (positive coroots) and j>0. The height of a coroot γ is

ht(γ)= γ,ρ, whereρ=12 αRα.

In the case that all the parameters are equal ( ti=t=qc fori=0,,n ) the values which appear in Theorem 3.1,

q-βk,ρc= qγ-jd,ρc= qjtht(γ), have positive exponents (in>0).

The set of dominant integral weights is

(𝔥*)+= { μ𝔥* μ,αi0 fori=1,,n } . (3.8)

Recall the notation for tw1/2 from (3.6). For μ(𝔥*)+, the symmetric Macdonald polynomial (see [14, Remarks after (6.8)]) is

Pμ=10Eμ where10= wW0 tw0w-12 Tw, (3.9)

so that Ti10= ti1/210 for i=1,2,n, and 10 has top term TW0 with coefficient 1. The symmetric Macdonald polynomials are W0–symmetric polynomials in Xμ which are eigenvectors for the action of W0–symmetric polynomials in the Yλ.

Let μ(𝔥*)+ and let Xμm= si1 si be a reduced word for the minimal length element Xμm in the coset XμW0. Let β,, β1 be as defined in (2.35) and let

𝒫(μ)= vW0 (v,w)

be the set of alcove walks of type w=(i1,,i) beginning at an element vW0. Then the symmetric Macdonald polynomial

Pμ= p𝒫(u) Xwt(p) tφ(p)12 tw0ι(p)-12 ( kf+(p) tβk-12 (1-tβk) 1- q-βk,ρc ) ( kf-(p) tβk-12 (1-tβk) q-βk,ρc 1- q-βk,ρc ) ,

where ι(p) is the initial alcove of the path p.

Proof.

The expression

10=vW0 tw0v-12 Xv,givesPμ 1=10Eμ1= vW0 tw0v-12 Xv τXμm1,

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 Pμ(0,t) and the Schur functions or Weyl characters are sμ=Pμ(0,0). 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 q=t=0, 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 λ–chain formulation of [9],[10].

When μ is not a regular weight, the formula for Pμ in Theorem 3.4 has an alternate formulation as a sum over paths whose initial alcove ι(p) is in the minimal coset representatives Wμ of W0/Wμ. To see this, suppose siμ=μ for some i{1,,n}. Then μ,αi=0 implies Y-αiEμ 1=ti-1Eμ 1. Further, let Xμm be the minimal length element in the coset XμW0. Then siXμm= Xsiμsim= Xμmsj for some j{1,,n}, so τiEμ1= τiτXμm 1=τXμm τj1=0.
Thus TiEμ1= ( τi- ti-12 -τi12 1- Y-αi ) Eμ1=ti12 Eμ1, and

Pμ1 = 10Eμ1= wW0 tw0w-12 TwEμ1= tw012 ( vWμ tw0v-12 Tv ) ( uWμ tw0u-12 Tu ) Eμ1 = Wμ(t) vWμ tw0v-12 Xv τXμm1.

Notes and References

This page is taken from a paper entitled A combinatorial formula for Macdonald Polynomials by Arun Ram and Martha Yip.

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