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 $E_{\mu }$ (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

$$\begin{array}{cc}ℂ\left[X\right]={\text{Ind}}_H^{\stackrel{\sim }{H}}\left(1\right)=ℂ\text{–span}\{q^k{X}^{\mu }1\mid k\in \frac{1}{2}\mathbb{Z},\mu \in {𝔥}_{\mathbb{Z}}^{*}\}& \text{(3.1)}\end{array}$$

with

$$\begin{array}{cc}{T}_{i}1=t_i^{1/2}1\text{and}g1=1,\text{for}g\in \Omega .& \text{(3.2)}\end{array}$$

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

$$\begin{array}{cc}\{E_{\mu }\mid \mu \in {𝔥}_{\mathbb{Z}}^{*}\},\text{where}{E}_{\mu }={\tau }_{X^{\mu }m}^{\vee }1& \text{(3.3)}\end{array}$$

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 ${𝔥}_{\mathbb{Z}}^{+}$ is the set of dominant integral coweights (analogous to ${\left({𝔥}_{\mathbb{Z}}^{*}\right)}^{+}$ defined in (3.8)), ${\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}^{+}$ and $Y^{{\lambda }^{\vee }}=s_{i_1}\dots s_{i_{\ell }}$ is a reduced word, then

$$Y^{{\lambda }^{\vee }}1=T_{i_1}\dots T_{i_{\ell }}1=t_{i_1}^{\frac{1}{2}}\dots t_{i_{\ell }}^{\frac{1}{2}}1=q^{\frac{1}{2}(c_{i_1}+\dots +c_{i_{\ell }})}1q^{\frac{1}{2}\sum _{\alpha \in R^{+}}{c}_{\alpha }⟨{\lambda }^{\vee },\alpha ⟩}1=q^{⟨{\lambda }^{\vee },{\rho }_c⟩}1,$$

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

$$\begin{array}{cc}{Y}^{{\lambda }^{\vee }}1=q^{⟨{\lambda }^{\vee },{\rho }_c⟩}1,\text{where}{\rho }_c=\frac{1}{2}\sum _{\alpha \in R^{+}}{c}_{\alpha }\alpha \text{.}& \text{(3.4)}\end{array}$$

More generally, if $X^{\mu }m$ is the minimal length element of the coset $X^{\mu }{W}_0$ then

$$\begin{array}{ccc}{Y}^{{\lambda }^{\vee }}{E}_{\mu }& =& Y^{{\lambda }^{\vee }}{\tau }_{X^{\mu }m}^{\vee }1={\tau }_{X^{\mu }m}^{\vee }{Y}^{m^{-1}{X}^{-\mu }{\lambda }^{\vee }}1={\tau }_{X^{\mu }m}^{\vee }{Y}^{m^{-1}({\lambda }^{\vee }+⟨{\lambda }^{\vee },\mu ⟩d)}1\\ & =& {\tau }_{X^{\mu }m}^{\vee }{Y}^{m^{-1}{\lambda }^{\vee }}{q}^{-⟨{\lambda }^{\vee },\mu ⟩}1=q^{⟨m^{-1}{\lambda }^{\vee },{\rho }_c⟩-⟨{\lambda }^{\vee },\mu ⟩}{\tau }_{X^{\mu }m}^{\vee }1=q^{⟨{\lambda }^{\vee },m{\rho }_c-\mu ⟩}{E}_{\mu }\\ & =& q^{⟨{\lambda }^{\vee },X^{-\mu }m.{\rho }_c⟩}{E}_{\mu },\end{array}$$

where, in the last line, the action of $W$ on ${𝔥}_{\mathbb{Z}}^{*}$ 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 $ℬ(v,\overrightarrow{w})$ denotes the set of alcove walks of type $\overrightarrow{w}=(i_1,\dots ,i_{\ell })$ beginning at $v\text{.}$ For $p\in ℬ(v,\overrightarrow{w})$ define the weight $\text{wt}\left(p\right)$ and the final direction $\phi \left(p\right)$ of $p$ by

$$\begin{array}{cc}{X}^{\text{end}\left(p\right)}=X^{\text{wt}\left(p\right)}{T}_{\phi \left(p\right)}^{\vee },\text{with}\text{wt}\left(p\right)\in {𝔥}_{\mathbb{Z}}^{*}\text{and}\phi \left(p\right)\in W_0\text{.}& \text{(3.5)}\end{array}$$

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

$$\begin{array}{cc}{t}_w^{1/2}=t_{i_1}^{1/2}\dots t_{i_{\ell }}^{1/2},\text{if}w=s_{i_1}^{\vee }\dots s_{i_{\ell }}^{\vee }\text{is a reduced word.}& \text{(3.6)}\end{array}$$

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),

$$Y^{-{\beta }_k^{\vee }}1=Y^{-(-{\gamma }^{\vee }+jd)}1=q^j{q}^{⟨{\gamma }^{\vee },{\rho }_c⟩}1,\text{if}{\beta }_k^{\vee }=-{\gamma }^{\vee }+jd$$

with ${\gamma }^{\vee }\in R^{\vee },j\in \mathbb{Z}\text{.}$ 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

$$\begin{array}{cc}{Y}^{-{\beta }_k^{\vee }}1=q^{⟨-{\beta }_k^{\vee },{\rho }_c⟩}1\text{.}& \text{(3.7)}\end{array}$$

Let $\mu \in {𝔥}_{\mathbb{Z}}^{*}$ and let $w=X^{\mu }m$ be the minimal length element in the coset $X^{\mu }{W}_0\text{.}$ Fix a reduced word $\overrightarrow{w}=s_{s_1}^{\vee }\dots s_{s_{\ell }}^{\vee }$ 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

$$E_{\mu }=\sum _{p\in ℬ\left(\overrightarrow{u}\right)}{X}^{\text{wt}\left(p\right)}{t}_{\phi \left(p\right)}^{\frac{1}{2}}\left(\prod _{k\in f^{+}\left(p\right)}\frac{t_{{\beta }_k^{\vee }}^{-\frac{1}{2}}(1-t_{{\beta }_k^{\vee }})}{1-q^{⟨-{\beta }_k^{\vee },{\rho }_c⟩}}\right)\left(\prod _{k\in f^{-}\left(p\right)}\frac{t_{{\beta }_k^{\vee }}^{-\frac{1}{2}}(1-t_{{\beta }_k^{\vee }})q^{⟨-{\beta }_k^{\vee },{\rho }_c⟩}}{1-q^{⟨-{\beta }_k^{\vee },{\rho }_c⟩}}\right),$$

where the sum is over the set $ℬ\left(\overrightarrow{\mu }\right)=ℬ(1,\overrightarrow{w})$ of alcove walks of type $i_1,\dots ,i_{\ell }$ beginning at 1.

		Proof.
	Since $E_{\mu }={\tau }_{X^{\mu }m}^{\vee }1,$ $$X^{\text{end}\left(p\right)}1=X^{\text{wt}\left(p\right)}{T}_{\phi \left(p\right)}^{\vee }1=X^{\text{wt}\left(p\right)}{t}_{\phi \left(p\right)}^{\frac{1}{2}}1\text{and}{Y}^{{\lambda }^{\vee }}1=q^{⟨{\lambda }^{\vee },{\rho }_c⟩}1,$$ 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 $ℬ\left(\overrightarrow{\mu }\right)$ 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 $j\in {\mathbb{Z}}_{>0}\text{.}$ The height of a coroot ${\gamma }^{\vee }$ is

$$\text{ht}\left({\gamma }^{\vee }\right)=⟨{\gamma }^{\vee },\rho ⟩,\text{where}\rho =\frac{1}{2}\sum _{\alpha \in R}\alpha \text{.}$$

In the case that all the parameters are equal $(t_i=t=q^c\text{for}i=0,\dots ,n)$ the values which appear in Theorem 3.1,

$$q^{⟨-{\beta }_k^{\vee },{\rho }_c⟩}=q^{⟨{\gamma }^{\vee }-jd,{\rho }_c⟩}=q^j{t}^{\text{ht}\left({\gamma }^{\vee }\right)},\text{have positive exponents (in}{\mathbb{Z}}_{>0}\text{).}$$

The set of dominant integral weights is

$$\begin{array}{cc}{\left({𝔥}_{\mathbb{Z}}^{*}\right)}^{+}=\{\mu \in {𝔥}_{\mathbb{Z}}^{*}\mid ⟨\mu ,{\alpha }_i^{\vee }⟩\ge 0\text{for}i=1,\dots ,n\}\text{.}& \text{(3.8)}\end{array}$$

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

$$\begin{array}{cc}{P}_{\mu }=1_0{E}_{\mu }\text{where}{1}_0=\sum _{w\in W_0}{t}_{w_{0}w}^{-\frac{1}{2}}{T}_w,& \text{(3.9)}\end{array}$$

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 $\mu \in \left({𝔥}_{\mathbb{Z}}^{*}\right)+$ 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

$$𝒫\left(\overrightarrow{\mu }\right)=\bigcup _{v\in W_0}ℬ(v,\overrightarrow{w})$$

be the set of alcove walks of type $\overrightarrow{w}=(i_1,\dots ,i_{\ell })$ beginning at an element $v\in W_0\text{.}$ Then the symmetric Macdonald polynomial

$$P_{\mu }=\sum _{p\in 𝒫\left(\overrightarrow{u}\right)}{X}^{\text{wt}\left(p\right)}{t}_{\phi \left(p\right)}^{\frac{1}{2}}{t}_{w_0\iota \left(p\right)}^{-\frac{1}{2}}\left(\prod _{k\in f^{+}\left(p\right)}\frac{t_{{\beta }_k^{\vee }}^{-\frac{1}{2}}(1-t_{{\beta }_k^{\vee }})}{1-q^{⟨-{\beta }_k^{\vee },{\rho }_c⟩}}\right)\left(\prod _{k\in f^{-}\left(p\right)}\frac{t_{{\beta }_k^{\vee }}^{-\frac{1}{2}}(1-t_{{\beta }_k^{\vee }})q^{⟨-{\beta }_k^{\vee },{\rho }_c⟩}}{1-q^{⟨-{\beta }_k^{\vee },{\rho }_c⟩}}\right),$$

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

		Proof.
	The expression $$1_0=\sum _{v\in W_0}{t}_{w_{0}v}^{-\frac{1}{2}}{X}^v,\text{gives}{P}_{\mu }1=1_0{E}_{\mu }1=\sum _{v\in W_0}{t}_{w_{0}v}^{-\frac{1}{2}}{X}^v{\tau }_{X^{\mu }m}^{\vee }1,$$ 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 $P_{\mu }(0,t)$ and the Schur functions or Weyl characters are $s_{\mu }=P_{\mu }(0,0)\text{.}$ 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 $i\in \{1,\dots ,n\}\text{.}$ 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 $j\in \{1,\dots ,n\},$ 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 $T_i{E}_{\mu }1=({\tau }_i^{\vee }-\frac{t_i^{-\frac{1}{2}}-{\tau }_i^{\frac{1}{2}}}{1-Y^{-{\alpha }_i^{\vee }}})E_{\mu }1=t_i^{\frac{1}{2}}{E}_{\mu }1,$ and

$$\begin{array}{ccc}{P}_{\mu }1& =& 1_0{E}_{\mu }1=\sum _{w\in W_0}{t}_{w_{0}w}^{-\frac{1}{2}}{T}_w{E}_{\mu }1=t_{w_0}^{\frac{1}{2}}\left(\sum _{v\in W^{\mu }}{t}_{w_{0}v}^{-\frac{1}{2}}{T}_v\right)\left(\sum _{u\in W_{\mu }}{t}_{w_{0}u}^{-\frac{1}{2}}{T}_u\right)E_{\mu }1\\ & =& W_{\mu }\left(t\right)\sum _{v\in W^{\mu }}{t}_{w_{0}v}^{-\frac{1}{2}}{X}^v{\tau }_{X^{\mu }m}^{\vee }1\text{.}\end{array}$$

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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