## Macdonald polynomials

Last update: 20 September 2012

## Macdonald polynomials

In this section we use Theorem 2.2 to give expansions of the nonsymmetric Macdonald polynomials ${E}_{\mu }$ (Theorem 3.1) and the symmetric Macdonald polynomials ${P}_{\mu }$ (Theorem 3.4).

Let $\stackrel{\sim }{H}$ be the double affine Hecke algebra (defined in (2.31)) and let $H$ be the subalgebra of $\stackrel{\sim }{H}$ generated by ${T}_{0},\dots ,{T}_{n}$ and $\Omega \text{.}$ The polynomial representation of $\stackrel{\sim }{H}$ is

$ℂ[X]= IndHH∼ (1)=ℂ–span { qkXμ1∣ k∈12ℤ ,μ∈𝔥ℤ* } (3.1)$

with

$Ti1= ti1/21 andg1=1, forg∈Ω. (3.2)$

The monomials ${X}^{\mu }1,\mu \in {𝔥}_{ℤ}^{*},$ form a $ℂ\left[{q}^{±1/e}\right]$–basis of $ℂ\left[X\right]$ is the basis of nonsymmetric Macdonald polynomials

${Eμ∣μ∈𝔥ℤ*}, whereEμ= τXμm∨1 (3.3)$

with ${X}^{\mu }m$ the minimal length element in the coset ${X}^{\mu }{W}_{0}\text{.}$ Note that ${\tau }_{w}^{\vee }1=0$ for $w\in {W}_{0}$ since ${\tau }_{i}^{\vee }1=0$ for $i=1,2,\dots ,n\text{.}$

If ${𝔥}_{ℤ}^{+}$ is the set of dominant integral coweights (analogous to ${\left({𝔥}_{ℤ}^{*}\right)}^{+}$ defined in (3.8)), ${\lambda }^{\vee }\in {𝔥}_{ℤ}^{+}$ and ${Y}^{{\lambda }^{\vee }}={s}_{{i}_{1}}\dots {s}_{{i}_{\ell }}$ is a reduced word, then

$Yλ∨ 1=Ti1 …Tiℓ1= ti112… tiℓ121= q 12 ( ci1+…+ ciℓ ) 1 q 12 ∑α∈R+ cα⟨λ∨,α⟩ 1= q⟨λ∨,ρc⟩ 1,$

since $⟨{\lambda }^{\vee },\alpha ⟩$ is the number of hyperplanes parallel to ${𝔥}^{\alpha }$ which are between ${Y}^{{\lambda }^{\vee }}$ and 1. If ${\lambda }^{\vee }\in {𝔥}_{ℤ}$ then ${\lambda }^{\vee }={\mu }^{\vee }-{\nu }^{\vee }$ for some ${\mu }^{\vee },{\nu }^{\vee }\in {𝔥}_{ℤ}^{+}$ and so, for all ${\lambda }^{\vee }\in {𝔥}_{ℤ},$

$Yλ∨1= q⟨λ∨,ρc⟩ 1,whereρc= 12∑α∈R+ cαα. (3.4)$

More generally, if ${X}^{\mu }m$ is the minimal length element of the coset ${X}^{\mu }{W}_{0}$ then

$Yλ∨Eμ = Yλ∨τXμm∨ 1=τXμm∨ Y m-1X-μ λ∨ 1=τXμm∨ Y m-1 ( λ∨+ ⟨λ∨,μ⟩d ) 1 = τXμm∨ Ym-1λ∨ q-⟨λ∨,μ⟩ 1= q ⟨m-1λ∨,ρc⟩ -⟨λ∨,μ⟩ τXμm∨1= q ⟨ λ∨,mρc-μ ⟩ Eμ = q ⟨ λ∨,X-μ m.ρc ⟩ Eμ,$

where, in the last line, the action of $W$ on ${𝔥}_{ℤ}^{*}$ is as in (2.15). Thus the ${E}_{\mu }$ are eigenvectors for the action of the ${Y}^{{\lambda }^{\vee }}$ on the polynomial representation $ℂ\left[X\right]\text{.}$

Retain the notation of (2.36-2.37) so that if $s={s}_{{i}_{1}}^{\vee }\dots {s}_{{i}_{\ell }}^{\vee }$ is a reduced word then $ℬ\left(v,\stackrel{\to }{w}\right)$ denotes the set of alcove walks of type $\stackrel{\to }{w}=\left({i}_{1},\dots ,{i}_{\ell }\right)$ beginning at $v\text{.}$ For $p\in ℬ\left(v,\stackrel{\to }{w}\right)$ define the weight $\text{wt}\phantom{\rule{0.2em}{0ex}}\left(p\right)$ and the final direction $\phi \left(p\right)$ of $p$ by

$Xend(p)= Xwt(p) Tφ(p)∨, withwt(p) ∈𝔥ℤ*and φ(p)∈W0. (3.5)$

In other words, $\text{wt}\phantom{\rule{0.2em}{0ex}}\left(p\right)$ is the "hexagon where $p$ ends". For $w\in W$ define

$tw1/2= ti11/2… tiℓ1/2, ifw=si1∨ …siℓ∨ is a reduced word. (3.6)$

If ${\beta }_{k}^{\vee }{s}_{{i}_{\ell }}^{\vee }\dots {s}_{{i}_{k+1}}^{\vee }{\alpha }_{{i}_{k}}$ are as defined in (2.35) then, by (3.4),

$Y-βk∨ 1= Y -(-γ∨+jd) 1=qj q⟨γ∨,ρc⟩ 1,ifβk∨ =-γ∨+jd$

with ${\gamma }^{\vee }\in {R}^{\vee },\phantom{\rule{0.2em}{0ex}}j\in ℤ\text{.}$ By (2.30) and the definition of ${\rho }_{c}$ in (3.4), the constant ${q}^{⟨{\gamma }^{\vee },{\rho }_{c}⟩}$ is a monomial in the symbols ${t}_{i}^{1/2}\text{.}$ To simplify the notation for these constants write ${q}^{j}{q}^{⟨{\gamma }^{\vee },{\rho }_{c}⟩}={q}^{⟨-{\beta }_{k}^{\vee },{\rho }_{c}⟩}$ so that

$Y-βk∨1= q⟨-βk∨,ρc⟩ 1. (3.7)$

Let $\mu \in {𝔥}_{ℤ}^{*}$ and let $w={X}^{\mu }m$ be the minimal length element in the coset ${X}^{\mu }{W}_{0}\text{.}$ Fix a reduced word $\stackrel{\to }{w}={s}_{{s}_{1}}^{\vee }\dots {s}_{{s}_{\ell }}^{\vee }$ for $w$ and let ${\beta }_{\ell }^{\vee },\dots \text{.}{\beta }_{1}^{\vee }$ be as defined in (2.35). With notations as in (3.5-3.7) the nonsymmetric Macdonald polynomial

$Eμ= ∑p∈ℬ(u→) Xwt(p) tφ(p)12 ( ∏k∈f+(p) tβk∨-12 (1-tβk∨) 1- q⟨-βk∨,ρc⟩ ) ( ∏k∈f-(p) tβk∨-12 (1-tβk∨) q⟨-βk∨,ρc⟩ 1- q⟨-βk∨,ρc⟩ ) ,$

where the sum is over the set $ℬ\left(\stackrel{\to }{\mu }\right)=ℬ\left(1,\stackrel{\to }{w}\right)$ of alcove walks of type ${i}_{1},\dots ,{i}_{\ell }$ beginning at 1.

 Proof. Since ${E}_{\mu }={\tau }_{{X}^{\mu }m}^{\vee }1,$ $Xend(p)1 =Xwt(p) Tφ(p)∨1= Xwt(p) tφ(p)12 1andYλ∨ 1= q⟨λ∨,ρc⟩ 1,$ applying the formula for ${\tau }_{{X}^{\mu }m}^{\vee }$ in Theorem 2.2 to $1$ gives the formula in the statement. $\square$

From the expansion of ${E}_{\mu }$ in Theorem 3.1, the nonsymmetric MAcdonald polynomial ${E}_{\mu }$ has top term ${t}_{m}^{1/2}{X}^{\mu },$ where ${X}^{\mu }m$ is the minimal length representative of the coset ${X}^{\mu }{W}_{0}\text{.}$ This term is the term corresponding to the unique alcove walk in $ℬ\left(\stackrel{\to }{\mu }\right)$ with no folds.

If $w={X}^{\mu }m={s}_{{i}_{1}}^{\vee }\dots {s}_{{i}_{\ell }}^{\vee }$ is a reduced word for the minimal length element of the coset ${X}^{\mu }{W}_{0}$ then ${w}^{-1}={s}_{{i}_{\ell }}^{\vee }\dots {s}_{{i}_{1}}^{\vee }$ is a walk from 1 to ${w}^{-1}$ which stays completely in the dominant chamber. This has the effect that the roots ${\beta }_{\ell }^{\vee },\dots ,{\beta }_{1}^{\vee }$ are all of the form $-{\gamma }^{\vee }+jd$ with ${\gamma }^{\vee }\in {\left({R}^{\vee }\right)}^{+}$ (positive coroots) and $j\in {ℤ}_{>0}\text{.}$ The height of a coroot ${\gamma }^{\vee }$ is

$ht(γ∨)= ⟨γ∨,ρ⟩, whereρ=12 ∑α∈Rα.$

In the case that all the parameters are equal $\left({t}_{i}=t={q}^{c}\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}i=0,\dots ,n\right)$ the values which appear in Theorem 3.1,

$q⟨-βk∨,ρc⟩= q⟨γ∨-jd,ρc⟩= qjtht(γ∨), have positive exponents (inℤ>0).$

The set of dominant integral weights is

$(𝔥ℤ*)+= { μ∈𝔥ℤ*∣ ⟨μ,αi∨⟩≥0 fori=1,…,n } . (3.8)$

Recall the notation for ${t}_{w}^{1/2}$ from (3.6). For $\mu \in {\left({𝔥}_{ℤ}^{*}\right)}^{+},$ the symmetric Macdonald polynomial (see [14, Remarks after (6.8)]) is

$Pμ=10Eμ where10= ∑w∈W0 tw0w-12 Tw, (3.9)$

so that ${T}_{i}{1}_{0}={t}_{i}^{1/2}{1}_{0}$ for $i=1,2,\dots n,$ and ${1}_{0}$ has top term ${T}_{{W}_{0}}$ with coefficient 1. The symmetric Macdonald polynomials are ${W}_{0}$–symmetric polynomials in ${X}^{\mu }$ which are eigenvectors for the action of ${W}_{0}$–symmetric polynomials in the ${Y}^{{\lambda }^{\vee }}\text{.}$

Let $\mu \in \left({𝔥}_{ℤ}^{*}\right)+$ and let ${X}^{\mu }m={s}_{{i}_{1}}^{\vee }\dots {s}_{{i}_{\ell }}^{\vee }$ be a reduced word for the minimal length element ${X}^{\mu }m$ in the coset ${X}^{\mu }{W}_{0}\text{.}$ Let ${\beta }_{\ell }^{\vee },\dots ,{\beta }_{1}^{\vee }$ be as defined in (2.35) and let

$𝒫(μ→)= ⋃v∈W0ℬ (v,w→)$

be the set of alcove walks of type $\stackrel{\to }{w}=\left({i}_{1},\dots ,{i}_{\ell }\right)$ beginning at an element $v\in {W}_{0}\text{.}$ Then the symmetric Macdonald polynomial

$Pμ= ∑p∈𝒫(u→) Xwt(p) tφ(p)12 tw0ι(p)-12 ( ∏k∈f+(p) tβk∨-12 (1-tβk∨) 1- q⟨-βk∨,ρc⟩ ) ( ∏k∈f-(p) tβk∨-12 (1-tβk∨) q⟨-βk∨,ρc⟩ 1- q⟨-βk∨,ρc⟩ ) ,$

where $\iota \left(p\right)$ is the initial alcove of the path $p\text{.}$

 Proof. The expression $10=∑v∈W0 tw0v-12 Xv,givesPμ 1=10Eμ1= ∑v∈W0 tw0v-12 Xv τXμm∨1,$ which is computed by the same method as in Theorem 2.2 and Theorem 3.1. $\square$

The Hall-Littlewood polynomials or Macdonald spherical functions are ${P}_{\mu }\left(0,t\right)$ and the Schur functions or Weyl characters are ${s}_{\mu }={P}_{\mu }\left(0,0\right)\text{.}$ 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 $\lambda$–chain formulation of [9],[10].

When $\mu$ is not a regular weight, the formula for ${P}_{\mu }$ in Theorem 3.4 has an alternate formulation as a sum over paths whose initial alcove $\iota \left(p\right)$ is in the minimal coset representatives ${W}^{\mu }$ of ${W}_{0}/{W}_{\mu }\text{.}$ To see this, suppose ${s}_{i}\mu =\mu$ for some $i\in \left\{1,\dots ,n\right\}\text{.}$ Then $⟨\mu ,{\alpha }_{i}^{\vee }⟩=0$ implies ${Y}^{-{\alpha }_{i}^{\vee }}{E}_{\mu }1={t}_{i}^{-1}{E}_{\mu }1\text{.}$ Further, let ${X}^{\mu }m$ be the minimal length element in the coset ${X}^{\mu }{W}_{0}\text{.}$ Then ${s}_{i}{X}^{\mu }m={X}^{{s}_{i}\mu }{s}_{i}m={X}^{\mu }m{s}_{j}$ for some $j\in \left\{1,\dots ,n\right\},$ so ${\tau }_{i}^{\vee }{E}_{\mu }1={\tau }_{i}^{\vee }{\tau }_{{X}^{\mu }m}^{\vee }1={\tau }_{{X}^{\mu }m}^{\vee }{\tau }_{j}^{\vee }1=0\text{.}$
Thus ${T}_{i}{E}_{\mu }1=\left({\tau }_{i}^{\vee }-\frac{{t}_{i}^{-\frac{1}{2}}-{\tau }_{i}^{\frac{1}{2}}}{1-{Y}^{-{\alpha }_{i}^{\vee }}}\right){E}_{\mu }1={t}_{i}^{\frac{1}{2}}{E}_{\mu }1,$ and

$Pμ1 = 10Eμ1= ∑w∈W0 tw0w-12 TwEμ1= tw012 ( ∑v∈Wμ tw0v-12 Tv ) ( ∑u∈Wμ tw0u-12 Tu ) Eμ1 = Wμ(t) ∑v∈Wμ tw0v-12 Xv τXμm∨1.$

## Notes and References

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