% Alain Lascoux
% Square-ice enumeration                   Modif 23 mai 99
% Seminaire Lotharingien vol. 42
% Plain Tex
% output 15 pages
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\auteurcourant={ALAIN LASCOUX}
\titrecourant={SQUARE-ICE ENUMERATION}

\vglue2cm
\centerline{SQUARE-ICE ENUMERATION}
\bigskip
\centerline{\bf Alain Lascoux}
\bigskip
\centerline{\sl Dedicated to George Andrews on the occasion
of his sixtieth birthday}
\bigskip
\abstract{Starting with plane partitions possessing certain type of
symmetries, many combinatorial objects came to the fore, the
enumeration of which was the subject of intensive studies during the
last twenty years, with of course, seminal contributions of George
Andrews. Thanks to a detour through two-dimensional ice models,
algebraic computations cristallised to the description of a certain
determinant of Cauchy type.  Dividing this determinant by some
straightforward factors, one is reduced to studying a symmetric
polynomial in two sets of variables.  We show how to
separate the variables with the help of divided differences, and
obtain the desired symmetric function as a product of two
rectangular matrices, each of them involving only one set of
variables. In the same run, we reduce the dimension by 1
and factorize the determinant  associated to the Bethe model of a
1-dimensional gas of bosons.}
\bigskip

\section 1. A combinatorial promenade|We first evoke some different
combinatorial objects related to the algebraic computations of
section~2.

The following figures appear in a forthcoming book by Bressoud [Br].
The elementary pieces (shown in Fig.~1) are the different models of
planar frozen water mol\-e\-cules.
$$
H-O-H\qquad
\petitematrice{H\cr
\trait\cr
O\cr
\trait\cr
H\cr}\qquad
\petitematrice{H&$--$&O\cr
&&\trait\cr
&&H\cr}\qquad
\petitematrice{&&H\cr
&&\trait\cr
H&$--$&O\cr
}\qquad
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\trait\cr
H\cr}\qquad
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\trait\cr
O&$--$&H\cr}
$$

\centerline{Fig. 1}

\bigskip
In Fig.~2 we have represented a display of those frozen water
molecules on a square grid. Keeping the oxygens and replacing each
hydrogen atom by an arrow pointing toward the oxygen to which it is
attached transforms Fig.~2 into an oriented graph shown in Fig.~3.
(One has added a top and bottom row of arrows, so that each vertex
has two incoming arrows and two outgoing ones. The ice becomes
electrically balanced.)

$$
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&&&&&&&&&&\trait&&&&&&&&&&\cr
&&H&&&&H&&&&H&&&&H&&&&H&&\cr
&&\trait&&&&\trait&&&&&&&&\trait&&&&\trait&&\cr
H&\tir&O&&H&\tir&O&&H&\tir&O&\tir
&H&&O&\tir&H&&O&\tir&H\cr}
$$

\centerline{Fig. 2}

\bigskip
\def\rarr{\rightarrow}\def \larr{\leftarrow}
\def\uparr{\uparrow}\def\dparr{\downarrow}

$$
\matrix{
&\uparr&&\uparr&&\uparr&&\uparr&&\uparr&\cr
\rarr&\bullet&\rarr&\bullet&\larr&\bullet&\larr&\bullet
&\larr&\bullet&\larr\cr
&\uparr&&\dparr&&\uparr&&\uparr&&\uparr&\cr
\rarr&\bullet&\larr&\bullet&\rarr&\bullet&\rarr&\bullet
&\larr&\bullet&\larr\cr
&\dparr&&\uparr&&\uparr&&\dparr&&\uparr&\cr
\rarr&\bullet&\rarr&\bullet&\larr&\bullet&\larr&\bullet
&\rarr&\bullet&\larr\cr
&\dparr&&\dparr&&\uparr&&\uparr&&\dparr&\cr
\rarr&\bullet&\rarr&\bullet&\rarr&\bullet&\rarr&\bullet
&\larr&\bullet&\larr\cr
&\dparr&&\dparr&&\uparr&&\dparr&&\dparr&\cr
\rarr&\bullet&\rarr&\bullet&\rarr&\bullet&\larr&\bullet
&\larr&\bullet&\larr\cr
&\dparr&&\dparr&&\dparr&&\dparr&&\dparr&\cr}
$$

\centerline{Fig. 3}

\bigskip
Instead of drawing arrows, one can write the numbers 0, 1 or $-1$
on the vertices of the preceding grid, according to the orientations
of the arrows, the precise coding of each of the six possible water
molecules being:
$1$ for the horizontal molecules,
$-1$ for the  vertical ones, and $0$ for the others.

A matrix thereby obtained has the property that non zero entries alternate
in each
row and column, always starting and finishing with~a~$1$
({\it alternating sign matrix}); continuing with the same example
we get
$$
\pmatrix{0&1&0&0&0\cr 1&-1&0&1&0\cr 0&1&0&-1&1\cr 0&0&0&1&0\cr 0&0&1&0&0\cr}.
$$

\setbox3=\hbox{ ${\s \kern -18pt time\kern 8pt  1\kern 8pt 2 \kern 8pt
3\kern 8pt 4\kern 8pt 5}$}
\setbox4=\hbox{
$\young{ 2 &3 &5 &5 &5\cr
     \blank & 1 &3 &4 &4\cr
\blank&\blank& 1 &2 &3\cr
\blank&\blank&\blank & 1 & 2\cr
\blank&\blank&\blank &\blank &1\cr }$}

Now, interpreting the alternating sign matrix as a history, according to
Viennot's paradigm (the entry $+1$ (resp. $-1$) in row $i$, column $j$, means
that the letter $i$ appears
(resp. disappears) at time $j$), we get a
staircase (for bats) Young tableau if we write the letters
which are present at time $j$ in decreasing order
$$\vbox{\box3 \box4}$$

The foregoing tableaux satisfy the usual conditions of being weakly
increasing in rows, and strictly decreasing in columns, but also the
condition that diagonals are
weakly decreasing~; conversely any such \lq\lq monotonous triangle"
(we shall simply say {\it triangle}) corresponds to an alternating sign matrix,
as well as to some ice state.

Enumerating triangles, or more general planar partitions has been studied
by Andrews [Axx], Mills, Robbins, Rumsey [MRR1], [MRR2], the story being told
by Bressoud in his book [Br].

\smallskip
The previous model is also related with the
``Ehresmann-Bruhat order" on the symmetric group (cf. [LS2]). It
stems from the remark  that the set of triangles of
a given order is a lattice (in fact a distributive lattice): the supremum
(boxwise) of two triangles is still a triangle. Now, it is easy to see that
the generators of the lattice are exactly those triangles
having exactly one entry with the property that the triangle remains
monotonous if that entry is reduced by 1.


For example,
$$\young{ 1&5&6 &6 &6 &6 &7\cr
\blank &1 &V &5 &5 &5 &6\cr
\blank&\blank & 1 &2 &3  &4 &5\cr
\blank&\blank&\blank& 1 &2 &3 &4\cr
\blank&\blank&\blank&\blank & 1 & 2 &3\cr
\blank&\blank&\blank&\blank &\blank &1 &2\cr
\blank &\blank&\blank&\blank&\blank &\blank &1\cr }$$
is such a generator (the special entry, which is allowed to decrease,
is denoted by $V$ and yields a monotonous triangle for $V=4$).

Now, the alternating sign matrix corresponding to such a triangle
is a {\it bigrassmannian permutation} matrix~:
The bigrassmannian permutations are exactly those permutations
obtained from the identity permutation  by cutting it
into four blocks, and exchanging the two middle blocks.
For instance, flipping $234$ and $56$
$$ [1\, | \, 234\, |\, 56\, | 7]  \mapsto
 [1\, |\, 56\, |\, 234\, |\, 7]$$
yields the bigrassmannian permutation associated with the previous triangle.

Of course, such permutations can be coded by the successive lengths of the
blocks, the two middle blocks being non empty ( in the preceeding case,
 the sequence of lengths is $1,2,3,1$,
and the permutation will be denoted by  $[[1,2,3,1]]$).

Taking the boxwise order on triangles into account, one gets
the following ordered set for
the generators of the lattice of triangles of dimension~4

\bigskip
\def\segment(#1,#2)\dir(#3,#4)\long#5{%
\leftput(#1,#2){\lline(#3,#4){#5}}}

\centerline{\vbox{
\offinterlineskip
%% Third level (top)
%
\centerput(4,0){$[[0,3,1,0]]$}
\centerput(26,0){$[[0,2,2,0]]$}
\centerput(48,0){$[[0,1,3,0]]$}
%
%
%% Second level
%
\centerput(-7,-20){$[[1,2,1,0]]$}
\centerput(15,-20){$[[1,1,2,0]]$}
\centerput(37,-20){$[[0,2,1,1]]$}
\centerput(59,-20){$[[0,1,2,1]]$}
%
%% First level (bottom)
%
\centerput(4,-40){$[[2,1,1,0]]$}
\centerput(26,-40){$[[1,1,1,1]]$}
\centerput(48,-40){$[[0,1,1,2]]$}
%
%% Tracing edges.
%
\segment(0,-2)\dir(-1,-2)\long{7}
\segment(8,-2)\dir(2,-1)\long{27}
\segment(22,-2)\dir(-3,-2)\long{21}
\segment(25,-2)\dir(-1,-2)\long{7}
\segment(28,-2)\dir(2,-3)\long{9}
\segment(31,-2)\dir(2,-1)\long{27}
\segment(42,-2)\dir(-3,-2)\long{21}
\segment(52,-2)\dir(2,-3)\long{9}
%
\segment(-9,-22)\dir(2,-3)\long{9}
\segment(-5,-22)\dir(2,-1)\long{27}
\segment(13,-22)\dir(-1,-3)\long{4.5}
\segment(17,-22)\dir(2,-3)\long{9}
\segment(35,-22)\dir(-1,-2)\long{7}
\segment(39,-22)\dir(1,-2)\long{7}
\segment(57,-22)\dir(-2,-1)\long{27}
\segment(61,-22)\dir(-2,-3)\long{9}
}\hskip5cm}

\vskip 4.5cm
Antichains in the preceeding set
(i.e. $k$-uples of non comparable bigrassmannian permutations) are
by definition in bijection with elements of the lattice of triangles of
size $n$,
 or alternating sign matrices of order $n$, or square ice configurations
with $n^2$ molecules. The reader may be willing to derive the $42$
alternating sign matrices of order~4 from the above figure; there are,
indeed, 42 antichains in this poset. For other connections, see [La]
and for more information  about the Ehresmann-Bruhat order on finite
Coxeter groups, see [GK].

\smallskip
The problem of enumerating alternating sign matrices, due to Mills, Robbins,
and Rumsey [MRR1], [MRR2] has resisted many attempts, till Zeilberger [Z1]
gave a
solution with the collaboration of his computer, the control of eighty
eight of
his friends (he has a lot more), and many references to the Bible. Shorter
proofs were given by Kuperberg [Ku] and [Z2] again, after realizing that
the partition function  of square ice models had already been obtained
by Korepin, Bogoliubov and Izergin [KBI]: it is expressed as a
determinant generalizing Cauchy determinant. We are now ready to
enumerate square ice models.

\section 2. Cauchy type determinants|
Given two sets of variables $X=\{ x\}$, $Y=\{y\}$, of the same
cardinality Cauchy introduced two matrices
(we do not burden variables with indices, because they play a symmetrical role;
however, to write matrices one chooses an arbitrary total order on the
$x$'s$\in X$ and the $y$'s $\in Y$)
$$M_0 = \left[ {1\over 1-xy}\right]_{x\in X, y\in Y}   \qquad
  {\rm and}\qquad  M_1 = \left[ {1\over x-y}\right]_{x\in X, y\in Y}   \ .$$
{}From them one can extract a scalar product on symmetric functions, as
well as several pairs of adjoint bases, the most conspicuous one being
the self-adjoint basis of Schur functions.

Pulling out from $\det(M_1)$ the Vandermonde determinants
$\Delta(X)$, $\Delta(Y)$ and
the product  $\prod_{x\in X, y\in Y} (1/(x-y))$, one is
left with a symmetric polynomial
$$ F_1(X,Y) =  \det(M_1) {\prod (x-y)\over \Delta(X)\, \Delta(Y)} \ . \eqno
(1)$$
One easily checks that $F_1$ is equal to 1, and this fact, in a sense,
could summarize the first chapter of Macdonald's book [Ma].

Borchardt [Bo] went one step further, and considered
 $$M_2 := \left[ {1\over (x-y)^2}\right]\ .$$
He proved that
$$\det(M_2)= \det(M_1)\, \Perm(M_1)\ ,$$
where
$$\Perm(M_1) =\sum_{\mu\in \S(X)} \left(
{1\over (x_1-y_1)\cdots (x_n-y_n)  }\right)^\mu$$
is the usual {\it permanent}, the summation being over all the permutations
of the $x$'s.
Therefore the  symmetric function
$$
F_2(X,Y) =\det(M_2) {\prod (x-y)^2\over \Delta(X)\,\Delta(Y)}\eqno(2)
$$
is expressed by a summation over the symmetric group.

\goodbreak
Now, thanks to some giants, the shoulders of which Zeilberger [Z2] escalated
to provide a short alternating-sign-matrices enumeration,
we know that we must polarize squares, the simplest versions being
$$
\eqalignno{(x-y)^2 &\longrightarrow (x-y) (qx-y);&\hbox{(ice)}\cr
(x-y)^2 &\longrightarrow (x-y) (x-y+\g) ;&\hbox{(Bethe)}\cr}
$$
with $q$ and $\g$ some fixed constants independent of the $x$'s and $y$'s.

Introducing another set of variables $Z=\{z\}$ of the same cardinality as
$X$ and $Y$, one can more generally polarize as follows
$$ (x_i -y_j) \longrightarrow (x_i-y_j)\, (z_i-y_j).$$

We intend to describe the symmetric polynomial
$$
\displaylines{\hfill
F_Z(X,Y) := \det\left( \left[ {1\over
(x_i-y_j)(z_i-y_j)}\right]
\right) {\prod(x-y)\, \prod (z-y)\over \Delta(X)\, \Delta(Y)}
\hfill\llap{(3)}\cr
\noalign{\hbox{in the two cases of physical interest}}
z_i= qx_i \qquad {\rm or}\qquad z_i=x_i +\g.\cr}
$$
In order to to this we will make use of the divided differences $\d_i$
involving pairs of variables
$x_i$, $z_i$
$$\d_i:  \ f \mapsto { f-f^{\sigma_i} \over x_i - z_i} \ ,$$
where $\sigma_i$ is the transposition exchanging $x_i$ and $z_i$.
We also need {\it complete functions} $S_k(A-B)$ of differences of sets
of variables ({\it alphabets}) that can be defined by the following
generating function~:
$$
\sum_0^\infty u^k\, S_k(A-B) := { \prod_{b\in B} (1-ub) \over
\prod_{a\in A} (1-ua) } \ . $$
We write $x_i+z_i$ for the alphabet $A=\{ x_i,z_i\}$ and
$S_{\a,\b}(A-B,C-D)$ for the two-row Schur function
$$ S_{\a,\b}(A-B,C-D) := \left|
\matrix{ S_\a(A-B) & S_{\a+1}(A-B) \cr
         S_{\b-1}(C-D) & S_\b(C-D) \cr}
     \right|. $$
In particular, $z_i^k\, \prod_{y\in Y}( x_i-y)$ can be written as
$S_{n,k}(x_i-Y, x_i+z_i)$.

However, the image of $\prod_{y\in Y}(1-y)/(1-x_i)$ under $\d_i$ is
$$\prod\nolimits_{y\in Y}(1-y)/(1-x_i)(1-z_i)\ .$$
In other words,
$\d_i$ sends $S_j(x_i-Y)$ onto $S_{j-1}(x_i+z_i-Y)$ for any $j\in \Z$.
Hence the image of $S_{n,k}(x_i-Y, x_i+z_i)$ under $\d_i$ is
$$S_{n-1,k}(x_i+z_i-Y, x_i+z_i) =
\left| \matrix{ S_{n-1}(x_i+z_i-Y) & S_n(x_i+z_i-Y)\cr
        S_{k-1}(x_i+z_i)   & S_{k}(x_i+z_i)}\right|  \eqno (4)$$
that can be developped by linearity in $Y$  into
$$\displaylines{
\sum_{h=-1}^{n-1} S_{h,k}(x_i+z_i, x_i+z_i)
\, S_{n-1-h}(-Y) \hfill (5)\cr
\noalign{\vskip-7pt}
\hfill{}=\sum_{h=k}^{n-1} S_{h,k}(x_i+z_i)
\, S_{n-1-h}(-Y) -\sum_{h=0}^{k-1} S_{k-1,h}(x_i+z_i)
\, S_{n-h}(-Y). \quad(6)\cr}
$$

On the other hand, the image of the Cauchy determinant
$|1/(z-y)|$ under the product $\d_1,\ldots,\, \d_n$ is
the determinant of
$$M_Z := \left[ {1\over (x_i-y_j)\, (z_i-y_j)}\right]\ ,$$
since each $\d_i$ acts on row $i$ only.
Write $R(X,Y)$ for the product of differences $\prod (x-y)$. The
Cauchy determinant $\left| 1/(z-y) \right|$ is equal to
$$ { \Delta(Y)\over R(X,Y)\, R(Z,Y)} \Delta(Z)\, R(X,Y)
 = { \Delta(Y)\over R(X,Y)\, R(Z,Y)} \, \det(M_3) \ , \eqno(7)  $$
with
$$ M_3:= \left[ S_{n,j-1}(x_i-Y, x_i+z_i) \right]$$

Now, the factor
$ \Delta(Y)\, (R(X,Y)\, R(Z,Y))^{-1}$ commutes with
the product of the $\d_i$, since it is symmetrical in the $x_i$ and $z_i$.
According to (4), the image of $\det(M_3)$ is therefore equal to
the determinant of
$$ M_4:= \left[ S_{n-1,j-1}(x_i+z_i-Y, x_i+z_i) \right] \ .  \eqno (8) $$

Expansion (5) exactly tells that
the matrix $M_4$ factorizes into a
product of a matrix of two-part Schur functions in $(x_i+z_i)$,
and a matrix of  $\Lambda_i:= S_{1^i}(Y)$ or zeroes.

For example, for $n=2$, one has
$$ \pmatrix{{s_{0}}(x_{1}+{z_1})& {s_{1}}(x_{1}+{z_1})& {s_{1, 1}}(
x_{1}+{z_1})\cr {s_{0}}(x_{2}+{z_2})& {s_{1}}(x_{2}+{z_2})& {
s_{1, 1}}(x_{2}+{z_2})}
\pmatrix{-\Lambda_{1}& -\Lambda_{2}\cr \Lambda_{0}& 0\cr
0& \Lambda_{0}}.$$

For $n=3$,
$$\left(\moyennematrice{ 1& {s_{1}}(x_{1}+{z_1})& {s_{2}}(x_{
1}+{z_1})& {s_{1, 1}}(x_{1}+{z_1})& {s_{2, 1}}(x_{1}+{z_1})& {s_{
2, 2}}(x_{1}+{z_1})\cr  1 & {s_{1}}(x_{2}+{
z_2})& {s_{2}}(x_{2}+{z_2})& {s_{1, 1}}(x_{2}+{z_2})& {s_{2, 1}
}(x_{2}+{z_2})& {s_{2, 2}}(x_{2}+{z_2})\cr  1
& {s_{1}}(x_{3}+{z_3})& {s_{2}}(x_{3}+{z_3})& {s_{1, 1}}(x_{
3}+{z_3})& {s_{2, 1}}(x_{3}+{z_3})& {s_{2, 2}}(x_{3}+{z_3})}\right)
$$
$$\times \kern 20pt \pmatrix{\Lambda_{2}& \Lambda_{3}& 0\cr -\Lambda_{1}& 0&
\Lambda_{3}\cr \Lambda_{0}& 0& 0\cr 0& -\Lambda_{1}& -
\Lambda_{2}\cr 0& \Lambda_{0}& 0\cr 0& 0& \Lambda_{0}}.
$$


Finally,
for $n=4$, writing $s_\lambda$ for the Schur functions of $x_i+z_i$,
$i=1,\ldots,\, 4$, the factorization is
$$[s_{0}, s_{1}, s_{2}, s_{3}, s_{1 1}, s_{2 1}, s_{22}, s_{
31}, s_{32}, s_{33}]
\pmatrix{-\Lambda_{3}& -\Lambda_{4}& 0& 0\cr \Lambda_{2}&
0& -\Lambda_{4}& 0\cr -\Lambda_{1}& 0& 0& -\Lambda_{4}\cr
\Lambda_{0}& 0& 0& 0\cr 0& \Lambda_{2}& \Lambda_{3}& 0\cr
0& -\Lambda_{1}& 0& \Lambda_{3}\cr 0& 0& -\Lambda_{1}& -\Lambda_{
2}\cr 0& \Lambda_{0}& 0& 0\cr 0& 0& \Lambda_{0}& 0\cr
0& 0& 0& \Lambda_{0}}.
$$

\medskip
Any specialization $z_i=g(x_i)$, $g$ being an arbitrary function
of one variable,  will allow to factor out a matrix of determinant
$\Delta(X)$ from the left  matrix, and thus to explicit the symmetric
function $F_Z(X,Y)$. The two-row Schur function $S_{\a,\b}(x_i+z_i)$
becomes a polynomial $f(x_i;\a,\b)$ of degree $n(\a,\b)$  that
can be formally written $S_{n(\a,\b)}(x_i - A_{\a,\b})$,  for some
alphabet $A_{\a,\b}$ of cardinality $n(\a,\b)$ (= the \lq\lq alphabet
of zeroes" of the polynomial $ f(x_i;\a,\b)$ ).

Now, for any integer $k$, one has  the expansion
$$\eqalign{
S_{k}(x_i - A_{\a,\b}) &= S_k((x_i-X)+ ( X - A_{\a,\b}))  \cr
 &=\sum_{0\leq j\leq n-1} S_{k-j}(x_i-X)\, S_j( X - A_{\a,\b}) \ . \cr}$$
Therefore, from the matrix $\left[ S_{n(\a,\b)}(x_i -A_{\a,\b})
\right]$, one can extract the left factor $\left[S_{j-1}(x_i-X)
\right]_{1\leq j,i\leq n}$  (the determinant of which is $\Delta(X))$
and one is left with the matrix
$$ \left[ S_{n(\a,\b)-i+1}(X -A_{\a,\b}) \right]_{
n-1\geq \a\geq\b\geq 0 ; 1\leq i\leq n  }\,.\eqno (9) $$

Let us be more precise in the case of Bethe's function (we refer to
[Ga] for many other expressions of this function).
For $z=x+\g$, the Schur function $S_{\a,\b}(x+z)$ specializes  to
$$\eqalignno{ {1\over\g}
\left| \matrix{ (x+\g)^{\a+1} & (x+\g)^\b \cr x^{\a+1} & x^\b \cr} \right|
&=(x+\g)^\b \sum_{i=0}^{\a-\b} x^{\a-i}\g^i {\a+1-\b \choose i+1 } & (10) \cr
&= \sum_0^\a\g^i x^{\a+\b-i}\,  \left( {\a+1 \choose i}- {i\choose \a+1-i}
\right) \cr}$$
 Let us write the preceding polynomial as
$$\sum_{0\leq i\leq \a+\b} c_{\a,\b}^i \g^i x^{\a+\b-i}  $$
and for any alphabet $X$, and any partition of length $\leq 2$,
define
$$S_{\a,\b}^k(X) := \sum_{0\leq i\leq \a+\b} c_{\a,\b}^i \g^i S_{\a+\b-i-k}(X)
\ .\eqno (11)$$
Write $\chi(a=b)$ for the Boolean function which tests equality.
The com\-bi\-na\-tion of (8) and (10) produces the following
factorization.

\medskip
\proclaim Theorem $\g$ .
Let $X$ and $Y$ be two alphabets of cardinality $n$, $\g$ an extra parameter.
Then the symmetric function
$$
\displaylines{\hfill
F^\g(X,Y) := \left| {1\over
(x-y)(x-y+\g)}\right|_{x\in X,y\in Y}
 {\prod(x-y)(x-y+\g)\over \Delta(X)\, \Delta(Y)}
\hfill\quad\llap{(12)}\cr
\noalign{\hbox{is the determinant of the product of matrices}}
\quad
\left[ S_{\a,\b}^{i-1} (X) \right]^{n-1\geq \a\geq\b\geq 0}_{1\leq i\leq n }
\hfill\cr
\hfill{}\times
\left[\vrule height10pt depth3pt width0pt \chi(\b\!=\!j\!-\!1)
S_{n-1-\a}(-\!Y)\!-\!\chi(\a\!=\!j\!-\!2)  S_{n-\b}(-\!Y)
  \right]_{n-1\geq \a\geq\b\geq 0}^{1\leq j\leq n} \,.\cr}
$$

The following factorizations and Bethe functions are
readily computed by ACE [AV], the output being randomly distributed by Maple.
Writing $\Lambda_i$ for $(-1)^i S_i(-Y)$, and $s_i$ for $S_i(X)$,
we get:
$$\displaylines{\quad\pmatrix{1& \g+2s_{1}& \g s_{1}+s_{2}\cr 0& 2&
\g+s_{1}}
\pmatrix{-\Lambda_{1}& -\Lambda_{2}\cr \Lambda_{0}& 0\cr
0& \Lambda_{0}} \hfill\cr
\hfill{}= (2\Lambda_2+g^{2}-g\Lambda_1)+(g-\Lambda_1)s_{1}+2s_{1, 1}\ ;
\quad\cr}$$


$$\displaylines{\noalign{\vskip-18pt}
\left(\matrix{
1& \g+2s_{1}& \g^{2}+3\g s_{1}+3s_{2}& \g s_{1}+s_{2}& \g^{
2}s_{1}+3\g s_{2}+2s_{3}\cr
0& 2& 3\g+3s_{1}& \g+s_{1}& \g^{2}+3\g s_{1}+2s_{2}\cr
0& 0& 3& 1& 3\g+2s_{1}\cr}\right.
\hfill\cr
\hfill
\left.\matrix{& \g^{2}s_{2}+2\g s_{3}+s_{4}\cr
& \g^{2}s_{1}+2\g s_{2}+s_{3}\cr
& \g^{2}+2\g s_{1}+s_{2}\cr}\right)
\times \pmatrix{\Lambda_{2}& \Lambda_{3}& 0\cr -\Lambda_{1}&
0&
\Lambda_{3}\cr \Lambda_{0}& 0& 0\cr 0& -\Lambda_{1}& -
\Lambda_{2}\cr 0& \Lambda_{0}& 0\cr 0& 0& \Lambda_{0}}\cr}
$$
${}=6s_{2, 2,
2}+(\g^{6}-2{\Lambda_1}\g^{5}+4{\Lambda_1}{\Lambda_3}\g^{2}
+2{\Lambda_2}^{2}\g^{2}
+3\g^{4}{\Lambda_2}+{\Lambda_1}^{2}\g^{4}+6{\Lambda_3}^{2}
-6{\Lambda_3}{\Lambda_2}\g-3{\Lambda_1}{\Lambda_2}\g^{3
}-3{\Lambda_3}\g^{3})+(-4{\Lambda_3}\g^{2}
-4{\Lambda_1}\g^{4}+2{\Lambda_2}^{2}\g+4{\Lambda_1}{\Lambda_3
}\g-4{\Lambda_1}\g^{2}{\Lambda_2}+2\g^{5}
-4{\Lambda_2}{\Lambda_3}+2{\Lambda_1}^{2}\g^{3}+4{\Lambda_2}\g^{3}
)s_{1}+(\g^{4}+{\Lambda_1}^{2}\g^{2}-{\Lambda_1}{\Lambda_2}\g
+2{\Lambda_1}{\Lambda_3}+{\Lambda_2}\g^{2}-2{
\Lambda_1}\g^{3}-3{\Lambda_3}\g)s_{2}+(-4{\Lambda_1}
+6\g)s_{2, 2, 1}+(3\g^{3}-3{\Lambda_3}-{
\Lambda_1}{\Lambda_2}-4{\Lambda_1}\g^{2}+3{\Lambda_2}\g
+{\Lambda_1}^{2}\g)s_{2, 1}+(2{\Lambda_2}^{2}+2{\Lambda_1
}^{2}\g^{2}-4{\Lambda_1}{\Lambda_2}\g-6{\Lambda_1}\g^{3}
+4\g^{4}+6{\Lambda_2}\g^{2})s_{1, 1}+
(12{\Lambda_3}-8{\Lambda_1}\g^{2}+6\g^{3}-4{\Lambda_1}{\Lambda_2}
+4{\Lambda_1}^{2}\g)s_{1, 1, 1}+(
2{\Lambda_2}+2\g^{2}-2{\Lambda_1}\g)s_{2, 2}
+(2{\Lambda_1}^{2}+6\g^{2}-6{\Lambda_1}\g)s_{2, 1, 1} $




\medskip
On the other hand, when the $z_i$ specialize to $qx_i$,
one has a more convenient factorization into smaller matrices. Indeed,
$$S_{\a,\b}(x+qx) = x^{\a+\b} (q^\b + q^{\b+1} +\cdots + q^\a)=
 x^{\a+\b} q^\b\, [\a-\b+1]_q \ , \eqno(13) $$
with $[k]_q:= 1+q+\cdots + q^{k-1}$.  Combining these values with expansion
(6), one gets the following property.

\medskip
\proclaim Theorem q.
Let $X$ and $Y$ be two alphabets of cardinality $n$, $q$ an extra parameter.
Then the symmetric function
$$ F_q(X,Y) := \left| {1\over (x-y)(qx-y)}\right|_{x\in X,y\in Y}
           \, {\prod(x-y)(qx-y)\over \Delta(X)\, \Delta(Y)}  \eqno(14) $$
is the determinant of the product of
$$\left[ S_{j-i}(X) \right]_{1\leq i\leq n, 1\leq j\leq 2n-1}\!\!
\left[ { q^{j-k+1} - q^{k-1} \over q-1}\, S_{n-j+k-1}(-Y) \right]_{ 1\leq j\leq
2n-1, 1\leq k\leq n}
 $$
The coefficient of any Schur function $S_\l(X)$, $\l_1\geq \cdots \geq
\l_n\geq0$, in the expansion of $F_q(X,Y)$ is equal to the minor
on rows $\l_n+1,\ldots,\, \l_1+n$, columns $1,\ldots,\, n$
of the matrix in $Y$.

\medskip
For example, for $n=2$, $F_q(X,Y)$ is the determinant of the product
$$\pmatrix{S_0 & S_1 & S_2\cr 0 &S_0 & S_1} \cdot
\pmatrix{-\Lambda_1 &-\Lambda_2\cr
1+q &0\cr  0& q}
= (1+q)\Lambda_2 -q(x_1+x_2)\Lambda_1
+ (q+q^2)\, x_1x_2  $$
still writing $S_k$ for $S_k(X)$, and $\Lambda_i$ for $(-1)^iS_i(-Y)$, i.e. for
the $i$-th elementary symmetric function of $Y$.

\goodbreak
For $n=3,4,5$, writing $[k]$ for the $q$-integer $[k]_q$,
ACE [AV] produces the right matrices
$$\displaylines{
\pmatrix{\Lambda_{2}& \Lambda_{3}& 0\cr -[2]\Lambda_{1}
& 0& [2]\Lambda_{3}\cr [3]& -q\Lambda_{1}& -q\Lambda_{
2}\cr 0& q[2]& 0\cr 0& 0& q^{2}\cr}, \quad
\pmatrix{-\Lambda_{3}& -\Lambda_{4}& 0& 0\cr [2]\Lambda_{
2}& 0& -[2]\Lambda_{4}& 0\cr -[3]\Lambda_{1}& q
\Lambda_{2}& q\Lambda_{3}& -[3]\Lambda_{4}\cr
 [4] & q[2]\Lambda_{1}& 0& q[2]\Lambda_{3}\cr
0& q [3]  & -q^{2}\Lambda_{1}& -q^{2}\Lambda_{2}\cr
0& 0& q^{2}[2]& 0\cr 0& 0& 0& q^{3}\cr },\cr
\pmatrix{\Lambda_{4}& \Lambda_{5}& 0& 0& 0\cr -[2]\Lambda_{
3}& 0& [2]\Lambda_{5}& 0& 0\cr [3]\Lambda_{2}
& -q\Lambda_{3}& -q\Lambda_{4}& [3]\Lambda_{5}& 0\cr
-[3]\Lambda_{1}& q [2] \Lambda_{2}& 0& -q[2]
\Lambda_{4}&  [4] \Lambda_{5}\cr
 [5]  & - q[3]\Lambda_{1}& q^{2}\Lambda_{2}& q^{
2}\Lambda_{3}& -q [3]  \Lambda_{4}\cr 0&
 q[4] & -q^2[2]\Lambda_1 &0 & q^2 [2]\Lambda_3 \cr
0& 0& q^{2}[3] & -q^{3}\Lambda_{1}& -q^{3}\Lambda_{2}
\cr 0& 0& 0& q^{3}[2]& 0\cr 0& 0& 0& 0& q^{4}\cr}\cr}
$$
and the expansion of $F_q(X,Y)$, for $n=3$, is (writing, for each Schur
function of $X$,
its coefficient in $Y$ as a sum of Schur functions in $Y$)

\medskip
{
\parindent 0mm
$ s_{0} (X):
(q+1)(q^{2}+q+1)(s_{1 1 1 1 1 1}+s_{2 1 1 1 1}+s_{2
 2 2}+s_{2 2 1 1})
$

$ s_{1} (X):
-q{(q+1)}^{2}(s_{1 1 1 1 1}+s_{2 2 1}+s_{2 1 1 1})
$

$ s_{2} (X):
q^{2}(s_{1 1 1 1}+s_{2 1 1})(q+1)
$

$ s_{1 1} (X):
q^{2}(s_{1 1 1 1}+s_{2 2}+s_{2 1 1})(q+1)
$

$ s_{2 1} (X):
-q^{2}(qs_{2 1}+2qs_{1 1 1}+s_{1 1 1}q^{2}+s_{1 1 1})
$

$ s_{2 2} (X):
q^{3}s_{1 1}(q+1)
$

$ s_{1 1 1} (X):
q{(q+1)}^{2}(s_{1 1 1}q^{2}-qs_{2 1}+s_{1 1 1})
$

$ s_{2 1 1} (X):
(q+1)q^{3}(s_{2}+s_{1 1})
$

$ s_{2 2 1} (X):
-s_{1}q^{3}{(q+1)}^{2}
$

$ s_{2 2 2} (X):
s_{}q^{3}(q+1)(q^{2}+q+1).
$

}

\medskip
Lastly, for $n=4$ one has the following expansion of the function
$F_q(X,Y)$ (that it is impossible to obtain directly from the expansion of (3),
the object being to large for Maple):

\smallskip
$ s_{0} (X):
(q^{2}+1)(q^{2}+q+1){(q+1)}^{2}(s_{1\ldots 1}+2s_{2 1\ldots 1}+3s_{2 2 1
1\ldots 1}+5s_{2 2 2 2 1\ldots 1}+4s_{2 2 2
 1\ldots 1}+3s_{2 2 2 2 2 1 1}+s_{3
1\ldots 1}+2s_{3 2 1\ldots 1}+4s_{3 2 2
 2 1 1 1}+3s_{3 2 2 1\ldots 1}+s_{2 2 2 2 2
2}+2s_{3 2 2 2 2 1}+s_{3 3 1\ldots 1}+3s_{3 3
 2 2 1 1}+2s_{3 3 2 1\ldots 1}+s_{3 3 3 3}+2s_{3
3 3 2 1}+s_{3 3 2 2 2}+s_{3 3 3 1 1 1})
$

\smallskip
$ s_{1} (X):
-q(q+1){(q^{2}+q+1)}^{2}(2s_{2 1\ldots 1}+s_{
1\ldots 1 1 1}+2s_{2 2 2 2 2 1}+s_{3
1\ldots 1 1}+2s_{3 2 1\ldots 1}+4s_{2 2
 2 1\ldots 1}+3s_{2 2 1\ldots 1}+4s_{2 2
2 2 1 1 1}+3s_{3 2 2 1\ldots 1}+3s_{3 2 2 2 1 1
}+s_{3 3 3 2}+s_{3 3 3 1 1}+s_{3 3 1\ldots 1}+2
s_{3 3 2 2 1}+2s_{3 3 2 1 1 1}+s_{3 2 2 2 2})
$

\goodbreak

$ s_{2} (X):
q^{2}(q^{2}+q+1){(q+1)}^{2}(s_{1\ldots 1 1}
+2s_{2 1\ldots 1 1}+3s_{2 2 1\ldots 1}+
3s_{2 2 2 2 1 1}+3s_{2 2 2 1\ldots 1}+s_{2 2 2 2
 2}+s_{3 1\ldots 1}+2s_{3 2 1\ldots 1}+2s_{
3 2 2 2 1}+2s_{3 2 2 1 1 1}+s_{3 3 1\ldots 1}+s_{
3 3 2 2}+s_{3 3 2 1 1})
$

\smallskip
$ s_{3} (X):
-q^{3}(q+1)(q^{2}+q+1)(s_{1\ldots 1}+2s_{2
1\ldots 1}+2s_{2 2 1\ldots 1}+2s_{2 2 2 2
 1}+2s_{2 2 2 1 1 1}+s_{3 1\ldots 1}+s_{3 2
1\ldots 1}+s_{3 2 2 2}+s_{3 2 2 1 1})
$

\smallskip
$ s_{1 1} (X):
q^{2}{(q+1)}^{2}(q^{2}s_{3 3 2 1 1}+q^{2}s_{3 2 2 1
1 1}+q^{2}s_{3 3 3 1}+q^{2}s_{2 2 2 1\ldots 1}+qs_{3
 3 3 1}+3qs_{2 2 1\ldots 1}+3qs_{2 2 2 2 1
1}+4qs_{2 2 2 1\ldots 1}+2qs_{2 1\ldots 1 1}
+qs_{1\ldots 1 1}+qs_{3 3 1\ldots 1}+qs_{
3 3 2 2}+2qs_{3 3 2 1 1}+qs_{2 2 2 2 2}+qs_{3
1\ldots 1}+2qs_{3 2 1\ldots 1}+2qs_{3 2 2 2
1}+3qs_{3 2 2 1 1 1}+s_{3 3 2 1 1}+s_{3 2 2 1 1 1}
+s_{3 3 3 1}+s_{2 2 2 1\ldots 1})
$

\smallskip
$ s_{2 1} (X):
-q^{3}{(q+1)}^{3}(s_{1\ldots 1}+2s_{2 1 1
1\ldots 1}+3s_{2 2 2 1 1 1}+3s_{2 2 1\ldots 1
}+2s_{2 2 2 2 1}+s_{3 1\ldots 1}+2s_{3 2 2 1
1}+2s_{3 2 1\ldots 1}+s_{3 2 2 2}+s_{3 3 2 1}+s_{3
3 1 1 1})
$

\smallskip
$ s_{3 1} (X):
q^{3}{(q+1)}^{2}(q^{2}s_{1\ldots 1 1}+q^{2}s_{2
1\ldots 1}+q^{2}s_{2 2 1\ldots 1}+q^{2}s_{2 2 2
2}+q^{2}s_{2 2 2 1 1}+qs_{1\ldots 1 1}+2qs_{2
 1\ldots 1}+2qs_{2 2 1\ldots 1}+qs_{2 2 2 2}+2
qs_{2 2 2 1 1}+qs_{3 1\ldots 1}+qs_{3 2 1 1 1}+
qs_{3 2 2 1}+s_{1\ldots 1 1}+s_{2 1\ldots 1
1 1}+s_{2 2 1\ldots 1}+s_{2 2 2 2}+s_{2 2 2 1 1})
$

\smallskip
$ s_{2 2} (X):
q^{4}{(q+1)}^{2}(s_{1\ldots 1 1}+2s_{2 1 1 1
1 1 1}+3s_{2 2 1\ldots 1}+s_{2 2 2 2}+2s_{2 2 2 1
 1}+2s_{3 2 1 1 1}+s_{3 3 1 1}+s_{3 1\ldots 1}+
s_{3 2 2 1})
$

\smallskip
$ s_{3 2} (X):
-q^{4}(q+1)(q^{2}s_{1\ldots 1}+q^{2}s_{2 1\ldots 1}
+q^{2}s_{2 2 1 1 1}+q^{2}s_{2 2 2 1}+qs_{3 1 1
 1 1}+qs_{3 2 1 1}+3qs_{2 2 1 1 1}+2qs_{2 2 2 1}
+2qs_{1\ldots 1}+3qs_{2 1\ldots 1}+s_{
 1\ldots 1}+s_{2 1\ldots 1}+s_{2 2 1 1 1}+s_{2 2 2 1})
$

\smallskip
$ s_{3 3} (X):
q^{5}{(q+1)}^{2}(s_{1\ldots 1}+s_{2 1\ldots 1}+s_{2 2 1 1})
$

\smallskip
$ s_{1 1 1} (X):
q^{2}(q+1)(-q^{2}s_{3 3 3}-2q^{2}s_{3 3 2 1}-q^{2}s_{3
3 1 1 1}+qs_{1\ldots 1}+2qs_{2  1\ldots 1}+qs_{2 2 2 1 1 1}
+2qs_{2 2 1\ldots 1}+
2qs_{2 2 2 2 1}+qs_{3 1\ldots 1}+qs_{3 2 1 1
1 1}-qs_{3 3 3}-qs_{3 3 2 1}+qs_{3 2 2 2}+q^{2}s_{3
 2 2 2}-q^{2}s_{3 2 2 1 1}+q^{2}s_{1\ldots 1
1 1 1}+2q^{2}s_{2 1\ldots 1}+q^{2}s_{2 2 1\ldots 1}
+2q^{2}s_{2 2 2 2 1}+q^{2}s_{3 1\ldots 1
}+2q^{4}s_{2 1\ldots 1}+2q^{4}s_{2 2 2 1 1
1}+2q^{4}s_{2 2 1\ldots 1}+2q^{4}s_{2 2 2 2 1}+q^{
4}s_{3 1\ldots 1}+q^{4}s_{3 2 2 1 1}+q^{4}s_{
1\ldots 1 1}+q^{4}s_{3 2 1\ldots 1}+q^{4}s_{3
2 2 2}+q^{3}s_{1\ldots 1}+2q^{3}s_{2
 1\ldots 1}+q^{3}s_{2 2 2 1 1 1}+2q^{3}s_{2 2
1\ldots 1}+2q^{3}s_{2 2 2 2 1}+q^{3}s_{3 1\ldots 1
}+q^{3}s_{3 2 1\ldots 1}-q^{3}s_{3 3 3}-q^{3}s_{3 3
 2 1}+q^{3}s_{3 2 2 2}+s_{3 1\ldots 1}+s_{3 2
1\ldots 1}+s_{3 2 2 2}+s_{3 2 2 1 1}+2s_{2 2 1\ldots 1}
+2s_{2 2 2 2 1}+2s_{2 2 2 1 1 1}+s_{1\ldots 1}+2s_{2 1\ldots 1})
$

\smallskip
$ s_{2 1 1} (X):
-q^{2}{(q+1)}^{2}(qs_{2 1\ldots 1}-2q^{2}s_{3 2 2
1}+qs_{2 2 1\ldots 1}+qs_{2 2 2 2}-q^{2}s_{3 3 2}+qs_{
1\ldots 1 1}-q^{2}s_{3 3 1 1}-q^{2}s_{3 1\ldots 1}
-2q^{2}s_{3 2 1 1 1}-q^{2}s_{2 2 2 1 1}+qs_{
2 2 2 1 1}+q^{2}s_{1\ldots 1 1}-q^{2}s_{2 2
1\ldots 1}+q^{2}s_{2 2 2 2}+q^{3}s_{2 2 2 1 1}+q^{3}
s_{1\ldots 1 1}+q^{3}s_{2 1\ldots 1}+q^{3
}s_{2 2 1\ldots 1}+q^{3}s_{2 2 2 2}+q^{4}s_{2 2 2 2
}+s_{1\ldots 1 1}+s_{2 1\ldots 1}+s_{2 2
 1\ldots 1}+s_{2 2 2 2}+s_{2 2 2 1 1}+q^{4}s_{2 2
2 1 1}+q^{4}s_{1\ldots 1 1}+q^{4}s_{2 2 1 1
1 1}+q^{4}s_{2 1\ldots 1})
$

\smallskip
$ s_{3 1 1} (X):
-q^{4}(q+1)(q^{2}s_{2 2 1 1 1}+q^{2}s_{2 2 2 1}+q^{2}
s_{1\ldots 1}+q^{2}s_{2 1\ldots 1}+2qs_{2 2
2 1}+qs_{3 2 1 1}+qs_{3 1\ldots 1}+2qs_{2 2 1 1 1}
+2qs_{2 1\ldots 1}+qs_{1\ldots 1}+qs_{3 2 2
}+s_{2 2 1 1 1}+s_{2 2 2 1}+s_{1\ldots 1}+s_{
2 1\ldots 1})
$

\smallskip
$ s_{2 2 1} (X):
-(q+1)q^{3}(-q^{4}s_{1\ldots 1}-q^{4}s_{2 1\ldots 1}
-q^{4}s_{2 2 1 1 1}-q^{4}s_{2 2 2 1}+q^{3}s_{3
 2 1 1}-q^{3}s_{1\ldots 1}+q^{3}s_{3 1\ldots 1}
-q^{3}s_{2 2 2 1}-q^{2}s_{1\ldots 1}+2q^{2}s_{
2 2 1 1 1}+q^{2}s_{3 2 2}+q^{2}s_{2 1\ldots 1}+2q
^{2}s_{3 1\ldots 1}+3q^{2}s_{3 2 1 1}+q^{2}s_{3 3 1}+
qs_{3 2 1 1}-qs_{1\ldots 1}+qs_{3 1\ldots 1}-
qs_{2 2 2 1}-s_{1\ldots 1}-s_{2 1\ldots 1}
-s_{2 2 1 1 1}-s_{2 2 2 1})
$

\goodbreak
\smallskip
$ s_{3 2 1} (X):
q^{5}(2s_{1\ldots 1}+2q^{2}s_{1\ldots 1}+3q^{2
}s_{2 1\ldots 1}+q^{2}s_{2 2 2}+2q^{2}s_{2 2 1 1}+5qs_{
2 1\ldots 1}+3qs_{1\ldots 1}+qs_{3 2 1}+2qs_{2 2
 2}+4qs_{2 2 1 1}+2qs_{3 1 1 1}+s_{3 1 1 1}+2s_{2
2 1 1}+q^{2}s_{3 1 1 1}+3s_{2 1\ldots 1}+s_{2 2 2})
$

\smallskip
$ s_{3 3 1} (X):
-q^{5}(q+1)(q^{2}s_{1\ldots 1}+q^{2}s_{2 1 1 1}+2qs_{
1\ldots 1}+qs_{2 2 1}+2qs_{2 1 1 1}+s_{1\ldots 1
}+s_{2 1 1 1})
$

\smallskip
$ s_{2 2 2} (X):
-q^{4}{(q+1)}^{2}(q^{2}s_{2 2 2}-qs_{3 1 1 1}-qs_{3 2
1}-qs_{2 2 1 1}-qs_{2 1\ldots 1}+s_{2 2 2})
$

\smallskip
$ s_{3 2 2} (X):
-q^{5}(q+1)(q^{2}s_{1\ldots 1}+q^{2}s_{2 1 1 1}+qs_{1
 1\ldots 1}+2qs_{2 1 1 1}+qs_{3 1 1}+qs_{2 2 1}+s_{
1\ldots 1}+s_{2 1 1 1})
$

\smallskip
$ s_{3 3 2} (X):
q^{5}{(q+1)}^{2}(q^{2}s_{1\ldots 1}+qs_{1\ldots 1}+qs_{2
1 1}+s_{1\ldots 1})
$

\smallskip
$ s_{3 3 3} (X):
-s_{1 1 1}q^{6}(q+1)(q^{2}+q+1)
$

\smallskip
$ s_{1\ldots 1} (X):
q{(q+1)}^{2}(q^{2}s_{3 3 2}+s_{1\ldots 1 1}+s_{
2 1\ldots 1}+s_{2 2 1\ldots 1}+s_{2 2 2 2}+s_{
2 2 2 1 1}-2q^{4}s_{3 2 1 1 1}-2q^{4}s_{3 1\ldots 1}
+q^{4}s_{3 3 2}-q^{4}s_{3 2 2 1}+2q^{2}s_{
1\ldots 1}+2q^{2}s_{2 2 2 2}+q^{2}s_{2 2 2 1 1}-q
^{5}s_{3 2 1 1 1}-q^{5}s_{3 1\ldots 1}-q^{5}s_{3 2
 2 1}+q^{6}s_{1\ldots 1}+q^{6}s_{2 1 1 1
1 1 1}+q^{6}s_{2 2 1\ldots 1}+q^{6}s_{2 2 2 2}+q^{6}
s_{2 2 2 1 1}+q^{3}s_{2 2 2 1 1}-q^{3}s_{3 2 1 1
1}-2q^{3}s_{3 1\ldots 1}+q^{3}s_{3 3 2}-q^{3}s_{3 2
2 1}+2q^{3}s_{2 2 2 2}+q^{3}s_{3 3 1 1}+q^{4}s_{2 2
2 1 1}+q^{5}s_{1\ldots 1}+q^{5}s_{2 2 2 2}
+2q^{3}s_{1\ldots 1}+q^{3}s_{2 2 1\ldots 1}+
2q^{4}s_{1\ldots 1}+2q^{4}s_{2 2 2 2}+qs_{
1\ldots 1}+qs_{2 2 2 2}-2q^{2}s_{3 1\ldots
1}-2q^{2}s_{3 2 1 1 1}-q^{2}s_{3 2 2 1}-qs_{3 1\ldots 1}
-qs_{3 2 1 1 1}-qs_{3 2 2 1})
$

\smallskip
$ s_{2 1 1 1} (X):
q^{3}{(q+1)}^{3}(2q^{2}s_{2 1\ldots 1}+q^{2}s_{
1\ldots 1}+q^{2}s_{3 1\ldots 1}+2q^{2}s_{2 2 1 1 1}+q
^{2}s_{3 2 1 1}+q^{2}s_{2 2 2 1}-qs_{3 2 2}-qs_{3 3
 1}-qs_{3 2 1 1}-qs_{2 2 2 1}-qs_{2 2 1 1 1}+2s_{
2 1\ldots 1}+s_{1\ldots 1}+s_{3 1\ldots 1}+
2s_{2 2 1 1 1}+s_{3 2 1 1}+s_{2 2 2 1})
$

\smallskip
$ s_{3 1 1 1} (X):
-q^{4}{(q+1)}^{2}(q^{2}s_{2 1\ldots 1}+q^{2}s_{3 1 1 1}
-qs_{3 2 1}-qs_{2 2 2}-qs_{2 2 1 1}+s_{2 1\ldots 1}
+s_{3 1 1 1})
$

\smallskip
$ s_{2 2 1 1} (X):
-q^{3}{(q+1)}^{2}(q^{4}s_{2 2 1 1}+q^{4}s_{1\ldots 1
1}+q^{4}s_{2 1\ldots 1}+2q^{3}s_{2 2 1 1}+2q^{3}s_{2 1
 1 1 1}+2q^{3}s_{1\ldots 1}+q^{2}s_{2 1\ldots 1}
+2q^{2}s_{1\ldots 1}-q^{2}s_{3 1 1 1}-q^{2}s_{3 3
}-2q^{2}s_{3 2 1}-q^{2}s_{2 2 2}+2qs_{2 2 1 1}+2qs_{2
 1\ldots 1}+2qs_{1\ldots 1}+s_{2 2 1 1}+s_{1 1
1\ldots 1}+s_{2 1\ldots 1})
$

\smallskip
$ s_{3 2 1 1} (X):
q^{4}(q+1)(q^{4}s_{1\ldots 1}+q^{4}s_{2 1 1 1}+q^{3}s_{
1\ldots 1}-q^{3}s_{2 2 1}+q^{3}s_{2 1 1 1}-q^{2}s_{
3 2}+q^{2}s_{1\ldots 1}-q^{2}s_{3 1 1}-3q^{2}s_{2 2
1}+qs_{1\ldots 1}-qs_{2 2 1}+qs_{2 1 1 1}+s_{1 1
1 1 1}+s_{2 1 1 1})
$

\smallskip
$ s_{3 3 1 1} (X):
q^{6}{(q+1)}^{2}(s_{1\ldots 1}+s_{2 2}+s_{2 1 1})
$

\smallskip
$ s_{2 2 2 1} (X):
q^{4}{(q+1)}^{3}(q^{2}s_{1\ldots 1}+q^{2}s_{2 2 1}+q^{
2}s_{2 1 1 1}-qs_{3 1 1}-qs_{3 2}-qs_{2 2 1}-qs_{2
1 1 1}+s_{1\ldots 1}+s_{2 2 1}+s_{2 1 1 1})
$

\smallskip
$ s_{3 2 2 1} (X):
-q^{4}{(q+1)}^{2}(s_{1\ldots 1}q^{4}+q^{3}s_{1\ldots 1}-q^{
2}s_{3 1}+q^{2}s_{1\ldots 1}-q^{2}s_{2 2}-2q^{2}s_{2 1
1}+qs_{1\ldots 1}+s_{1\ldots 1})
$

\smallskip
$ s_{3 3 2 1} (X):
-q^{6}{(q+1)}^{3}(s_{2 1}+s_{1 1 1})
$

\smallskip
$ s_{3 3 3 1} (X):
s_{1 1}q^{6}(q^{2}+q+1){(q+1)}^{2}
$

\smallskip
$ s_{2 2 2 2} (X):
q^{3}{(q+1)}^{2}(q^{6}s_{1\ldots 1}+s_{1\ldots 1}q^{5}-q^{
5}s_{2 1 1}+2s_{1\ldots 1}q^{4}-q^{4}s_{2 1 1}+q^{4}s_{
3 1}-q^{3}s_{2 1 1}+q^{3}s_{3 1}+q^{3}s_{2 2}+2q^{3}s_{
1\ldots 1}+2q^{2}s_{1\ldots 1}-q^{2}s_{2 1 1}+q^{2}s_{3
1}+qs_{1\ldots 1}-qs_{2 1 1}+s_{1\ldots 1})
$

\smallskip
$ s_{3 2 2 2} (X):
q^{5}(q+1)(q^{4}s_{1 1 1}-q^{3}s_{3}+q^{3}s_{1 1 1}-q^{3
}s_{2 1}-q^{2}s_{3}-2q^{2}s_{2 1}+q^{2}s_{1 1 1}-qs_{3}+
qs_{1 1 1}-qs_{2 1}+s_{1 1 1})
$

\smallskip

$ s_{3 3 2 2} (X): q^{6}{(q+1)}^{2}(s_{2}q^{2}+qs_{2}+qs_{1 1}+s_{2})
$

\smallskip
$ s_{3 3 3 2} (X): -s_{1}q^{6}(q+1){(q^{2}+q+1)}^{2}
$

\smallskip
$ s_{3 3 3 3} (X): q^{6}(q^{2}+q+1)(q^{2}+1){(q+1)}^{2}
$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip\bigskip\bigskip
\centerline{\bf References}
\medskip
\parindent 0mm

[AV]\ ACE, S. Veigneau, ACE. {\it  an algebraic environment for the computer
algebra system MAPLE},

{\tt http://phalanstere.univ-mlv.fr/$\sim$ ace} (1998).

[A77]\ G. Andrews.  Plane Partitions II:
The equivalence of the Bender-Knuth and Mac Mahon conjecture.
{\it Pacific J.} {\bf 72} (1977) 283--291.

[A78]\ G. Andrews. Plane Partitions I: The Mac Mahon conjecture.
{\it Studies in Foundations and Combinatorics, Advances in Maths Supplementary
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[A79]\ G. Andrews. Plane Partitions III: The weak Macdonald conjecture.
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[A87]\ G. Andrews. Plane Partitions IV: A conjecture of Mills-Robbins-Rumsey.
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[Bo]\ C.W. Borchardt. Bestimmung der symmetrischen Verbindungen ihrer
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[Br]\ D. Bressoud. Proofs and Confirmations. The story of the alternating sign
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[Ga]\ M. Gaudin. {\it La fonction d'onde de Bethe}, Masson (1983).

[GK]\ M. Geck, S. Kim.  Bases for the Bruhat-Chevalley order on
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[KBI]\ V. Korepin, N. Bogoliubov, A. Izergin. {\it Quantum inverse scattering
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[Ku]\ G. Kuperberg. Another proof of the refined alternating sign
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[La]\  A. Lascoux.  Ordonner le groupe sym\'etrique: pourquoi
utiliser l'al\-g\`ebre de Iwahori-Hecke ?  {\it ICM
Berlin 1998, Documenta Mathematica,  volume III}, (1998) 355--364.


[LS1]\  A. Lascoux \& M.P. Sch\"utzenberger,
 Alg\`ebre des diff\'erences divis\'ees,
{\it Discrete Maths} {\bf  99} (1992) 165-179.

[LS2]\  A. Lascoux \& M.P. Sch\"utzenberger,
{\it Treillis et bases des groupes de Coxeter},
Electronic Journal of Combinatorics  {\bf 3} (1996) R27

[Ma]\ I.G. Macdonald. {\it Symmetric functions and Hall polynomials}
Oxford University Press (1995).

[MRR1]\ W. Mills, D. Robbins, H. Rumsey. Proof of the Macdonald conjecture.
{\it Inventiones M.}{\bf 66} (1982) 73--87.

[MRR2]\ W. Mills, D. Robbins, H. Rumsey. Alternating sign matrices and
descending plane partitions. {\it J.Comb. Theory A} {\bf 34} (1983) 340--359.


[Z1]\ D. Zeilberger. Proof of the alternating sign
matrix conjecture. {\it Electronic J. Comb.} {\bf 3} (1996) R 13.

[Z2]\ D. Zeilberger. Proof of the refined alternating sign
matrix conjecture.  {\it New York J. Math} {\bf 2} (1996) 59--68.



\vskip 15mm
{\obeylines
\hfill                  C.N.R.S., Institut Gaspard Monge   \hfill
\hfill                  Universit\'e de Marne-la-Vall\'ee,  \hfill
\hfill                   5 Bd Descartes, Champs sur Marne,   \hfill
\hfill                   77454 Marne La Vall\'ee Cedex 2 FRANCE  \hfill
\hfill \tt                      Alain.Lascoux@univ-mlv.fr
\hfill                http://phalanstere.univ-mlv.fr/$\sim$al
}

\bye