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\begin{document}

\title{\bf Architectonique des formules \\
pr\'ef\'er\'ees d'Alain Lascoux}


\author{Piotr Pragacz\thanks{Supported by KBN
grant 2P03A 024 23.}\\
\small Institute of Mathematics of Polish Academy of Sciences\\
\small \'Sniadeckich 8, 00-956 Warszawa, Poland\\
\small P.Pragacz@impan.gov.pl}
\date{ }

\maketitle


\begin{abstract}
This article, written in December 2004, is an expanded version
of the author's lecture opening the LascouxFest at the S\'eminaire
Lotha\-rin\-gien
de Combinatoire in Domaine Saint-Jacques, Ottrott,
March 29-31, 2004. We discuss here some aspects of the work of
Alain Lascoux (and some of his coworkers), related to symmetric functions
and, more generally,
Schubert polynomials. We illustrate some of the techniques he uses:
determinants, transformations of alphabets, reproducing kernels,
planar displays, divided differences, and vertex operators.
The aim of this article is to show to the reader
working in Algebraic Combinatorics (and others!) what we can
learn from Alain to make our computations more efficient and more
exciting.
\end{abstract}

%First page headline in LaTeX for S\'eminaire Lotharingien de Combinatoire
%--first part
\thispagestyle{myheadings}
\font\rms=cmr8 
\font\its=cmti8
\font\bfs=cmbx8

\markright{\its S\'eminaire Lotharingien de
Combinatoire \bfs 52 \rms (2005), Article~B52d\hfill}
\def\thepage{}


\section{Introduction}\label{intro}


I\footnote{I believe that Marcel-Paul Sch\"utzenberger would be much more
competent (than me) to tell you about the work of Alain. He was the closest
coworker of Alain for many years. Also, Bernard Leclerc and Jean-Yves
Thibon would surely accomplish this job better than me -- their collaboration
with Alain, though shorter in time, was/is more intensive. I will not
cover here all aspects of Alain's work. For example,
I will not touch the Kostka-Foulkes, Kazhdan-Lusztig, and
Macdonald polynomials, noncommutative methods, and
the singularities of Schubert varieties. I will concentrate on my
25 year collaboration with Alain and on topics that seem to be
in the ``intersection'' of the interests of the audience of SLC and those
of myself.} met Alain in Spring 1978 during his first visit to Poland.
This was just after his famous Thesis \cite{LasThes} had been passed.
He taught us the combinatorics of syzygies
of determinantal ideals, using the combinatorics of
the Bott theorem on cohomology of homogeneous bundles on flag varieties.
His 1977 Thesis introduced to Commutative Algebra and Combinatorics
{\it Schur functors}, a functorial version of Schur functions,
allowing one to express the syzygies of determinantal ideals
(previously only symmetric and exterior powers were used, as well as
{\it hook} Schur functors -- the latter by D. Buchsbaum and
D. Eisenbud \cite{BE}).

Apart from the Bott theorem (for $GL_n$)\footnote{Alain discovered
it independently by himself (cf. \cite{Lg} and \cite{Lassyz}, Addendum).},
he used a functorial version of
{\it the Cauchy formula} for symmetric and exterior powers of
the tensor product of two vector bundles. I believe that the Cauchy formula
is for Alain
a prototype of a ``good" algebraic formula. Therefore, in the next section
we discuss several incarnations of it. In particular, the Cauchy formula
gives us a certain ``reproducing kernel". In various places of this
article, I shall show other reproducing kernels -- one of the leitmotivs
of Alain's algebraic way of thinking.

Already in early Alain's papers (cf. \cite{Lbul}, \cite{Lg}), some
determinants show up. Determinants and more generally, minors, play an
important role in Alain's work. Often he studies them via various
deformations, and
especially by modifying the alphabets that are arguments of Schur functions.
We discuss determinantal techniques and some determinantal expressions
in Section 3.

%First page headline in AmS-LaTeX for S\'eminaire Lotharingien de Combinatoire
%--restoring the headers and pagenumbering
\pagenumbering{arabic}
\addtocounter{page}{1}
%\markboth{\SMALL PIOTR PRAGACZ}{\SMALL ARCHITECTONIQUE DES FORMULES}
%
%

In the beginning of the 80's, Alain discovered with
Marcel-Paul Sch\"utzen\-berger that the classical Schur functions
are a very particular
case of {\it (simple) Schubert polynomials} defined by using divided
differences (of Newton). Even more natural are {\it (double) Schubert
polynomials} which, defined originally algebro-combinatorially, have a
transparent geometric interpretation: they are cohomology classes
of flagged degeneracy loci. There is an important analogue of the Cauchy
formula for Schubert polynomials (cf.\ Theorem \ref{TCL} in Section 4).

Operator techniques become more and more apparent in Alain's work. Apart
from divided differences, used in the theory of Schubert polynomials,
and their variations (e.g.\ {\it isobaric} divided differences used
in the theory of\break {\it Grothendieck polynomials}), he uses also
{\it vertex operators}. In Section 5, I will tell you how vertex operator
were helpful in the theory of $\widetilde Q$-polynomials.

It would be totally banal to say to the audience of SLC that planar
displays are useful to see better some algebro-combinatorial properties.
However, in Section 6, I wish to show you some recent examples
of the use of planar displays of divided differences.

Already from Alain's Thesis, I learned about the $\lambda${\it-rings of
Grothendieck}. The $\lambda$-ring structure played -- I believe --
a large role in Alain's algebro-combinatorial thinking: it allows one
to treat
symmetric functions as operators on polynomials.
It is truly amazing that the techniques he developed, under the general
``umbrella'' of $\lambda$-rings, provide a uniform approach to numerous
classical polynomials (e.g.\ symmetric, orthogonal) and formulas
(e.g.\ interpolation formulas or those of representation theory
of general linear groups and symmetric groups). The polynomials
and formulas are often related to the names such as:
E. B\'ezout, A. Cauchy, A. Cayley, P. Chebyshev, L. Euler, C.F. Gauss,
C.G. Jacobi, J. Lagrange, E. Laguerre, A-M. Legendre, I. Newton,
I. Schur, T.J. Stieltjes, J. Stirling, J.J. Sylvester,
J.M. Hoene-Wro\'nski, and others. See \cite{LCBMS}.

In fact in Section 7, I will tell you about my current work with
Alain on some aspects of Euclid's algorithm. You will see again
some determinants, the Lagrange interpolation, and reproducing kernels.

The last section will contain some ``concluding remarks" and pictures
from Nankai University in Tianjin, where Alain worked during the
last years.

I will not repeat in this article the stories and jokes that I told during
my talk in Ottrott, because of the rule that I try to follow:
``Jokes should not be repeated."

\section{The Cauchy formula}

The Cauchy formula is one of Alain's favourite formulas.
He told me this
already during his very first lesson on symmetric functions
in Paris, in November 1978. As any important formula, it has many 
incarnations.

\medskip

\noindent
{\bf I. (Cauchy kernel)} Let $\A$, $\B$ be two alphabets. We have
\begin{equation}
\prod_{a\in A, b\in B} \frac{1}{1-ab}=\sum_I S_I(\A)S_I(\B)=:K(\A,\B)\,,
\end{equation}
where the sum is over all partitions $I$ \footnote{In this section, we
use French partitions in order not to offend Cauchy. The reader not 
familiar with Schur functions will find their definition in the 
next section. The notation for partitions, used in the present
article, is that of \cite{LCBMS}.}.

Consider the ring $\mathfrak{Sym}$ of symmetric functions in a countable
alphabet of variables. Let $( \ , \ )$ be the scalar product on
$\mathfrak{Sym}$ such that $\{S_I\}$ is an orthonormal basis. Given two
such alphabets $\A$ and $\B$ and $f\in \mathfrak{Sym}$, we have
\begin{equation}
(f(\A), K(\A,\B)) = f(\B)
\end{equation}
-- a fundamental {\it reproducing} property.

\smallskip

\noindent
{\bf II. (Resultant)}  Let $\card(\A)=m$, $\card(\B)=n$. We have
\begin{equation}
\prod_{a\in \A,b\in \B}(a-b)=\sum_{I \subset n^m} S_I(\A) S_{{n^m}/I}(-\B).
\end{equation}

\smallskip

\noindent
{\bf III. (Diagonal and Specht modules)} Let ${\cal S}^I$ be the irreducible
representation of the symmetric group $\mathfrak{S}_n$ indexed by
the partition $I$, $|I|=n$. Then for the diagonal embedding
$$
\mathfrak{S}_n \hookrightarrow \mathfrak{S}_n \times \mathfrak{S}_n\,,
$$
we have
\begin{equation}
\Ind({\cal S}^{(n)}) = \oplus_{|I|=n} {\cal S}^I \otimes {\cal S}^I\,,
\end{equation}
where ${\cal S}^{(n)}$  is the one dimensional representation of
${\mathfrak S}_n$ with the trivial action.

\smallskip
\noindent
{\bf IV. (Bitableaux)} Let $X=(x_{ij})_{m\times n}$ be a matrix of
indeterminates. We search for a ``determinantal'' additive basis of
the polynomial ring ${\bf Z}[X]$ (instead of a familiar monomial basis).
Set
\begin{equation}
(i_1,\ldots,i_k |j_1,\ldots,j_k):=\det(x_{{i_p}{j_q}})_{1\le p,q \le k}\,.
\end{equation}

In the following, when speaking about {\it tableaux} and {\it standard
tableaux} we mean the corresponding planar objects explained in 
\cite{LCBMS}, p.176. ``Transposing'' will mean taking reflection in
the main diagonal.

A {\it bitableau} is a pair of {\it transposed tableaux}. These transposed
tableaux must have the same shape, called the {\it shape of a bitableau}.
We display planarly a bitableau as:
$$
\left(\,\CD
a_1 & a_2 & a_3 & {} & {} & {} & {} & {} \\
c_1 & c_2 & c_3 & c_4 & c_5 & {} & {} & {} \\
e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 & e_8 \\
\endCD
 \ \ \ , \ \ \ \CD
b_1 & b_2 & b_3 & {} & {} & {} & {} & {} \\
d_1 & d_2 & d_3 & d_4 & d_5 & {} & {} & {} \\
f_1 & f_2 & f_3 & f_4 & f_5 & f_6 & f_7 & f_8 \\
\endCD \, \right)
$$
(here the shape is 358), where $a_i, c_j, e_k \le m$ and $b_i, d_j, f_k
\le n$.

Here is an example of a bitableau of shape 245:

$$
\left(\,\CD
3 & 4 & {} & {} & {} \\
2 & 4 & 5 & 9 & {} \\
2 & 3 & 5 & 6 & 7 \\
\endCD
 \ \ \ , \ \ \ \CD
7 & 8 & {} & {} & {} \\
2 & 3 & 6 & 7 & {} \\
1 & 3 & 6 & 8 & 9 \\
\endCD \, \right)\,.
$$

With a bitableau, we associate the product of minors:
\begin{multline*}
(a_1, a_2, a_3|b_1, b_2, b_3) \
(c_1, c_2, c_3, c_4, c_5|d_1, d_2, d_3, d_4, d_5)\\
\times (e_1, e_2, e_3, e_4, e_5, e_6, e_7, e_8|f_1, f_2, f_3, f_4, f_5, f_6,
f_7, f_8)\,.
\end{multline*}


Now a basic theorem says that {\it products of minors associated with
bi\-tab\-leaux form an additive basis of} ${\bf Z}[X]$. 
Several mathematicians
contributed to this result; the ``most combinatorial''
references, containing the {\it Rota straight\-ening formula},
are \cite{DRS} and \cite{DKR}.

Alain in his Thesis considers a characteristic zero variant of this
result: given two vector spaces or vector bundles,
\begin{equation}
S^n(E\otimes F)=\oplus_{|I|=n} V_I(E)\otimes V_I(F)\,,
\end{equation}
where $V_I(E)$ is the polynomial representation of $GL(E)$, called by Alain
the {\it Schur functor}, corresponding to the weight $I$ (and invented
in Schur's 1901 Thesis \cite{Schur}).

In fact, in Alain's Thesis the following parallel formula
plays an even more important role:
\begin{equation}
\wedge^n(E\otimes F)=\oplus_{|I|=n} V_I(E)\otimes V_{I^{\sim}}(F)\,,
\end{equation}
where $I^{\sim}$ is the conjugate partition of $I$.
The importance of this formula comes from the fact that it describes
the {\it Koszul syzygies} of the ideal generated by the entries
of a generic matrix, and then via suitable derived functors
(following a method introduced in G.~Kempf's 1971 thesis) 
allow one to
describe syzygies of determinantal ideals \cite{LasThes}, \cite{Lassyz}.

In terms of characters, the last decomposition corresponds to
\begin{equation}
\prod_{a\in \A, b\in \B}(1+ab)=\sum_I S_I(\A) S_{I^{\sim}}(\B)\,.
\end{equation}

\noindent
{\bf V. (Robinson-Schensted-Knuth correspondence)} This is a favourite
way of seeing the Cauchy formula by combinatorialists. Its simplest variant
asserts a bijection between permutations in ${\mathfrak S}_n$ and pairs of
standard tableaux of the same shape of weight $n$.

For example, the R-S-K correspondence associates with the permutation
$[3,6,4,1,7,2,5]$ \footnote{Here we follow the convention of writing
a permutation $\sigma \in {\mathfrak S}_n$ by its sequence of values:
$\sigma=[\sigma_1, \sigma_2,\ldots,\sigma_n]$.}
the last pair of standard tableaux of shape 133 in the following display:

%%%%  diagram
{\arraycolsep0pt
\let\lra\longrightarrow\let\mi\mathit
$$
\begin{array}{cclclclcl}
3\ ,\quad \mathit1&\lra&\begin{array}{l}36\end{array}&\ ,\quad &
\begin{array}{l}\mi{12}\end{array}&\quad\lra\quad&
\begin{array}{l}6\\34\end{array}&\ ,\quad &
\begin{array}{l}\mi3\\\mi{12}\end{array}\\[20pt]
&\quad\lra\quad&\begin{array}{l}6\\3\\14\end{array}&\ ,\quad &
\begin{array}{l}\mi4\\\mi3\\\mi{12}\end{array}&\quad\lra\quad&
\begin{array}{l}6\\3\\147\end{array}&\ ,\quad &
\begin{array}{l}\mi4\\\mi3\\\mi{125}\end{array}\\[32pt]
&\quad\lra\quad&\begin{array}{l}6\\34\\127\end{array}&\ ,\quad &
\begin{array}{l}\mi4\\\mi{36}\\\mi{125}\end{array}&\quad\lra\quad&
\begin{array}{l}6\\347\\125\end{array}&\ ,\quad &
\begin{array}{l}\mi4\\\mi{367}\\\mi{125}\end{array}
\end{array}
$$}

To be more precise, what corresponds to the Cauchy formula is a similar
bijection between monomials in commutative {\it biletters} and pairs of
tableaux of the same shape (cf. \cite{Kn}).

According to Alain, the Cauchy formula gathers many (if not all) features
of a ``good'' algebraic formula: firstly, because it gives us some
useful reproducing kernel; secondly, because of its functorial character;
thirdly, because it is an interesting result on determinants having a natural
``bijective interpretation'' with the help of planar displays.
In the forthcoming sections, I will show you several variations
on these themes.

We also signal that a variant of a ``Cauchy formula''
for {\it loop groups} was used by G. Faltings in his derivation of
the famous {\it Verlinde formula} (for the moduli space of vector
bundles on a curve) -- see Theorem 2.4 in \cite{Fal}.


\section{Determinants}

Everybody (since Jacobi) knows that Schur functions are given by
determinants in {\it complete} or {\it elementary} symmetric functions.
One of the very first things on symmetric functions that I learned from
Alain was that the former are more important that the
latter. This was also claimed by J.M. Hoene-Wro\'nski
(1778-1853), a Polish-French universal scientist admired by Alain
(cf.\ \cite{Lwr}).

\bigskip
\centerline{\includegraphics{kor.pdf}}
\smallskip
\centerline{\small Castle K\'ornik near Pozna\'n in Poland, containing a 
large collection}
\centerline{\small of the {\it original} manuscripts of Wro\'nski. They
probably contain}
\centerline{\small interesting mathematical things still to be
(re)discovered.}
\bigskip

Working with a determinantal presentation of Schur functions, Alain
always considers them as minors of a large matrix:
$$
\CD
S_0 \ & \ S_1 \ & \ S_2 \ & \ S_3 \ & \  S_4 \ & \ ... \\
0 & S_0 & S_1 & S_2 & S_3 & ... \\
0 & 0 & S_0 & S_1 & S_2 & ... \\
0 & 0 & 0 & S_0 & S_1 & ... \\
0 & 0 & 0 & 0 & S_0 & ... \\
\vdots & \vdots & \vdots & \vdots & \vdots & \\
\endCD
$$
\noindent
Usually $S_i=S_i(\A)$ where $\A$ is an {\it alphabet} or even a
{\it virtual alphabet}. More precisely, given $I=(i_1,\ldots,i_n),
J=(j_1,\ldots,j_n)\in {\bf Z}^n$, he sets
\begin{equation}
S_{J/I}(\A):=\det(S_{j_k-i_h+k-h}(\A))_{1\le h,k \le n}\,.
\end{equation}
If $I$ consists of zeros, such a determinant is, up to sign, a Schur
function indexed by a French partition:
$$
S_{5,7,2,0,3}
=\left|\,\CD
S_5 \ & \ S_8 \ & S_4 \ & \ S_3 \ &  \ S_7 \\
S_4 & S_7 & S_3 & S_2 & S_6 \\
S_3 & S_6 & S_2 & S_1 & S_5 \\
S_2 & S_5 & S_1 & 1 & S_4 \\
S_1 & S_4 & 1 & 0 & S_3 \\
\endCD\,\right|
=\left|\,\CD
S_3 \ & \ S_4 \ & S_5 \ & \ S_7 \ &  \ S_8 \\
S_2 & S_3 & S_4 & S_6 & S_7 \\
S_1 & S_2 & S_3 & S_5 & S_6 \\
1 & S_1 & S_2 & S_4 & S_5 \\
0 & 1 & S_1 & S_3 & S_4 \\
\endCD\,\right|=S_{3,3,3,4,4}\,.
$$
Among minors of a given matrix, there are many relations:
Laplace expansions, Pl\"ucker relations, etc. that can be
used to get interesting relations for Schur functions.
Alain recommended, many times, that I go through Muir's treatment
of determinants in five volumes \cite{Muir}. For him \cite{Muir}
is an example of an extremely useful ``encyclopaedic'' piece of work,
unfortunately rather rare in Mathematics.\footnote{Most interesting
and useful determinantal relations were ``systematized'' by
B. Leclerc in \cite{Lec}.}

In particular, a familiar {\it Binet-Cauchy formula} for the minors
of a product of matrices (cf. \cite{Muir}, 1812, chap. IV), implies
\begin{equation}
S_{J/I}(\A+\B)=\sum_K S_{J/K}(\A)S_{K/I}(\B)\,.
\end{equation}

I want now to discuss some applications of the {\it Bazin-Sylvester
identity} for minors.
Suppose that an $\infty\times n$ matrix is given.
For a subset $A\subset\N$ with $\card(A)=n$, we denote by
$[A]$ the $n\times n$ minor of
the matrix taken on rows indexed by $A$. Now suppose that three subsets
$A,B,C\subset \N$ are given with $\card(A)=\card(C)=p\le n$ and
$\card(A)+\card(B)=n$. Then
\begin{equation}
\det([ABc\setminus a])_{a\in A,c\in C} = [AB]^{p-1}[BC]\,.
\end{equation}
In \cite{LPsh} this identity was used to obtain a generalization to
{\it skew}
Schur functions of the classical {\it Schubert-Giambelli identity}
presenting a Schur function in terms of {\it hook} Schur functions.

Afterwards, Alain and the author used it to derive in \cite{LPr} 
the following determinantal formula for a Schur function in terms of
{\it ribbon} Schur functions.
Take the partition $J=11444677$. The following picture shows
the ``ribbon-decomposition" of the diagram of $J$:

%\vskip30pt


%%% Pierwsza macierz
\setbox0=\hbox{$\odot$\ }
\unitlength=\wd0
\def\z#1{\ifcase#1\odot\or\square\or\blacksquare\or\star\or\aleph\fi}
\def\p#1{\hbox{$\z#1$}}
\begin{center}\begin{picture}(8,8)\small
\put(0,0){\p1}\put(1,0){\p3}\put(2,0){\p3}\put(3,0){\p3}
\put(4,0){\p2}\put(5,0){\p2}\put(6,0){\p0}
\put(0,1){\p4}\put(1,1){\p1}\put(2,1){\p2}\put(3,1){\p2}
\put(4,1){\p2}\put(5,1){\p0}\put(6,1){\p0}
\put(0,2){\p4}\put(1,2){\p3}\put(2,2){\p1}\put(3,2){\p0}
\put(4,2){\p0}\put(5,2){\p0}
\put(0,3){\p3}\put(1,3){\p3}\put(2,3){\p2}\put(3,3){\p1}
\put(0,4){\p2}\put(1,4){\p2}\put(2,4){\p2}\put(3,4){\p0}
\put(0,5){\p0}\put(1,5){\p0}\put(2,5){\p0}\put(3,5){\p0}
\put(0,6){\p0}\put(0,7){\p0}
\put(4,4){\line(1,1)4}
\put(5,4){\hbox{d}}\put(5.6,4.5){\hbox{i}}\put(6,5){\hbox{a}}
\put(6.5,5.5){\hbox{g}}\put(7,6){\hbox{o}}\put(7.5,6.5){\hbox{n}}
\put(8,7){\hbox{a}}\put(8.5,7.5){\hbox{l}}
\end{picture}
\end{center}

%\vskip30pt

Consider now the matrix:
$$\hbox{\small$
\left[\,\CD
S_1 \ & \ S_2 \ & \ S_6 \ & \ S_7 \ & \ S_8 \ & \ S_{11} \ &
 \ S_{13} \ & \ S_{14} \\
S_0 & S_1 & S_5 & S_6 & S_7 & S_{10} & S_{12} & S_{13} \\
\bullet & S_0 & S_4 & S_5 & S_6 & S_9 & S_{11} & S_{12} \\
\bullet & \bullet & S_3 & S_4 & S_5 & S_8 & S_{10} & S_{11} \\
\bullet & \bullet & S_2 & S_3 & S_4 & S_7 & S_9 & S_{10} \\
\bullet & \bullet & S_1 & S_2 & S_3 & S_6 & S_8 & S_9 \\
\bullet & \bullet & S_0 & S_1 & S_2 & S_5 & S_7 & S_8 \\
\bullet & \bullet & \bullet & S_0 & S_1 & S_4 & S_6 & S_7 \\
\bullet & \bullet & \bullet & \bullet & S_0 & S_3 & S_5 & S_6 \\
\bullet & \bullet & \bullet & \bullet & \bullet & S_0 & S_2 & S_3 \\
\bullet & \bullet & \bullet & \bullet & \bullet & \bullet & S_0 & S_1 \\
\bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & S_0 \\
\endCD\,\right]\,,$}
$$
where we use $\bullet$ for $S_i$, $i<0$. Let $A=\{9,12,14,15\}$,
$B=\{{\it 2,3,7,8}\}$, $C=\{{\bf 1,4,5,6}\}$. Then Bazin-Sylvester's
formula expresses $S_{11444677}$ as the following determinant of
$8\times 8$-minors:
$$\hbox{\small$
\left|\,\CD
[{\it 2 3} {\bf 6} {\it 78},12,14,15] \ & \ [{\it 23}{\bf 6}{\it 789},14,15]
\ & \ [{\it 23}{\bf 6}{\it 789},12,15] \ & \ [{\it 23}{\bf 6}{\it 789},12,14]
 \\
[{\it 23}{\bf 5}{\it 78},12,14,15] & [{\it 23}{\bf 5}{\it 789},14,15] &
[{\it 23}{\bf 5}{\it 789},12,15] & [{\it 23}{\bf 5}{\it 789},12,14] \\
[{\it 23}{\bf 4}{\it 78},12,14,15] & [{\it 23}{\bf 4}{789},14,15] &
[{\it 23}{\bf 4}{\it 789},12,15] & [{\it 23}{\bf 4}{\it 789},12,14] \\
[{\bf 1}{\it 2378},12,14,15] & [{\bf 1}{\it 23789},14,15] &
[{\bf 1}{\it 23789},12,15] & [{\bf 1}{\it 23789},12,14] \\
\endCD\,\right|=$}
$$

\smallskip

{\arraycolsep0pt
\def\h{\hphantom{\square}}
$$%\scriptsize
\hbox{\tiny$
=\left|\begin{array}{rrrr}
\ \begin{array}{c}
\z4\\\z4\\\z1\\\ \\\ \\\
\end{array} & \qquad
\begin{array}{ccc}
\z4&\h&\h\\\z4\\\z1\\\z3&\z3&\z3\\\ \\\
\end{array} & \qquad
\begin{array}{cccc}
\z4\\\z4\\\z1\\\z2&\z2&\z2\\&&\z2&\z2\\\
\end{array} & \qquad
\begin{array}{cccc}
\z4\\\z4\\\z1\\\z0&\z0&\z0\\&&\z0&\z0\\&&&\z0
\end{array} \quad \\[3pt]
\begin{array}{cc}
\z3&\z3\\\h&\z3\\&\z1\\\ \\\ \\\
\end{array} & \qquad
\begin{array}{cccc}
\z3&\z3&\h&\h\\\h&\z3\\&\z1\\&\z3&\z3&\z3\\\ \\\
\end{array} & \qquad
\begin{array}{ccccc}
\z3&\z3\\\h&\z3\\&\z1\\&\z2&\z2&\z2\\&&&\z2&\z2\\\
\end{array} & \qquad
\begin{array}{ccccc}
\z3&\z3\\\h&\z3\\&\z1\\&\z0&\z0&\z0\\&&&\z0&\z0\\&&&&\z0
\end{array} \quad \\[3pt]
\begin{array}{ccc}
\z2&\z2&\z2\\&&\z2\\&&\z1\\\ \\\ \\\
\end{array} & \qquad
\begin{array}{ccccc}
\z2&\z2&\z2&\h&\h\\&&\z2\\&&\z1\\&&\z3&\z3&\z3\\\ \\\
\end{array} & \qquad
\begin{array}{cccccc}
\z2&\z2&\z2\\&&\z2\\&&\z1\\&&\z2&\z2&\z2\\&&&&\z2&\z2\\\
\end{array} & \qquad
\begin{array}{cccccc}
\z2&\z2&\z2\\&&\z2\\&&\z1\\&&\z0&\z0&\z0\\&&&&\z0&\z0\\&&&&&\z0
\end{array} \quad \\[3pt]
\quad \begin{array}{cccc}
\z0\\\z0\\\z0&\z0&\z0&\z0\\&&&\z0\\&&&\z1\\\ \\\ \\\
\end{array} & \qquad
\begin{array}{cccccc}
\z0\\\z0\\\z0&\z0&\z0&\z0&\h&\h\\&&&\z0\\&&&\z1\\&&&\z3&\z3&\z3\\\ \\\
\end{array} & \qquad
\begin{array}{ccccccc}
\z0\\\z0\\\z0&\z0&\z0&\z0\\&&&\z0\\&&&\z1\\&&&\z2&\z2&\z2\\&&&&&\z2&\z2\\\
\end{array} & \qquad
\begin{array}{ccccccc}
\z0\\\z0\\\z0&\z0&\z0&\z0\\&&&\z0\\&&&\z1\\&&&\z0&\z0&\z0\\&&&&&\z0&\z0\\&&&&&&\z0
\end{array} \quad
\end{array}
\right|\,.$}
$$}
With this ``generic example'' the reader easily sees a general pattern.
It is also possible to get the ribbon determinant by a suitable deformation
of the Schubert-Giambelli hook determinant (cf.\ \cite{LPr} and Section 6).

A generalization of this result to ``super-Schur'' functions has recently
been obtained by W.Y.C. Chen, G-G. Yan, and A.L.B. Yang \cite{CYY}.

Ribbon Schur functions admit a lift to the noncommutative
situation, and play a basic role in the theory of {\it noncommutative}
symmetric functions (cf., e.g., \cite{Th2}).

For a more systematic development of the theory of symmetric functions,
the reader is referred to Macdonald's book \cite{Mcd1}. Alain's recent 
book
\cite{LCBMS} exploits a more ``$\lambda$-ring approach'' to symmetric
functions which allows one to treat them as operators on polynomials.
Among the techniques used most frequently, we mention ``transformations 
and specializations of alphabets'' (see also \cite{LSform}, \cite{LPs}).

\smallskip

Schur functions admit a useful generalization to (no more symmetric)
{\it multi-Schur functions}.
Given  $I=(i_1,i_2, \ldots, i_r) \in  \N^r $, and alphabets\break
$\A_1,\ldots ,\A_r$, $\B_1,\ldots ,\B_r$, the {\it multi-Schur function}
$S_I(\A_1\moins \B_1,\ldots,\A_r\moins \B_r)$ is
\begin{equation}
S_I(\A_1\moins\B_1,\ldots, \A_r\moins \B_r) := \det(
     S_{i_k+k-h}(\A_k\moins \B_k))_{1\leq h,k \leq r}  \ .
\end{equation}
For example, $S_{5,7,2,0,3}(\A\moins\B,\C\moins\D,\E\moins\F,\G\moins\H,
\K\moins\L)$ equals

\smallskip
$$\hbox{\small$
\left|\,\CD
S_5(\A\moins\B) \ & \ S_8(\C\moins\D) \ & \ S_4(\E\moins\F) \ & \
S_3(\G\moins\H) \ & \ S_7(\K\moins\L) \\
S_4(\A\moins\B) & S_7(\C\moins\D) & S_3(\E\moins\F) & S_2(\G\moins\H) & 
S_6(\K\moins\L)\\
S_3(\A\moins\B) & S_6(\C\moins\D) & S_2(\E\moins\F) & S_1(\G\moins\H) & 
S_5(\K\moins\L)\\
S_2(\A\moins\B) & S_5(\C\moins\D) & S_1(\E\moins\F) & 1 & S_4(\K\moins\L)\\
S_1(\A\moins\B) & S_4(\C\moins\D) & 1 & 0 & S_3(\K\moins\L)\\
\endCD\,\right|\,.$}
$$

\smallskip
The following simple lemma is masterfully exploited by Alain in numerous
situations (cf., e.g., \cite{LSform}, \cite{LCBMS}):

\begin{lemma}[Jacobi-Lascoux Lemma] Let \ $\D_0,\D_1,\ldots,\D_{n-1}$ be
alphabets such that $\card(\D_i)\le i$. Then
\begin{equation}
S_I(\A_1\moins \B_1,\ldots,\A_r\moins \B_r)=
\det(S_{j_k-i_h+k-h}(\A_k\moins\B_k\moins\D_{r-h}))_{1\leq h,k \leq r}\,.
\end{equation}
In other words, we can replace, in row $h$, $\A_{\bullet}\moins\B_{\bullet}$
by $\A_{\bullet}\moins\B_{\bullet}\moins\D_{r-h}$ without changing 
the determinant.
\end{lemma}
This is easily seen from the relation:
\begin{multline}
S_i(\A\moins\B\moins\D_h)=S_i(\AB)+S_1(-\D_h)S_{i-1}(\AB)+\cdots\\
{}\cdots+S_h(-\D_h)S_{j-h}(\AB)\,,
\end{multline}
by subtracting from successive rows of the LHS-determinant,
appropriate combinations of its lower rows.

\bigskip

For example,
$$\hbox{\small$
\left|\,\CD
S_i(\A) \ & \ S_{j+1}(\B) \ & \ S_{k+2}(\C)\\
S_{i-1}(\A) & S_j(\B) & S_{k+1}(\C)\\
S_{i-2}(\A) & S_{j-1}(\B) & S_k(\C)\\
\endCD\,\right|
=
\left|\,\CD
S_i(\A\moins y\moins z) \ & \ S_{j+1}(\B\moins y\moins z) \
& \ S_{k+2}(\C\moins y\moins z)\\
S_{i-1}(\A\moins x) & S_j(\B\moins x) & S_{k+1}(\C\moins x)\\
S_{i-2}(\A) & S_{j-1}(\B) & S_k(\C)\\
\endCD\,\right|\,.$}
$$

In particular, the Jacobi-Lascoux Lemma allows one to present
a monomial in a determinantal way:
$$
a^5b^3c^2=S_{2,3,5}(a+b+c,a+b,a)\,,
$$
which sometimes appears to be useful.


\section{Schubert polynomials}

Schur functions and multi-Schur functions are a very particular
case of a family of {\it Schubert polynomials} invented by Alain
and M-P. Sch\"utzen\-berger in 1982 (\cite {LSschub}). We shall follow
the convention (used in \cite{LSschub}) that operators acts on their 
{\it left}.

Let $n$ be a fixed positive integer.
The symmetric group ${\mathfrak S}_n$ is the group with generators
$s_1,\ldots,\, s_{n-1}$, where
\begin{equation}
s_i:=[1,\ldots,i-1,i+1,i,i+2,\ldots,n]\,,
\end{equation}
subject to the relations:
$$
s_i^2=1\,, \ \ s_{i-1} s_i s_{i-1} =s_i s_{i-1} s_i\,, \ \ \hbox{and} \ \
s_is_j =s_js_i \quad \forall i,j: |i-j|>1 \,.
$$

Let $\A=(a_1,\ldots,a_n)$ be an alphabet of variables. The group
${\mathfrak S}_n$ acts on the polynomial ring ${\bf Z}[\A]$ by permuting
the variables. We define, following Newton, a (simple) divided difference
$\d_i: {\bf Z}[\A] \to {\bf Z}[\A]$, by
\begin{equation}
f \d_i = \frac{f-f^{s_i}}{a_i-a_{i+1}}
\end{equation}
for $f\in {\bf Z}[\A]$, where $f^{\sigma}:=f(a_{\sigma_1},\ldots,
a_{\sigma_n})$ for $\sigma \in {\mathfrak S}_n$.

\smallskip

The $\d_i$'s satisfy the Moore-Coxeter relations.
Thus for $\sigma \in {\mathfrak S}_n$, given its reduced decomposition
$\sigma=s_{i_1}\, s_{i_2}\, \cdots s_{i_{\ell}}$, the operator
\begin{equation}\label{d}
\d_{\sigma}:= \d_{i_1}\, \d_{i_2}\, \cdots \d_{i_{\ell}}
\end{equation}
is well defined (i.e. does not depend on a reduced decomposition of
$\sigma$).

Let $\B=(b_1,\ldots,b_n)$ be another alphabet of variables.
The definition of Schubert polynomials $\X(\A,\B)$
(often called {\it double Schubert polynomials}\footnote{It seems that
we need a deeper reflection on ``double'' versus ``single'' in various
mathematical situations.})
of Alain and M-P. Sch\"utzen\-berger goes as follows.
For $\omega=[n,n-1,\ldots,1]$,
\begin{equation}
\X_{\omega}(\A,\B)= \prod_{i+j \le n}(a_i-b_j)\,,
\end{equation}
and for arbitrary $\sigma \in {\mathfrak S}_n$,
\begin{equation}
\X_\sigma(\A,\B)=\X_{\omega}(\A,\B) \d_{\omega \sigma}\,.
\end{equation}
(In fact, it makes sense to speak about Schubert polynomials for
any pair of alphabets $\A, \B$ of cardinality $\ge n$, cf. \cite{LCBMS}.)
Here we use
divided differences w.r.t. the variables $\A$, but the variables
from $\B$ appear just as systematically because of the equation:
\begin{equation}
X_\sigma(\A,\B)= (-1)^{l(\sigma)} X_{{\sigma}^{-1}}(\B,\A)\,.
\end{equation}

Schubert polynomials satisfy the following fundamental {\it vanishing
property}: all positive degree Schubert polynomials vanish under
the specialization $\A=\B$.

When we specialize $\B$ with zeros, we get {\it simple Schubert polynomials}
$X_{\sigma}(\A)$; note that $X_\omega(\A)=
a_1^{n-1}a_2^{n-2}\cdots a_{n-1}$. For a didactic exposition of Schubert
polynomials, see \cite{Mcd2}. {\it Schubert functors} having characters
equal to the corresponding simple Schubert polynomials were constructed
by W. Kra\'skiewicz and the author in \cite{KP} (see also \cite{LSf}).

There is an associated reproducing kernel in this story. Consider the ring:
${\mathfrak Pol}={\bf Z}[\A,\B]/{\cal I}$\,,
where $\cal I$ is the ideal generated by all $f(\A)-f(\B)$ where $f$ is
symmetric. On ${\mathfrak Pol}$ there is a scalar product with values
in ${\mathfrak Sym}(\A)$, defined by
\begin{equation}\label{sc}
(f,g):= f g \d_\omega
\end{equation}
for $f,g\in {\mathfrak Pol}$. Note that the {\it Jacobi symmetrizer}
$\d_\omega$ acts on ${\bf \Z}[\A]$ as follows:
\begin{equation}
f \mapsto \sum_{\sigma\in {\mathfrak S}_n} \Bigl(\frac{f}{\prod_{i<j}
(a_i-a_j)}\Bigr)^{\sigma}\,.
\end{equation}

\smallskip

Set
\begin{equation}
{\bf K}(\A,\B)
:=\prod_{1\le i< j\le n}(b_i-a_j)
=\pm \X_{\omega}((a_n,\ldots,a_1),\B)\,.
\end{equation}
For any polynomial $f$ in $n$ variables, in $\mathfrak Pol$ we have
\begin{equation}
(f(\A),{\bf K}(\A,\B))=f(\B)\,.
\end{equation}

\smallskip

There is the following nonsymmetric Cauchy-type formula for Schubert
polynomials:
\begin{theorem}[Cauchy-Lascoux Formula]\label{TCL}  Let $\A,\B,\C$ be three
alphabets of cardinality $\ge n$. Then for $\sigma \in {\mathfrak S}_n$,
\begin{equation}\label{CL}
\X_{\sigma}(\A,\B)= \sum_{\tau, \eta} \X_{\tau}(\C,\B) \X_{\eta}(\A,\C)\,,
\end{equation}
where the sum is over $\sigma=\tau \eta$ with $l(\sigma)=l(\tau)+l(\eta)$.
\end{theorem}
(Cf. \cite{LCBMS}, Theorem 10.2.6.)
The most remarkable thing about the Cauchy-Lascoux formula is that we can
take an arbitrary(!) alphabet $\C$.
To be just modest, for $\C=(0,\ldots)$, Eq. (\ref{CL}) gives
a quadratic expression of a Schubert polynomial in terms of simple Schubert
polynomials in $\A$ and $\B$.
When $\sigma=\omega$, we get an expression for the total Chern
class of the full flag manifold (cf.\ \cite{Lch} -- the paper where
(double) Schubert polynomials originally appeared).
When $\sigma=[n+1,\ldots,n+m,1,\ldots,m]$, we get an expression
for the total Chern class of the Grassmannian (i.e., the Cauchy formula
for the resultant from Section 1).

In 1978, during his first visit to Poland, Alain said to me a sentence which
impressed me a lot at that time:

\smallskip

\centerline{``The Schubert variety is a Schur function.'' \footnote{By
extending the message of this sentence, I was happy (in 1986) to
communicate to Alain that ``The Lagrangian Schubert class is a
Schur $Q$-function.'' (cf.\ \cite{Prag1}, \cite{Prag2}).}}

\smallskip

\noindent
This was probably the fastest introduction to the cohomology ring of
a Grassmannian. I am pretty sure that what Alain does in Algebra or
Combinatorics is often deeply motivated by Geometry (of Grassmannians,
flag varieties, Schubert varieties etc.) This is also the case
for Schubert polynomials. In \cite{F}, W. Fulton showed that the polynomials
$\X_\sigma(\A,\B)$ are the cohomology classes of {\it flagged degeneracy
loci} associated with a morphism of vector bundles with {\it Chern roots}
$\A$ and $\B$. The key point is that divided differences admit a
geometric interpretation as {\it correspondences} in flag bundles
({\it loc.cit}). This is also a proper place to mention the results
of I.N. Bernstein, I.M. Gelfand and S.I. Gelfand \cite{BGG},
and M. Demazure \cite{Dem1},
\cite{Dem2} from the 70's, giving formulas for Schubert classes
for flag manifolds in terms of divided differences (for any semisimple
group); however in the $SL_n$-case their expressions were more
complicated than the ones given by Schubert polynomials.
In \cite{P}, one can find more details on these and related issues.

As far as I know, one of the main motivations for introducing Schubert
polynomials was the authors' fascination for {\it Newton interpolation}.
In \cite{LSint} (see also \cite{LCBMS}, pp. 148--149), the following
formula was given. Suppose that $\A$
and $\B$ are two alphabets of variables of cardinalities $n$ and $\infty$,
respectively.
Let ${\mathfrak S}_{\infty}$ be the infinite symmetric group acting
in a natural way on $\B$. For any polynomial $f$ in $n$ variables,
we have
\begin{equation}\label{int}
f(\A)=\sum_\sigma f(b_1,\ldots,b_n) \d_{{\sigma}^{-1}} \
\X_{\sigma}(\A,\B) \ ,
\end{equation}
where the sum is over $\sigma \in {\mathfrak S}_{\infty}$
satisfying $\sigma(n+1)<\sigma(n+2)< \ldots$\,, and the operators
$\d_{\sigma}$ act on the variables from $\B$.

For $n=1$, the sum is over the identity permutation and permutations
$\sigma^{(i)}=s_i s_{i-1} \cdots s_1$, where $i\ge 1$. We have
$$
\X_{\sigma^{(i)}}(\A,\B)=(a-b_1)(a-b_2) \cdots (a-b_i)\,.
$$
Therefore (\ref{int}) reads:
\begin{multline*}
f(a)=f(b_1)+f(b_1)\d_1 \cdot (a-b_1)+f(b_1)\d_1\d_2 \cdot (a-b_1)(a-b_2) \\
+f(b_1)\d_1\d_2\d_3 \cdot (a-b_1)(a-b_2)(a-b_3) + \cdots \ ,
\end{multline*}
which is the Newton Interpolation Formula.
Note that Newton's interpolation and that of Lagrange are used
in \cite{FuL}, written with Amy M. Fu, to deduce some new $q$-identities.

Alain and M-P. Sch\"utzenberger invented also {\it Grothendieck polynomials}
which evaluate K-theoretic classes of the structure sheaves of Schubert
varieties. This was done with the help of {\it isobaric divided
differences} and led to Combinatorial K-theory (cf., e.g., \cite{Buch}).
Other important polynomials defined with the help of isobaric divided
differences are {\it Key polynomials}. 
Let $\A$ be a countable alphabet of variables.
Given a permutation $\sigma$,
let $c=(c_1,c_2,\ldots)$ denote its {\it code}, where
\begin{equation}
c_i=\card\{j>i: \sigma_j<\sigma_i\}
\end{equation}
for $i=1,2\ldots$. The group ${\mathfrak S}_\infty$ acts on codes by permuting
their components.

Now, if $\sigma$ is {\it dominant} (i.e., $c$ is weakly decreasing), then
\begin{equation}
K_\sigma(\A):=X_\sigma(\A)=a^c=a_1^{c_1}a_2^{c_2} \cdots \ \,,
\end{equation}
a monomial.
Otherwise, if $c$ and $i$ are such that $c_i<c_{i+1}$, then
\begin{equation}
K_{cs_i}(\A)=K_c(\A)\pi_i\,,
\end{equation}
where $\pi_i$ is a (simple) isobaric divided difference defined for
$f\in {\bf Z}[\A]$ by
\begin{equation}
f \pi_i = \frac{a_if - a_{i+1}f^{s_i}}{a_i - a_{i+1}}\,.
\end{equation}

The Schubert polynomial is a {\it nonnegative} combination of Key
polynomials. For example,
$$
X_{2154367\ldots}=K_{30010\ldots}+K_{10210\ldots}+K_{2020\ldots}\,.
$$
Since Key polynomials also have a geometric meaning (they are related
to sections of line bundles on Schubert varieties (cf.\ \cite{FL})),
this nonnegativity should have a geometric explanation.

Finally, note that Key polynomials are used in \cite{Lschur} to give a
``Cauchy-type'' quadratic expression for the product:
$$
\prod_{i+j\le n+1}(1-a_ib_j)^{-1}\,.
$$

\medskip

The Jacobi symmetrizer $\d_\omega$, used above to define the
scalar product, is a ``queen mother'' of many other {\it symmetrizing
operators}. Apart from their use in interpolation theory, numerical analysis,
and cohomology theory of flag manifolds, such operators are useful
in algebraic computations. Consider the operator $\pi=\pi_\omega$
(defined using isobaric divided differences and a reduced decomposition
of $\omega$).
Using symmetrization, its action on the ring of series in $a_1,\ldots,a_n$
is described as follows:
\begin{equation}
f \mapsto \sum_{\sigma\in {\mathfrak S}_n} \Bigl(f \cdot \prod_{i<j}
\bigl(1-\frac{a_j}{a_i}\bigr)^{-1}\Bigr)^{\sigma}\,.
\end{equation}
Two basic properties of $\pi$ are:
\begin{equation}\label{5.2}
f \ \hbox{symmetrical} \Rightarrow fg \pi=f(g \pi)\,,
\end{equation}
\begin{equation}
a_1^{i_1}\cdots a_n^{i_n} \pi = S_I(\A) \ \hbox{when} \ I\in {\N}^n\,.
\end{equation}
Schur functions are {\it eigenfunctions} of $\pi$, i.e. for $m\le n$,
$I\in \N^m$, $J\in \N^{n-m}$, we have
\begin{equation}\label{5.4}
S_I(a_1+\cdots+a_m) \ S_J(a_{m+1}+\cdots+a_n) \ \pi=S_{IJ}(\A)\,.
\end{equation}
where $IJ$ is the element $(i_1,\ldots,i_m,j_1,\ldots,j_{n-m})$ of ${\N}^n$.
For $n\ge 2$,
from Eq. (\ref{5.4}) with $m=n-1$, $I=j0\ldots0$, $J=0$,
it follows that
\begin{equation}\label{5.5}
\prod_{i\le n-1}(1\moins a_i)^{-1} \pi=\prod_{i\le n}(1\moins a_i)^{-1} \,.
\end{equation}
More directly, writing
$$
\prod_{i\le n-1}(1\moins a_i)^{-1} = (1\moins a_n) \prod_{i\le n}(1\moins a_i)^{-1},
$$
we can use (\ref{5.2}) and recover Eq. (\ref{5.5}):
$$
\prod_{i\le n-1}(1\moins a_i)^{-1} \pi= 1\moins a_n \ \pi \cdot
\prod_{i\le n}(1\moins a_i)^{-1}=\prod_{i\le n}(1\moins a_i)^{-1}\,.
$$

It is not hard to check the following identities:
\begin{equation}\label{5.6}
\prod_{i\le n-1}(1\moins a_ia_n)\pi=1 \ \ \hbox{(resp.}
\ 1\moins a_1\cdots a_n)
\ \ \hbox{for} \ n \ \hbox{odd \ (resp. even)}\,,
\end{equation}

\begin{equation}\label{5.7}
a_n \prod_{i\le n-1}(1\moins a_ia_n)\pi=0 \ \ \hbox{(resp.}
\ a_1\cdots a_n)
\ \ \hbox{for} \ n \ \hbox{even \ (resp. odd)}\,.
\end{equation}

We illustrate on a classical formula how to use the operator $\pi$. Let
us consider
$$
F(n\moins 1)=\prod_{i\le n-1} (1\moins a_i)^{-1} \prod_{i<j\le n-1}
(1\moins a_ia_j)^{-1}\,.
$$
Then
$$
F(n\moins 1)\pi =(1\moins a_n)(1\moins a_1a_n)\cdots
(1\moins a_{n-1}a_n)F(n)\pi=
(1\moins a_1\cdots a_n)F(n)
$$
thanks to Eqs. (\ref{5.2}), (\ref{5.6}), and (\ref{5.7}). Assuming now
by induction that $F(n\moins 1)=\sum S_I(a_1+\cdots+a_{n-1})$, summed over
all partitions $I=(i_1,\ldots,i_{n-1})$, we transform this identity
with the help of $\pi$ into
$$
(1\moins a_1\cdots a_n)F(n)= \sum S_I(a_1+\cdots+a_n)\,,
$$
the sum over the same $I$'s. Dividing by
$(1\moins a_1\cdots a_n)$, we finally obtain the identity of Schur:
\begin{equation}
\prod_{i\le n} (1\moins a_i)^{-1} \prod_{i<j\le n} (1\moins a_ia_j)^{-1}=
\sum S_J(a_1+\cdots+a_n)\,,
\end{equation}
with the sum now over all partitions $J=(j_1,\ldots,j_n)$.

Also other identities for {\it $S$-function series} can be proven
in a similar way.
We refer the reader to \cite{LPs}, where, e.g., the $S$-function expansion
of
$$
\prod_{i\le n}(1-a_i+a_i^2)^{-1}\prod_{i<j\le n}(1-a_ia_j)^{-1}
$$
is given using the same method.

\section{Divided differences versus vertex operators}

In this section, I will tell you about the biggest mathematical surprise
that came to me from Alain (about 1995).

Let $n$ be a fixed positive integer.
Besides the symmetric group (i.e. the Weyl group of
type A) ${\mathfrak S}_n$, we shall consider two other Weyl groups.

The hyperoctahedral group (i.e. the Weyl group of type C)
$\mathfrak C_n$, is an extension of $\mathfrak S_n$
by an element $s_0$ such that $s_0^2=1$, $s_0 s_1 s_0 s_1=s_1 s_0 s_1 s_0$,
and $s_0 s_i=s_i s_0$ for $i\ge 2$\,.

The Weyl group $\mathfrak D_n$ of type D is an extension of
$\mathfrak S_n$ by an element $\ss$ such that $\ss^2=1$,
$s_1 \ss\ =\ss s_1$, $\ss s_2 \ss = s_2 \ss s_2$, and
$\ss s_i=s_i \ss$ for $i>2$.

Let $\A=(a_1,\ldots,a_n)$ be an alphabet of variables.
The groups $\mathfrak S_n$, $\mathfrak C_n$,
and $\mathfrak D_n$ act on ${\bf Z}[\A]$: $s_i(a_i)=a_{i+1}$,
$s_0(a_1)=-a_1$, $\ss(a_1)=-a_2$.

\smallskip

Apart from Newton's divided differences:
$$
f\, \d_i :=  \frac{f-f(\ldots,a_{i+1},a_i,\ldots)}{a_i-a_{i+1}}\,,
$$
we have two other divided differences associated to simple roots
of type C and D:
\begin{equation}
f\, \d_0 :=  \frac{f-f(-a_1,a_2,\ldots)}{2a_1}\,,
\end{equation}
and
\begin{equation}
f\, \dd := \frac{f-f(-a_2,-a_1,a_3,\ldots)}{a_1 + a_2}\,.
\end{equation}
The $\d_i,\, \d_0,\, \dd$ satisfy the Coxeter relations, together
with the relations
\begin{equation}
\dd^2= 0 =   \d_i^2 \ \ \ \hbox{for} \ \ \ 0\leq i <n \ .
\end{equation}
Therefore, to any element $w$ of the groups $\mathfrak C_n$
and $\mathfrak D_n$, there corresponds
a {\it divided difference} $\d_w$. Any reduced decomposition
$s_{i_1}\, s_{i_2}\, \cdots s_{i_{\ell}}= w$ gives rise to a factorization
$\d_{i_1}\, \d_{i_2}\, \cdots \d_{i_{\ell}}$ of $\d_w$ (\cite{BGG},
\cite{Dem1}).

We now invoke single Schubert polynomials. Here it will be more handy
to index them by the codes of permutations, and for a didactic
reason we repeat their definition in this setting. Write
$\rho$ for the sequence $(n-1,\ldots, 1,0)$. One defines recursively
Schubert polynomials $Y_\alpha$, for any sequence 
$\alpha=(\alpha_1, \ldots, \alpha_n) \in \N^n$
such that $\alpha_j\le n-j$ for $j=1,\ldots,n$, by
\begin{equation}
Y_\alpha\, \d_i = Y_{\beta}
 \ , \ \ \ \hbox{if} \ \ \alpha_i >\alpha_{i+1}\,,
\end{equation}
where
\begin{equation}
\beta = (\alpha_1,\ldots,\alpha_{i-1}, \alpha_{i+1},
\alpha_i-1, \alpha_{i+2},\ldots, \alpha_n)\,,
\end{equation}
starting from $Y_\rho= x^\rho$\,.

If $\alpha\in \N^n$ is weakly decreasing,
then $Y_\alpha$ is equal to the monomial $x^\alpha$ -- a property already
mentioned.
If, on the contrary,
$\alpha_1\leq \cdots \leq \alpha_k $ and $\alpha_{k+1}=\cdots =\alpha_n=0$,
for some $k\le n$,
then
$Y_\alpha$ coincides with the Schur polynomial $S_I(a_1+\cdots+a_k)$,
where $I=(\alpha_1,\ldots, \alpha_k)$.

\smallskip

Besides Schubert polynomials, we shall need the following
$\Q$--{\it polynomials} of \cite{PR}.
We set $\Q_i:= \Lambda_i=\Lambda_i(\A)$,
the $i$th elementary symmetric polynomial in $\A$\,.
Given two nonnegative integers $i \ge j$, we put
\begin{equation}
\Q_{i,j}:=\Q_i \Q_j+2\sum\limits^j_{p=1}(-1)^p\Q_{i+p} \Q_{j-p}\,.
\end{equation}

We have $\Q_{i,i}(\A) = \Lambda_i(a_1^2+\cdots+a_n^2)$.

Given any English partition\footnote{In this section, we use English
partitions in order not to offend Ron King who is in the audience, and
D.E. Littlewood whose result will be used here.}
$I=(i_1 \ge \ldots \ge i_k)$,
where we can assume $k$ to be even, we set
\begin{equation}
\Q_I:= \hbox{Pfaffian} (M)\,,
\end{equation}
where $M=(m_{p,q})$ is the $k\times k$ skew-symmetric matrix
with $m_{p,q}= \Q_{i_p,i_q}$  for $1 \le p < q \le k$.

\medskip

\noindent
{\bf For special attention of the audience of SLC:} $\Q$-polynomials form
an interesting basis of the ring of symmetric polynomials. A $\Q$-polynomial
is not, in general, a nonnegative combination of Schur $S$-polynomials.
It is, however, a nonnegative combination of monomials. It would be
useful to establish a combinatorial rule for the coefficients of this
expansion. Also, it is worth studying other combinatorial properties
of $\Q$-polynomials (cf. \cite{PR}, \cite{Tam}).

\bigskip

Here is a sign that the $\Q_I$'s are well suited to divided
differences. Consider the following operator:
\begin{equation}
\nabla:= (\d_0 \d_1 \cdots \d_{n-1}) \cdots (\d_0 \d_1) \d_0\,.
\end{equation}
Denote by
\begin{equation}
\langle \ ,\ \rangle : {\mathfrak Sym}(\A) \times {\mathfrak Sym}(\A) \to
{\mathfrak Sym}(a_1^2,\ldots,a_n^2)
\end{equation}
the scalar product defined for $f, g\in {\mathfrak Sym}(\A)$ by
\begin{equation}
\langle f,g \rangle := f g \, \nabla\,.
\end{equation}
Let $\rho(n)$ denote the strict partition $(n,n-1,\ldots,1)$.
We have the
following {\it orthogonality property} of $\Q$-polynomials: For strict
partitions $I,J\subseteq \rho(n)$,
\begin{equation}
\langle \Q_I,\Q_{\rho(n) \smallsetminus J} \rangle
= \pm \delta_{I J},
\end{equation}
where $\rho(n) \smallsetminus J$ is the strict partition whose parts
complement the parts of $J$ in $\{n,n-1,\ldots, 1\}$.
Given another alphabet $\B$ of $n$ variables, define
\begin{equation}
\Q(\A, \B):= \sum \Q_I(\A)\, \Q_{\rho(n) \smallsetminus I}(\B) \ ,
\end{equation}
where the summation is over all strict partitions $I\subseteq \rho(n)$.
The polynomial
$\Q(\A,\B)$ is a reproducing kernel for $\langle \ ,\ \rangle$:
for a symmetric polynomial $f$ in $n$ variables,
\begin{equation}
\langle f(\A)\,, \Q(\A,\B) \rangle = \pm f(\B) \ .
\end{equation}
In the beginning of 90's, computing with SYMMETRICA \cite{KKL},
J. Ratajski and the author
obtained evidence that the following two identities should be true
for a strict partition $I\subset \rho(n)$:

Firstly,
\begin{equation}\label{Q1}
\Q_I \, \d_0 \d_1 \cdots \d_{i_1-1}
=\pm  \Q_{(i_2,i_3,\ldots)}\,.
\end{equation}

\smallskip

Secondly, using standard barred-permutation notation for elements
of the group ${\mathfrak C}_n$,
we have for $w(I):=[i_1,\ldots,i_{\ell(I)},\overline{j_1}, \ldots ,
\overline{j_h}]$\,,
\begin{equation}\label{Q2}
x^{\rho} \Q_{\rho(n)} \, \partial_{w(I)} = \pm  \Q_I\,.
\end{equation}
We tried to find conceptual proofs of these identities.
More generally, let us consider the following operator for $k\le n$,
\begin{equation}
\nabla_k(n):= (\d_0 \d_1 \cdots \d_{n-1}) \cdots (\d_0 \d_1
\cdots \d_{n-k})\,.
\end{equation}
The problem is, how to compute with $\nabla_k(n)$ ?
Here is an answer.
\begin{theorem}[\cite{LPadv}]\label{Tc} Let $k\leq n$ and let
$\alpha =(\alpha_1 \leq \cdots \leq \alpha_k) \in \N^{k}$
with $\alpha_k \le n-k$. Suppose that $I\subseteq \rho(n)$
is a strict partition.
Then the image of $\Q_I \, Y_\alpha$ under $\nabla_k(n)$ is 0 unless
$n-0-\alpha_1,\ldots, n-(k-1) -\alpha_k$ are parts of $I$. In this case,
the image is $\pm \Q_J$, where $J$ is the strict partition with parts
$\{i_1,\ldots, i_{\ell(I)} \} \smallsetminus
\{ n-0-\alpha_1,\ldots, n-(k-1)-\alpha_k \}
$.
\end{theorem}

For example, for $n=7$ and $k=2$\,,
$$
\aligned
&\Q_{(5,4,3,2,1)} \, Y_{(2,5)} \, \nabla_2(7) \cr
&=\Q_{(5,4,3,2,1)} \, Y_{(2,5)}\, (\d_0 \d_1 \d_2 \d_3 \d_4 \d_5 \d_6)
(\d_0 \d_1 \d_2 \d_3 \d_4\d_5)= \Q_{(4,3,2)}\,; \cr
\endaligned
$$
and for $k=3$\,, we have
$$
\Q_{(7,5,4,3,1)} \, Y_{(2,3,4)} \nabla_3(7) = \Q_{(7,4)}\,.
$$

This theorem implies (\ref{Q1}) and (\ref{Q2}) ({\it loc.cit.}).

In fact the ``key case'' is the case $k=1$.
Suppose $n\ge p>0$. Let  $IpJ\subseteq \rho(n)$ be a strict partition.
Then
\begin{equation}\label{key1}
x_1^{n-p}\Q_{IpJ} \, \d_0 \d_1 \cdots \d_{n-1}=\pm \Q_{IJ}\,.
\end{equation}
If $H\subseteq \rho(n)$ is a strict partition with no part equal to $p$, 
then
\begin{equation}\label{key2}
x_1^{n-p}\Q_H \, \d_0 \d_1 \cdots \d_{n-1} = 0\,.
\end{equation}

\smallskip

After digesting our computations, Alain conceived the idea of using
{\it vertex operators} \footnote{Perhaps for J-Y. Thibon \cite{Th}
and some
other participants of SLC, this is ``bread and butter'', but for us
this was fantastic and absolutely genial!}. These are some (formally)
infinite differential operators (mixing multiplication and differentiation)
born in the theory of Kac-Moody algebras and inspired by Mathematical
Physics. In the proof below, where operators act on symmetric functions,
they serve to give ``long'' but useful expressions for $s_0$ and $\d_0$\,.

\bigskip

\noindent
{\bf Sketch of proof of (\ref{key1}) and (\ref{key2})} \ Let $( \ , \ )$
be the standard inner
product on $\mathfrak Sym$, the ring of symmetric functions in a countable
set of variables $a_1,a_2,\ldots$. We shall use the {\it Foulkes'
derivative} $D_f$ for $f\in {\mathfrak Sym}$:
$$
(g\cdot D_f, h) = (g, f\cdot h)\,.
$$
We shall treat polynomials as operators acting by multiplication. 
We have (cf. \cite{LPadv}, Lemma 5.6):
\begin{equation}
s_0 = 1 - 2D_{P_1}a_1 + 2D_{P_2}a_2-2D_{P_3}a_3+\cdots \,.
\end{equation}
Here, $P_i$ is the sum of all hook Schur functions of degree $i$ (called
Schur's $P$-functions, well-established among the audience of SLC
\cite{LLTp}).
Thus
\begin{equation}
\d_0= D_{P_1} - D_{P_2}a_1 + D_{P_3}a_1^2 - \cdots \,.
\end{equation}
Using this equality and
\begin{equation}
a_1^p\d_1 \cdots \d_{n-1}
=S_{p-n+1}(a_1+\cdots+a_n)\,,
\end{equation}
we compute for $0<k\le n$,
\begin{equation}
a_1^{n-k} \d_0\d_1\cdots \d_{n-1}
= D_{P_k}-D_{P_{k+1}}S_1 +D_{P_{k+2}}S_2 - \cdots \,.
\end{equation}
(after restriction to the first $n$ variables).
Now define
\begin{equation}
U_k^S:= D_{P_k} - D_{P_{k+1}}S_1 + D_{P_{k+2}}S_2 - \cdots\,.
\end{equation}
and
\begin{equation}
U_k^{\Lambda}:= D_{P_k} - D_{P_{k+1}}\Lambda_1 +
D_{P_{k+2}}\Lambda_2 - \cdots \,.
\end{equation}
What is to be shown is:
\begin{equation}
\Q_{IkJ} \ U^S = \pm \Q_{IJ} \ \ \ \hbox{or}  \ \ \  0\,.
\end{equation}
Note that $U_k^S$ is dual to $U_k^{\Lambda}$ for the involution:
\ $S_I \to S_{{I^{\sim}}}$.
So if we define $Q_I'$\footnote{These functions were studied on another
occasion by Alain, B. Leclerc, and J-Y. Thibon \cite{LLT}.}
as duals to $\Q_I$, then
it is required to show:
\begin{equation}
Q_{IkJ}' \ U_k^{\Lambda} = \pm Q_{IJ}' \ \ \  \hbox{or} \ \ \ 0\,.
\end{equation}
We invoke now a result of Littlewood: {\it The basis $\{Q_J'\}$,
where $J$ runs over strict partitions, is conjugate to $\{P_J\}$,
the basis of Schur's $P$-functions}.
Define
\begin{equation}
V_k^{\Lambda}:= P_k-D_{\Lambda_1}P_{k+1}+D_{\Lambda_2}P_{k+2}-\cdots \,.
\end{equation}
So finally
we must show:
\begin{equation}
P_J \ V_k^{\Lambda}= P_{kJ}
\end{equation}
-- but this follows from:
\begin{equation}
P_{kJ}= \sum_{j\ge 0}(-1)^j \ P_J D_{\Lambda_j} \ P_{k+j}\,.
\end{equation}
We refer to \cite{LPadv} for details. We do not know proofs of Eqs.
(\ref{Q1}) and (\ref{Q2}) without using vertex operators.

The formulas discussed here are important for cohomology classes of
Lagrangian Schubert classes and degeneracy loci \cite{PR}, \cite{LPadv},
and, consequently, for quantum cohomology of Lagrangian Grassmannians
(cf., e.g., \cite{Tam}).

For Weyl groups of type D, there are similar results. As ${\mathfrak D}_n$
is a subgroup of ${\mathfrak C}_n$, they can be, in fact, deduced from 
Theorem \ref{Tc} with the help of some additional computations with
divided differences of type C. It is convenient to perform them using 
some planar displays which is the subject of the next section.


\section{Planar displays}

First we want to show another derivation of the ribbon determinant
from Section 3 with the help of some deformations of diagrams.
Given a skew diagram, its {\it right corner} is the box at the extreme right
of the bottom row, and its {\it left corner} is the upper box of the left
column. Given two skew diagrams $H, K$, let $\aleph$ be the right corner
of $H$ and $\vrule width6.5pt height6.5pt depth0pt$ the left corner of $K$.
We define
$H \blacktriangleright K$ to be the skew diagram obtained by gluing the
two diagrams by their corners, $\aleph$, 
$\vrule width6.5pt height6.5pt depth0pt$
being on the same horizontal, and $H \blacktriangledown K$ to be the skew diagram
obtained by gluing the two corners on a vertical:

{\arraycolsep0pt
\def\h{\hphantom{\square}}
$$%\scriptsize
H\blacktriangleright K=
{\hbox{\small$\begin{array}{cccccccccccc}
\z0&\z0\\
\z0&\z0&\z0&\z0\\
\h&\h&\z0&\z0&\z0&\z0&\h&\h&\h&\h&\h&\h\\
\h&\h&\h&\z0&\z0&\z4&\z2&\z3&\z3&\z3&\z3&\z3\\
\h&\h&\h&\h&\h&\h&\h&\h&\z3&\z3&\z3&\z3\\
\h&\h&\h&\h&\h&\h&\h&\h&\h&\h&\z3&\z3
\end{array}$}}\qquad
H\blacktriangledown K=
{\hbox{\small$\begin{array}{cccccccccccc}
\z0&\z0&\h&\h&\h&\h&\h&\h&\h&\h&\h\\
\z0&\z0&\z0&\z0\\
&&\z0&\z0&\z0&\z0\\
&&&\z0&\z0&\z4\\
&&&&&\z2&\z3&\z3&\z3&\z3&\z3\\
&&&&&&&\z3&\z3&\z3&\z3\\
&&&&&&&&&\z3&\z3
\end{array}$}}
$$}


\begin{lemma}\label{MacMah} If $H$ and $K$ are two skew diagrams, then
\begin{equation}\label{Mah}
S_H \cdot S_K = S_{H \blacktriangleright K} + S_{H \blacktriangledown K}\,.
\end{equation}
\end{lemma}
This equality (known already to P. MacMahon for ribbon Schur functions) 
has a simple justification.
The Schur function $S_H$ is the sum $\sum t$ of all the tableaux of diagram
$H$, and similarly $S_K$ is the sum $\sum t'$ of all the tableaux of diagram
$K$. For any pair of tableaux $t,t'$, let $y$ be the letter in the box
$\aleph$ of $t$ and $z$ the letter in the box 
$\vrule width6.5pt height6.5pt depth0pt$ of $t'$.
According as $y\le z$ or $y>z$, the product of (monomials of) tableaux
is a tableau of the diagram $H \blacktriangleright K$ or $H \blacktriangledown K$
and conversely, cutting into two pieces all the pairs of tableaux
of diagrams $H \blacktriangleright K$ or $H \blacktriangledown K$, one obtains
all the pairs of tableaux of respective diagrams $H,K$ (all this is
a trivial consequence of the {\it Jeu de Taquin} which allows one
to move the parts of tableaux (cf.\ \cite{Schut}).

Jeu de Taquin is a favourite ``jeu" of Alain; it is ultimately related
to the notion of {\it Le mono\"\i de plaxique} -- an important subject
developed with M-P. Sch\"utzen\-berger, which offers us, among other things,
a noncommutative lift of Schur functions (cf.\ the original
paper \cite{LSpl} and \cite{LLTpl} which contains a more didactic
exposition of these issues). The ``plactic point of view'' gives,
according to Alain, the simplest,
most natural, and most transparent proof of the famous
{\it Littlewood-Richardson rule} ({\it loc.cit.}).

We now show another way of obtaining the ribbon determinant
expressing a Schur function. Take, e.g., $I=13557$.
The decompositions of the diagram of $I$ into respectively hooks
and ribbons are:
{\arraycolsep0pt
\def\h{\hphantom{\square}}
$$
{\hbox{\small$\begin{array}{ccccccc}
\z0&\h&\h&\h&\h&\h&\h\\
\z0&\z3&\z2\\
\z0&\z3&\z1&\z2&\z2\\
\z0&\z1&\z3&\z3&\z3\\
\z1&\z0&\z0&\z0&\z0&\z0&\z0
\end{array}$}}
\qquad \ \ \ \hbox{and} \qquad \ \ \ \ \
{\hbox{\small$\begin{array}{ccccccc}
\z0&\h&\h&\h&\h&\h&\h\\
\z0&\z0&\z0\\
\z3&\z3&\z1&\z0&\z0\\
\z2&\z1&\z3&\z3&\z0\\
\z1&\z2&\z2&\z3&\z0&\z0&\z0
\end{array}$}}
$$}

We have
\begin{multline*}
S_{13557}=\left|\,\begin{matrix} S_{13}  &   S_{14}  &  S_{17} \\
S_{113}  &  S_{114}  &  S_{117} \\
S_{1113} & S_{1114} & S_{11117}\end{matrix}\, \right|\\
{}=\left|\, \begin{matrix}
S_{13}  & S_{13\blacktriangledown 1} & S_{13\blacktriangledown 1 \blacktriangledown 3} \\
S_{113} & S_{113\blacktriangledown 1} & S_{13\blacktriangledown 1\blacktriangledown 3} \\
S_{11113} & S_{11113\blacktriangledown 1} & S_{11113\blacktriangledown1 \blacktriangledown 3} \end{matrix}\,
\right|=
\left|\, \begin{matrix}
S_{13}  &  S_{13\blacktriangledown 1}  & S_{13\blacktriangledown 1 \blacktriangledown 3} \\
S_{1\blacktriangleright 13} & S_{1\blacktriangleright 13\blacktriangledown 1} &
S_{1\blacktriangleright 3\blacktriangledown 1\blacktriangledown 3} \\
S_{11\blacktriangleright 1\blacktriangleright 13} & 
S_{11\blacktriangleright 1 \blacktriangleright 13 \blacktriangledown 1}
& S_{11\blacktriangleright 1\blacktriangleright 13 \blacktriangledown1 \blacktriangledown 3}
\end{matrix}\, \right|\,,
\end{multline*}
which is the desirable ribbon determinant. This is
a direct consequence of Lemma \ref{MacMah} combined with elementary
operations on columns and rows of the determinants involved. We refer to
\cite{LPr} for more applications of such deformations of planar diagrams.

During our mathematical discussions, Alain often mentioned the
Gessel-Viennot ``planar'' interpretation of binomial determinants
in terms of nonintersecting paths in $\N \times \N$ (cf. \cite{GV}).

\medskip

Now we want to discuss reduced decompositions and divided differences.
Relations between reduced decompositions in the Weyl groups can be
represented planarly. Let us explain it through the example of the
symmetric group ${\mathfrak S}_n$.
By definition, a planar display will be identified with its reading from
left to right and top to bottom ({\it row-reading}). We shall also use
{\it column-reading}, that is, reading successive columns downwards,
from left to right.

For example, we shall write
$$\begin{matrix} 2 \\ 1  &  2 \end{matrix}  \ \equiv\ \
\begin{matrix} 1 & 2 \\ & 1 \end{matrix}
$$
for the following equality for simple transpositions:
$$
s_2\, s_1\, s_2 = s_1\, s_2\, s_1 \ .
$$

Suppose that a rectangle is filled
row-wise from left to right, and column-wise from bottom to top
with consecutive numbers from $\{1,\ldots, n-1\}$.

Then one easily checks that
its row-reading and column-reading produce
two words which, interpreted as words in the $s_i$, are congruent
modulo the Coxeter relations.

\def\h{\hphantom{1111}}
Here is an example of such a congruence:
$$
\begin{matrix} 
3456\h\h\\\h2345\h \\ \h\h1234 \end{matrix} \equiv
\begin{matrix} 3 \\ 2 \\ 1 \end{matrix} \cdot
\begin{matrix} 4 \\ 3 \\ 2 \end{matrix} \cdot 
\begin{matrix} 5 \\ 4 \\ 3 \end{matrix} \cdot 
\begin{matrix} 6 \\ 5 \\ 4 \end{matrix}
$$
the congruence class being conveniently denoted by the rectangle
$$
\begin{matrix}
3456 \\ 2345 \\ 1234 
\end{matrix}
$$

\smallskip

More generally, the planar arrays that we shall write, will have the
property that their row-reading and column-reading are congruent
modulo Coxeter relations \ (this is a ``Jeu de Taquin"
for reduced decompositions).
In this notation, we have, for any integers $a,b,c,d,k$ such that
$$
1\leq a<b,c<d\leq n, \ \  a+d=b+c, \ \ k<d-b,
$$
the congruence
\begin{equation}\label{con1}
\begin{matrix}  b+1 & \cdots & b+k \\
          b  & \cdots & \cdots & \cdots & d \\
	               \vdots &     &        &      & \vdots \\
		              a   & \cdots & \cdots & \cdots & c
\end{matrix}
\quad \equiv \quad
\begin{matrix}  b  & \cdots & \cdots & \cdots & d \\
		\vdots &     &        &      & \vdots \\
   			a   & \cdots & \cdots & \cdots & c \\
						& & c-k & \cdots & c-1
\end{matrix}
\end{equation}

\smallskip

We shall display divided differences of type C planarly according
to the same conventions as for products of $s_i$'s.
For example, the divided difference:
$$
\d_0\d_1\d_2\d_3 \d_0\d_1\d_2\d_0\d_1
$$
will be displayed as
$$
\begin{array}{r} \d_0  \d_1  \d_2  \d_3 \\
              \d_0  \d_1  \d_2 \\
                  \d_0  \d_1
\end{array}
$$
As said before, the displays that we write have the property that their
row-reading is congruent to their column-reading, and thus the
preceding one encodes the equality
$$
\d_0 \d_1 \d_0 \d_2 \d_1 \d_0 \d_3 \d_2 \d_1
= \d_0\d_1\d_2\d_3 \d_0\d_1\d_2\d_0\d_1\,.
$$

\smallskip

When one wants to get an analog of Theorem \ref{Tc} for type D,
the following relation is, e.g., needed (cf.\ \cite{LPmmj}):
\begin{equation}\label{ne}
\arraycolsep0pt
\begin{array}{ccccccc} \d_0 &\d_1 &\d_2 &\d_3 &\d_4 &\d_5 &\d_6 \\
     & \d_0 &\d_1 &\d_2 &\d_3 &\d_4 &\d_5 \\
      && \d_0 &\d_1 &\d_2 &\d_3 &\d_4 \\
       &&&& \d_1 &\d_2 &\d_3
\end{array} =
\begin{matrix}
\d_4\d_5\d_6 \,
\begin{array}{ccccccc} \d_0 &\d_1 &\d_2 &\d_3 &\d_4 &\d_5 &\d_6 \cr
	     & \d_0 &\d_1 &\d_2 &\d_3 &\d_4 &\d_5 \cr
	     && \d_0 &\d_1 &\d_2 &\d_3 &\d_4 \cr
	     \end{array}
\end{matrix}
\end{equation}
Now, thanks to the relations (\ref{con1}) written in terms of divided
differences, we have
\begin{multline*}
\begin{array}{r} \d_0 \d_1 \d_2 \d_3 \d_4 \d_5 \d_6 \\
           \d_0 \d_1 \d_2 \d_3 \d_4 \d_5 \\
	   \d_0 \d_1 \d_2 \d_3 \d_4 \\
          \d_1 \d_2 \d_3 \\
\end{array}\\
\arraycolsep0pt
{}=\left( \begin{array}{r} \d_0 \d_1 \d_2\\
	   \d_0 \d_1\\ \d_0 \\
\end{array}
\right) \,
\left( \begin{array}{r} \d_3 \d_4 \d_5 \d_6 \\
\d_2 \d_3 \d_4 \d_5 \\
\d_1 \d_2 \d_3 \d_4\\
\d_1 \d_2 \d_3
\end{array}
\right)
= \left(\begin{array}{r} \d_0 \d_1 \d_2\\  \d_0 \d_1\\
	  \d_0
\end{array}
\right)
\left(
\begin{array}{l} \d_4 \d_5 \d_6 \\
	  \d_3 \d_4 \d_5 \d_6 \cr\d_2 \d_3 \d_4 \d_5 \\
	  \d_1 \d_2 \d_3 \d_4
	  \end{array}
	  \right)\\
\arraycolsep0pt
{}=\left(\begin{array}{r} \d_0 \d_1 \d_2\cr  \d_0 \d_1\cr \d_0
	  \end{array}
	  \right)
	  \left(\d_4\d_5\d_6\right)
	  \left(\begin{array}{c} \d_3 \d_4 \d_5 \d_6 \\
	  \d_2 \d_3 \d_4 \d_5 \\
	  \d_1 \d_2 \d_3 \d_4 \\
	  \end{array}
	  \right)
\end{multline*}
Since $\d_4\d_5\d_6$ commutes with the divided differences on its left,
the last expression may be rewritten as
$$\hbox{\small$
\d_4\d_5\d_6 \,
\begin{array}{r} \d_0 \d_1 \d_2 \d_3 \d_4 \d_5 \d_6 \cr
	      \d_0 \d_1 \d_2 \d_3 \d_4 \d_5 \cr
	      \d_0 \d_1 \d_2 \d_3 \d_4 \cr
	     \end{array}$}
$$
which is what we need for (\ref{ne}). You see that with planar displays
it is much easier to compute with reduced decompositions and divided
differences for Weyl groups.

Also a permutation can be displayed in a planar way using its {\it Rothe
diagram}. This leads to Alain's favourite combinatorial
presentation of Schubert polynomials with the help of {\it Weintrauben}
of A. Kohnert \cite{Koh}.


\section{Euclid's algorithm}

Already during his first visit to Poland in 1978, Alain said me:

\smallskip

\centerline{``Even in elementary Mathematics there are interesting}
\centerline{things still to be discovered!''}

\smallskip
\noindent
The present section seems to be a confirmation of this sentence.
I will tell you about our recent work \cite{LPsylv}, \cite{LPbez}
on some aspects of Euclid's algorithm.

\bigskip

Let $\A$ and $\B$ be two alphabets of cardinalities $m$ and $n$.
(We can think of $\A$ and $\B$ as multi-sets of complex numbers.)
Consider two monic polynomials $f(x)=S_m(x-\A)$ and
$\varphi(x)=S_n(x-\B)$. Assume that $m \ge n$. Let us look at the
iterated division of $f$ and $\varphi$:
\begin{equation}
f = * \ \varphi + c_1 {\cal R}_1  \quad  , \quad \  \varphi = * \ {\cal R}_1
+ c_2{\cal R}_2 \quad , \quad \
{\cal R}_1= * \ {\cal R}_2 + c_3 {\cal R}_3 \quad , \quad \ldots \ .
\end{equation}
The successive coefficients ``\ ${*}$\ " are the unique polynomials
such that
$$
n>\deg {\cal R}_1 > \deg {\cal R}_2 > \deg {\cal R}_3 > \cdots \ .
$$
Instead of the usual Euclidean algorithm, where $c_1=c_2=c_3=\cdots=1$ and
where ${\cal R}_i$ are rational functions in the roots $\A$ and $\B$
of $f$ and $\varphi$, we choose the constants $c_i$ in such a way that the
successive remainders ${\cal R}_i$ are polynomials in $\A$ and $\B$ with
the top coefficient equal to $S_{i^{m-n+i}}(\A-\B)$. The problem is to give
explicit expressions for such {\it normalized polynomial remainders}
in terms of roots.

An interesting solution to the above problem was proposed
by Sylvester about 160 years ago, who found ``the successive residues,
divested of their allotrious factors \dots " that is, who also normalized
the remainders in such a way as to obtain polynomials in the roots,
the last ``residue'' being the resultant.
To state Sylvester's solution, we need the following notation.
For two finite sets $\A$ and $\B$ of elements in a commutative ring, we set
\begin{equation}
R(\A,\B):= \prod _{a \in \A, b \in \B} (a-b)\,.
\end{equation}
Let now $\A=(a_1,\ldots,a_m)$, $\B=(b_1,\ldots,b_n)$ be two alphabets
of (commuting) variables.
Let $0\le p\le m$, $0\le q\le n$ be two integers.
Define, after Sylvester, the following double sum:
\begin{equation}
{\Sylv}^{p,q}(\A,\B;x)
:=\sum_{\A'\subset \A}  \ \sum_{\B'\subset \B} R(x,\A')
R(x,\B') \frac{R(\A',\B')R(\A\moins \A',\B\moins \B')}
{R(\A',\A\moins \A')R(\B',\B\moins \B')}\,,
\end{equation}
the sum being over all subsets $\A'$ of cardinality $p$ and $\B'$
of cardinality $q$. Sometimes, we will also use the notation
$\Sylv^{p,q}(x)$ or $\Sylv^{p,q}$ for this sum. The Sylvester sum
$\Sylv^{p,q}(\A,\B;x)$ is a polynomial in $\A$, $\B$,
and $x$; moreover $\deg_x \Sylv^{p,q}=p+q$\,.
Observe that $\Sylv^{0,0}=R(\A,\B)$ and if $p=0$ or $q=0$ then the
Sylvester sum reduces to a single sum.
For example,
\begin{equation}
{\Sylv}^{0,q}(\A,\B;x)= \sum_{\B'\subset \B} R(x,\B')
\frac{R(\A,\B\moins \B')}{R(\B',\B\moins \B')}\,,
\end{equation}
the sum being over all subsets $\B'$ of cardinality $q$.
In this case, the assertion of the next theorem was already proved
by C.W. Borchardt in 1860.

Note that
\begin{equation}
R(\A,\B\moins \B')=\prod_{b\in \B \smallsetminus \B'} f(b)\,.
\end{equation}
For $q=n-1$, one recovers Lagrange interpolation (cf., e.g., \cite{LCBMS}):
\begin{equation}
\pm {\cal R}_1=\sum_{i=1}^n f(b_i)\prod_{j \ne i}
{\frac{x-b_j}{b_i-b_j}}\,.
\end{equation}

\smallskip

Let us come back to double Sylvester sums.
For $m=3$, $n=2$, the Sylvester sum $\Sylv^{1,1}(\A,\B;x)$ equals:
$$
\aligned
 \frac{(a_1-b_1)(a_2-b_2)(a_3-b_2)}{(a_1-a_2)(a_1-a_3)(b_1-b_2)}
(x-a_1)(x-b_1) &\\
{}+\frac{(a_1-b_2)(a_2-b_1)(a_3-b_1)}{(a_1-a_2)(a_1-a_3)(b_2-b_1)}
(x-a_1)(x-b_2) &\\
{}+\frac{(a_2-b_1)(a_1-b_2)(a_3-b_2)}{(a_2-a_1)(a_2-a_3)(b_1-b_2)}
(x-a_2)(x-b_1) &\\
{}+\frac{(a_2-b_2)(a_1-b_1)(a_3-b_1)}{(a_2-a_1)(a_2-a_3)(b_2-b_1)}
(x-a_2)(x-b_2) &\\
{}+\frac{(a_3-b_1)(a_1-b_2)(a_2-b_2)}{(a_3-a_1)(a_3-a_2)(b_1-b_2)}
(x-a_3)(x-b_1) &\\
{}+\frac{(a_3-b_2)(a_1-b_1)(a_2-b_1)}{(a_3-a_1)(a_3-a_2)(b_2-b_1)}
(x-a_3)(x-b_2) & \,.
\endaligned
$$
This is equal to \ $-2(x-b_1)(x-b_2)$, so you see that these sums
are quite tricky.
Summations of this type play nowadays an important role, for example, in
representation theory, and in the description of Gysin maps
(called also ``integrations over a fiber") for fibrations with
homogeneous spaces as fibers.

\smallskip

The main result connecting the polynomial remainders and Sylvester sums
is the following theorem.
\begin{theorem}[\cite{LPsylv}]\label{Tsylv}  If $p+q<n$, then specializing
$\A$ and $\B$ to the roots of the polynomials $f$ and $\varphi$
respectively, we have
\begin{equation}
{\Sylv}^{p,q} = \pm \binom{p+q}p {\cal R}_{n-(p+q)}\,.
\end{equation}
\end{theorem}
This result was stated without complete proof in Sylvester's papers
(cf., e.g., \cite{Sylv}).
Our proof uses the following presentation of ${\cal R}_{n-d}$
for $d=0,1,\ldots,n-1$:
$$
{\cal R}_{n-d} = \pm S_{1^d;(m-d)^{n-d}}(\B-x;\B-\A)\,.
$$
(Cf. \cite{LCBMS}, \cite{LPsylv}.)
This determinantal presentation will allow us to run induction (on $n$).
Before giving a sketch of the proof of the theorem, we state some results
that we shall need. Suppose that $\B=(b_1,\ldots,b_n)$ is an alphabet
of variables. In the following, divided differences act on the variables
from $\B$.

\smallskip

\noindent
1) \ For every $k\in \N$, and an alphabet $\A$ which is independent of $\B$,
we have
\begin{equation}\label{dS}
S_k(\B_i -\A) \d_i =S_{k-1}(\B_{i+1}-\A)\,.
\end{equation}

\smallskip

\noindent
2) \ The first remainder ${\cal R}_1$ of the division of $f(x)=S_m(x-\A)$
by $\varphi(x)=S_n(x-\B)$ equals
\begin{equation}\label{rl}
{\cal R}_1 = \bigl( f(b_1)R(x,\B-b_1)\bigr) \d_1 \cdots \d_{n-1}\,.
\end{equation}
This is again an incarnation of the {\it Lagrange interpolation}
in the points $a \in \A$.
For example, for $n=3$, ${\cal R}_1$ equals
\begin{equation}
f(b_1)\frac{(x\moins b_2)(x\moins b_3)}{(b_1\moins b_2)(b_1\moins b_3)}
+f(b_2)\frac{(x\moins b_1)(x\moins b_3)}{(b_2\moins b_1)(b_2\moins b_3)}
+f(b_3)\frac{(x\moins b_1)(x\moins b_2)}{(b_3\moins b_1)(b_3\moins b_2)}\,.
\end{equation}

\smallskip

\noindent
3) \ The specialization ${\Sylv}^{p,q}(\A,\B;b)$, $b\in \B$, is equal to
$\pm R(b,\A) {\mathbf c}$,
where ${\mathbf c}$ is the top coefficient of ${\Sylv}^{p,q}(\A,\B-b;x)$.

\medskip
\noindent
{\bf Sketch of proof of the theorem} \  Write $d:=p+q$.
It is necessary to show that
\begin{equation}
{\Sylv}^{p,q}(\A,\B;x) = \pm \binom{d}p
S_{1^{d};(m-d)^{n-d}}(\B-x;\B-\A)
\end{equation}
if $d<n$. I will give you a rather detailed computation because it gathers
together
several tricks useful while calculating with multi-Schur functions.

\smallskip

Since $\deg_x{\Sylv}^{p,q}<n$, ${\Sylv}^{p,q}$ coincides with its first
remainder modulo $R(x,\B)$. By Eq. (\ref{rl}),
\begin{equation}
{\Sylv}^{p,q}(x)= \bigl({\Sylv}^{p,q}(b_1) R(x,\B-b_1)\bigl)
\d_1\cdots\d_{n-1}\,.
\end{equation}
By property 3), we have
\begin{equation}
{\Sylv}^{p,q}(b_1)
=\pm R(b_1,\A) \times
\hbox{the  top  coefficient} \ {\mathbf c} \ \hbox{in} \
{\Sylv}^{p,q}(\A,\B-b_1;x)\,.
\end{equation}
If $d<n-1$, then by the induction assumption,
\begin{equation}
{\mathbf c}=\pm \binom{d} p S_{\square} (\B-b_1-\A)\,,
\end{equation}
with $\square = (m-d)^{n-d-1}$. This is also true if $d=n-1$ (cf.\
\cite{LPsylv}).
We are left with the computation of
$$
\bigl(R(x,\B-b_1)R(b_1,\A)\bigl) S_{\square}(\B-b_1-\A)\d_1 \cdots \d_{n-1}
\,.
$$
In this computation, we use three simple identities:
\begin{equation}
R(x,\B-b_1)=
\pm \sum_{i=0}^{n-1} (-1)^i S_{1^{n-i-1}}(\B-x)b_1^i\,,
\end{equation}
\begin{equation}
b_1^iR(b_1,\A)=S_{m+i}(b_1-\A)\,,
\end{equation}
\begin{equation}
S_{m+i}(b_1-\A) S_{\square}(\B-b_1-\A) = S_{\square ; m+i}(\B-\A;b_1-\A)\,.
\end{equation}
Finally, using (\ref{dS}),
$$
{\Sylv}^{p,q}(b_1)R(x,\B-b_1)=\pm \binom{d}p \sum_{i=0}^{n-1}(-1)^i
S_{1^{n-1-i}}(\B-x)S_{\square;m+i}(\B-\A; b_1-\A)
$$
is sent via \ $\d_1 \cdots \d_{n-1}$ \ to
\begin{equation}\label{sum}
\pm \binom{d} p \sum_{i=0}^{n-1}(-1)^i S_{1^{n-1-i}}(\B-x)
S_{\square; m+i-(n-1)}(\B-\A)\,.
\end{equation}
In the sum (\ref{sum}), the terms for $i=0, \ldots, n-d-2$ disappear.
In the remaining sum for $i=n-d-1,\ldots,n-1$:
$$
\sum_{j=0}^d (-1)^j S_{1^{d-j}}(\B-x) S_{\square ; m-d+j}(\B-\A)\,,
$$
one recognizes a Laplace expansion of \ $S_{1^d;(m-d)^{n-d}}(\B-x;\B-\A)$\,.

This computation shows the importance of having a {\it proper} determinantal
expression (here for a normalized polynomial remainder) that allows us to
use induction.

\bigskip

Perform now the Euclidean algorithm with nonstandard signs:
$$
f=\cQ_0\phi-\cR_1, \ \ \phi=\cQ_1\cR_1-\cR_2, \ \ \cR_1=\cQ_2\cR_2-\cR_3 \ ,...
$$
$$
..., \cR_{n-2}=\cQ_{n-1}\cR_{n-1}-\cR_n, \ \ \cR_{n-1}=\cQ_n\cR_n\,.
$$

We may rewrite the first equation as follows:
$$
\frac{\varphi}{f}=\frac{1}{\cQ_0-\frac{\cR_1}{\varphi}}\,.
$$
Iterating, we get a continued fraction:
$$  \frac{\varphi}{f} =  \cfrac{1}{\cQ_0 -
      \cfrac{1}{\cQ_1 -
       \cfrac{1}{\ddots}
        \cfrac{}{
        \cfrac{1}{\cQ_n}}}} \ \ .  $$

For $i=1,\ldots,n$, we denote by $\cN_i$ and $\cD_i$ the numerator
and denominator of the $i$th convergent of this continued fraction.
For $i=1,\ldots,n$, we have (cf., e.g., \cite{LPbez})
\begin{equation}\label{Ri1}
\cR_i=\phi \cD_i - f \cN_i\,.
\end{equation}

Suppose, from now on, that $\A$ and $\B$ (the multi-sets of roots
of $f$ and $\varphi$) are alphabets of cardinalities $n+1$ and $n$.
One finds in the book of Brioschi \cite{Brioschi1},
p.~167, the following identities: for a fixed $i$,

\begin{equation}\label{42}
\sum_{a\in\A}\cD_i(a) \cN_i(a) \frac{\phi(a)}{ R(a,\A \moins a)} =0\,,
\end{equation}
\begin{equation}\label{43}
\sum_{a\in\A}\cD_i(a) \cD_{i-1}(a) \frac{\phi(a)}{ R(a,\A \moins a)} =0\,,
\end{equation}
\begin{equation}\label{44}
\sum_{a\in\A} \cD_i^2(a) \cQ_{i}(a) \frac{\phi(a)}{ R(a,\A \moins a)} =0\,.
\end{equation}
These three identities make use of the following functional on $\C[x]$:
\begin{equation}\label{45}
\mu: g(x) \mapsto
 \sum_{a\in\A}  g(a) \frac{\phi(a)}{R(a,\A\moins a)} \ .
\end{equation}
The functional $\mu$ is an incarnation of the Lagrange interpolation
in the points $a\in\A$. It is characterized by the fact that it sends
each $x^i$, $i\in \N$, onto the complete function $S_i(\AB)$.

Given a linear functional, it is known how to write an orthogonal
basis in terms of \emph{moments}:
\begin{proposition}\label{Ph}
The Schur polynomials
$S_{i^i}(\AB\moins x)$ for $i=0,1,\ldots,n$, form a unique
(up to normalization) orthogonal basis with respect to the functional $\mu$,
of polynomials of respective degrees $0,1,\ldots,n$.
\end{proposition}
(More precisely, this is a rewriting, in terms of Schur functions, of the
classical expressions of orthogonal polynomials in terms of
{\it Hankel determinants} involving moments.)

By combining Eq. (\ref{Ri1}) with
\begin{equation}\label{con2}
{\cal R}_i \equiv \phi \ S_{i^i}(\AB\moins x) \ \mod \ f \,,
\end{equation}
(cf.\ \cite{LCBMS}, p. 49), we thus get the following basic result.
\begin{proposition}\label{P17}
For any fixed $0\le i < j \le n$, the following relations hold:
\begin{equation}\label{47}
\sum_{a\in\A}  a^i \cD_j(a) \frac{\phi(a)}{R(a,\A\moins a)} =0  \,,
\end{equation}
\begin{equation}\label{48}
  \sum_{a\in\A}   \cD_i(a) \cD_j(a) \frac{\phi(a)}{R(a,\A\moins a)} =0 \,.
\end{equation}
\end{proposition}
Brioschi's relation (\ref{43}) is just a particular case
of Eq. (\ref{48}) with consecutive $i$ and $j$. Similarly,
the two remaining relations follow from the proposition
(cf.\ \cite{LPbez}).

In terms of Euclidean remainders, the proposition is rewritten as follows:
for any fixed $0\le i < j \le n$,
\begin{equation}\label{51}
\sum_{a\in\A}  a^i \cR_j(a) \frac{1}{\phi(a) R(a,\A\moins a)} =0  \,,
\end{equation}
\begin{equation}\label{52}
\sum_{a\in\A}   \cR_i(a) \cR_j(a) \frac{1}{\phi(a)
R(a,\A\moins a)} =0 \,.
\end{equation}

We want now to discuss Christoffel-Darboux kernels associated with
these orthogonal polynomials. This will be done by investigating
some bivariate polynomials called {\it B\'ezoutians}.

Suppose that $f, \varphi$ is an arbitrary pair of univariate polynomials.
Denoting by $\d_{xy}$ the Newton divided difference acting on the variables
$x$ and $y$, we set (after B\'ezout \cite{Bezout}):
\begin{equation}\label{7}
\Bez(f,\phi) := f(x) \phi(y) \partial_{xy}\,,
\end{equation}
and call this bivariate polynomial (in $x$ and $y$) the {\it B\'ezoutian}
(of $f$ and $\phi$).

In \cite{LPbez} the following formula (missed by classics) was proved:
\begin{theorem}\label{T3} With the above notation,
\begin{equation}\label{14}
\Bez(f,\phi)= p_0 \phi(x)\phi(y)
+\sum_{i=1}^n p_i \cR_i(x)\cR_i(y)\,,
\end{equation}
where $p_i:=\cQ_i(x) \d_{x,y}$.
\end{theorem}

We say that a pair $(f,\phi)$ is {\it general} if the Euclidean quotients
$\cQ_i$ are of degree $1$ for $i=1,\ldots,n$. From now on, we assume that
$(f,\phi)$ is a general pair of monic polynomials of degrees $(n\plus 1,n)$
with alphabets of roots $\A$ and $\B$.
The theorem gives:
\begin{equation}\label{17}
\Bez(f,\phi)= \phi(x)\phi(y) +\sum_{i=1}^n \alpha_i \cR_i(x)\cR_i(y)\,,
\end{equation}
where $\alpha_i$ is the coefficient of $x$ in $\cQ_i(x)$.

Using Eq. (\ref{17}), the congruence (\ref{con2}), and an expression for
$\alpha_i$ from \cite{LCBMS},
we infer (cf.\ \cite{LPbez}):
\begin{theorem}\label{T14} With the above assumptions,
we have the following congruence modulo the ideal $(f(x), f(y))$:
\begin{equation}\label{38}
\Bez(\A,\B)
\equiv \phi(x)\phi(y)
\Bigl(1 + \sum_{i=1}^n (-1)^i \frac {S_{i^i}(\AB\moins x)
S_{i^i}(\AB\moins y)}{S_{(i-1)^i}(\AB)S_{i^{i+1}}(\AB)} \Bigr)\,.
\end{equation}
\end{theorem}

We recall now an important notion from the theory of {\it orthogonal
polynomials}.
Given a family of orthogonal polynomials $P_0(x),\ldots, P_n(x)$
associated with a linear functional $\mu$, then the
{\it Christoffel-Darboux kernel} is:
\begin{equation}\label{55}
K(x,y):=  \sum_{i=0}^n P_i(x) P_i(y) / \bigl( P_i(x)^2 \ \mu\bigr)\,.
\end{equation}

To compute the Christoffel-Darboux kernel in the present situation,
we record the following {\it normalization property}:
\begin{lemma}\label{L100}  With the above notation,
\begin{equation}\label{55.1}
S_{i^i}(\AB\moins x)^2 \ \mu=
(-1)^i S_{(i\moins 1)^i}(\AB)S_{i^{i+1}}(\AB)\,.
\end{equation}
\end{lemma}

By combining the congruence (\ref{38}), Proposition \ref{Ph} and
Lemma \ref{L100}, we infer the following congruence modulo the
ideal $(f(x),f(y))$:
\begin{equation}\label{56}
 \Bez(\A,\B) \equiv \phi(x)\phi(y)\, K(x,y)\,.
\end{equation}
This congruence suggests that $\Bez(\A,\B)$ (similarly to $K(x,y)$)
should have a {\it reproducing} property.
\begin{theorem}[\cite{LPbez}]
For a polynomial $g(x)$,
\begin{equation}\label{57}
g(x)\, \Bez(\A,\B) \ \mu \equiv \phi(y)^2\,  g(y)
          \ \mod f(y) \,.
\end{equation}
\end{theorem}
\noindent
{\bf Sketch of proof} \ It follows from the {\it Leibniz rule} that 
$\Bez(\A,\B)$ is congruent to
$f(x)\partial_{xy} \cdot \phi(x)$ modulo $f(y)$.
Recall that $f(x)\partial_{xy}=S_n(x+y-\A)$.
For any $i\ge 0$, we have, modulo $f(y)$,
$$
x^i\, \phi(y)\, S_n(x+y-\A) \ \mu
\equiv \phi(y)S_{i+n}(\A-\B+y-\A)= \phi(y)^2 \, y^i\,.
$$

So we have arrived at another reproducing property, this time
for B\'ezou\-tians. There are similar results for a (general) pair
of monic polynomials of the same degree (cf. \cite{LPbez}).

\section{Concluding remarks}

I hope that I will not offend much the audience of SLC by saying that
Alain is rather {\it algebraist} than combinatorialist.
To be a bit more
precise, I think that Alain is doing {\it Combinatorial Algebra}.
His favourite mathematicians (read: algebraists) are mentioned already
J.M. Hoene-Wro\'nski and the Italian algebraic geometer G.Z. Giambelli
(1879-1953).
They both did very original mathematics that, to a large extent,
was not accepted by the mathematical establishment of their time
(cf.\ \cite{Lwr} and \cite{Lak}).
This is often the price you must pay for your originality:

\smallskip

\centerline{``To reach the source you must go against the stream.''}

\smallskip

Alain is now, and has been for several years, very much involved in 
teaching young Chinese mathematicians at the Center of Combinatorics
of Nankai University in Tianjin
in the People Republic of China. The following picture of him and some
of his students was taken on the occasion of his $59$th Birthday in Tianjin:

\smallskip
\centerline{\includegraphics{tian.pdf}}
\smallskip

Here are two other pictures from Nankai University. The first one
is with S.S. Chern\footnote{S.S. Chern passed away in Tianjin on December 3,
2004 at the age of 93.}, a mathematician who translated many geometric
problems to Algebra by inventing the famous {\it Chern characteristic
classes}:

\smallskip
\centerline{\includegraphics[width=8cm]{ch.pdf}}
\smallskip

\noindent
Since, by the {\it splitting principle} the Chern classes of a vector
bundle are elementary symmetric functions in its Chern roots, many
computations of Chern classes amount to calculations of symmetric
functions. Given a vector bundle $E$ and partition $I$, we define $S_I(E)$
to be the Schur polynomial $S_I(\A)$ with $\A$ specialized 
to the Chern roots of $E$.
The following problem is central here.
Determine the integer coefficients $c_K$ in the Schur polynomial
expansion: \ $S_I(V_J(E))=\sum c_K S_K(E)$, where $V_J$ stands for a Schur
functor. For the Chern (resp. {\it Segre}) classes of $S^2(E)$ 
and $\wedge^2(E)$
this was done in \cite{Lt} (resp. \cite{Prag1}, \cite{LaLaT}).
By \cite{P}, $\S$7, we know that $c_K \ge 0$. Note that the expansion of
the {\it top} Chern class of $V_J(E)$ determines those of all other Chern
classes (cf. \cite{P}, Proposition 2.1).

\medskip

The second picture is with Sir M. Atiyah:

\smallskip
\centerline{\includegraphics{at.pdf}}
\smallskip

\noindent
Atiyah said in 1993 (on the occasion of Hirzebruch's $65$th Birthday):

\smallskip

\centerline{``It is reassuring to know that the algebraic virtuosity}
\centerline{of our ancestors is still present in our genes.''}

\bigskip\medskip

\noindent
{\large I wish heartily the S\'eminaire Lotharingien de Combinatoire, and in
particular Alain, that this will be the case for Alain Lascoux's ingenious
computations in Combinatorial Algebra!}

\bigskip\medskip

\noindent
{\bf Acknowledgments}{\small \ I thank the organizers of SLC 52 for their
invitation to attend this event.
My sincere thanks go to Jean-Yves Thibon whose constant encouragement
allowed me to bring my talk at the LascouxFest to a paper.
I am also greatly indebted to Ron King for a number of comments and 
suggestions that helped to ameliorate its exposition.
Jan Krzysztof Kowalski from our institute in Warsaw kindly helped me 
with some mysteries of LaTeX.
The pictures from Tianjin are courtesy of Bill Chen from the Center
of Combinatorics of Nankai University.}

\bigskip

\begin{thebibliography}{}\small



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\emph{Schubert cells and cohomology of the spaces $G/P$,}
Russian Math. Surveys, {\bf 28:3} (1973), 1--26.


\bibitem{Bezout} M. B\'ezout,
\emph{Recherches sur le degr\'e des \'equations r\'esultants de
l'evanouisse\-ment des inconnues, et sur les moyens qu'il convient
d'employer pour trouver ces \'equations},
M\'em. Acad. Roy. Sci. Paris (1764), 288--338.


\bibitem{Brioschi1} F. Brioschi, \emph{Th\'eorie des d\'eterminants,}
 appendix to the French edition, Paris, Mallet-Bachelier 1856.


\bibitem{Buch} A.S. Buch,
\emph{Combinatorial K-theory,}
in: ``Topics in cohomological studies of algebraic varieties''
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\emph{Generic free resolutions and a family of generically perfect
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Adv. Math. {\bf 18} (1975), 245--301.

\bibitem{CYY} W.Y.C. Chen, G-G. Yan, A.L.B. Yang,
\emph{The skew Schubert polynomials,}
Europ. J. Comb. {\bf 25/8} in honor of Alain Lascoux (2004),
1181--1196.


\bibitem{Dem1} M. Demazure,
\emph{Invariants sym\'etriques entiers des groupes de Weyl et torsion,}
Invent. Math. {\bf 21} (1973), 287--301.


\bibitem{Dem2} M. Demazure,
\emph{D\'esingularisation des vari\'et\'es de Schubert g\'eneralis\'ees,}
Ann. Scient. \'Ecole Norm. Sup. {\bf 7} (1974), 53--88.


\bibitem{DKR} J. D\'esarm\'enien, J. Kung, G-C. Rota,
\emph{Invariant theory, Young bitableaux, and combinatorics,}
Adv. Math. {\bf 27} (1978), 63--92.


\bibitem{DRS} P. Doubilet, G-C. Rota, J. Stein,
\emph{Combinatorial methods in invariant theory,}
Stud. Appl. Math. {\bf 53} (1974), 185--216.


\bibitem{Fal} G. Faltings,
\emph{A proof for the Verlinde formula,}
J. Alg. Geom. {\bf 3} (1994), 347--374.


\bibitem{FuL} A.M. Fu, A. Lascoux,
\emph{$q$-identities from Lagrange and Newton interpolation,}
Adv. Appl. Math. {\bf 31} (2003), 527--531.


\bibitem{F} W. Fulton,
\emph{Flags, Schubert polynomials, degeneracy loci, and determinantal
formulas,}
Duke Math. J. {\bf 65} (1992), 381--420.


\bibitem{FL} W. Fulton, A. Lascoux,
\emph{A Pieri formula in the Grothendieck ring of a flag bundle,}
Duke Math. J. {\bf 76} (1994), 711--729.


\bibitem{GV} I. Gessel, X. Viennot,
\emph{Binomial determinants, paths, and hook length formulae,}
Adv. Math. {\bf 58} (1985), 300--321.


\bibitem{KKL} A. Kerber, A. Kohnert, A. Lascoux,
\emph{SYMMETRICA, an object oriented computer-algebra system for the
symmetric group,}
J. Symb. Comp. {\bf 14} (1992), 195--203.


\bibitem{Kn} D.E. Knuth,
\emph{Permutations, matrices, and generalized Young tableaux,}
Pacific J. Math. {\bf 34} (1970), 709--727.

\bibitem{Koh} A. Kohnert,
\emph{Weintrauben, Polynome, Tableaux,}
Bayreuther Math. Schriften {\bf 38} (1991), 1--97.


\bibitem{KP} W. Kra\'skiewicz, P. Pragacz,
\emph{Schubert functors and Schubert polynomials,}
Europ. J. Comb. {\bf 25/8} in honor of Alain Lascoux (2004),
1327--1344.

\bibitem{Lak} D. Laksov,
\emph{Remarks on Giovanni Zeno Giambelli's work and life,}
(A. Brigaglia et al. eds.),
Rend. Circ. Mat. Palermo, serie {\bf II} no. {\bf 36} (1994), 207--218.


\bibitem{LaLaT} D. Laksov, A. Lascoux, A. Thorup,
\emph{On Giambelli's theorem on complete correlations,}
Acta Math. {\bf 162} (1989), 143--199.


\bibitem{Lbul} A. Lascoux,
\emph{Puissances ext\'erieures, d\'eterminants et cycles de Schubert,}
Bull. Soc. Math. France {\bf 102} (1974), 161--179.


\bibitem{Lg} A. Lascoux,
\emph{Fonctions de Schur et Grassmanniennes,}
C. R. Acad. Sc. Paris {\bf 281} (1975), 813--815, 851--854.


\bibitem{LasThes} A. Lascoux,
\emph{Th\`ese,}
Universit\'e Paris 7, 1977.


\bibitem{Lt} A. Lascoux,
\emph{Classes de Chern d'un produit tensoriel,}
C. R. Acad. Sc. Paris {\bf 286} (1978), 385--387.


\bibitem{Lassyz} A. Lascoux,
\emph{Syzygies des vari\'et\'es d\'eterminantales,}
Adv. Math. {\bf 30} (1978), 202--237.


\bibitem{Lch} A. Lascoux,
\emph{Classes de Chern des vari\'et\'es de drapeaux,}
C. R. Acad. Sc. Paris {\bf 295} (1982), 393--398.


\bibitem{Lwr} A. Lascoux,
\emph{Wro\'nski's factorization of polynomials,}
in: ``Topics in Algebra'' (S. Balcerzyk et al. eds.), Banach Center
Publications {\bf 26(2)} (1990), 379--386.


\bibitem{Lschur} A. Lascoux,
\emph{Double crystal graphs,}
in: ``Studies in memory of Issai Schur'' (A. Joseph et al. eds.), Progress
Math. {\bf 210}, 95--114, Birkh\"auser 2003.

\bibitem{LCBMS} A. Lascoux,
\emph{Symmetric functions and combinatorial operators on polynomials,}
CBMS/AMS Lectures Notes {\bf 99}, Providence, 2003.


\bibitem{LLT} A. Lascoux, B. Leclerc, J-Y. Thibon,
\emph{Fonctions de Hall-Littlewood et polyn\^o\-mes de Kostka-Foulkes
aux racines de l'unit\'e,}
C. R. Acad. Sc. Paris {\bf 316} (1993), 1--6.


\bibitem{LLTp} A. Lascoux, B. Leclerc, J-Y. Thibon,
\emph{Une nouvelle expression des fonctions $P$ de Schur,}
C. R. Acad. Sc. Paris {\bf 316} (1993), 221--224.


\bibitem{LLTpl} A. Lascoux, B. Leclerc, J-Y. Thibon,
\emph{The Plactic Monoid,}
in: ``Combinatorics on Words'', ed. by M. Lothaire, Cambridge University
Press 2001.


\bibitem{LPsh} A. Lascoux, P. Pragacz,
\emph{Equerres et fonctions de Schur,}
C. R. Acad. Sc. Paris {\bf 299} (1984), 955--958.

\bibitem{LPr} A. Lascoux, P. Pragacz,
\emph{Ribbon Schur functions,}
Europ. J. Comb. {\bf 9} (1988), 561--574.


\bibitem{LPs} A. Lascoux, P. Pragacz,
\emph{$S$-functions series,}
J. Phys. A: Math. Gen. {\bf 21} (1988), 4105--4114.


\bibitem{LPadv} A. Lascoux, P. Pragacz,
\emph{Operator calculus for $\Q$-polynomials and Schubert polynomials,}
Adv. Math. {\bf 140} (1998), 1--43.


\bibitem{LPmmj} A. Lascoux, P. Pragacz,
\emph{Orthogonal divided differences and Schubert polynomials,
$\widetilde P$-polynomials, and vertex operators,}
Michigan Math. J. {\bf 48} in honor of William Fulton (2000), 417--441.


\bibitem{LPsylv} A. Lascoux, P. Pragacz,
\emph{Double Sylvester sums for subresultants and multi-Schur functions,}
J. Symb. Comp. {\bf 35} (2003), 689--710.


\bibitem{LPbez} A. Lascoux, P. Pragacz,
\emph{Bezoutians, Euclidean algorithm, and orthogonal polynomials,}
preprint (2004).


\bibitem{LSpl} A. Lascoux, M-P. Sch\"utzenbeger,
\emph{Le mono\"\i de plaxique,}
in: `` Non-commuta\-tive structures in algebra and geometric combinatorics'',
Napoli 1978, Quaderni de ``La Ricerca Scientifica'' {\bf 109}, C.N.R. Roma
(1981), 129--156.


\bibitem{LSschub} A. Lascoux, M-P. Sch\"utzenberger,
\emph{Polyn\^omes de Schubert,}
C. R. Acad. Sc. Paris {\bf 294} (1982), 447--450.


\bibitem{LSint} A. Lascoux, M-P. Sch\"utzenberger,
\emph{Interpolation de Newton \`a plusieurs variables,}
S\'eminaire d'Alg\`ebre Dubreil--Malliavin 1983--1984 (M-P. Malliavin ed.),
Springer Lecture Notes in Math. {\bf 1146} (1985), 161--175.



\bibitem{LSkey} A. Lascoux, M-P. Sch\"utzenberger,
\emph{Keys and standard bases,}
in: ``Invariant Theory and Tableaux'' (D. Stanton ed.), Springer 1990,
125--144.


\bibitem{LSform} A. Lascoux, M-P. Sch\"utzenberger,
\emph{Formulaire raisonn\'e de fonctions sym\'e\-triques,}
Universit\'e Paris 7, 1985.


\bibitem{LSf} A. Lascoux, M-P. Sch\"utzenberger,
\emph{Fonctorialit\'e des polyn\^omes de Schubert,}
in: ``Invariant Theory'' (R. Fossum et al. eds.), Contemporary
Mathematics A.M.S. {\bf 88} (1989), 585--598.


\bibitem{Lec} B. Leclerc,
\emph{On identities satisfied by minors of a matrix,}
Adv. Math. {\bf 100} (1993), 101--132.


\bibitem{Mcd1} I.G. Macdonald,
{\emph Symmetric functions and Hall polynomials,}
2nd ed., Oxford Univ. Press 1995.


\bibitem{Mcd2} I.G. Macdonald,
\emph{Notes on Schubert polynomials,}
Publ. LACIM {\bf 6}, UQUAM, Montr\'eal, 1991.


\bibitem{Muir} T. Muir,
\emph{History of determinants},
5 volumes, Dover reprints 1960.


\bibitem{Prag1} P. Pragacz,
\emph{Enumerative geometry of degeneracy loci,}
Ann. Scient. \'Ec. Norm. Sup. {\bf 21} (1988), 413--454.

\bibitem{Prag2} P. Pragacz,
\emph{Algebro-geometric applications of Schur $S$- and $Q$-polynomials,}
S\'eminaire d'Alg\`ebre Dubreil--Malliavin 1989--1990 (M-P. Malliavin ed.),
Springer Lecture Notes in Math. {\bf 1478} (1991), 130--191.


\bibitem{P} P. Pragacz,
\emph{Symmetric polynomials and divided differences
in formulas of intersection theory,}
in: ``Parameter Spaces'',
Banach Center Publications {\bf 36} (1996), 125--177.


\bibitem{PR} P. Pragacz, J. Ratajski,
\emph{Formulas for Lagrangian and orthogonal degeneracy loci;
$\widetilde Q$-polynomial approach,}
Compositio Math. {\bf 107} (1997), 11--87.

\bibitem{Schur} I. Schur,
\emph{\"Uber eine Klasse von Matrizen die sich einer gegebenen Matrix
zuordnen lassen}, Dissertation, Berlin, 1901; see: Ges. Abhandlungen
{\bf 1}, 1--72.


\bibitem{Schut} M-P. Sch\"utzenberger,
\emph{La correspondance de Robinson,}
in: ``Combinatoire et R\'epresentation du Groupe Sym\'etrique''
(D. Foata ed.),
Springer Lecture Notes in Math. {\bf 579} (1977), 59--135.


\bibitem{Sylv} J.J. Sylvester,
\emph{A theory of the syzygetic relations of two rational integral
functions}, Phil. Trans. Royal Soc. London vol. CXLIII, Part III (1853),
407--548. Also: \emph{The collected mathematical papers}, Cambridge
(1904), vol.~1, paper 57, pp.~429--586.


\bibitem{Tam} H. Tamvakis,
\emph{Gromov-Witten invariants and quantum cohomology of Grassmannians,}
in: ``Topics in cohomological studies of algebraic varieties''
(P. Pragacz ed.), Trends in Mathematics, Birkh\"auser 2005 -- to appear.


\bibitem{Th} J-Y. Thibon,
\emph{Hopf algebras of symmetric functions and tensor products of
symmetric group representations,}
Internat. J. Alg. Comp. {\bf 2} (1991), 207--221.

\bibitem{Th2} J-Y. Thibon,
\emph{Lectures on noncommutative symmetric functions,}
in: `` Interaction of combinatorics and representation theory'',
MSJ Memoirs {\bf 11} (2001), 39--94.

\medskip

For Alain's explicit computations the following reference is relevant:

\bibitem{V} S. Veigneau,
\emph{ACE, an algebraic combinatorics environment for the computer
algebra system MAPLE,} 1998.

\bigskip

And other books and papers \ldots

\end{thebibliography}

\end{document}
