%  YangRota.tex  corrections 8 fev 02  

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amsfonts}

\title{ Yang-Baxter graphs, Jack and Macdonald polynomials}

\author{\small Alain Lascoux\\
\small CNRS, Institut Gaspard Monge, Universit\'e de Marne-la-Vall\'ee\\
\small 77454 Marne-la-Vall\'ee Cedex, France\\
\small  Alain.Lascoux@univ-mlv.fr }

\def\be{\begin{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}

\def\s{\scriptstyle }
\def\a{\alpha}
\def\b{\beta}
\def\l{\lambda}
\def\L{\Lambda}
\def\e{\epsilon}
\def\g{\gamma}
\def\ss{\sigma}
\def\d{\partial}

\def\S{{\mathfrak S}}
\def\Sym{{\mathfrak Sym}}


\def\N{{\mathbb N}}
\def\Z{{\mathbb Z}}
\def\C{{\mathbb C}}

\def\cH{{\mathcal H}}
\def\E{{\mathcal E}}
\def\Pol{{\mathcal Pol}}
\def\JJ{\widehat J}
\def\SS{\widehat \S}
\def\MM{\widehat M}
\def\xx{\widehat \xi}
\def\MMS{\widehat M^{\S}}
\def\bt{\overline \tau }
\def\bG{\overline G}
\def\bT{\overline T}
\def\bp{\overline \pi}
\def\bksi{\overline \xi}
\def\hphi{\widehat{\Phi}}

\def\d{\partial}
\def\ket{\hbox{\large $\triangleright$}}
\def\D{\diamondsuit}
\def\EP{\Upsilon}
\def\CAR{CAR}
\def\qed{\hfill QED}
\def\QED{\hfill QED}

\def\moins{\raise 1pt\hbox{{$\scriptstyle -$}}}
\def\plus{\raise 1pt\hbox{{$\scriptstyle +$}} }

\def\hfl#1{{\stackrel{#1}{\hbox to 22mm{\rightarrowfill}}}}
\def\vfl#1{\llap{$#1$}
        \left\downarrow{\vbox to 6mm{}}\right.}

%%%%%%%%%%%%%%%%%%%%%%
\newtheorem{theorem}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{Def}[theorem]{Definition}
\def\proof{\noindent{\it Proof.}\ }


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}
\date{}
\maketitle
\begin{abstract}
We describe properties of the affine graph underlying the recursions
between the different varieties  of non symmetric Macdonald and
Jack polynomials. We use an arbitrary function of one variable in
the definition of affine edges, and of Cherednik's elements,
to unify the different theories. We describe the symmetrizing operators 
furnishing the symmetric polynomials from the non symmetric ones.
\end{abstract}
\hfill{\emph{D\'edi\'e \`a Gian-Carlo Rota, ami et ma\^{\i}tre \`a
pens\'ees}

\section{Introduction}

There are several families of mathematical objects related to the symmetric
group which are obtained by simple recursions involving ``local'' operators
\cite{Vershik}.

All these constructions can be summarized as follows~: one has a directed
graph, with an origin $\ket$. Edges are labelled by elements $\theta$ of an
associative algebra. One also chooses an initial element 
$Y_{\ket}$ in a right module
over this algebra. The {\it evaluation} of a path from the origin
$\ket$ to a given vertex $v$  is the product $\theta_1\theta_2\cdots \theta_k$
of the labels encountered along the path. 

We shall say that a labelling is {\it consistent} if all the paths from the
origin to any vertex $v$ give an element 
$Y_v:= Y_{\ket}\, \theta_1 \cdots \theta_k$ depending only upon $v$.

For example, Young orthogonal idempotents for a given partition $\l$
can be described this way. The graph is the partially ordered set of tableaux
of shape the diagram of $\l$, with origin the ``Yamanouchi tabeau'',
and one needs only to know the idempotent corresponding to this tableau,
which will play the r\^ole of $Y_{\ket}$ (and that Young denotes $PNP$),
to generate all the other idempotents, the labels of edges being of the type
$c(s_i + 1/k)$, $c\in \C, \, k\in\Z$.  We shall recall later that matrices
representing the symmetric group can be described in the 
same fashion.

Other examples are associated to the {\it permutohedron} 
\big(i.e. the {\em Cayley graph} of the symmetric
group $\S_n$, edges $v\mapsto vs_i$ corresponding to multiplication
on the right by simple transposition, whenever length increases:
$\ell(vs_i)=\ell(v)+1$ \big).

Labelling edges by divided differences (resp. isobaric divided 
differences, to be defined later), and starting with 
$$  Y_{\ket} = x_1^{n-1}\cdots x_n^0 \quad \text{or}\quad 
 Y_{\ket} =\prod_{i+j\leq n} (x_i-y_j) $$
(resp. $  Y_{\ket} = \prod_{i+j\leq n} (1-y_j/x_i) $)
one generates polynomials $Y_v$ which are called 
{\it Schubert polynomials} \cite{LS2}, \cite{mac2}
(resp. {\it Grothendieck polynomials}, \cite{LS2},\cite{LS8}).


We shall call a  data~:

\centerline{ permutohedron + initial  $Y_{\ket}$ + consistent labelling}

\noindent 
a {\it Yang-Baxter graph} for the symmetric group, and
 the elements $Y_v$, $v\in \S_n$, {\it Yang-Baxter elements}.
This notion underlies Young's constructions \cite{Yo}, consistency
arising from {\it Yang-Baxter relations}.

For example, writing $Y_v$ on the graph instead of $v$, here is the 
Yang-Baxter graph giving Schubert polynomials for $\S_3$ \cite{LS8},
\cite{mac2}~: 

\begingroup
\setbox31=\hbox{$ \s \d_1 $}
\setbox32=\hbox{$\, \swarrow$}
\setbox3=\hbox{\kern 4mm{\raise 3mm\copy31}\box32}
\setbox41=\hbox{$ \searrow$}
\setbox42=\hbox{$\,\s \d_2   $}
\setbox4=\hbox{{\box41 \raise 3mm\copy42}\kern 4mm}
\setbox7=\hbox{$ {\s \d_2} \, \bigg\downarrow$}
\setbox8=\hbox{$\bigg\downarrow \, {\s \d_1} $}
\setbox11=\hbox{\kern 4mm$ {\s \d_1}  \searrow$}
\setbox12=\hbox{$ \swarrow  {\s \d_2} $\kern 4mm}

$$ \begin{matrix}
   &{\s (x_1-y_1)(x_1-y_2)(x_2-y_1)} \cr
     \box3      &                        &\box4 \cr  
\noalign{\vskip 1mm}
{\s(x_1-y_1)(x_2-y_1) }  & & {\s (x_1-y_1)(x_1-y_2)}
 \cr\noalign{\vskip 1mm}\box7  &
&\box8 \cr\noalign{\vskip 1mm}
{\s x_1-y_1}  & & {\s x_1+x_2-y_1-y_2} \cr \noalign{\vskip 2mm}
\box11 & &\box12 \cr &1 \cr
\end{matrix} $$
\endgroup

\medskip
Our main graph will be associated to the affine symmetric group.
In fact, we shall take only "half" of the extended affine symmetric group,
i.e. we shall use the monoid $\SS_n$ generated by 
$$ \{ s_1,\ldots, s_{n-1},\, \tau : 1\leq i \leq n-1 \} $$
satisfying relations 

\begin{gather}
 s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1} \ , \
s_i s_j = s_j s_i \ |i-j|>1  \label{eq:Cox1} \\
 s_i\, \tau = \tau\, s_{i-1}\, ,\, 2\leq i\leq n-1 \ ; \
   s_1\, \tau^2 = \tau^2\, s_{n-1} \ .  \label{eq:Cox2}
\end{gather}

The elements of $\SS_n$ can be thought as acting 
 both on infinite vectors $v= [\ldots, v_i, \ldots]$, $i\in \Z$
and on an infinite set of indeterminates $x_i$, $i\in \Z$~:
the generator $s_i$ 
transposes in $v$ components $v_{i+kn}, v_{i+1+kn}$ for all $k\in \Z$,
and transposes all $x_{i+kn}, x_{i+1+kn}$ simultaneously;
the generator $\tau$ acts as a shift on $v$:
$(v\tau)_i =v_{i+1}$ and  $x_i\, \tau= x_{i-1}$ for all $i\in \Z$.  

Using also the inverse $\bt :=\tau^{-1}$ would give a 
faithful representation of
the extended affine symmetric group.

Let us take the origin vector 
\be  \ket :\quad  \ket_1=0=\cdots = \ket_n \quad,  \quad  
\ket_{i\pm n} =\ket_i \pm 1\ \forall i\in \Z \ .
\end{equation}
 
Define  $\Gamma_n$ to be the following graph:
vertices  are  the orbit of $\ket$  
under $\SS_n$;  the (oriented) edges, labelled $s_i$ or $\tau$, 
join  pairs $v,\, v\, s_i$ or $v,\, v\tau$. 
The connected components obtained
by erasing $\tau$-edges will be called  {\it cells}.
Since the vectors $v$ are "periodic" in the sense that $v_{i+n}=v_i+1$,
$\forall i\in \Z$, one will denote them by the restriction to their
components $v_1,\ldots, v_n$. Thus $\ket$ will also be written 
$[0,\ldots, 0]$, and $v$ will be written $[v_1,\ldots, v_n]$. 


\begin{Def}  
An {\em affine  Yang-Baxter graph} for $\SS_n$ is a consistent
labelling of the edges of 
$\Gamma_n$ by elements of an associative algebra, together with
a choice of an initial element $Y_{\ket}$ in a right module for this algebra.
The elements $Y_v$, $v\in \SS_n$, are called {\em Yang-Baxter elements}.
\end{Def}

By default, we shall suppose that the algebra has a unit $1$ and that
$Y_{\ket}=1$.
 
To find consistent labelling of edges, 
we shall label vertices  by a second infinite vector $c(v)$,
called the vector of 
{\it spectral parameters}. One chooses an initial vector
$c(\ket)$, and one takes 
 its orbit under $\SS_n$~: as before, 
the generator $s_i$ simultaneously transpose components of $c(v)$ of index
$i+kn$, $i\plus 1 +kn$, $\forall k\in \Z$,
and the generator  $\tau$ acts by translation : 
$c_i(v\tau)= c_{i+1}(v),\ \forall i\in\Z$.

For example, if one takes $c(\ket)$ such that $c_i(\ket)$ 
is the remainder of $i\moins 1 \mod n$, then $\e(v):=c(v)$ will be periodic:
$\e_{i+n}(v)= \e_i(v) \forall i\in\Z$, and 
$$ \e_1(v),\ldots, \e_n(v) $$
is the {\it standardization} of $v_1,\ldots, v_n$ (i.e. one 
replaces multiple occurrence of the same value $\a$
in $[v_1,\ldots, v_n]$ by  different letters $\a,\a',\a''\ldots$, 
$\a>\a'>\a''>\cdots$, 
reading from left to right; 
then one replaces these new letters decreasingly by $0,1,\ldots,n\moins 1$;
this is an operation much used in the theory of Young tableaux,
since the article of Schensted). In the case where 
$v_1\geq v_2\geq \cdots\geq v_n$, then $\e(v)= [0,1,\ldots, n\moins 1]$.
 
The vectors $c(v)$ that we shall choose will simply be  a 
combination of $v$ and $\e(v)$. 

For example, introducing extra parameters $\a, q,t= -t_2/t_1$,
we shall take
$$ c(\ket)= \ldots \, [\, 0,\moins 1,\ldots, \moins n\plus 1 \,], \, 
 \a, \a\moins 1, a\moins 2, \ldots, 2\a ;\,  2\a\moins 1, \ldots $$
i.e. $c(\ket)=[0,\moins 1,\ldots, \moins n\plus 1]$ with periodicity 
$c_{i+n}(\ket)=\a +c_i(\ket)$. Then 
\be  c(v) =\a v -\e(v) \quad \text{i.e.}\quad c_i(v)= \a v_i -\e_i(v) \ .
\end{equation}
Taking on the other hand $c(\ket)= [t^0,t^1,\ldots, t^{n-1}]$ with
periodicity  $c_{i+n}(\ket)= q c_i(\ket)$ entails
\be  c_i(v) = q^{v_i} t^{\e_i(v)} \ .
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\large Cells and the group algebra of the symmetric group}

Let us first detail Yang-Baxter graphs for the symmetric group,
taking, as a first algebra to index edges, the group algebra of $\S_n$ itself.

In fact, already at the beginning of             
the century Young gave an example of such a graph to describe
representations of the symmetric group (for a modern approach, cf.
\cite{OV}).
Young \cite{Yo} 
defined matrices $\ss_i$  representing simple transpositions $s_i$.
Relations $\ss_i\ss_j=\ss_j\ss_i$ are immediate in his construction, 
and thus, he had essentially 
to check that his construction worked for $\S_3$.
In that case, he had 
 three parameters $c_1,c_2,c_3$
(the {\it contents} of three consecutive letters in an arbitrary Young
tableau). For simplicity of notation, let us specialize 
$c_1=0,\, c_2=2,\, c_3=5$.  
Young considered a vector space of dimension 6, with basis 
indexed by the permutations of $c(v^0)= [0,2,5]$~: 
$$[0,2,5],\, [0,5,2],\, [2,0,5],\, [2,5,0],\, [5,0,2],\, [5,2,0]\ , $$
and wrote the two matrices
$$\hskip -1cm \ss=
\begin{bmatrix}
-{1\over 2} & 0 & {\sqrt{3}\over 2} & 0 & 0 &0\cr
         0   & -{1\over5} &0 &0 &{2\sqrt{6}\over5} & 0\cr
{\sqrt{3}\over2} &0 &{1\over2} &0 &0 &0\cr
0&0&0 &-{1\over3} &0 &{2\sqrt{2}\over3} \cr
0&{2\sqrt{6}\over5} & 0&0 &{1\over5} &0\cr
0&0&0 &{2\sqrt{2}\over3} &0 &{1\over3}\cr
\end{bmatrix}\, , \  \ss' =
\begin{bmatrix}
-{1\over3} & {2\sqrt{2}\over3}  &0&0&0 &0\cr
{2\sqrt{2}\over3}  & {1\over3} &0&0&0 &0\cr
0&0 &-{1\over5} & {2\sqrt{6}\over5} &0&0\cr
0&0   & {2\sqrt{6}\over5} &{1\over5} &0&0 \cr
0&0&0&0   &-{1\over2} & {\sqrt{3}\over2} \cr
0&0&0&0  & {\sqrt{3}\over2}  &{1\over2}\cr
\end{bmatrix}$$

\def\hfl#1{{\stackrel{#1}
             {\hbox to 29pt {\rightarrowfill}}}}

These two matrices are made of $2\times 2$ sub-matrices, 
$$\begin{bmatrix}
 -\rho & \sqrt{1 -\rho^2} \cr \sqrt{1 -\rho^2} & \rho
\end{bmatrix}  \ , $$
each of which 
depends upon a single parameter $\rho$ ($=1/2, 1/3$ or $1/5$)
and codes a representation
of $\S_2$ on a space of dimension 2.     
Now the matrix $\ss$ 
 is a patch of three elementary matrices and 
can be encoded by the arrows:  
$$ [0,2,5]\ \hfl{s+\frac{1}{2}}\ [2,0,5] \ , \
 [0,5,2]\ \hfl{s+\frac{1}{5}}\ [5,0,2] \ , \ 
 [2,5,0]\ \hfl{s+\frac{1}{3}}\ [5,2,0] \ , $$
and the matrix $\ss'$  can similarly be encoded  by: 
$$ [0,2,5]\ \hfl{s'+\frac{1}{3}}\ [0,5,2] \ , \
 [2,0,5]\ \hfl{s'+\frac{1}{5}}\ [2,5,0] \ , \
 [5,0,2]\ \hfl{s'+\frac{1}{2}}\ [5,2,0] \ . $$


\setbox31=\hbox{$ s+\frac{1}{2} $}
\setbox32=\hbox{$\, \swarrow$}
\setbox3=\hbox{\kern 4mm{\raise 3mm\box31}\box32}
\setbox41=\hbox{$ \searrow$}
\setbox42=\hbox{$\, s'+\frac{1}{3}  $}
\setbox4=\hbox{{\box41 \raise 3mm\box42}\kern 4mm}
\setbox5=\hbox{$[2,0,5]$}
\setbox6=\hbox{$[2,5,0] $}
\setbox7=\hbox{$ s'+\frac{1}{5}  \, \bigg\downarrow$\kern 8.8mm}
\setbox8=\hbox{\kern 8.8mm$\bigg\downarrow \, s+\frac{1}{5} $}
\setbox9=\hbox{$[0,5,2]$}
\setbox10=\hbox{$[5,0,2]$}
\setbox11=\hbox{\kern 4mm$ s+\frac{1}{3} \ \searrow$}
\setbox12=\hbox{$ \swarrow \ s'+\frac{1}{2}  $\kern 4mm}

Thus the two matrices can be encoded at the same time by the graph 
$$ \begin{matrix}
   &[0,2,5]\cr\box3 & &\box4 \cr  \noalign{\vskip 1mm}
\box5  & &\box6 \cr\noalign{\vskip 1mm}\box7  &
&\box8 \cr\noalign{\vskip 1mm}\box9  & &\box10 \cr \noalign{\vskip 1mm}
\box11 & &\box12 \cr &[5,2,0] \cr  
\end{matrix} $$
which describes their action on the six-dimensional vector space with
basis $[0,2,5],\ldots, [5,2,0]$. 

Relation $\ss\ss' \ss= \ss'\ss\ss'$ becomes
\begin{multline*}
 (s_1+{1\over 2}) {1\over \sqrt{ 1-{1\over 2^2}}} 
(s_2+{1\over 5}) {1\over \sqrt{ 1-{1\over 5^2}}}
 (s_1+{1\over 3}){1\over \sqrt{ 1-{1\over 3^2}}} = \\
=  (s_2+{1\over 3}){1\over \sqrt{ 1-{1\over 3^2}}}
(s_1+{1\over 5})  {1\over \sqrt{ 1-{1\over 5^2}} }
(s_2+{1\over 2}) {1\over \sqrt{ 1-{1\over 2^2}}} \ ,
\end{multline*}
but now, it is an identity in the group algebra of $\S_3$, 
instead of being a matrix identity, and the graph itself,
with $Y_{v^0}=1$, is a Yang-Baxter graph for $\S_3$.

\medskip
Yang \cite{Ya}
 solved the same problem as Young did, but formulated it directly  
in terms of group algebra instead of matrices of representation: 
he wanted to label the edges of the
permutohedron by elements of the group algebra of the symmetric group
in such a way that the product of elements from the origin to a given
permutation $v$ be independent of the path.  He found the solution 
of choosing a  vector of "generic" parameters
for the identity permutation
(the mutual differences of these parameters 
are only required to be different from 0 and 1),
generating vectors $c(v)$ by simple transpositions of components, 
and labelling the edge $[v \rightarrow v\, s_i]$ by 
\be  s_i + \frac{1}{ c_{i+1}(v) - c_i(v)} \ .
\end{equation}
Since the permutohedron is paved by squares $s_i s_j =s_j s_i$, $|i-j|>1$,
and hexagons $s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}$, consistency of Yang's
solution lies upon {\it Yang-Baxter equation} 
\be
 ( s_i + {1\over\a}) (s_{i+1} + {1\over \a\plus \b }) ( s_i + {1\over\b }) =
( s_{i+1} + {1\over\b }) (s_i + {1\over \a\plus \b}) ( s_{i+1} + {1\over\a }) 
\ , \label{eq:YB} 
\end{equation}
the  commutations 
\be   ( s_i + {1\over\a}) ( s_j + {1\over\b }) =
            ( s_j + {1\over\b }) ( s_i + {1\over\a}) \ , \ |i-j| >1
     \label{eq:YB2} 
\end{equation} 
being evident.

Thus, consistency  arises from additivity : 
$$ c_{i+2}-c_i= (c_{i+2}-c_{i+1}) + (c_{i+1}-c_i) \ , $$
as in the case of Young where parameters were axial distances
(= contents).

\setbox31=\hbox{$ s+\frac{1}{\a} $}
\setbox32=\hbox{$\, \swarrow$}
\setbox3=\hbox{\kern 4mm{\raise 3mm\box31}\box32}
\setbox41=\hbox{$ \searrow$}
\setbox42=\hbox{$\, s'+\frac{1}{\b}  $}
\setbox4=\hbox{{\box41 \raise 3mm\box42}\kern 4mm}
\setbox5=\hbox{${}^{\hspace{7pt}[2,1,3]}_{[\a,0,\a+\b]}$}
\setbox6=\hbox{$ {}^{\hspace{7pt}[2,3,1]}_{[\a,\a+\b,0]} $}
\setbox7=\hbox{$ s'+\frac{1}{\a+\b}  \, \bigg\downarrow$\kern 8.8mm}
\setbox8=\hbox{\kern 8.8mm$\bigg\downarrow \, s+\frac{1}{\a+\b} $}
\setbox9=\hbox{${}^{\hspace{7pt}[1,3,2]}_{[0,\a+\b,\a]} $}
\setbox10=\hbox{${}^{\hspace{7pt}[3,1,2]}_{[\a+\b,0,\a]}$}
\setbox11=\hbox{\kern 4mm$ s+\frac{1}{\b} \ \searrow$}
\setbox12=\hbox{$ \swarrow \ s'+\frac{1}{\a}  $\kern 4mm}

Writing on the vertices of the graph $c[v]$ under $v$, and taking
$c([1,2,3])=[0,\a,\a\plus \b] $, then a cell with edges in the group algebra
of $\S_3$ looks like
$$ \begin{matrix}
   &{}^{\hspace{7pt}[1,2,3]}_{[0,\a,\a+\b]}\cr
\box3  & &\box4 \cr  \noalign{\vskip 1mm}
\box5  & &\box9 \cr\noalign{\vskip 1mm}
\box7  & &\box8 \cr\noalign{\vskip 1mm}
\box6  & &\box10 \cr \noalign{\vskip 1mm}
\box11 & &\box12 \cr 
&{}^{\hspace{7pt} [3,2,1]\ }_{[\a+\b,\a,0]} \cr
\end{matrix} $$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\large Cells and the degenerate affine Hecke algebra} 
 
We have some freedom for what concerns choosing the $Y_v$ as operators 
on polynomials in the $x_i$'s. Indeed, any family of operators 
satisfying the braid relations can be used. 

We can keep the relations $s_i^2=1$, but deform the transpositions of
variables. 
For example, we can take 
\be 
\ss_i = s_i +  \d_i \ , \, 1\leq i \leq n-1 \ ,  \label{eq:cell2}
\end{equation}
where $\d_i$ is the divided difference \cite{BGG} 
\be 
 f \mapsto f\, \d_i := { f- f^{s_i}\over x_i -x_{i+1}} \label{eq:op1}
\end{equation}
(one could introduce an extra parameter $\gamma$ and take 
$\ss_i := s_i + \gamma \, \d_i $; this is the homogeneous version 
of the preceding one, and thus not more general).

The $\ss_i$ still represent the symmetric group 
(they satisfy (\ref{eq:Cox1})) and generate an algebra
with linear basis $\{ \ss_w : w\in \S_n \}$ (cf. \cite{LS2}).   

Labelling of  edges
uses the spectral parameters in the same way as in the preceding section,
since the $\ss_i$'s satisfy the same relations as the $s_i$'s :
the edge $ [v \mapsto v\, \ss_i]$ is labelled 
\be
  \ss_i + { 1\over c_{i+1}(v) -c_i(v)} = 
    s_i+  \d_i + { 1\over c_{i+1}(v) -c_i(v)} \label{eq:cell4}
\end{equation}

Relations between indeterminates $x_i$, $1\leq i\leq n$ (considered as 
\lq\lq multiplication by $x_i$")  and the $\ss_i$ or $\d_i$ 
are given by {\em Leibniz rule}:
\be 
   x_i \ss_i= \ss_i x_{i+1} +1\ , \ x_{i+1} \ss_i= \ss_i x_{i} -1 \ , 
    \ x_j\ss_i = \ss_i x_j , j\neq i,i+1 \label{eq:Leibniz}
\end{equation}
The algebra generated by the $\ss_i$, $1\leq i \leq n-1$ and $x_i$, 
 $1\leq i \leq n$ is called the {\it affine degenerate Hecke algebra}. 

The limit case $\ss_i=\d_i$ gives also rise to a Yang-Baxter graph
because the $\d_i$'s themselves satisfy the braid relations. Thus for
any permutation $v$ exists a well defined divided difference $\d_w$,
as well as a $\ss_w$.

In fact, since $1-s_i =\d_i\, (x_i -x_{i+1})$ 
the affine Hecke algebra is a sub-algebra
of the {\it algebra of divided differences} generated by the $\d_i$,
and rational functions in the  $x_j$ (see \cite{LS2},\cite{LS5}).
This last algebra is a free module over the ring of rational functions
in the $x_i$'s, with  basis the divided differences $\d_w$, $w\in \S_n$. 
Expansions of the $\ss_w$ or $w$  in the basis of $\d_w$ involve 
 Schubert polynomials  (cf. \cite{LLT1}, \cite{LLT2}, \cite{FK}). 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\large Cells and the affine Hecke algebra}  

One has more general operators on polynomials  
satisfying the braid relations (see \cite{LS2}, \cite{LS5}).  

In particular,  let $q$ be an extra parameter, and let $x_{i+n}= q x_i$,
$\forall i\in \Z$.  Then one defines 
{\it isobaric divided differences} $\bp_i$, $i=1,\ldots, n-1$ acting on the 
quotient ring  $\Z[x_i : i\in \Z]/( x_{i+n}= q x_i)$ 
 as twisted derivations:
$$ fg \bp_i = f ( g\bp_i) + f\bp_i (g s_i) $$
satisfying the conditions 
\bea
 x_j \bp_i &=& 0 \ , \ j\not \equiv i,i\plus 1\  \mod(n) \ ,\notag  \\
 x_j\bp_i &=& x_{j+1} \ ,\ j\equiv i \ , \label{eq:op2} \\
 \bp_i^2 &=&  - \bp_i \ .  \notag 
\eea

\begin{lemma}
  The $\bp_i$, $1\leq i\leq n-1$, satisfy the braid relations (\ref{eq:Cox1}).
\end{lemma}

Indeed, the image of $\bp_i$ under the isomorphism of 
the quotient ring onto the space of 
polynomials in $x_1,\ldots, x_n$, is the usual isobaric divided difference
(that we shall denote by the same letter)
\be
 \bp_i : f \mapsto (f - f^{s_i}) \frac{x_{i+1}}{x_i - x_{i+1}}  \
                      ,\ 1\leq i\leq n-1 \ . \label{eq:op3}
\end{equation}

These operators commute with multiplication by symmetric functions in 
$x_1,\ldots, x_n$ and braid relations can directly be checked on an 
appropriate basis of the space of polynomials
(as a module over symmetric polynomials
in $x_1,\ldots, x_n$).

Demazure \cite{Dem} defined such operators for any root system. 

Let us notice that the operators $\pi_i:= \bp_i +1$ also satisfy
the braid relations together with $\pi_i^2= \pi_i$.


It has been noticed in \cite{LS2}( where, more generally  one uses
$D_i:= t_1\pi_i+t_2\d_i+t_3 s_i$) 
and \cite{Lu} that the operators
\be
 T_i = T_{x_i,x_{i+1}} := \pi_i (t_1+t_2) - t_2\, s_i \label{eq:op4}
\end{equation}
 represent the  Hecke algebra $\cH_n$  of the
symmetric group $\S_n$.
The generators $T_i$ satisfy the braid relations 
(\ref{eq:Cox1}) together with
\be (T_i -t_1)\, (T_i-t_2) = 0 \ . \label{eq:op5}
\end{equation}


Reversing the order on $x_1,\ldots, x_n$ give other generators
\be
\bT_i := -t_1t_2 T_i^{-1} = \bp_i\, (t_1+t_2) - t_2\, s_i = T_i -(t_1+t_2)
\ , \ 1\leq i\leq n-1 \ , \label{eq:op6}
\end{equation}
the Hecke relation becoming in that case 
\be (\bT_i +t_1)\, (\bT_i +t_2) = 0 \ . \label{eq:op7}
\end{equation}

We prefer keeping two parameters $t_1,t_2$, because all cases 
$[t_1,t_2]=[t,-1]$, 
$[-1,t],[t,1/t],[1,t],[t,1]$ occur in the literature.
We shall also have to use the involutions 
$[t_1,t_2] \leftrightarrow [\pm t_2, \pm t_1]$.

Leibniz formula (\ref{eq:Leibniz}) implies the following commutations 
\bea   x_i\, \bT_i &=& T_i\, x_{i+1} \ ,\ 1\leq i\leq n\moins 1 
                      \label{eq:op8} \\
  x_j T_i  &=& T_i\, x_j \ ,\  j\neq i,i\plus 1\ ,\ 1\leq j\leq n \ . \notag
\eea 

One still has a Yang-Baxter equation,

\begin{multline}
 ( T_i + {t_1\plus t_2\over\a \moins 1})\, 
(T_{i+1} + {t_1\plus t_2\over \a\b\moins 1 })\, 
( T_i + {t_1\plus t_2\over\b\moins 1  }) =  \\
( T_{i+1} + {t_1\plus t_2\over\b\moins 1 })\, 
(T_i + {t_1\plus t_2\over \a\b\moins 1})\, 
( T_{i+1} + {t_1\plus t_2\over\a\moins 1 }) \ .
\label{eq:YB3}
\end{multline}

As in the preceding section, one chooses for the identity permutation
$v^0$ a vector of parameters $c(v^0)=[c_1,\ldots, c_n]$ 
(all different, and different from 0) 
and define $c(v)$, $v\in \S_n$, 
to be $[c_{v_1},\ldots, c_{v_n}]$.

Thanks to Yang-Baxter relation (\ref{eq:YB3}), 
 consistency will be now insured by taking, on an edge $v\mapsto v\, s_i$,
 the label
\be T_i +\frac{t_1+t_2}{c_{i+1}/c_i(v) -1} \ .  \label{eq:edge}
\end{equation}

In particular, given two positive integers $a,b$, taking the spectral
parameters
$[(-t_2)^{a+b}, t_1^a (-t_2)^b, t_1^{a+b}]$,
that is $\a=(-t_1/t_2)^a$, $\b=(-t_1/t_2)^b$, 
 the left hand side of (\ref{eq:YB3}) becomes
\begin{multline*}
 \bigl( T_i + \frac{(-t_2)^a}
   {t_1^{a-1} -t_1^{a-2}t_2 +\cdots + (-t_2)^{a-1}}  \bigr) \\
\bigl( T_{i+1} + \frac{(-t_2)^{a+b}}
   {t_1^{a+b-1} -t_1^{a+b-2}t_2 +\cdots + (-t_2)^{a+b-1}}  \bigr)\\
\bigl( T_i + \frac{(-t_2)^b}
   {t_1^{b-1} -t_1^{b-2}t_2 +\cdots + (-t_2)^{b-1}}  \bigr)  
\end{multline*} 


When dealing with reduced decompositions, it is convenient to use planar
displays, Young tableaux, and jeu de taquin, to handle more easily 
the braid relations (cf. \cite{LS6} for the nilplactic monoid).  
All the displays
that we shall write will have the property that the words obtained by
reading them column-wise, from left to right,
are congruent modulo the braid relations to the words
obtained by reading them row-wise (from top to
bottom). For example, writing $i$ for $T_i$, then the planar object 
$\begin{smallmatrix} 2 & 3 &4\\[2pt] 1 &2&3 \\ \end{smallmatrix}$
 can be read $(T_2T_3T_4)(T_1T_2T_3)$ as well as 
$ (T_2T_1)(T_3T_2)(T_4T_3)$.

With these conventions, a recursive application of braid relations leads
to the following identities.

\begin{lemma} Let $i,j,k$ be three positive integers, $i<j$, $i<k$. Then
the following two words are congruent modulo the braid relations:
\be  \begin{matrix} 
  k &\cdots &j\plus k\moins i\\ \vdots &&\vdots\\[2pt]  i&\cdots &j\\[2pt]
            && j\moins 1 \\ &&\vdots \\ && i\\  \end{matrix} \ \equiv \
\begin{matrix}
k\plus j\moins i\\ \vdots\\[2pt] k\plus 1\\[2pt]
 k &\cdots &j\plus k\moins i\\ \vdots &&\vdots\\[2pt]  i&\cdots &j\\
\end{matrix} \ . \label{eq:plax1}
\end{equation}
Moreover 
\be  \begin{matrix}
  k &\cdots &j\plus k\moins i\\ \vdots &&\vdots\\[2pt]  i&\cdots &j
\end{matrix}\, \bT_i\cdots \bT_{j-1} \ \equiv \  
\bT_{k+1}\cdots \bT_{j+k-i}\ 
\begin{matrix}
  k &\cdots &j\plus k\moins i\\ \vdots &&\vdots\\[2pt]  i&\cdots &j
\end{matrix}  \ .       \label{eq:plax2}
\end{equation}
\end{lemma}
For example, 
$$\begin{smallmatrix} 2 & 3 &4\\[2pt] 1 &2&3 \\[2pt]&&2\\[2pt] &&1
\end{smallmatrix}   \equiv
\begin{smallmatrix} 4\\[2pt] 3\\[2pt] 2 & 3 &4\\[2pt] 1 &2&3 
\end{smallmatrix} \qquad \text{and}\qquad
 \begin{smallmatrix} 2 & 3 &4\\[2pt] 1 &2&3\end{smallmatrix}
\, \bT_1\bT_2 \equiv \bT_3\bT_4 
\begin{smallmatrix} 2 & 3 &4\\[2pt] 1 &2&3\end{smallmatrix} \ .
$$

\section{\large  Symmetrizations}

Each of the algebras considered in the preceding sections contains 
elements (that we shall call {\it symmetrizing operators})
sending polynomials in $x_1,\ldots,x_n$ onto symmetric polynomials,
or to products of symmetric polynomials by a fixed factor of
Vandermonde type. 

The first, and most fundamental, of symmetrizing operators, 
is due to Cauchy and Jacobi:
\be
 f  \mapsto \sum_{w\in \S_n} (-1)^{\ell(w)} w \, \frac{1}{\Delta}  \ ,
   \label{eq:sym1}
\end{equation}
where $\Delta:=\prod_{i<j\leq n}(x_i-x_j)$ is the {\it Vandermonde}.
It sends monomials to $\pm $ a Schur function, or to $0$. 
It can be written as a product of divided differences. In fact, it is equal
to $\d_\omega$, where $\omega:=[n,\ldots,1]$ is the permutation of maximal
length in $\S_n$. Its multiple $\d_\omega/n!$ 
 can be interpreted as the 1-dimensional idempotent corresponding to the
alternating representation of the symmetric group.

In the Hecke algebra, there are two 1-dimensional (quasi)-idempotents
which similarly can be expressed as a sum of linear generators,
or expressed with divided differences.

Define 
\bea
\nabla_\omega &:=& \sum_{w\in \S_n} (-t_1)^{\ell(w\omega)} T_w \ ,
  \label{eq:sym2}\\
\square_\omega &:=& \sum_{w\in \S_n} (-t_2)^{\ell(w\omega)} T_w \ .
   \label{eq:sym3}
\eea

\begin{theorem}
1) The element $\nabla_\omega$, as an operator on polynomials, factorizes:
$$ \nabla_\omega =\d_\omega\, \prod_{1\leq i<j\leq n}(t_2x_i+t_1 x_j) \ .$$

2) It factorizes inside the Hecke algebra:
$$ \nabla_\omega= \nabla_{\omega'} \bigl(
 T_{n-1}\cdots T_1 -t_2 T_{n-1}\cdots T_2 +\cdots +
   (-t_2)^{n-2}T_{n-1}+ (-t_2)^{n-1} \bigr)\ ,$$
where $\omega' =[n\moins 1,\ldots,1,n]$.

3) It is a Yang-Baxter element $Y_\omega$, for the symmetric group $\S_n$,
for the initial vector of parameters 
$[t_1^{n-1}, -t_1^{n-2}t_2,\ldots, (-t_2)^{n-1}]$.

4) It is equal to another sum 
$$\nabla_\omega:= \sum_{w\in \S_n} t_2^{\ell(w\omega)} \bT_w \ .$$

5) The element $\square_\omega$ is obtained from $\nabla_\omega$ by interchange
of $t_1$ and $t_2$ in the expressions 2,3,4. The involution 
$t_1\mapsto -t_2$, $t_2\mapsto -t_1$,
 $T_i \leftrightarrow \bT_i$,$i=1,\ldots, 
n\moins 1$, preserves $\nabla_\omega$ and $\square_\omega$. 
Moreover,
$$\square_\omega:=\prod_{1\leq i<j\leq n}(t_1x_i+t_2 x_j)\, \d_\omega \ .$$
\end{theorem}

A proof of this theorem is given in \cite{DKLLST}. Since all the operators
appearing in theorem 4 commute with multiplication by symmetric functions,
theorem 4 essentially reduces to check that all elements, but one, of
the Schubert basis for $\S_n$ are sent to 0 by $\nabla_\omega$. 

The last factorization appear in geometry, $\square_\omega$ 
being interpreted
as an Euler-Poincar\'e characteristic for a relative flag manifold
(cf. \cite{Hi}, first chapter of \cite{La77}, 
\cite{La}).  

One has similar factorizations in the degenerate affine Hecke algebra 
 generated by the $\ss_i=s_i+ \gamma\d_i$ , as shows the following 
theorem (cf. \cite{LLT1}):   

\begin{theorem} 1) The sum of all the linear generators of the 
 degenerate affine Hecke algebra factorizes (as operators on polynomials)
\be
 \widetilde{\square}_\omega:=\sum_{w\in \S_n} \ss_w 
= \prod_{i<j\leq n}(x_i-x_j+\gamma) \, \d_\omega \ . \label{eq:sym7}
\end{equation}

2) It is a Yang-Baxter element for the parameters $[1,2,\ldots,n]$:
$$ \widetilde{\square}_\omega
 = (\ss_1+1) \Bigl((\ss_2+\frac{1}{2})(\ss_1+1)  \Bigr)
\cdots \Bigl((\ss_{n-1} +\frac{1}{n\moins 1}) \cdots 
(\ss_2+\frac{1}{2})(\ss_1+1)  \Bigr) \ .$$
\end{theorem} 

\section{\large Affine edges} 

We have shown that labelling of edges of a given cell are determined by a
single vector of spectral parameters.
We have only retained three algebras, the group algebra of the symmetric group,
that we shall see as the specialization $\gamma=0$ of the  
degenerate affine Hecke algebra, and the affine Hecke algebra.

It remains to define the labels of the affine 
edges connecting cells and to check
the compatibility between the labels of different cells. 
We shall see that it amounts to introduce 
\lq\lq periodicity" conditions on the $x_i$'s and on the 
initial vector  $c(\ket)$ of spectral parameters.

To treat the different varieties of Jack and Macdonald polynomials at the
same time, we shall use an arbitrary function $G:=G_n$ of $x_n$,
and define the label of all
affine edges of the Yang-Baxter graph to be
$$  \Phi:= \tau\, G \ $$
(recall that $\tau$ is the shift: $x_i\, \tau = x_{i-1}\, \tau$).

We could even be more general and take for example 
$G$ to be a differential operator
on functions of $x_n$. At this stage, we need only the commutation
of $G$ with $x_1,\ldots, x_{n-1}$, $s_1,\ldots, s_{n-2}$. 
   
Let us write   
$$ G_i:= s_i\cdots s_{n-1} G_n s_{n-1}\cdots s_i \ ,\ 1\leq i\leq n-1 \ .$$
 

In the case of the degenerate affine Hecke algebra with generators
$\ss_i =s_i+\gamma \d_i$,  we used  
differences $c_{i+1}-c_i$ and $x_{i+1}-x_i$, and 
consistency requires them to be
invariant under $i\mapsto i+n$. For that purpose, we introduce
two more parameters  $\a, \b$:
\be
    x_{i+n}= x_i +\b \ , \ c_{i+n}(\ket) =  c_{i}(\ket) +\a\ 
                      \forall i\in\Z \ .  \label{eq:aff1}
\end{equation}


The subgraph 
$\moyennematrice{ v &\rightarrow &v s_{i+1} \cr
          \downarrow & & \downarrow\cr
         v\tau &\rightarrow &v\tau s_i &= v  s_{i+1}\tau }$
give rise to the following commutative diagram 
(writing $c(v)$ instead of $v$, and
1,2,3 instead of $i,i+1,i+2$):

\def\hfl#1{{\stackrel{#1}
             {\hbox to 30mm{\rightarrowfill}}}}
\be \begin{matrix}
 [\ldots c_1,c_2,c_3\ldots] 
  &\hfl{s_2\plus \gamma\d_2 +\frac{1}{c_3-c_2}} 
 & [\ldots c_1,c_3,c_2\ldots] \cr
\noalign{\vskip 2mm}\vfl{\Phi}&& \vfl{\Phi} \cr
   [\ldots c_2,c_3,c_4\ldots] &
    \hfl{s_1\plus \gamma\d_1 +\frac{1}{c_3-c_2}}  i
            &  [\ldots c_3,c_2,c_4\ldots]  \cr 
\end{matrix}  \label{eq:aff2}
\end{equation}
which says nothing more than 
$x_2\tau =x_1$, $x_3\tau=x_2$, $\d_2\tau= \tau\d_1$,
$s_2\tau= \tau s_1$.

The commutativity of

\def\hfl#1{{\stackrel{#1}{\hbox to 33mm{\rightarrowfill}}}}
\be  \begin{matrix}
 [\ldots c_{n-1},c_n,c_{n+1},c_{n+2}\ldots] &
  \hfl{s_1\plus \gamma\d_1 +{1\over c_2-c_1}} & 
[\ldots  c_{n-1},c_n,c_{n+2},c_{n+1} \ldots] \cr
\noalign{\vskip 2mm}\vfl{\Phi\Phi}&& \vfl{\Phi\Phi} \cr
   [\ldots  c_{n+1},c_{n+2},c_{n+3},c_{n+4} \ldots] &
 \hfl{s_{n-1} \plus \gamma\d_{n-1} +{1\over c_2-c_1}}  &  
[\ldots c_{n+2}, c_{n+1},c_{n+3},c_{n+4} \ldots]
   \cr 
\end{matrix}   \label{eq:aff3}
\end{equation}
is also immediate, once noticing that $\tau G_n\tau G_n = \tau^2 G_{n-1}G_n$,
and that $G_{n-1}G_n$, being symmetrical in $x_{n-1}$ and $x_n$,
 commutes with $s_{n-1}$, as well as with $\d_{n-1}$.

Braid relations, together with these two types of diagrams allow to deform
any path $\ket \mapsto v$ into any other one with the same end points.
Thus conditions (\ref{eq:cell4}) and (\ref{eq:aff1})  furnish
a Yang-Baxter graph with edges in the degenerate Hecke algebra.


\medskip 
In the case of the  affine Hecke algebra,  we used
 $c_{i+1}/c_i-1$ and $1- x_{i+1}/x_i$ in the labels of edges of cells, 
and we need them to be invariant under $i\mapsto i+n$.
We thus introduce two extra parameters $q,q'$ (in the following sections, 
we shall restrict to $q=q'$) and put
\be  x_{i+n}= q x_i \ , \ c_{i+n}(\ket) = q' c_i(\ket) \ , 
\ \forall i\in \Z \label{eq:aff4}
\end{equation} 

Now, since $\pi_i\tau = \tau\pi_{i-1}$, $s_i\tau = \tau s_{i-1}$,
$1<i<n$, and since $G_{n-1} G_n$ commutes with $T_{n-1}$, one  still has 
commutative diagrams (where $T_1,T_2$ stand 
for $T_i,T_{i+1}$, $1\leq i \leq n-2$ in the first diagram)~: 

\def\hfl#1{{\stackrel{#1}{\hbox to 26mm{\rightarrowfill}}}}
\begin{equation} 
\begin{matrix}
 [\ldots c_1,c_2,c_3\ldots] &\hfl{T_2 +\frac{t_1+t_2}{c_3/c_2-1}} 
& [\ldots c_1,c_3,c_2\ldots] \\[6pt]
   \vfl{\Phi}&& \vfl{\Phi} \\[0pt]
[\ldots c_2,c_3,c_4\ldots] &\hfl{T_1 + \frac{t_1+t_2}{c_3/c_2-1}}
 &  [\ldots c_3,c_2,c_4\ldots] \\ 
\end{matrix}     \label{eq:aff5}
\end{equation}
and
\be \begin{matrix}
 [\ldots c_{n-1},c_n,c_{n+1},c_{n+2}\ldots] &
 \hfl{T_1 +{t_1+t_2\over c_2/c_1-1}  } &
  [\ldots  c_{n-1},c_n,c_{n+2},c_{n+1} \ldots] \\[6pt]
                        \vfl{\Phi\Phi}&& \vfl{\Phi\Phi} \\[0pt]
   [\ldots  c_{n+1},c_{n+2},c_{n+3},c_{n+4} \ldots] &
 \hfl{T_{n-1} +{t_1+t_2\over c_2/c_1-1}}  &
             [\ldots c_{n+2}, c_{n+1},c_{n+3},c_{n+4} \ldots]  
\end{matrix}  \label{eq:aff6} 
\end{equation}

Conditions (\ref{eq:edge}) and (\ref{eq:aff4})  provide
a Yang-Baxter graph with edges in the  Hecke algebra.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\large Cherednik's Algebras}  

The double affine Hecke algebra $\C[t_1,t_2,q][T_1,\ldots, T_{n-1},
\tau^{\pm1}, x_1^{\pm1}, \ldots x_n^{\pm 1}]$ 
admits some maximal commutative subalgebra
which was found by Cherednik in his study of Macdonald's polynomials.
We need a more general algebra than the original
Cherednik's algebra, and for that purpose, 
we shall use the element $G_n$ introduced in the preceding section, 
supposing only for the moment that it is an  invertible morphism of
the space of functions in $x_n$. 

Let us write $\bG:= G^{-1}$,
$\bt = \tau^{-1}$,  
$\bT_i:= -t_1t_2\,  T_i^{-1} $, $1\leq i \leq n-1$.  Recall 
that $x_i\, \bt= \bt\,  x_{i+1}$, and that 
 $\Phi= \tau G_n$ label the affine edges of the Yang-Baxter graph.. 


For $1\leq i\leq n$, define the {\it Cherednik element} $\xi_i$ by
\be 
    \xi_i := T_i\dots T_{n-1} \, x_n\, \bG_n \, \bt\, 
        \bT_1\cdots \bT_{i-1} \ . \label{eq:cher1}
\end{equation}

\begin{proposition}
 The $\xi_i$ generate a commutative algebra 
and satisfy the relations
\begin{gather}
 \xi_{i+1} T_i =\bT_i\, \xi_i \quad
  \& \quad (\xi_i+\xi_{i+1})T_i =T_i(\xi_i+\xi_{i+1}) 
\ , 1\leq i \leq n-1 \ , \notag\\
 \xi_j T_i = T_i \xi_j \ , j\neq i,i+1    \label{eq:cher2}\\
 \xi_{i+1} \, \Phi = \Phi\, \xi_i \ , \  i=1,\ldots, n-1 \ . \notag
\end{gather}

Moreover, if $G_n$ commutes with multiplication by $x_n$, then
\be   q\xi_1\, \Phi= \Phi\, \xi_n \ . \label{eq:cher3}
\end{equation}
\end{proposition}

\proof
One  checks without difficulty the relations  between the $\xi_i$ and $T_j$
and the commutativity of the $\xi_i$.

Thanks to these relations, the compatibilities between $\Phi$ and the
 $\xi_i$ is reduced to show that $\xi_2\, \Phi = \Phi\, \xi_1$, i.e.
 $$ T_2\cdots T_{n-1} x_n \bG_n \bt\, \bT_1 \tau G_n =
\tau G_n T_1\cdots T_{n-2} T_{n-1} x_n \bG_n\, \bt \ .$$

Now the factor  $T_1\cdots T_{n-2}$ commutes with $G_n$ and cancels with 
the factor $ T_2\cdots T_{n-1}$ on the left hand side;  
we are left to show that 
$$  x_n \bG_n \bt\, \bT_1 \tau G_n = \tau G_n\, T_{n-1}  x_n \bG_n\, \bt \ .$$

Let us take the indeterminates $x_2,\ldots,x_n, x_{n+1}=qx_1$
instead of $x_1,\ldots x_n$. It allows us to use $G_{n+1}:= \tau\, G_n\, \bt$. 
Let us moreover write $T_n:= \bt\, T_1\, \tau = \tau\, T_{n-1}\, \bt$;
concretely $T_n$ operates on $x_n, x_{n+1}$ according to (\ref{eq:op4}) 
and leaves $x_2,\ldots, x_{n-1}$ invariant. 

Introducing factors
$\bG_{n+1} G_{n+1}$ and $  \bt \tau$, one has to show
$$ x_n (G_{n+1} \bG_{n+1})\, \bG_n \bt \bT_1\,  \tau G_n =
\tau G_n\,   (\bt \tau)  T_{n-1}\, 
(\bt \tau)\,  x_n (\bt \tau)\, \bG_n \bt \ .$$

Taking into account that  $\bG_{n+1} \bG_n$ 
 commutes with $\bt \bar T_1 \tau = T_{x_n,x_{n+1}}$, we are 
finally reduced to make sure that 
 $$ x_n G_{n+1} \bT_n \bG_{n+1} \bG_n G_n = 
 G_{n+1} T_n x_{n+1} \bG_{n+1} \ , $$
that is 
 $$ x_n\,  \bT_n = T_n x_{n+1} \ , $$
which is no other than relation (\ref{eq:op8}) for the pair $(x_n,x_{n+1})$. 

One similarly checks that $T_1\cdots T_{n-1}\, \bt \tau\, x_n = 
x_1 \bT_1\cdots \bT_{n-1} = \tau x_0 \bt \bT_1\cdots \bT_{n-1} = \tau x_n \xi_n
/q$, that is 
\be  \xi_1\, \tau\, x_n = \tau\, x_n \, \xi_n/q  \ . \label{eq:cher4}
\end{equation}
 

There remains the case of 
$\Phi\, \xi_n = \tau G_n \, x_n G_n \bT_1\cdots \bT_{n-1}$. 
If $G_n$ commutes with $x_n$, then 
$$\Phi\, \xi_n= \tau x_n \bt\, 
\bT_1\cdots \bT_{n-1} = q T_1\cdots T_{n-1}\, x_n=
q \left( T_1\cdots T_{n-1}\, x_n \bG_n \bt \right)\, (\tau G_n) \ , $$ 
that is 
$$  \Phi\, \xi_n  =  q \xi_1\, \Phi  \ . $$ \QED


\smallskip  The $\xi_i$'s satisfy the same commutation relations with the
$T_i$'s as the $x_i$'s do. In fact, one can interpolate between 
$\xi_i$ and $x_i$ without losing a commutative algebra. 

\begin{theorem}
Let $\b\in \C$; for $i=1,\ldots, n-1$, let 
$$  \xi_i^\b := \xi_i +\b x_i  \ .$$
Then the $\xi_i^\b$'s generate a commutative algebra, and satisfy the
relations
\begin{gather}
 \xi_{i+1}^\b T_i =\bT_i\, \xi_i^\b \quad 
  \& \quad (\xi_i^\b+\xi_{i+1}^\b)T_i =T_i(\xi_i^\b+\xi_{i+1}^\b)\ ,
                                  1\leq i \leq n-1 \ , \label{eq:cher5}\\
 \xi_j^\b T_i = T_i \xi_j^\b \ , j\neq i,i+1
                                                        \label{eq:cher6}\\
 \xi_{i+1}^\b \, \Phi = \Phi\, \xi_i^\b \ , \  i=1,\ldots, n-1 \ .
                                                          \label{eq:cher7}
\end{gather}

Moreover, if $G_n$ commutes with multiplication by $x_n$, then
\be  q\xi_1^\b\, \Phi= \Phi\, \xi_n^\b \ . \label{eq:cher8}
\end{equation}
\end{theorem}

\proof Commutations relations (\ref{eq:cher5},\ref{eq:cher6},\ref{eq:cher7})
   are a consequence of those for $\b=0$.
Given $1\leq i<j\leq n$, the commutator 
$ [ \xi_i^\b,\, \xi_j^\b]$ is equal to 
$\b [ x_i,\, \xi_j^\b] -\b [ \xi_i^\b,\, x_j]$.  Explicitly,
$$x_j\xi_i^\b =  x_j \underbrace{T_i\cdots T_{j-2} }_A\,  T_{j-1} \,
\underbrace{T_j\cdots T_{n-1}\, x_n\, \bG_n }_B\,  \bt\,
      \underbrace{\bT_1\cdots \bT_{i-1}}_C   \ .            $$        
Now, $x_j$ commutes with $A$; $x_jT_{j-1}= \bT_{j-1} x_{j-1}$; 
$x_{j-1}$ commutes with $B$; passing through $\bt$, it  is transformed 
into $x_j$ again and commutes with $C$.

Therefore 
$$[ \xi_i^\b,\, x_j]=  A \bigl( T_{j-1} -\bT_{j-1}\bigr) B\, \bt\, Cx_j
= A(t_1+t_2) \, B\, \bt\, Cx_j \ . $$
Similarly, starting from 
$$ x_i\xi_j^\b = x_i\, B\, \bt\, C\,  \bT_i \underbrace{
\bT_{i+1}\cdots \bT_{j-1} }_D \ ,$$
one finds that 
$$ [\xi_j^\b ,\, x_i] = B\, \bt\, C  x_{i+1} D (t_1+t_2) \ .$$
Finally, one remarks that $x_{i+1} D = T_{i+1}\cdots T_{j-1}\, x_j
:=E x_j$, and that $A$ commutes with $B$,  becomes equal to $E$
when passing through $\bt$ and commutes with $C$ at the end. 
Therefore the $\xi_i^\b$'s generate a commutative algebra.   \QED

\medskip
One checks, for any $1\leq i<n$ the commutations 
\be
 T_i(\xi_i^\b + \xi_{i+1}^\b)= (\xi_i^\b + \xi_{i+1}^\b) T_i \quad \& \quad
T_i(\xi_i^\b\xi_{i+1}^\b)= (\xi_i^\b\xi_{i+1}^\b)T_i \ ,
                                          \label{eq:cher9}
\end{equation}
and this ensures the commutations of each $T_i$ with the elementary
symmetric functions in the Cherednik elements.
Therefore one has:  

\begin{lemma}
The Hecke algebra generated by the $T_i$'s commutes with the ring of
symmetric functions in the $\xi_i^\b$.
\end{lemma} 

For computations, it is convenient to write monomials  
 in the $\xi_i$'s (without repetition) in a canonical form. 
Write $\theta$ for 
$x_n\, \bG_n\, \bt$. Then one checks, from the commutation relations
(\ref{eq:cher2}),  
together with $T_i\bT_i =(-t_1t_2)$,  that:

\begin{lemma} Let 
$k$ be an integer, $n\geq i_1>i_2>\cdots >i_k\geq 1$ a decreasing sequence.
Then the monomial $ \xi_{i_1} \xi_{i_2} \cdots \xi_{i_k}
(t_1t_2)^{-k(k-1)}$ is equal to
$$  \begin{matrix}
  & & i_1 &\cdots & n\moins 1 \\[3pt]
  & i_2 &\cdots &\cdots & n\moins 2\\  
  & \vdots    &       &       &  \vdots \\[2pt]
 i_k  &\cdots &\cdots &\cdots & n\moins k \\
\end{matrix}  \quad \theta^k \quad
\begin{matrix}
  k         & \cdots  & \cdots & \cdots &i_1\moins 1 \\[3pt]
  k\moins 1 & \cdots & \cdots &i_2\moins 1  \\
  \vdots    & \cdots &\vdots \\[2pt]
  1         & \cdots & i_k\moins 1  \\
\end{matrix}  \ ,          $$
where the planar display on the left must be read (columwise or rowwise)
as a product of $T_j$'s, and the display on the right, 
as a product of $\bT_j$'s.
\end{lemma}

For example, for $n=8$, 
$$ \xi_6 \xi_3 \xi_2  (t_1t_2)^{-6} =
 \begin{smallmatrix} 
  . & . & 6 & 7\\[2pt]
  3 & 4 &5 &6 \\[2pt]
  2 & 3 &4 &5\\  \end{smallmatrix} \ \theta^3 \ 
\begin{smallmatrix}  3 &4&5\\[2pt] 2 & . &.\\[2pt]
  1 & . &.\\  \end{smallmatrix} =
  \left(T_6T_7 T_3 T_4 T_5 T_6 T_2 T_3 T_4 T_5 \right) \, \theta^3 \, 
\left(\bT_3 \bT_4 \bT_5 \bT_2\bT_1\right)\ . 
$$
 
Exchanging left and right, one gets another commutative algebra 
generated by the 
$$ T_{i-1}\cdots T_1\, \tau \frac{1}{x_n}G_n\, \bT_{n-1}\cdots \bT_i 
+\frac{\b}{x_i} \ ,\ 1\leq i\leq n\ .$$
In the case of interest for vanishing Macdonald polynomials, one takes
$G_n= x_n -(-t_2)^{n-1}$. The following lemma
(due to Knop \cite{knop2}) shows that in that case, one can choose an
appropriate $\b$ in order to get operators preserving polynomials.

\begin{lemma}
Let  
\be \xx_i := T_{i-1}\cdots T_1\, \tau  
\bigl( 1- \frac{(-t_2)^{n-1}}{x_n} \bigr)\bT_{n-1}\cdots \bT_i
+ \frac{(-t_1t_2)^{n-1}}{x_i} ,\ 1\leq i\leq n\ . \label{eq:cher10}
\end{equation}
Then the $\xx_i$'s generate a commutative algebra which preserves
polynomials in $x_1,\ldots, x_n$.
\end{lemma} 

\proof
Because of the commutation relations (\ref{eq:cher2}), we have only to check 
that $\xx_n$ preserves polynomials. 
Recall that $x_{i+1}T_i= \bT_i x_i$, which implies, because $x_i+x_{i+1}$
commute with $T_i$, that 
$$ x_iT_i = T_ix_{i+1} + (t_1+t_2)x_i \ .$$
Therefore, for $1\leq i\leq n-1$, one has
$$ x_i T_{n-1}\cdots T_1 = T_{n-1}\cdots T_1 x_{i+1} + (t_1+t_2)
 T_{n-1}\cdots T_{i+1} \, \bT_{i-1}\cdots \bT_1\, x_1 \ ,$$
and 
\begin{multline}
 x_i \xx_n = T_{n-1}\cdots T_1\, \tau (1- \frac{(-t_2)^{n-1}}{x_n}) x_i 
                                + \frac{(-t_1t_2)^{n-1}x_i}{x_n} \\ 
  +(t_1+t_2) T_{n-1}\cdots T_{i+1} \, \bT_{i-1}\cdots \bT_1\, \tau\, 
\bigl( x_0- \frac{(-t_2)^{n-1}x_0}{x_n} \bigr) \notag
\end{multline}
and this last expression is equal to
$ \xx_n x_i$ plus a polynomial (because $x_0= x_n/q$).

On the other hand, 
$$ x_n \xx_n = \bT_{n-1}\cdots \bT_1\tau 
   \bigl( x_0- \frac{(-t_2)^{n-1}x_0}{x_n}
                     \bigr) + (-t_1t_2)^{n-1} \ . $$
In all, $\xx_n$ commutes with polynomials, modulo addition of polynomials.
Checking that 
$$ 1\, \xx_n= t_1^{n-1} \bigl( 1- \frac{(-t_2)^{n-1}}{x_n} \bigr) +
\frac{(-t_1t_2)^{n-1}}{x_n} = t_1^{n-1} \ ,$$
one concludes that $\xx_n$ preserves polynomials. \QED 

\medskip
There are also several Cherednik algebras relative to the degenerate
affine Hecke algebra. We just state the analogs of the results that
we described for the Hecke algebra, keeping the same notations for Cherednik's
elements because they will be used in a different context. 

The first Cherednik's elements are the isobaric version of Dunkl's
elements (\cite{Dunkl}). 

Let $D_i$ be the isobaric derivative $f\mapsto x_i \frac{d}{dx_i}(f)$,
and let $\bp_{i,j}$ denote $\bp_{x_i,x_j}$.
Define, following \cite{cher1},  
\be   \xi_i := \a D_i +\sum_{j\neq i,j=1}^n \bp_{i,j} +1-i \ , \ 1\leq i\leq n
\ .  \label{eq:cher11}
\end{equation}

A second variety of elements involves the arbitrary function  $G_n$
of $x_n$
(it can be obtained by degeneration
of the $\xi_i^\b$'s). Define
\be   \xx_i := x_i -\ss_{i-1}\cdots \ss_1\, \tau\, G_n\,
   \ss_{n-1}\cdots \ss_i \ 1\leq i\leq n \ ,  \label{eq:cher12}
\end{equation}
with  $\ss_i=s_i +\d_i$ 
 and $\tau$ still such that $x_i\tau=x_{i-1}$,
$x_0= x_n-\a$.

Define  moreover 
$$\Phi = s_1\ldots s_{n-1}\, x_n  \quad ,\quad 
\hphi = \tau\, G_n \ .$$
Then one has (cf. \cite{cher1},\cite{KS2} for the first part,
\cite{knop2} for the second part when $G_n=x_n+n-1$):
\begin{theorem}
The $\xi_i$'s (resp. the $\xx_i$'s), $1\leq i\leq n$,
generate a commutative algebra.

They satisfy
\begin{gather}
 \xi_{i+1} s_i =s_i\, \xi_i -1 \quad
  \& \quad (\xi_i+\xi_{i+1})s_i =s_i(\xi_i+\xi_{i+1})
\ , 1\leq i \leq n-1 \ , \notag\\
 \xi_j s_i = s_i \xi_j \ , j\neq i,i+1    \label{eq:cher13}\\
 \xi_{i} \, \Phi = \Phi\, \xi_{i-1} \ , \  i=1,\ldots, n,\
\xi_0=\xi_n-\a \ . \notag
\end{gather}
\begin{gather}
 \xx_{i+1} \ss_i =\ss_i\, \xx_i -1 \quad
  \& \quad (\xx_i+\xx_{i+1})\,\ss_i =\ss_i(\xx_i+\xx_{i+1})
\ , 1\leq i \leq n-1 \ , \notag\\
 \xx_j\, \ss_i = \ss_i\, \xx_j \ , j\neq i,i+1    \label{eq:cher14}\\
 \xx_{i} \, \hphi = \hphi\, \xx_{i-1} \ , \  i=1,\ldots, n,\ 
\xx_0=\xx_n-\a \ . \notag
\end{gather}

\end{theorem}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
                   
\section{\large Propagation of vanishing properties, and of eigenvectorness}

Each edge of a cell in a Yang-Baxter graph is an operator on the ring of
polynomials in only two variables, say $x,y$, as a rank 2 free-module
over $\Sym(x,y)$. We shall easily check, in this section, that
if a vertex is an eigenvector for Cherednik elements, or vanishes 
for some values of the $x_i$'s, then its 
neighbours in a cell have similar properties. 

Choosing appropriate functions $G_n$ of $x_n$, the same 
will be true for neighbours connected by an affine edge. 

\begin{lemma}
Let $f$ be a function of $x_1,\ldots,x_n$, and $i$ be an integer:
$1\leq i\leq n-1$.  Then

1) if $f$ is an eigenfunction of $\xi_i^\b, \xi_{i+1}^\b$, of respective
 eigenvalues $a,b$, then $g:= f\, \bigl(T_i+ (t_1+t_2)/(b/a-1) \bigr)$
 is an eigenvector of $\xi_i^\b, \xi_{i+1}^\b$
 with eigenvalues $b,a$. 

2) if $f$ is an eigenfunction of $\xi_1^\b,\ldots, \xi_n^\b$  of respective
 eigenvalues $c_1,\ldots, c_n$, then all elements in the linear span 
 of $\{ f,\, f\,T_i\}$ are eigenfunctions of \\
$\xi_1^\b,\ldots,\xi_{i-1}^\b,\, 
 \xi_{i+2}^\b,\ldots, \xi_n^\b$ of respective
 eigenvalues $c_1,\ldots, c_{i-1}, c_{i+2},\ldots, c_n$.
\end{lemma}

\proof 
$$ f \xi_{i+1}^\b T_i =b f T_i = b f (T_i+ {t_1+t_2\over b/a -1} -
  {t_1+t_2\over b/a -1})= b g -b  {t_1+t_2\over b/a -1}f $$
Similarly,
$$   f\bT_i \xi_i^\b= f( T_i + {t_1+t_2\over b/a -1} +{b(t_1+t_2)
  \over a -b}) \xi_i^\b = g \xi_i^\b + f {ab(t_1+t_2) \over a -b} \
.$$
Equality $\xi_{i+1}^\b T_i = \bT_i \xi_i^\b$ implies that
$$g \xi_i^\b = g\, b \ , \ g  \xi_{i+1}^\b=g\, a \ .$$ 
This settles the first point, the second being a direct consequence
of the commutation relations (\ref{eq:cher5},\ref{eq:cher6},\ref{eq:cher7}).
\QED

\begin{corollary} Let 
$f$ be an eigenfunction of $\xi_1^\b,\ldots, \xi_n^\b$  of respective
 eigenvalues $c_1,\ldots, c_n$. Then all elements of the space
$f\, \cH_n$ (in particular the symmetric function $f\square_\omega $)
are eigenfunctions of $\prod_{i=1}^n (1+\g \xi_i)$ with eigenvalue
$\prod( 1+\g c_i)$, for any $\g\in\C$.
\end{corollary}

\begin{lemma} Let $f(x,y)$ be a function of two variables.

1) Suppose that there exist $a,b$: $a\neq b$ such that $f(a,b)=f(b,a)=0$.
Then $fT_{x,y},\, fs_{x,y},\, f\d_{x,y}$ also vanish in 
$[x,y]=[a,b]$ and $[x,y]=[b,a]$.

2)  Suppose that $f(a,b)=0$, with $b=(-t_2/t_1)a$. Then 
$fT_{x,y}$ also vanish in $[x,y]=[a,b]$. 

  Suppose that $f(a,b)=0$, with $b=a-\g$. Then $f ( s_{x,y}+\g\d_{x,y})$
vanish in $[x,y]=[a,b]$.

3) Suppose that $f(b,a)=0$, $a\neq b$. Then 
$$ f\bigl( T_{x,y} +\frac{t_1+t_2}{b/a-1} \bigr) \quad\text{and} \quad 
 f\bigl( s_{x,y}+\g\d_{x,y} +\frac{\g}{b-a}  \bigr) $$
vanish in $[x,y]=[a,b]$ 
\end{lemma}

\proof Write $f=f_1+yf_2$, with $f_1,f_2\in \Sym(x,y)$. 

If $f(a,b)=f(b,a)=0$ and $a\neq b$, then both $f_1$ and $f_2$ vanish 
in $[a,b]$ and $[b,a]$, and so do $fT_{x,y}$, $fs_{x,y}$,
$f\d_{x,y}$. This proves the first point.  

The expression $fT_{x,y}=t_1f_1 -t_2xf_2$ specializes, for $[x,y]=[a,b]$
into $t_1 f_1(a,b)-t_2af_2(a,b)= t_1f(a,b)$, which is null in the second case.
Similarly, $f ( s_{x,y}+\g\d_{x,y})= f_1+xf_2-\g f_2$ 
specializes into $f_1(a,b)+(a-u)f_2(a,b)$, which is null when $f(a,b)=0$.

In the third case, 
$$ f\bigl( T_{x,y} +\frac{t_1+t_2}{b/a-1} \bigr)=
 \bigl(t_1+\frac{t_1+t_2}{b/a-1}  \bigr)f_1(x,y)
+ \bigl( -t_2x+\frac{t_1+t_2}{b/a-1}y \bigr)f_2(x,y) $$
specializes, for $[x,y]=[a,b]$, into 
$$ \bigl(\frac{t_1b/a +t_2}{b/a-1}  \bigr)f_1(a,b)
+ \bigl(\frac{t_1b +t_2a}{b/a-1}y \bigr)f_2(a,b) = af(b,a) \ , $$
that is, vanishes.  

Similarly 
$$ f\bigl( s_{x,y}+\g\d_{x,y} +\frac{\g}{b-a}  \bigr)=
f_1 \bigl(1+\frac{\g}{b-a} \bigr) + f_2 \bigl(x-\g+\frac{\g y}{b-a} \bigr) $$
specializes, for $[x,y]=[a,b]$, into $af(b,a)$.
\QED 

Propagation of vanishing properties and of eigenvectors
along affine edges are insured by the
choice of appropriate $\Phi$'s, as shows the following lemma~:

\begin{lemma} 1) Let $f$ be an eigenfunction of the Cherednik elements
$\xi_1,\ldots, \xi_n$, with eigenvalues $c_1,\ldots, c_n$. 
Let $\Phi=\tau G_n$, with $G_n$ commuting with multiplication by $x_n$.
Then $f\Phi$ is an eigenfunction of $\xi_1,\ldots, \xi_n$,
with eigenvalues $c_2,\ldots, c_n, qc_1$ (in the case of the Hecke algebra)
or $c_2,\ldots, c_n, c_1+\a$ (in the case of the degenerate Hecke algebra).

2) Let $k$ be an integer, and let $f$ be a function vanishing in all
$c(u)$, $|u|\leq k$. Let $\Phi=\tau G_n$, with $G_n( (-t_2)^{n-1})=0$
(in the case of the Hecke algebra) and $G_n(1-n)=0$ 
(in the case of the degenerate Hecke algebra). Then  $f\, \Phi$ 
vanishes in all $c(u)$, $|u|\leq k+1$.
\end{lemma}

\proof Commutation rules of Cherednik's elements with $\Phi$ 
insure the first point.  If $f$ vanishes in $c(u)$, then $f\tau$ 
vanishes in $c(u\tau)$. The vectors $u :\, |u|\leq k+1$ which are not 
in the image of $\tau$ are such that $u_n=0$, in which case
$c_n(u) = (-t_2)^{n-1}$ or $1-n$, depending on the algebra. In that case,
the vanishing of $G_n$ implies the vanishing of $f\, \Phi$. \QED 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\large Macdonald polynomials} 

Take Yang-Baxter graph with
\begin{gather}
\ss_i:= T_i \ ,\ 1\leq i\leq n-1 \ ,\ c_i(\ket)= t_1^{n-i} (-t_2)^{i-1}
\ , \ 1\leq i\leq n \ ,  G_n=x_n \ , \notag \\
 x_{i+n}= qx_i \ , \ c_{i+n}(\ket)= q c_i(\ket) \ ,\ \forall i \in  \Z
  \label{eq:ma1}
\end{gather}

\begin{Def}
 The   {\em Macdonald polynomial} $M_v$,  $v\in \N^n$,
 is the image of the polynomial $1$
 under the element $Y_v$ associated to the 
Yang-Baxter graph satisfying (\ref{eq:ma1}). In other words, one has:
$$M_v:= 1\, Y_v \ , \ \forall v\in \N^n \ . $$
\end{Def}

The easiest characterization  of these polynomials 
(cf. \cite{mac3}, \cite{cher3}) is that they are eigenfunctions of
Cheredenik's elements $\xi_i$ (for the same $G$), that is for 
$$ \xi_i := T_i\dots T_{n-1} \, \bt\, \bT_1\cdots \bT_{i-1} \ .$$

\begin{theorem}
 Let $v\in \N^n$. Then the Macdonald polynomial $M_v$ is a simultaneous
eigenfunction of the $\xi_i$'s, i.e.  one has 
   $$M_v\, \xi_j =  M_v\, c_j(v)\ , \ j: 1\leq j \leq n\ . $$
\end{theorem}

\proof Relations (\ref{eq:cher2},\ref{eq:cher3}) ensure that $M_v$ is an
eigenfunction, with eigenvalue $c(v)$, if it is true for the origin 
$v=\ket$. Indeed, $1\, T_i= t_1$, $1\, \bT_i=-t_2$ implies that
$1\, \xi_i= t_1^{n-i} (-t_2)^{i-1}= c_i(\ket)$. \QED 

\bigskip
Mimachi and Noumi give in \cite{MN} a reproducing kernel for the 
Macdonald polynomials.

The labelling of the affine edges of the Yang-Baxter graph by $\Phi= \tau\,
x_n$ is due to  Sahi \cite{sahi1} and Knop \cite{knop2}.
They call $\Phi$ a ``creation operator''.
Baker and Forrester \cite{forr97b} similarly define a {\it lowering operator}
$\hphi$ :
\begin{equation}
 \hphi =\bigl(T_{n-1}\cdots T_1\, \tau -t_1^{n-1}\bigr) 
\frac{1}{x_n}\, \bT_{n-1}\cdots \bT_1 \ . 
\end{equation}

\begin{lemma}
 $\hphi$ preserves polynomials and sends the Macdonald polynomial 
$M_v$ onto a multiple of $M_{v\bt}$ if $v_n>0$, and onto 0 otherwise.
\end{lemma}

\proof  Let $\bksi_n:=T_{n-1}\cdots T_1\, \tau$. Then 
 $\bksi_n\, \xi_n=(-t_1t_2)^{n-1}$ and 
$\bksi_n-t_1^{n-1}$ annihilates all $M_v$, with $v_n=0$ (because in that case, 
$c_n(v)=(-t_2)^{n-1}$).  Therefore  $\hphi$ annihilates those 
Macdonald polynomials. 

 Since $\Phi$ has zero kernel, we shall evaluate $\Phi\hphi$ to
determine the action of $\hphi$ on the other Macdonald polynomials.

\bea 
\Phi\hphi &=& \tau\, x_n\, T_{n-1}\cdots T_1 \tau \frac{1}{x_n}\, 
 \bT_{n-1}\cdots \bT_1 - t_1^{n-1}\tau\, x_n \frac{1}{x_n} \,
 \bT_{n-1}\cdots \bT_1 \notag\\
&=& \tau\, \bT_{n-1}\cdots \bT_1\, ( x_1\tau \frac{1}{x_n})
   \,\bT_{n-1}\cdots \bT_1
    - t_1^{n-1}\tau\,\bT_{n-1}\cdots \bT_1  \notag\\
&=& \frac{1}{q}\, \bksi_1^2 -  t_1^{n-1} \bksi_1 \ , \notag
\eea
 with $\bksi_1 := \tau\, \bT_{n-1}\cdots \bT_1$.  
Therefore $\hphi$ preserves polynomials, and sends $M_v$ 
on a non-zero multiple of $M_{v\bt}$ if $v_n\neq 0$.                                   \QED 

A simpler lowering operator is
\be \widehat{\hphi} := \bigl(T_{n-1}\cdots T_1\, \tau -t_1^{n-1}\bigr)
\frac{1}{x_n}\, \bt \ .
\end{equation}

\medskip 
Since the eigenvalues of $M_v$ under the $\xi_j$'s are permuted
of those of $M_{vs_i}$, 
any linear combination of $M_v$, where the $v$'s are in the orbit under $\S_n$,
is an eigenfunction of any symmetric function in the $\xi_j$. In particular, 
let $\Xi:= \xi_1+\cdots +\xi_n$. One has the following theorem
(\cite{mac3}, \cite{cher3}).

\begin{theorem}
 For each partition $\l \in \N^n$, there exists a unique 
polynomial $M_\l^\S$, symmetrical  in $x_1,\ldots, x_n$, 
whose expansion in the basis of Schur functions 
is of the type 
$M_\l^\S = S_\l + \sum_{\mu <\l} c_\l^\mu S_\mu$ 
and  which is an eigenfunction of $\Xi$
(the sum being over all partitions smaller than $\l$ 
with respect to the dominance order).

Given any $v$ in the $\S_n$-orbit of $\l$, then 
up to multiplication by a scalar (we write $\dot{=}$),  one has 
\begin{equation*}
 M^\S_\l \dot{=} M_v \sum_{w\in \S_n} (-t_2)^{\binom{n}{2}-\ell(w)}T_w = 
M_v  \prod_{1\leq i<j<\leq n} (t_1x_i +t_2x_j) \,
\d_\omega = M_v\, \square_\omega \ . 
\end{equation*}
\end{theorem}

\proof Since $\Xi$  commutes with $\square_\omega$, $M_v\, \square_\omega$
 is a symmetric eigenfunction of $\Xi$ with eigenvalue 
$ q^{\l_1}t_1^{n-1} + \cdots + q^{\l_n} (-t_2)^{n-1}$ which 
characterizes the partition $\l$.  
One can check that the $\xi_i$'s have a triangular action on polynomials
by putting an appropriate order on monomials, related to the Bruhat order
on the affine symmetric group. We refer for that point to \cite{KS2} and
\cite{knop1}. Thus $M^\S_\l$ is not only characterized by an eigenvalue,
but also by a leading term.   \QED

\smallskip
The first polynomials, in the basis of Schur functions  are, 
for two variables
\begin{gather}
M_{0,0}^\S=1,\, M_{1,0}^\S= S_1,\, 
M_{2,0}^\S \dot{=} (qt_1+t_2)S_2 +(qt_2+t_1)S_{1,1},\, 
M_{1,1}^\S= S_{1,1} \notag\\
 M_{3,0}^S \dot{=} (t_2+q^2t_1) S_3 - (q+1)(t_1+qt_2) S_{2,1}, \notag 
\end{gather}
\begin {multline} 
M_{4,0}^\S \dot{=}  (t_2+q^3t_1)(t_2+q^2t_1) S_4  \\
 + (1+q+q^2)(t_2+q^2t_1)(t_1+qt_2) S_{3,1}
+(q+q^3)(t_1+t_2)(t_1+qt_2) S_{2,2} \notag
\end{multline}

\medskip
From the expression of the $\xi_i$'s one sees that all 
$\square_\omega\, \xi_i\, \square_\omega$, and thus also 
$\square_\omega\, \Xi\, \square_\omega$, $\square_\omega\, \Xi$,
 are proportional to $\square_\omega\, \bt\, \square_\omega\ $.
But now,  $\square_\omega$ acts as a scalar on symmetric function, and 
therefore the action of $\Xi (1\, \square_\omega)$ 
on symmetric functions is just
\begin{multline}
 f\mapsto f\,\tau\, \square_\omega  =
  f\,\tau\, \prod_{1\leq i<j<\leq n} (t_1x_i +t_2x_j) \, \d_\omega \\
  = f(x_2,\ldots, x_n, qx_1)\, \prod_{1\leq i<j<\leq n} (t_1x_i +t_2x_j) \,
\d_\omega \ ,
\end{multline}
which is the usual expression of the  
Debiard-Sekiguchi-Macdonald operator. 

Creation (or raising) operators on symmetric 
Macdonald polynomials are similar operators
(\cite{LLM1})
$$ B_k:= \bigl(t_1\tau -(-t_2)\bigr)\cdots \bigl(
t_1^k\tau-(-t_2)^k\bigr) \, x_1\cdots x_k\, \square_\omega \ .$$
The image of $M_\l^{\S}$, when $\ell(\l)\leq k$, is proportional
to  $M_{\l+1^k}^{\S}$ .
Other creation operators are given in \cite{LV2}, \cite{Kajihara},
 \cite{KN1}, \cite{KN2}. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\large Vanishing Macdonald polynomials}

Take Yang-Baxter graph with
\begin{gather} 
\ss_i:= T_i \ ,\ 1\leq i\leq n-1 \ ,\ c_i(\ket)= t_1^{n-i} (-t_2)^{i-1} 
\ , \ 1\leq i\leq n \ ,  G_n=x_n -(-t_2)^{n-1} \ , \notag\\
 x_{i+n}= qx_i \ , \ c_{i+n}(\ket)= q c_i(\ket) \ ,\ \forall i \in  \Z
\label{eq:van1}
\end{gather}


\begin{Def}
 The {\em vanishing  Macdonald polynomial} $\MM_v$,  $v\in \N^n$,
 is the image of $1$
 under the element $Y_v$ associated to the Yang-Baxter graph
satisfying (\ref{eq:van1}):
$$\MM_v:= 1\, Y_v \ . $$
\end{Def}

The appropriate Cherednik's elements are  \cite{knop2}
$$ \xx_i := T_{i-1}\cdots T_1\, \tau
\bigl( 1- \frac{(-t_2)^{n-1}}{x_n} \bigr)\bT_{n-1}\cdots \bT_i
+ \frac{(-t_1t_2)^{n-1}}{x_i} ,\ 1\leq i\leq n\ $$

\begin{theorem}
 Let $v\in N^n$, and $k=|v|$. Let $\Pol(v)$ be the 
subspace of polynomials $f$
of degree $\leq |v|$ such that 
 $$\forall u\in \N^n,\ |u|\leq |v|,\ u\neq v 
                               \Rightarrow f(c(u))=0 \ .$$
Then $\Pol(v)$ is 1-dimensional and contains  $\MM_v$.

The polynomial $\MM_v$ is an eigenfunction of the $\xx_i$'s 
with eigenvalues  
$$(-t_1t_2)^{n-1}/c_1(v),\ldots, (-t_1t_2)^{n-1}/c_n(v) \ .$$
The component of top degree of $\MM_v$ is $M_v$. 
\end{theorem}


The proof is the same as for Macdonald polynomials for what concerns
eigenfunctions. One checks that $1$ is an eigenfunction of the
$\xx_i$'s with eigenvalues
$(-t_2)^{n-1}, (-t_2)^{n-2}t_1,\ldots, t_1^{n-1}$, because
$1T_i=t_1$, $1\bT_i =-t_2$. 
On the other hand, it is not difficult to 
see that $\Pol(v)$ is 1-dimensional. 
The choice 
of $G_n= x_n -(-t_2)^{n-1}$ ensures that vanishing propagates along 
an affine edge. 

Restricting $G_n$ to its top-degree component $x_n$ gives $M_v$
instead of $\MM_v$.


In the case $n=2$, the first polynomials, and the vectors $c(v)$ are 
\setbox2=\hbox{${\stackrel{\Phi}{\hbox to 5mm{\rightarrowfill}}} $}
\setbox3=\hbox{${\stackrel{\s}{\hbox to 5mm{\rightarrowfill}}} $}
$$\petitematrice{[0,0]\cr [t_1,-t_2]\cr 1} \copy2 
\petitematrice{[0,1]\cr [-t_2,qt_1]\cr x_{2}+t_2}  \copy3
\petitematrice{[1,0]\cr [qt_1,-t_2]\cr 
\moins {({t_1} \plus  {t_2})t_2(x_{2}+t_2)\over
q{t_1}+t_2}\moins t_2(x_{1}\moins {t_1 })} \copy2
 \petitematrice{[0,2]\cr [-t_2,q^2t_1]\cr
t_2(x_{2}\plus {t_2})( t_1\moins {x_2\over q}
\moins {t_1+t_2\over qt_1+t_2}
(x_1\plus t_2))  }
$$

Given any $v\in \N^n$, there is a unique symmetric polynomial
(up to normalization) in the linear span of $\MM_{v\, w}$, $w\in \S_n$.
This polynomial is obtained as the image of $\MM_{v}$
under 
$$\square_\omega =
\sum_{w\in \S_n} (-t_2)^{\binom{n}{2}-\ell(w)} T_w
=  \prod_{1\leq i<j<\leq n} (t_1x_i +t_2x_j) \, \d_{\omega} $$
 and clearly inherits from $\MM_{v}$  
vanishing and eigenfunction properties, which are stated in the
next theorem.

\begin{theorem}
Let $n$ be an integer, $\l\in \N^n$ be a partition, $v$ any vector in its
orbit under $S_n$. Then 
$$ \MMS_v:=  \MM_v\, \prod_{1\leq i<j<\leq n} (t_1x_i +t_2x_j) \,
\d_{\omega} $$
is, up to a scalar, the unique symmetric polynomial vanishing in all 
$c(\mu)$, $\mu$ partition $\in \N^n$, $|\mu|\leq |\l|$, $\mu \neq \l$.

It is also the unique eigenfunction of $\xx_1+\cdots +\xx_n$
with eigenvalue  
$$ (-t_2)^{n-1} q^{-\l_1} + (-t_2)^{n-2}t_1 q^{-\l_2} + \cdots 
+ t_1^{n-1} q^{-\l_n} \ .$$
\end{theorem}
 
 
For $n=2$, one has the following polynomials,
 in terms of Schur functions:
$$\MMS_{0,0}= 1 \ , \ \MMS_{1,0}= S_1 +t_2-t_1\ , \ \MMS_{1,1} = 
S_{1,1} +t_2S_1+t_2^2 \ , $$ 
$$ \MMS_{2,0} \dot{=} (qt_1+t_2) S_2 + (qt_2+t_1) S_{1,1} + 
(q+1) (t_2^2-qt_1^2)S_1 + q(t_1-t_2)(qt_1^2-t_2^2)\ . $$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\large Jack polynomials} 

Take Yang-Baxter graph with
\begin{gather}
\ss_i:=s_i \ ,\ 1\leq i\leq n-1 \ ,\ c_i(\ket)= 1-i 
\ , \ 1\leq i\leq n \, ,\,  G_n=x_n \ , \notag \\  
 x_{i+n}=x_i \ , \ c_{i+n}(\ket)= c_i(\ket) +\a \ ,\ \forall i \in  \Z
\label{eq:jac1}
\end{gather}
 
\begin{Def}
 The {\em non symmetric Jack polynomial} $J_v$,  $v\in \N^n$,  
 is the image of $1$ under the Yang-Baxter element $Y_v$ given by the 
 graph satisfying (\ref{eq:jac1}):   
$$J_v:= 1\, Y_v \ . $$
\end{Def}

For example, for $n=2$, one has the following sequence of polynomials,
writing  $v, c(v), J_v$ on top of each other as before, and writing 
$\ss$ for $\ss_1=s_1$:

\setbox2=\hbox{${\stackrel{\Phi}{\hbox to 5mm{\rightarrowfill}}} $}
\setbox3=\hbox{${\stackrel{\ss}{\hbox to 5mm{\rightarrowfill}}} $}
$$\petitematrice{[0,0]\cr [0,\moins 1]\cr 1} \copy2 
\petitematrice{[0,1]\cr [\moins 1,\a]\cr x_2} \copy3
\petitematrice{[1,0]\cr [\a,\moins 1]\cr x_1+{x_2\over 1+\a}} \copy2 
 \petitematrice{[0,2]\cr 
[\moins 1,2\a]\cr x_2(x_2+{x_1\over 1+\a})} \copy3
 \petitematrice{[2,0]\cr [2\a,\moins 1]\cr 
x_1^2+{2x_1x_2+x_2^2\over 1+2\a}} \copy2 \cdots
$$

The symmetric polynomials are obtained, up to  normalization, as images 
of non-symmetrical ones under
$\Delta(X)\d_\omega= \sum_{w\in\S_n} w$, i.e. for any partition $\l\in \N$,
any $v$ in the orbit of $\l$ under $\S_n$, one has 
$$ J_\l^{\S} = J_v\, \Delta(X)\, \d_\omega = \sum_{w\in\S_n}
    \bigl(J_v \bigr)^w  \ .$$

One checks that 1 is an eigenvector of the Cherednik elements
(defined in (\ref{eq:cher11}))  
\be   \xi_i := \a D_i +\sum_{j\neq i,j=1}^n \bp_{i,j} +1-i \ , 
\ 1\leq i\leq n \ ,  \notag
\end{equation}
 with eigenvalues $0,\moins 1,\ldots, \moins n\plus 1$.
These values are our initial spectral parameters, and therefore
the commutation relations (\ref{eq:cher13})
 imply the following theorem due to Opdam
( \cite{opdam}, \cite{KS2}).

\begin{theorem} For any  $v\in \N^n$, the polynomial $J_v$ is a simultaneous
eigenfunction of $\xi_1,\ldots, \xi_n$, with eigenvalues $c_1(v),\ldots,
c_n(v)$.  

For any partition $\l\in \N^n$, and any symmetric function
$f\in \Sym(n)$, the Jack polynomial $J_\l^{\S}$ is an eigenvector of
$f(\xi_1,\ldots, \xi_n)$ with eigenvalue 
$f(c_1(\l),\ldots, c_n(\l))$.
\end{theorem}

Baker and Forrester \cite{forr97a}   give another symmetrization,
using Sahi's scalar product. Introduce another set of variables
$Y=\{y_1,\ldots, y_n\}$.  Then, according to Sahi \cite{sahi2}, the kernel
$$ \Omega(X,Y):=  \prod_{i=1}^n (1-x_iy_i)^{-1} \prod_{i=1}^n\prod_{j=1}^n 
   (1-x_iy_j)^{-1/\a}$$
diagonalizes in the basis of Jack polynomials: there exist non-zero constants
$m_v$ such that
$$ \Omega(X,Y)= \sum_{v\in \N^n} m_v J_v(X) J_v(Y) \ . $$

Recall that, thanks to the Cauchy determinant,  
$$ \prod_{i=1}^n (1-x_iy_i)^{-1} \d_\omega^X = \Delta(Y) \prod_{i,j}
  (1-x_iy_j)^{-1} \ ;$$
this last function is sent, 
under $\d_\omega^Y$ onto $n!\prod_{i,j} (1-x_iy_j)^{-1}$. 

Therefore, the image of $\Omega(X,Y)$ under $\d_\omega^X\d_\omega^Y$ is
$n! \prod_{i,j} (1-x_iy_j)^{-1 -1/\a}$. This last function diagonalizes
in the basis of symmetric Jack polynomials, 
but this time for the parameter $1/(1+1/\a) = \a/(1+\a)$.

Therefore, all $J_{vw} \d_\omega$, for a fixed $v\in \N^n$, coincide, up
to non zero-scalars, and are proportional to the same symmetric Jack
polynomial ( one has to check that the polynomial of index a permutation
of $[\l_1\plus n\moins 1,\ldots, \l_n\plus 0]$ 
is sent to the polynomial of index
the (decreasing) partition $\l$. As for normalization, one can check that, 
for  a weakly increasing $u\in \N^n$,
$$ J_{u_1,u_2+1,\ldots, u_n+n-1} \d_\omega= 
  \sum\nolimits^{'}_{w} (J_u)^w \ ,$$
where the sum on the right is over cosets representatives of $\S_n$ 
modulo the stabilizer of $u$, and the polynomial $J_u$ is for the parameter
$\a/(1+\a)$. 
For example,
$$ J_{014;\, \a} \d_{321} = J_{002;\, \a/(1+\a)} (1+s_2+s_2s_1)  \ .$$


\medskip
Knop and Sahi \cite{KS2} show that symmetric Jack polynomials can be
obtained through still another symmetrization~:

\begin{lemma}
Let $\l$ be a partition of length $k\leq n$, and $v$ be any permutation 
of $[\l_1-1,\ldots, \l_k-1, 0^{n-k}]$. Then 
$$ J_\l^{\S} = \sum_{w\in\S_n} \bigl(J_v\, x_1\cdots x_k \bigr)^w \ . $$ 
\end{lemma}
\proof  
$$ J_{0^{n-k},\l_1,\ldots, \l_k} = J_{\l_1-1,\ldots, \l_k-1, 0^{n-k}}\,
\tau^k\, x_n\cdots x_{n-k+1} = J_{\l_1-1,\ldots, \l_k-1, 0^{n-k}}\
 x_k\cdots x_1 \, \tau^k .$$
The image of the left hand-side under $\sum w$ is $J_\l^{\S}$.
However, 
$$ \sum_{w\in \S_k\times \S_{n-k}}  (J_{\l_1-1,\ldots, \l_k-1, 0^{n-k}})^w 
= \sum_{w\in \S_k\times \S_{n-k}}  (J_v)^w $$
and this allows to replace $J_{\l_1-1,\ldots, \l_k-1, 0^{n-k}}$
by $J_v$ in the summation on $\S_n$. \QED 


There is an abundant literature about symmetric Jack polynomials
(cf. the references given in \cite{mac}). They can be obtained
through ``creation operators'' \cite{LV1}.  
Their simplest explicit expression seems to be a determinant
given in \cite{LLM2}.
They are eigenfunctions of the {\it Sekiguchi operator}
$$ \bigl( D_1 +(n\moins 1)\b +t\bigr) \bigl( D_2 +(n\moins 2)\b +t\bigr) 
\cdots \bigl( D_n +0\b +t\bigr)\, x_1^{n-1}\cdots x_n^0\, \d_\omega \ ,$$
$D_i$ still being $f\mapsto x_i \frac{d}{dx_i}(f)$,
and $\b=1/\a$.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\large Vanishing Jack polynomials} 


Take Yang-Baxter graph with
\begin{gather}
\ss_i:=s_i+\d_i \ ,\ 1\leq i\leq n-1 \ ,\ c_i(\ket)=1-i 
\ , \ 1\leq i\leq n \ ,  G_n=x_n +n-1 \ , \notag\\
 x_{i+n}=x_i +\a \ , \ c_{i+n}(\ket)= c_i(\ket) +\a \ ,\ \forall i \in  \Z
\label{eq:jav1}
\end{gather}
and let the Cherednik algebra be generated by the 
\be \xx_i := x_i - \ss_{i-1}\cdots \ss_1\, \tau\, (x_n\plus n\moins 1)\, 
  \ss_{n-1}\cdots \ss_i \ , \ 1\leq i\leq n \ .
\end{equation}

\begin{Def}
 The {\em vanishing  Jack polynomial} $\JJ_v$,  $v\in \N^n$,
 is the image of $1$ under the Yang-Baxter element $Y_v$ 
associated to graph satisfying (\ref{eq:jav1}):
$$\JJ_v:= 1\, Y_v \ . $$
\end{Def}

For example, for $n=2$, one has the following sequence of polynomials~:
$$\kern -10pt
\petitematrice{[0,0]\cr [0,\moins 1]\cr 1} \copy2 
\petitematrice{[0,1]\cr [\moins 1,\a]\cr x_2\plus 1} \copy3
\petitematrice{[1,0]\cr [\a,\moins 1]\cr x_1\plus {x_2+1\over 1+\a}} \copy2
 \petitematrice{[0,2]\cr [-1,2\a]\cr (x_2\plus 1)
(x_2\moins \a\plus {x_1+1\over 1+\a})} \copy3
 \petitematrice{[2,0]\cr [2\a,\moins 1]\cr x_1(x_1\moins \a)\plus  
{ (x_2+1)(2x_1+x_2+1-3\a)\over 1+2\a}} 
$$

Knop and Sahi \cite{KS1}
 have  shown that these polynomials can be characterized by vanishing 
conditions. Let $\Pol_k$ be the space of polynomials in $x_1,\ldots, x_n$
of degree $\leq k$. 

\begin{theorem}
 Let $v\in N^n$, and $k=|v|$. Let $\Pol(v)$ be the
subspace of polynomials $f$
of degree $\leq |v|$ such that
 $$\forall u\in \N^n,\ |u|\leq |v|,\ u\neq v
                               \Rightarrow f(c(u))=0 \ .$$
Then $\Pol(v)$ is 1-dimensional and contains  $\JJ_v$.

The polynomial $\JJ_v$ is an eigenfunction of the $\xx_i$'s
with eigenvalues $c(v)$.

The restriction of $\JJ_v$ to its component of top degree
( $=|v|$) is equal to $J_v$.
\end{theorem}

\proof  The dimension of $\Pol_k$ is equal to the number of equations $+1$,
this accounts for the existence of a vanishing polynomial, 
unicity follows by putting an appropriate order
on monomials.   Moreover,
restricting $s_i +\d_i$ and $G_n= x_n +n-1$ to their terms of highest degree
shows that the leading term of $\JJ_v$ is $J_v$.

The labels on the edges of cells have been chosen to propagate
vanishing conditions, at the same time as eigenvector properties. 
The choice of $G=x_n+n-1$ insures that 
for any $f$, $f\Phi$ vanishes in $c(v)$ for any $v$ such that 
$v_n=0$ (in which case $c_n(v)=1-n$).

Because of the commutation relations (\ref{eq:cher14}), 
it is sufficient to test
that $1$ is an eigenfunction for $\xx_n$, with the right eigenvalue, 
to get that all $\xx_i$ diagonalize.
Indeed 
$$ 1\, \xx_n= x_n - 1\ss_{n-1}\cdots \ss_1\, \tau\, (x_n+n-1) = -n+1\ .$$
\QED 


\smallskip
Given any $v\in \N^n$, the linear span of $\JJ_{vw},\, w\in\S_n$ 
contains, up to  normalization, a unique symmetric polynomial (\ref{eq:sym7}) 
$$ \JJ_v^{\S} :=  \sum_{w\in \S_n} \JJ_v\, \ss_w =
\JJ_v \prod_{1\leq i,j\leq n} (1+x_i-x_j)\, \d_\omega $$
(Sahi and Knop give another
symmetrization), and this symmetric polynomial inherits from $\JJ_v$
the properties stated by the following theorem~:

\begin{theorem}
 Let $n$ be an integer, $\l\in \N^n$ be a partition, $v$ any vector in its
orbit under $\S_n$ and let $\g$ be an indeterminate.  Then
$$ \JJ^\S_v:=  \JJ_v\, \prod_{1\leq i<j<\leq n} (1+x_i-x_j) \,
\d_\omega $$
is, up to a scalar, the unique symmetric polynomial vanishing in all
$c(u)$, $u\in \N^n$, $|u|\leq |v|$, $u$ not a permutation of $v$.

It is also the unique eigenfunction of $(\g +\xx_1) \cdots (\g+ \xx_n)$ with
eigenvalue 
$$ (\g+\a\l_1 -0)(\g + \a\l_2-1) \cdots (\g+ \a\l_n -n\plus 1) \ .$$
\end{theorem}

For $n=2$ the first polynomials, in the basis of Schur functions, are 
$$\JJ^\S_{0,0}= 1 \ , \JJ^\S_{1,0}= S_1 +1\ , \ 
\JJ^\S_{1,1} = S_{1,1} +S_1+ 1 \ , $$
$$ \JJ^\S_{2,0} = (\a+1)S_2 +(1-\a)S_{1,1} +((1-\a)(2+\a)S_1 -(\a^2 +2\a-1) $$
\begin{multline}
 \JJ^\S_{3,0}=
 (2\a+1)S_{3} -2(\a-1)S_{2, 1}-3(\a+1)(2\a-1)S_2 +3(2\a-1)(\a-1)S_{1, 1}
 \\
 +(-12\a+4\a^{3}+2\a^{2}+3)S_{1} +(2\a-1)(2\a^{2}+5\a-1)\ . \notag
\end{multline}

The symmetric Jack polynomials specialize, for $\a=1$, into Schur functions. 
Similarly, vanishing symmetric Jack polynomials specialize into some 
Schubert polynomials. Indeed, Schubert polynomials are polynomials 
in two sets of variables, indexed 
by permutations, or by integral vectors (which are the codes of the
permutations), which have  vanishing properties related to the Bruhat order
\cite{LS8}.
In the special case that we need now, let $\l\in \N^n$ be a partition and 
$X_\l$ be the Schubert polynomial in $x_1,\ldots, x_n$, $y_1,\ldots,
y_\infty$, of index the increasing reordering of $\l$ 
(it is  a {\it Grassmannian Schubert polynomial} \cite{mac2}). 
Then $X_\l$ vanishes in every  partition $\mu\in \N$: $|\mu|\leq |\l|$,
$\mu\neq \l$, i.e $X_\l$, evaluated in   
$$ x_1=\mu_1+n-1,\ldots, x_n=\mu_n+0  \quad , \quad 
                   y_1=0,y_2=1,y_3=2,\ldots, y_i=i\moins 1, \ldots $$
is null. 
Therefore, one has the following lemma (stated by Knop and Sahi \cite{KS1}
in terms of {\it factorial Schur functions}; in the literature,
 $X_\l$ is rather denoted
 ${\mathbb Y}_{\l_n,\ldots,\l_1}$).

\begin{lemma} For any partition $\l\in \N^n$, the specialization $\a$=1 of 
$\JJ^\S_\l$ is equal to the  Schubert polynomial
$X_\l$ evaluated in $\{x_1+n-1,\ldots, x_n+n-1\}$, 
$\{0,1,2,\ldots \}$. 
\end{lemma}


\bibliographystyle{plain}
\begin{thebibliography}{10}

\bibitem{forr97a}
T.~H. Baker and P.~J. Forrester.
\newblock Symmetric {Jack} polynomials from non-symmetric theory.
\newblock {\em Annals of Combinatorics}, {\bf 3} (1999) 159--170. 

\bibitem{forr97b}
T.~H. Baker and P.~J. Forrester.
\newblock A $q$-analogue of the type {$A$} {Dunkl} operator and integral
  kernel.
\newblock {\em Int. Math. Res. Not.}, {\bf 14}(1997) 667--686.

\bibitem{forr00}
T.~H. Baker and P.~J. Forrester.
\newblock Isomorphisms of type $A$ affine Hecke algebras and multivariate
orthogonal polynomials.
\newblock {\em Pacific J.M.}, {\bf 194} (2000) 19--41. 

\bibitem{BGG}
I.N.~Bernstein, I.M.~Gelfand, S.~Gelfand.
\newblock{ Schubert cells and cohomology of the spaces $G/P$ }, 
\newblock{Russ. Math. Survey}, {\bf28} (1973) 1--26.

\bibitem{cher1}
I.~Cherednik.
\newblock A unification of the {Knizhnik--Zamolodchikov} and {Dunkl} operators
  via affine {Hecke} algebras.
\newblock {\em Inv. Math.}, {\bf 106}(1991)   411--432.

\bibitem{cher2}
I.~Cherednik.
\newblock Integration of quantum many--body problems by affine
  {Knizhnik--Zamolodchikov} equations.
\newblock {\em Commun. Math. Phys.}, {\bf 106} (1994) 65--95.

\bibitem{cher3}
I.~Cherednik.
\newblock Non-symmetric {Macdonald} polynomials.
\newblock {\em Int. Math. Res. Not.}, {\bf 10} (1995) 483--515.

\bibitem{Dem}
M.~Demazure.
\newblock Une formule des caract\`eres , 
\newblock {\em Bull. Sc. Math.}, {\bf 98} (1974) 163--172.

\bibitem{DKLLST}
 G.~Duchamp, D.~Krob, A.~Lascoux, B.~Leclerc, T.~Scharf, J.Y.~Thibon.
\newblock{  Euler-Poincar\'e characteristics and polynomial
representations of Iwahori-Hecke algebras}
\newblock{\em  Pub. RIMS  (Kyoto)}, {\bf 31} (1995) 179--201. 


\bibitem{Dunkl}
C.~F. Dunkl.
\newblock Differential-difference operators associated to reflection groups.
\newblock {\em Trans. Amer. Math. Soc.}, {\bf 311} (1989) 167--183.

\bibitem{FK}
 S. Fomin and A. Kirillov.
\newblock {The Yang-Baxter equation, symmetric functions and Schubert
polynomials},
\newblock{\em     Discrete Math.}, {\bf 153} (1996) 123--143.

\bibitem{Hi}
F.~Hirzebruch. 
\newblock{\em Gesammelte Abhandlungen},
\newblock Springer (1987).

\bibitem{Kaneko}
J.~Kaneko.
\newblock {$q$-Selberg} integrals and {Macdonald} polynomials.
\newblock {\em Ann. Sci. \'Ec. Norm. Sup. $4^e$ s\'erie}, {\bf 29}
(1996) 1086--1110.

\bibitem{Kajihara}
Y. Kajihara and M. Noumi
\newblock {Raising operators of row type for Macdonald polynomials}
\newblock {\em Compositio Math.}, {\bf 120} (2000) 119--136. 

\bibitem{KN1}
A. Kirillov and M. Noumi.
\newblock {Affine Hecke algebra and raising operators for
Macdonald polynomials},
\newblock {\em Duke Math. J.}, {\bf 93} (1998) 1--39.

\bibitem{KN2}
A. Kirillov and M. Noumi.
\newblock { q-difference raising operators for Macdonald polynomials
and the integrality of transition coefficients},
\newblock {\em CRM Proceedings and Lect. Notes}, {\bf 22} (1999) 227--243.

\bibitem{knop1}
F.~Knop.
\newblock Integrality of two variable {Kostka} functions.
\newblock {\em J. Reine Ang. Math.}, {\bf 482} (1997) 177--189.

\bibitem{knop2}
F.~Knop.
\newblock Symmetric and non-symmetric quantum
Capelli polynomials. 
\newblock {\em Comment. Math. Helvet.}, {\bf 72} (1997) 84--100.

\bibitem{KS1}
F.~Knop and S.~Sahi.
\newblock Difference equations and symmetric polynomials defined by their
  zeros.
\newblock {\em Int. Math. Res. Not.}, {\bf 10} (1996) 473--486.

\bibitem{KS2}
F.~Knop and S.~Sahi.
\newblock A recursion and combinatorial formula for {Jack} polynomials.
\newblock {\em Inv. Math.}, {\bf 128} (1997) 9--22.

\bibitem{La77}
A. Lascoux.
\newblock Polyn\^omes sym\'etriques, Foncteurs de Schur et Grassmanniennes. 
\newblock{\em  Th\`ese, Universit\'e Paris 7} (1977).

\bibitem{La}
A. Lascoux.
\newblock About the "$y$" in the $\chi\sb y$-characteristic of Hirzebruch.
\newblock{\em Algebraic geometry: Hirzebruch 70 (Warsaw, 1998),
Contemp. Math.}, {\bf 241}(1999) 285--296.

\bibitem{LLM1} 
L.~Lapointe, A.~Lascoux, J.~Morse.
\newblock{ Determinantal expressions for Macdonald polynomials}
\newblock{\em Int. Math. Res. Notices}, {\bf 18} (1998) 957--978.

\bibitem{LLM2}
L.~Lapointe, A.~Lascoux, J.~Morse.
\newblock{Determinantal expressions for Jack polynomials},
\newblock{\em   Electronic Journal of Combinatorics}, {\bf 7} 1n (2000).

\bibitem{LV1}
L.~Lapointe, L.~Vinet.
\newblock A Rodrigues formula for the Jack polynomials and Stanley conjecture,
\newblock {\em Int. Math. Res. Notices} {\bf9} (1995) 419--424. 

\bibitem{LV2}
L.~Lapointe, L.~Vinet.
\newblock{Creation operators for the Macdonald and Jack polynomials}
\newblock{\em Letters Math. Phys.}, {\bf 40} (1997) 269--286. 


\bibitem{LLT1}
A. Lascoux, B. Leclerc and J.Y. Thibon.
\newblock {Twisted Action of the symmetric group on the cohomology
of a flag manifold}, 
\newblock{\em Banach Center Publications}, {\bf 36} (1996) 111--124.

\bibitem{LLT2}
A. Lascoux, B. Leclerc and J.Y. Thibon.
\newblock { Flag Varieties and the Yang-Baxter Equation },
\newblock{\em Letters in Math. Phys.}, {\bf 40} (1997) 75--90.

\bibitem{LS2}
A. Lascoux \& M.P. Sch\"utzenberger.
\newblock { Symmetry and Flag manifolds},
\newblock {\em Invariant Theory, Springer L.N.}, {\bf 996} (1983) {118--144}.

\bibitem{LS5}
A. Lascoux \& M.P. Sch\"utzenberger.
\newblock { Symmetrization operators on polynomial rings},
\newblock{\em Funct.Anal.}, {\bf 21} (1987) {77--78}.

\bibitem{LS6}
A. Lascoux \& M.P. Sch\"utzenberger.
\newblock { Fonctorialit\'e des polyn\^omes de Schubert}
\newblock{\em in  Invariant Theory,
Contemporary Maths}, {\bf 88} (1989) 585--598.

\bibitem{LS8}
A. Lascoux \& M.P. Sch\"utzenberger.
\newblock { Alg\`ebre des diff\'erences divis\'ees},
\newblock{\em Discrete Maths}, {\bf  99} (1992) {165--179}.

\bibitem{Lu} 
G. Lusztig.
\newblock Affine Hecke algebras and their graded version.
\newblock {\em J. Amer. Math. Soc.}, {\bf 2} (1989) 599-685.

\bibitem{mac2}
I.~G. Macdonald.
\newblock {\em Notes on Schubert polynomials}
\newblock LACIM, Publi. Universit\'e Montr\'eal, (1991).

\bibitem{mac}
I.~G. Macdonald.
\newblock {\em Symmetric functions and {Hall} polynomials}.
\newblock Oxford University Press, Oxford, 2nd edition (1995).

\bibitem{mac3}
I.~G. Macdonald.
\newblock Affine {Hecke} algebras and orthogonal polynomials.
\newblock {\em S\'eminaire Bourbaki, 47\`eme ann\'ee}, {\bf 797} (1995).



\bibitem{MN}
K.~Mimachi and M.~Noumi.
\newblock A reproducing kernel for nonsymmetric {Macdonald} polynomials.
\newblock q-alg/9610014.

\bibitem{OV}
 A.~Okounkov,  A.~Vershik,  
\newblock A new approach to representation
theory of symmetric groups, 
\newblock{\em Selecta Math.}, {\bf 2} (1996)  581--605.

\bibitem{opdam}
E.~M. Opdam.
\newblock Harmonic analysis for certain representations of graded {Hecke}
  algebras.
\newblock {\em Acta Math.}, {\bf 175} (1995) 75--121.

\bibitem{sahi1}
S.~Sahi.
\newblock Interpolation, integrality and a generalization of Macdonald's
polynomials.
\newblock {\em Int. Math. Res. Not.}, {\bf 10} (1996) 457--471.

\bibitem{sahi2}
S.~Sahi.
\newblock A new scalar product for nonsymmetric {Jack} polynomials.
\newblock {\em Int. Math. Res. Not.}, {\bf 20} (1996) 997--1004.

\bibitem{sahi97a}
S.~Sahi.
\newblock The binomial formula for nonsymmetric {Macdonald} polynomials.
\newblock q-alg/9703024.

\bibitem{Vershik}
A. Vershik.
\newblock Local algebras and a new version of Young's orthonormal form
\newblock{\em Topics in Algebra, Banach Center Pub.}, {\bf 26} (1990)
467--473. 

\bibitem{Ya} 
C.N.~Yang.
\newblock Some exact results for the many-body
problem in one dimension with repulsive delta-function interaction,
\newblock Phys. Rev. Lett., {\bf 19} (1967), 1312-1315.

\bibitem{Yo} 
A.~Young.
\newblock {\em The Collected Papers of Alfred Young},
\newblock  University of Toronto Press (1977). 
\end{thebibliography}


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