Examples of Macdonald polynomials

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

Last update: 25 September 2012

Type $A_2$

The Weyl group $W_0=⟨s_1,s_2\mid s_1^2=s_2^2=1,s_1{s}_2{s}_1=s_2{s}_1{s}_2⟩$ acts on the lattices

$$\begin{array}{cc}{𝔥}_{\mathbb{Z}}=\mathbb{Z}{\omega }_1^{\vee }+\mathbb{Z}{\omega }_2^{\vee }\text{and}{𝔥}_{\mathbb{Z}}^{*}=\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2,& \text{(4.7)}\end{array}$$

where $s_1$ and $s_2$ are the reflections in the hyperplanes determined by

$$\begin{array}{cc}{\alpha }_1^{\vee }=2{\omega }_1^{\vee }-{\omega }_2^{\vee },{\alpha }_2^{\vee }=-{\omega }_1^{\vee }+2{\omega }_2^{\vee },{\alpha }_1=2{\omega }_1-{\omega }_2,\text{and}{\alpha }_2=-{\omega }_1+2{\omega }_2,& \text{(4.8)}\end{array}$$

with $⟨{\omega }_i^{\vee },{\alpha }_j⟩={\delta }_{ij},$ and $⟨{\omega }_i^{\vee },{\alpha }_j^{\vee }⟩={\delta }_{ij}\text{.}$ In this case,

$$\begin{array}{cc}{\phi }^{\vee }={\alpha }_1^{\vee }+{\alpha }_2^{\vee },\text{and}\phi ={\alpha }_1+{\alpha }_2\text{.}& \text{(4.9)}\end{array}$$

The double affine braid group $\stackrel{\sim }{ℬ}$ is generated by $T_0,T_1,T_2,g,X^{{\omega }_1},X^{{\omega }_2},$ and $q^{1/3},$ with relations

$$\begin{array}{cc}\begin{array}{c}{T}_i{T}_j{T}_i=T_j{T}_i{T}_j,\text{for}i\ne j,\\ X^{\mu }{X}^{\lambda }=X^{\mu +\lambda }=X^{\lambda }{X}^{\mu },\text{for}\mu ,\lambda \in {𝔥}_{\mathbb{Z}}^{*},\\ T_1{X}^{{\omega }_2}=X^{{\omega }_2}{T}_1,T_2{X}^{{\omega }_1}=X^{{\omega }_1}{T}_2,T_1{X}^{{\omega }_1}{T}_1=X^{-{\omega }_1+{\omega }_2},T_2{X}^{{\omega }_2}{T}_2=X^{{\omega }_1-{\omega }_2},\\ g^3=1,g{X}^{{\omega }_1}=q^{1/3}{X}^{-{\omega }_1+{\omega }_2}g,g{X}^{{\omega }_2}=q^{2/3}{X}^{-{\omega }_1}g,\\ g{T}_0{g}^{-1}=T_1,g{T}_1{g}^{-1}=T_2,g{T}_2{g}^{-1}=T_0\text{.}\end{array}& \text{(4.10)}\end{array}$$

The formula (2.27) gives

$$\begin{array}{cc}g=Y^{{\omega }_1^{\vee }}{T}_1^{-1}{T}_2^{-1},g^2=Y^{{\omega }_2^{\vee }}{T}_2^{-1}{T}_1^{-1},T_0=Y^{{\phi }^{\vee }}{T}_1^{-1}{T}_2^{-1}{T}_1^{-1},& \text{(4.11)}\end{array}$$ $\begin{array}{cc}{g}^{\vee }=X^{{\omega }_1}{T}_1^{\vee }{T}_2^{\vee },{\left(g^{\vee }\right)}^2=X^{{\omega }_2}{T}_2^{\vee }{T}_1^{\vee },{\left(T_0^{\vee }\right)}^{-1}=X^{\phi }{T}_1^{\vee }{T}_2^{\vee }{T}_1^{\vee }\text{.}& \text{(4.12)}\end{array}$

At this point, the following Proposition, which is the type $A_2$ case of Theorem 2.1, is easily proved by direct computation.

(Duality). Let $Y^d=q^{-1}\text{.}$ The double affine braid group $\stackrel{\sim }{ℬ}$ is generated by $T_0^{\vee },T_1^{\vee },T_2^{\vee },g^{\vee },Y^{{\omega }_1^{\vee }},Y^{{\omega }_2^{\vee }},$ and $q^{1/3},$ with relations

$$\begin{array}{c}{\left(T_1^{\vee }\right)}^{-1}{Y}^{{\omega }_1^{\vee }}{\left(T_1^{\vee }\right)}^{-1}=Y^{-{\omega }_1^{\vee }+{\omega }_2^{\vee }},{\left(T_2^{\vee }\right)}^{-1}{Y}^{{\omega }_2^{\vee }}{\left(T_2^{\vee }\right)}^{-1}=Y^{{\omega }_1^{\vee }-{\omega }_2^{\vee }},\\ {\left(T_1^{\vee }\right)}^{-1}{Y}^{{\omega }_2^{\vee }}=Y^{{\omega }_2^{\vee }}{\left(T_1^{\vee }\right)}^{-1},{\left(T_2^{\vee }\right)}^{-1}{Y}^{{\omega }_1^{\vee }}=Y^{{\omega }_1^{\vee }}{\left(T_2^{\vee }\right)}^{-1},\\ {\left(g^{\vee }\right)}^3=1,g^{\vee }{Y}^{{\omega }_1^{\vee }}=q^{-1/3}{Y}^{-{\omega }_1^{\vee }+{\omega }_2^{\vee }}{g}^{\vee },g^{\vee }{Y}^{{\omega }_2^{\vee }}=q^{-2/3}{Y}^{-{\omega }_1^{\vee }}{g}^{\vee },\\ g^{\vee }{T}_0^{\vee }{\left(g^{\vee }\right)}^{-1}=T_1^{\vee },g^{\vee }{T}_1^{\vee }{\left(g^{\vee }\right)}^{-1}=T_2^{\vee }\text{and}{g}^{\vee }{T}_2^{\vee }{\left(g^{\vee }\right)}^{-1}=T_0^{\vee }\text{.}\end{array}$$

To give a concrete example of Theorem 3.4 let us compute the symmetric Macdonald polynomial $P_{\rho }$ where $\rho ={\alpha }_1+{\alpha }_2\text{.}$ Since

$$1_0=X^{s_1{s}_2{s}_1}+t^{-1/2}{X}^{s_1{s}_2}+t^{-1/2}{X}^{s_2{s}_1}+t^{-2/2}{X}^{s_1}+t^{-2/2}{X}^{s_2}+t^{-3/2},$$

and $X^{\rho }m=s_0^{\vee }$ is the minimal length element of the coset $X^{\rho }{W}_0,$

$$\begin{array}{ccc}{P}_{\rho }& =& 1_0{E}_{\rho }=1_0{\tau }_0^{\vee }1\\ & =& (X^{s_1{s}_2{s}_1}+t^{-1/2}{X}^{s_1{s}_2}+t^{-1/2}{X}^{s_2{s}_1})(T_0^{\vee }+\frac{t^{-1/2}(1-t)}{1-Y^{-{\alpha }_0^{\vee }}})1\\ & & +(t^{-2/2}{X}^{s_1}+t^{-2/2}{X}^{s_2}+t^{-3/2})({\left(T_0^{\vee }\right)}^{-1}+\frac{t^{-1/2}(1-t)Y^{-{\alpha }_0^{\vee }}}{1-Y^{-{\alpha }_0^{\vee }}})1\\ & =& (X^{s_1{s}_2{s}_1{s}_0}+t^{-1/2}{X}^{s_1{s}_2{s}_0}+t^{-1/2}{X}^{s_2{s}_1{s}_0}+t^{-2/2}{X}^{s_1{s}_0}+t^{-2/2}{X}^{s_2{s}_0}+t^{-3/2}{X}^{s_0})1\\ & & +(X^{s_1{s}_2{s}_1}+t^{-1/2}{X}^{s_1{s}_2}+t^{-1/2}{X}^{s_2{s}_1})\frac{t^{-1/2}(1-t)}{1-Y^{-{\alpha }_0^{\vee }}}1\\ & & +(t^{-2/2}{X}^{s_1}+t^{-2/2}{X}^{s_2}+t^{-3/2})\frac{t^{-1/2}(1-t)Y^{-{\alpha }_0^{\vee }}}{1-Y^{-{\alpha }_0^{\vee }}}1\text{.}\end{array}$$

Since $Y^{-{\alpha }_0^{\vee }}1=Y^{{\phi }^{\vee }-d}1=q{Y}^{{\alpha }_1^{\vee }+{\alpha }_2^{\vee }}1=t^{2}q1,$

$$\begin{array}{ccc}{P}_{\rho }& =& \left(\begin{array}{c}{X}^{w_0\rho }+t^{-1/2}{X}^{s_1{s}_2\rho }{T}_2^{\vee }+t^{-1/2}{X}^{s_2{s}_1\rho }{T}_1^{\vee }\\ +t^{-2/2}{X}^{s_1\rho }{T}_2^{\vee }{T}_1^{\vee }+t^{-2/2}{X}^{s_2\rho }{T}_1^{\vee }{T}_2^{\vee }+t^{-3/2}{X}^{\rho }{T}_1^{\vee }{T}_2^{\vee }{T}_1^{\vee }\end{array}\right)1\\ & & +(T_1^{\vee }{T}_2^{\vee }{T}_1^{\vee }+t^{-1/2}{T}_1^{\vee }{T}_2^{\vee }+t^{-1/2}{T}_2^{\vee }{T}_1^{\vee })\frac{t^{-1/2}(1-t)}{1-t^{2}q}1\\ & & +(t^{-2/2}{T}_1^{\vee }+t^{-2/2}{T}_2^{\vee }+t^{-3/2})\frac{t^{-1/2}(1-t)t^{2}q}{1-t^{2}q}1\\ & =& (X^{w_0\rho }+X^{s_1{s}_2\rho }+X^{s_2{s}_1\rho }+X^{s_1\rho }+X^{s_2\rho }+X^{\rho })1\\ & & +(t^{3/2}+t^{1/2}+t^{1/2})\frac{t^{-1/2}(1-t)}{1-t^{2}q}1+(t^{-1/2}+t^{-1/2}+t^{-3/2})\frac{t^{-1/2}(1-t)t^{2}q}{1-t^{2}q}1\\ & =& (X^{w_0\rho }+X^{s_1{s}_2\rho }+X^{s_2{s}_1\rho }+X^{s_1\rho }+X^{s_2\rho }+X^{\rho }+(t+2+2tq+q)\frac{1-t}{1-t^{2}q})1\text{.}\end{array}$$

The set $𝒫\left(\overrightarrow{\rho }\right)$ contains 12 alcove walks,

The Hall-Littlewood polynomial and the Weyl character are

$$P_{\rho }(0,t)=m_{\rho }+(2+t)(1-t)\text{and}{s}_{\rho }=P_{\rho }(0,0)=m_{\rho }+2,$$

where $m_{\rho }=X^{w_0\rho }=X^{s_1{s}_2\rho }+X^{s_2{s}_1\rho }+X^{s_1\rho }+X^{s_2\rho }+X^{\rho }\text{.}$

The expression $X^{s_1{s}_2\rho }{s}_2=s_1^{\vee }{s}_2^{\vee }{s}_0^{\vee }$ is a reduced word for the minimal length element in the coset $X^{s_1{s}_2\rho }{W}_0$ and Theorem 2.2 is illustrated by

$$\begin{array}{ccc}{\tau }_1^{\vee }{\tau }_2^{\vee }{\tau }_0^{\vee }& =& (T_1^{\vee }+\frac{t^{-1/2}(1-t)}{1-Y^{-{\alpha }_1^{\vee }}}){\tau }_1^{\vee }{\tau }_0^{\vee }=(T_1^{\vee }{\tau }_2^{\vee }+{\tau }_2^{\vee }\frac{t^{-1/2}(1-t)}{1-Y^{-s_2{\alpha }_1^{\vee }}}){\tau }_0^{\vee }\\ & =& (T_1^{\vee }{T}_2^{\vee }+T_1^{\vee }\frac{t^{-1/2}(1-t)}{1-Y^{-{\alpha }_2^{\vee }}}+T_2^{\vee }\frac{t^{-1/2}(1-t)}{1-Y^{-s_2{\alpha }_1^{\vee }}}+\frac{t^{-1/2}(1-t)}{1-Y^{-{\alpha }_2^{\vee }}}\frac{t^{-1/2}(1-t)}{1-Y^{-s_2{\alpha }_1^{\vee }}}){\tau }_0^{\vee }\\ & =& T_1^{\vee }{T}_2^{\vee }{\tau }_0^{\vee }+T_1^{\vee }{\tau }_0^{\vee }\frac{t^{-1/2}(1-t)}{1-Y^{-s_0{\alpha }_2^{\vee }}}+T_2^{\vee }{\tau }_0^{\vee }\frac{t^{-1/2}(1-t)}{1-Y^{-s_0{s}_2{\alpha }_1^{\vee }}}+{\tau }_0^{\vee }\frac{t^{-1/2}(1-t)}{1-Y^{-s_0{\alpha }_2^{\vee }}}\frac{t^{-1/2}(1-t)}{1-Y^{-s_0{s}_2{\alpha }_1^{\vee }}}\\ & =& T_1^{\vee }{T}_2^{\vee }{T}_0^{\vee }+T_1^{\vee }{T}_2^{\vee }\frac{t^{-1/2}(1-t)}{1-Y^{-{\alpha }_0^{\vee }}}+T_1^{\vee }{\left(T_0^{\vee }\right)}^{-1}\frac{t^{-1/2}(1-t)}{1-Y^{-s_0{\alpha }_2^{\vee }}}\\ & & +T_1^{\vee }\frac{t^{-1/2}(1-t)Y^{-{\alpha }_0^{\vee }}}{1-Y^{-{\alpha }_0^{\vee }}}\frac{t^{-1/2}(1-t)}{1-Y^{-s_0{\alpha }_2^{\vee }}}+T_2^{\vee }{\left(T_0^{\vee }\right)}^{-1}\frac{t^{-1/2}(1-t)}{1-Y^{-s_0{s}_2{\alpha }_1^{\vee }}}\\ & & +T_2^{\vee }\frac{t^{-1/2}(1-t)Y^{-{\alpha }_0^{\vee }}}{1-Y^{-{\alpha }_0^{\vee }}}\frac{t^{-1/2}(1-t)}{1-Y^{-s_0{s}_2{\alpha }_1^{\vee }}}+{\left(T_0^{\vee }\right)}^{-1}\frac{t^{-1/2}(1-t)}{1-Y^{-s_0{\alpha }_2^{\vee }}}\frac{t^{-1/2}(1-t)}{1-Y^{-s_0{s}_2{\alpha }_1^{\vee }}}\\ & & +\frac{t^{-1/2}(1-t)Y^{-{\alpha }_0^{\vee }}}{1-Y^{-{\alpha }_0^{\vee }}}\frac{t^{-1/2}(1-t)}{1-Y^{-s_0{\alpha }_2^{\vee }}}\frac{t^{-1/2}(1-t)}{1-Y^{-s_0{s}_2{\alpha }_1^{\vee }}},\end{array}$$

where the eight terms in this expansion correspond to the eight alcove walks in $ℬ(1,s_1^{\vee }{s}_2^{\vee }{s}_0^{\vee })=ℬ\left(\overrightarrow{s_1{s}_2\rho }\right)$ pictured below. Applying the expansion of ${\tau }_1^{\vee }{\tau }_2^{\vee }{\tau }_0^{\vee }$ to $1$ and using

$$\begin{array}{cc}\begin{array}{c}{Y}^{-{\alpha }_0^{\vee }}1=Y^{{\phi }^{\vee }-d}1=t^{2}q1,Y^{-s_0{\alpha }_2^{\vee }}1=Y^{{\alpha }_1^{\vee }-d}1=tq1,\\ \text{and}{Y}^{-s_0{s}_2{\alpha }_1^{\vee }}1=Y^{{\phi }^{\vee }-2d}1=t^2{q}^{2}1,\end{array}& \text{(4.13)}\end{array}$$

computes

$$\begin{array}{ccc}{E}_{s_1{s}_2\rho }& =& (X^{s_1{s}_2\rho }{t}^{1/2}+t\frac{t^{-1/2}(1-t)}{1-t^{2}q}+X^{s_1\rho }t\frac{t^{-1/2}(1-t)}{1-tq}+t^{1/2}\frac{t^{-1/2}(1-t)t^{2}q}{1-t^{2}q}\frac{t^{-1/2}(1-t)}{1-tq}\\ & & +X^{s_2\rho }t\frac{t^{-1/2}(1-t)}{1-t^2{q}^2}+t^{1/2}\frac{t^{-1/2}(1-t)t^{2}q}{1-t^{2}q}\frac{t^{-1/2}(1-t)}{1-t^2{q}^2}\\ & & +X^{\rho }{t}^{3/2}\frac{t^{-1/2}(1-t)}{1-tq}\frac{t^{-1/2}(1-t)}{1-t^2{q}^2}+\frac{t^{-1/2}(1-t)t^{2}q}{1-t^{2}q}\frac{t^{-1/2}(1-t)}{1-tq}\frac{t^{-1/2}(1-t)}{1-t^2{q}^2})1\text{.}\\ & =& t^{1/2}(X^{s_1{s}_2\rho }+\frac{(1-t)}{1-t^{2}q}+X^{s_1\rho }\frac{(1-t)}{1-tq}+t\frac{(1-t)q}{1-t^{2}q}\frac{(1-t)}{1-tq}+X^{s_2\rho }\frac{(1-t)}{1-t^2{q}^2}\\ & & +t\frac{(1-t)q}{1-t^{2}q}\frac{(1-t)}{1-t^2{q}^2}+X^{\rho }\frac{(1-t)}{1-tq}\frac{(1-t)}{1-t^2{q}^2}+\frac{(1-t)q}{1-t^{2}q}\frac{(1-t)}{1-tq}\frac{(1-t)}{1-t^2{q}^2})1\text{.}\end{array}$$

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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