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                      \carrel55{5.5}\carrel45{4.5} \, }


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                      \carrel55{5.5}\carrel45{4.5}}

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\def\bnab{\overline{\nabla}}


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\def\I{{\cal I}}
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\title{\bf TWISTED ACTION OF THE SYMMETRIC GROUP \\
ON THE COHOMOLOGY OF A FLAG MANIFOLD\thanks{Supported by PRC Math-Info
and EEC grant  n$^0$ ERBCHRXCT930400.}
}


\author{\sc ALAIN LASCOUX \hskip 15mm  
BERNARD LECLERC\\ 
{\ft L.I.T.P, Universit\'e Paris 7} \\ 
{\ft 2, Place Jussieu}\\
{\ft 75251 Paris cedex 05, France}
\and 
JEAN-YVES THIBON \\
{\ft Institut Gaspard Monge} \\
{\ft Universit\'e de Marne-la-Vall\'ee} \\
{\ft 2, rue de la Butte-Verte}\\
{\ft 93166 Noisy-le-Grand cedex, France} 
}

\date{}



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\begin{document}

\newfont{\rml}{cmu10}
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\maketitle


\begin{abstract}
Classes dual to Schubert cycles constitute  a basis
on the cohomology ring of the flag manifold $\F$, self-adjoint
up to indexation with respect to the
intersection form. Here, we study the bilinear form
$$
(X\, ,\,Y) := \<X\cdot Y\, ,\, c(\F)\>
$$
where $X,Y$ are cocycles, $c(\F)$ is the total
Chern class of $\F$ and $\<\ ,\ \>$ is the intersection form. 
This form is related to a twisted action of the symmetric
group of the cohomology ring, and to the degenerate affine Hecke algebra.
We give a distinguished basis for this form, which is a deformation of the usual
basis of Schubert polynomials, and apply it to the computation of
the Schubert cycle expansions of Chern classes of flag manifolds.
\end{abstract}


% ***************************************************************************
% ***************************************************************************
\newpage
\section{Introduction and preliminaries}

Let $V$ be a complex vector space of dimension $n$, and $\F=\F(V)$ be
the variety of complete flags in $V$. It is well known that the cohomology
ring $H^*(\F\, ,\,\C)$ is the quotient of the polynomial ring
$\C[X]= \C[x_1,x_2,\ldots,x_n]$ by the ideal $\I^+$ of 
symmetric polynomials without constant term.

Let $\sigma_i$, $i=1,\ldots,n-1$ be the simple transposition exchanging
$x_i$ and $x_{i+1}$. Denote by $\partial_i$ the linear operator
on $\C[x_1,\ldots,x_n]$ defined by
\begin{equation}
\partial_if := {f - \sigma_i f \over x_i -x_{i+1}}
\end{equation}
(Newton's divided difference). The operators $\partial_1,\ldots,\partial_{n-1}$
induce operators on $H^*(\F)$. 

According to \cite{BGG} and \cite{Dem}, the basis of Schubert cycles can be
obtained from the class of a point 
$P={1\over n!}\prod_{i<j}(x_i-x_j)$
by successive applications of divided difference operators.
Taking as representative of $P$ the polynomial $X:=x_1^{n-1}x_2^{n-2}\cdots x_1^0$,
one obtains polynomials $X_\mu$, $\mu\in\S_n$, called Schubert polynomials,
which represent the Schubert subvarieties in the cohomology ring \cite{LSschub1}.
A detailed account of the algebraic theory of Schubert polynomials can be found
in Macdonald's treatise \cite{Mcd}.

Divided differences satisfy the braid relations
\begin{equation}\label{BRAID}
\left\{ \matrix{
\d_i\d_{i+1}\d_i = \d_{i+1}\d_i\d_{i+1} \cr
\d_i\d_j = \d_j\d_i\qquad {\rm for}\quad |i-j|>1 \cr
}\right.
\end{equation}
but the squares $\d_i^2$ are null. These relations allow to define
operators $\d_\mu$ for any permutation $\mu\in\S_n$: if $\mu=
\sigma_{i_1}\sigma_{i_2}\cdots\sigma_{i_m}$ is a reduced decomposition of
$\mu$, one sets $\d_\mu=\d_{i_1}\d_{i_2}\cdots \d_{i_m}$. The result
does not depend on the choice of a particular reduced decomposition of $\mu$.

To recover an action of the symmetric group, one can take any $q\in\C$ and
define
\begin{equation}
D_i := \sigma_i + q\d_i \ ,\qquad 1\le i\le n-1 \ .
\end{equation}
These operators still satisfy the braid relations 
\begin{equation}\label{BRAID1}
\left\{\matrix{
D_iD_{i+1}D_i = D_{i+1}D_i D_{i+1} \cr
D_iD_j = D_j D_i \qquad (|i-j|>1)\cr}\right. 
\end{equation}
together with
\begin{equation}
D_i^2 = 1 \ ,
\end{equation}
so that they generate a representation of the symmetric group $\S_n$ on 
the polynomial ring $\C[x_1,\ldots,x_n]$, as well as on the
cohomology ring $H^*(\F\, ,\, \C)$.
These operators have been considered by 
Cherednik  and Bernstein (\cf \cite{Che1}\cite{Che2}). 
Similar operators, acting on the equivariant $K$-theory of flag
manifolds, have been used by Lusztig \cite{Lus}.
More general operators satisfying braid relations  have
been given in \cite{LS}.

As $\d_i$ decreases degrees by $1$, all $q\not=0$ will
give equivalent representations of $\S_n$,
and by homogeneity, the
general case can be recovered from the case $q=1$. For simplicity,
we set $q=1$, and write
\begin{equation}
s_i := \sigma_i + \d_i \ .
\end{equation}
We denote as above by $s_\mu$ the product of operators $s_i$ corresponding
to a permutation $\mu$. Remark that the operator algebra generated by the $s_i$
and the variables $x_j$ (interpreted as operators $f\mapsto x_jf$)
is isomorphic to the degenerate affine Hecke algebra considered
in \cite{Che1}.

Schubert calculus for other classical groups can be found in the work of
 Fulton \cite{Ful} and 
of Pragacz and Ratajski \cite{PR}.


This paper is organized as follows. We first define certain elements
(Yang-Baxter operators) of the degenerate affine Hecke algebra. Then
we use them to define a bilinear form on the cohomology of a flag
manifold. We exhibit a distinguished basis, called {\it
affine Schubert polynomials}, and compute its adjoint basis.  
We then apply this formalism to the computation of the Schubert
expansions of Chern classes.







\medskip
{\bf Acknowledgements} 
The preparation of this paper has been facilitated by the use of the
program system SYMMETRICA \cite{KKL} and of the Maple package SP \cite{Vei}.





\section{Yang-Baxter operators}

We shall define inductively operators $\carre_\mu$ and
$\nabla_\mu$ associated with any permutation $\mu$ in $\S_n$.
Set $\carre_{12\ldots n}=1$, $\nabla_{12\ldots n}=1$, and, if
$\mu=\sigma_i\alpha$ with $\ell(\mu)=\ell(\alpha)+1$, and $\beta=\alpha^{-1}$,
\begin{equation}\label{carnab}
\left\{\matrix{
\carre_\mu =  \left( s_i + {1\over \beta_{i+1}-\beta_i}\right)\carre_\alpha \cr
\nabla_\mu =  \left( s_i - {1\over \beta_{i+1}-\beta_i}\right)\nabla_\alpha \cr
}\right.
\end{equation}

Using the braid relations (\ref{BRAID1}), one can check that
this definition is consistent, \ie does not depend on the
chosen factorization (see  \cite{Che1,Che2} and \cite{DKLLST}). This
follows in fact from a classical solution of the Yang-Baxter equation.
In \cite{Yang}, C.N. Yang observed that the operators defined by
$Y_i(u)=u^{-1} +\sigma_i$, where $u$ is a scalar parameter and $\sigma_i$
the transposition $(i,i+1)$ satisfy the ``Quantum Yang-Baxter Equation
with spectral parameter'':
\begin{equation}
Y_i(u-v)Y_{i+1}(u-w)Y_{i}(v-w) =
Y_{i+1}(v-w) Y_i(u-w) Y_{i+1} (u-v)
\end{equation}
It follows that  given a $n$-uple of
parameters ${\bf u}=(u_1,\ldots,u_n)$, one can define for any permutation
$\mu\in \S_n$ an operator $R_\mu({\bf u})$
by the following prescription: $R_\mu({\bf u})= 
Y_i(u_{\beta(i+1)}-u_{\beta(i)})R_\alpha({\bf u})$,
where, as above, $R_{12\ldots n}=1$, $\mu=\sigma_i\alpha$, $\ell(\mu)=\ell(\alpha)+1$
and $\beta=\alpha^{-1}$. Then, our operators (\ref{carnab}) are respectively
$R_\mu({\bf u})$ and $R_\mu(-{\bf u})$, where ${\bf u}=(1,2,\ldots,n)$ and
$\sigma_i$ is interpreted as $s_i$. 

For the maximal element $\omega=(n,n-1,\ldots,1)$ of $\S_n$, one
has the following factorization property (given in \cite{DKLLST} for
the case of the Hecke algebra):

\begin{proposition}\label{FACT}
Define polynomials 
$\theta=\displaystyle \prod_{1\le i<j\le n}(1+x_i-x_j)$ and
$\theta^*=\displaystyle \prod_{1\le i<j \le n}(1-x_i+x_j)$.
Then, for any polynomial $f$,

{\rm (i)} $\nabla_\omega f = \theta^*\, \d_\omega f$ 

\medskip
{\rm (ii)} $\carre_\omega f = \d_\omega(\theta f)$.
\end{proposition}

\Proof
Recall that the classes of the Schubert polynomials $X_\mu$, $\mu\in\S_n$,
form a basis of $H^*(\F)=\C[X]/\I^+$. Given $\mu$ and $i$ such that
$\ell(\mu\sigma_i)>\ell(\mu)$, the polynomial $X_\mu$ is symmetrical
in $x_i$ and $x_{i+1}$. As such, it is sent to $0$ by the operator
$\nabla_{\sigma_i}=\sigma_i+\d_i-1$.

Now, for any permutation $\mu\not=\omega$, there exists an $i$ such
that $\ell(\mu\sigma_i)>\ell(\mu)$. If we choose a reduced decomposition
of $\omega$ ending by $\sigma_i$, $\omega=\nu\sigma_i$, say, we see that $X_\mu$ is sent
to $0$ by $\d_\omega=\d_\nu\d_{\sigma_i}$ and  by $\nabla_\omega=
\nabla_\mu\nabla_{\sigma_i}$.

Thus, $\nabla_\omega$ as well as $\d_\omega$ annihilate all Schubert
polynomials $X_\mu$ for $\mu\not=\omega$. Finally, $X_\omega=x_1^{n-1}\ldots x_n^0$
is sent to $1$ by $\d_\omega$. To conclude, it remains to prove
that
$$
\nabla_\omega (X_\omega) = \prod_{1\le i<j \le n}(1-x_i+x_j) \ .
$$ 
This formula can be proved by induction on $n$ using the factorization
$$
\omega_n = \sigma_1\sigma_2\cdots\sigma_{n-1}\omega_{n-1}\ ,
$$
which gives
$$
\nabla_{\omega_n} =\Bigl(s_1-1\Bigr)\cdots\Bigl(s_{n-1}-{1\over n-1}\Bigr)
 \nabla_{\omega_{n-1}}\ .
$$
\cqfd









\section{Quadratic form}

Recall that the intersection form of the cohomology ring $H^*(\F\, ,\, \C)$
is induced by the form on $\C[X]$ 
\begin{equation}
\<f\, ,\,g\> = \d_\omega(fg)\mid_0 = \d_\omega (fg\mid_{\ell(\omega)} )
\end{equation}
where $f\mid_k$ denotes the homogeneous component of
degree $k$   of $f$ (\cf \cite{BGG}, \cite{Dem1}). 
With respect to this form, the Schubert polynomials satisfy 
\begin{equation}
\<X_\mu\, ,\, X_\nu\> =
\left\{\matrix{
1 & {\rm if} \ \ \nu=\omega\mu \cr
0 & {\rm otherwise}
}\right.
\end{equation}

The tangent bundle $T\F$ of the flag manifold has a composition
sequence $\{L_iL_j^{-1}\}_{i<j}$ where $L_1,L_2,\ldots,L_n$ are the
tautological line bundles on $\F$. The total Chern class of $L_i$ being 
$c(L_i)=1+x_i$, the total Chern class of the tangent bundle
of $\F$ is
\begin{equation}
c(\F)=\prod_{i<j}(1+x_i-x_j)
\end{equation}
(see {\it e.g.} \cite{Hir}, our convention is $L_i=\xi_i^*$
in the notation of \cite{Hir}). Consider now the following
quadratic form on $\C[X]$:

\begin{definition}
$$
(f\, ,\, g) := \carre_\omega(fg)\mid_0 \ .
$$
\end{definition}

Thus, in the cohomology ring, we see from Proposition \ref{FACT} that
\begin{equation}
(f\, ,\, g) = \<f\, ,\, g\, c(\F)\> = \<f\, c(\F)\, ,\, g\> \ .
\end{equation}

\begin{lemma}\label{SELFAD}
The operators $\carre_i$ are self-adjoint with respect to
the quadratic form $(\ , \ )$.
\end{lemma}

\Proof For any $i$, $\carre_\omega\,\carre_i = 2\,\carre_\omega$,
since $\carre_i^2= 2\,\carre_i$ and since one can find a reduced 
decomposition of $\omega$ ending with $\sigma_i$. Now,
$$
(\carre_i f\, ,\, g) = \carre_\omega\left((\carre_i f)g\right)\mid_0
$$
$$
={1\over 2}\, \carre_\omega\, \carre_i \left((\carre_i f) g\right)\mid_0
={1\over 2}\, \carre_\omega \left( (\carre_i f)(\carre_i g)\right)\mid_0
$$
since $\carre_i f$ is a scalar for $\carre_i$, being symmetrical
in $x_i,x_{i+1}$. The last expression being symmetrical in
$f,g$, this proves that $(\carre_i f\, ,\,g)=(f\, ,\, \carre_i g)$
\cqfd






\section{Affine Schubert polynomials}\label{AFSCHUB}

Let $\H_n$ be the linear subspace of $\C[x_1,\ldots,x_n]$ generated
by the monomials $x^I=x_1^{i_1}x_2^{i_2}\cdots x_n^{i_n}$
such that $i_k \le n-k$. Let $\Pi$ be the projector from
$\C[X]$ onto $\H_n$ associating to a polynomial $P$ the unique representative
in $\H_n$
of its class $\overline{P}\in \C[X]/\I^+$.


\begin{definition}
Let $\mu\in\S_n$. The {\rm affine Schubert polynomial} of index $\mu$
is defined by
$$
Z_\mu = \Pi \bigl(\carre_{\mu^{-1}\omega}Z_{\omega}\bigr)
$$
where $Z_\omega:=X_\omega=x_1^{n-1}x_2^{n-2}\cdots x_n^0$.
\end{definition}


\begin{example}{\rm
For $n=3$, 
\begin{eqnarray*}
Z_{321} & = & x_1^2 x_2    \\ 
Z_{312} & = & x_1^2 \\
Z_{231} & = & x_1x_2 \\
Z_{213} & = & x_1 - { 1/2}x_1x_2 -x_1^2 \\
Z_{132} & = & x_1+x_2 - x_1x_2 -{ 1/ 2} x_1^2 \\
Z_{123} & = & 1 \\
\end{eqnarray*}
}
\end{example}

In general, one has $Z_\omega=X_\omega$, $Z_\mu=X_\mu + $(terms of
degree $>\ell(\mu)$), and $Z_{id}=1$, the last identity being due
to the fact that $\carre_\omega(Z_\omega)$ is symmetrical with
term of lowest degree $X_{id}=1$.

\begin{example}{\rm For $n=3$,
\begin{eqnarray*}
Z_{321} & = & X_{321} \\
Z_{312} & = & X_{312} \\
Z_{231} & = & X_{231} \\
Z_{213} & = & X_{213} - {1/2}X_{231}-X_{312} \\
Z_{132} & = & X_{132}-X_{231}-{1/2}X_{312} \\
Z_{123} & = & X_{123} \\
\end{eqnarray*}
}
\end{example}


\begin{theorem}\label{BASIS}
The  polynomials $Z_\mu$, $\mu\in\S_n$, form a basis
of $\H_n$. The quadratic form $(\ , \ )$ is positive definite, and
the adjoint basis of $\{Z_\mu\}$ is $\{\zz_\mu\}$
where $\zz_\mu = \Pi(\nabla_{\mu^{-1}\omega} X_\omega)$. 
\end{theorem}


\Proof $Z_\mu$ is a non-homogeneous polynomial with the
Schubert polynomial $X_\mu$ as its term of smallest degree.
Since the classes of the Schubert polynomials form a basis
of $H^*(\F)$, the same is true for the $Z_\mu$.

The polynomials $\zz_\omega Z_\mu \theta$ (for $\mu\not= id$) have
no component of degree $\ell(\omega)$. Therefore, their
images under $\d_\omega$ are symmetric polynomials without constant
term, which 
proves that for all $\mu\not= id$, $(\zz_\omega\, ,\, Z_\mu)=0$.
On the other hand, 
$$(\zz_\omega\, ,\, Z_{12\ldots n})=(\zz_\omega\, ,\, 1)$$
$$=\d_\omega\left( X_\omega\theta \right)\mid_0
=\d_\omega\left( (X_\omega\theta)|_{\ell(\omega)} \right)
$$
$$=\d_\omega X_\omega = X_{12\ldots n} = 1\ . $$
For the general case of a $\zz_\nu$, one uses induction on the
length of $\nu$. Let $\nu$ and $i$ be such that 
$\ell(\nu\sigma_i)<\ell(\nu)$. Then, for any $\mu$,
$$
(Z_\mu\, ,\, \zz_\nu) =
(Z_\mu\, ,\, (s_i - k)\zz_\nu)
$$
(where $k$ is an appropriate constant)
$$
=( (s_i-k)Z_\mu \, ,\, \zz_\nu) 
=k' (Z_\mu\, ,\, \zz_\nu) + k'' (Z_{\mu\sigma_i}\, ,\, \zz_\nu)
$$
(for some other scalars $k',k''$). By induction, one can suppose $(Z_\mu\, ,\,\zz_\nu)
=0$ for $\mu\nu^{-1}\not=\omega$.

One is thus reduced to study the case
$$
\mu\sigma_i\nu^{-1} =\omega\ , \ \ \ell(\mu\sigma_i) > \ell(\mu) \ .
$$
In that case,
$$
Z_{\mu\sigma_i} = \left(s_i + {1\over r}\right) Z_\mu \ \ {\rm and} \ \
\zz_{\mu\sigma_i}=\left(s_i - {1\over r}\right) \zz_\mu
$$
for a certain integer $r$. Then, we check
$$
(Z_{\mu\sigma_i}\, ,\, \zz_{\nu\sigma_i})
=\left( \left(s_i+{1\over r}\right) Z_\mu\, ,\, 
        \left(s_i-{1\over r}\right) \zz_\mu \right)
$$
$$
=\left( \left( s_i^2 - {1\over r^2}\right) Z_\mu\, ,\, \zz_\nu\right) = 0 \ ,
$$
and
$$
(Z_\mu\, ,\, \zz_{\nu\sigma_i}) =
\left( \left( s_i + {1\over r}-{2\over r} \right) Z_\mu\, ,\, \zz_\nu \right)
$$
$$
=(Z_{\mu\sigma_i}\, ,\, \zz_\nu) - {2\over r}(Z_\mu\, ,\, \zz_\nu) = 1 - 0\ .
$$
\cqfd












\begin{example}{\rm Again for $n=3$,
\begin{eqnarray*}
\zz_{321} & = & x_1^2 x_2 \\
\zz_{312} & = & x_1^2 - 2x_1^2x_2 \\
\zz_{231} & = & x_1x_2 -2x_1^2x_2 \\
\zz_{213} & = & x_1 -  {3/2}x_1x_2 -3x_1^2+3x_1^2x_2 \\
\zz_{132} & = & x_1+x_2-3x_1x_2- {3/2}x_1^2 +3x_1^2x_2 \\
\zz_{123} & = & 1-4x_1-2x_2 +6x_1x_2 +6x_1^2 -6x_1^2x_2 \\
\end{eqnarray*}
}
\end{example}


\section{Change of basis}

The operators $\d_i$ are self-adjoint with respect to $\<\ ,\ \>$, but
$\sigma_i$ is adjoint to $-\sigma_i$. This implies that $-s_i$ is
adjoint to $\bar s_i:=\sigma_i-\d_i$.

Let us define $\bcar_\mu$, $\bnab_\mu$ to be the images of $\carre_\mu$ and
$\nabla_\mu$ under the replacement $s_i\mapsto \bar s_i$. We also define
$$
\bz_\mu:= (-1)^{\ell(\omega\mu)}\Pi\,\Bigl(\bcar_{\mu^{-1}\omega} Z_\omega
\Bigr)
$$
$$
\bzz_\mu:=(-1)^{\ell(\omega\mu)}\Pi\,\Bigl(\bnab_{\mu^{-1}\omega} Z_\omega
\Bigr)
$$
Then, $(-1)^{\ell(\mu)}\bz_\mu$ is obtained from $Z_\mu$ under the transformation
$x_i\mapsto -x_i$, since signs in the expansion of $Z_\mu$ correspond
to degree.

\begin{lemma} \label{ADJ2}
$\{\bz_{\omega\mu}\}$ is the adjoint basis of $\{Z_\mu\}$ with respect to $\<\ ,\ \>$,
{\rm i.e.} one has $\<\bz_{\omega\mu}\, ,\,Z_\mu\>=1$ and
$\<\bz_{\omega\mu}\, ,\, Z_\nu\>=0$ for $\nu\not=\mu$.

Similarly, $\{\bzz_{\omega\mu}\}$ is the adjoint basis of $\{\zz_\mu\}$ for $\<\ ,\ \>$.
\end{lemma}

\Proof As in section \ref{AFSCHUB}, the lemma is proved by induction on the
length of $\mu$, starting from the case
$$
\<Z_{\omega\mu}\, ,\, \bz_\omega\>=0 \ \ {\rm if} \ \ \mu\not=\omega \,.
$$

Take $i$ such that $\ell(\mu\sigma_i)>\ell(\mu)$.
Then,
$$
\<Z_{\omega\mu\sigma_i}\, ,\, \bz_\nu\>
=\left\< \left( s_i+{1\over r}\right) Z_{\omega\mu}\, \bz_\nu \right\>
=\left\< Z_{\omega\mu}\, ,\, \left(-\bar{s}_i+{1\over r}\right)\bz_\nu\right\>
\ .
$$
Since $(-\bar{s}_i+{1\over r})\bz_\nu$ is a linear combination
of $\bz_\nu$ and $\bz_{\nu\sigma_i}$,
the nullity of the scalar products $\bz Z_{\omega\mu\sigma_i}\, ,\, \bz_\nu\>$
follows from those of $\<Z_{\omega\mu}\, ,\, Z_\nu\>$ for $\nu\not=\mu$
and $\nu\not =\mu\sigma_i$. In the special case $\nu=\mu\sigma_i$, one
has 
$$
\<Z_{\omega\mu\sigma_i}\, ,\, \bz_{\mu}\> =
\left\< Z_{\omega\mu}\, ,\,
\left( -\bar{s}_i+{1\over r}\right)
\left( -\bar{s}_i-{1\over r}\right) Z_{\mu\sigma_i}\right\>
$$
$$
= \left\< Z_{\omega\mu}\, ,\,
\left(1-{1\over r^2}\right) Z_{\mu\sigma_i} \right\>
$$
which is null. \cqfd



\begin{example}
$$
\< Z_{23514}\, , \, \bz_{41352} \> 
=\left\<
\Bigl( s_2+{1\over 2}\Bigr) Z_{2{\bf 53}14}\, ,\,
\Bigl(-\bar{s}_2-{1\over 2}\Bigr) \bz_{4{\bf 31}52}
\right\>
$$
$$
=\left\<
\Bigl(s_2-{1\over 2}\Bigr)\Bigl(s_2+{1\over 2}\Bigr)Z_{25314}\, ,\,
\bz_{43152}
\right\>
$$
$$
=\left\<
\Bigl(1-{1\over 4}\Bigr) Z_{25314}\, ,\, \bz_{43152}
\right\> = 0 \ .
$$
\end{example}

Let $\{A_\mu\}$ and $\{B_\nu\}$ be two bases of $\H_n$. We denote
by $M(A,B)$ the transition matrix from the basis $\{A_\mu\}$ to
the basis $\{B_\nu\}$, with the convention
\begin{equation}
A_\mu = \sum_\nu M(A,B)_{\mu\nu} B_\nu \ .
\end{equation}

For example, $M(Z,X)_{\mu\nu}=\<Z_\mu\, ,\, X_{\omega\nu}\>$
and $M(X,Z)_{\mu\nu}=(X_\mu\, ,\, \zz_{\omega\nu})$. These
matrices have a symmetry property , thanks to the following
property of the scalar product:
\begin{equation}
\<\omega P\, ,\, \omega Q\>
=(-1)^{\ell(\omega)} \<P\, ,\, Q\> \ .
\end{equation}
Indeed, taking into account the two identities
\begin{equation}
\omega(X_\mu)=(-1)^{\ell(\mu)} X_{\omega\mu\omega} \ , \ \
\omega(Z_\mu)=(-1)^{\ell(\mu)} \bz_{\omega\mu\omega}
\end{equation}
we see that the four matrices
$$M(Z,X) \ , \ M(X,Z)\ ,\ M(\zz ,X)\ {\rm and}\ M(X,\zz) $$
possess the symmetry
\begin{equation}
M_{\mu\nu} = M_{\omega\mu\omega, \omega\nu\omega} \ .
\end{equation}

Furthermore, we have the following relation between these matrices
and their inverses:

\begin{theorem} 
The inverse of $M(Z,X)$ is a matrix with nonnegative entries, given by
$$ M(X,Z)_{\mu\nu} = | M(Z,X)_{\nu\omega,\mu\omega} | \ . $$
Similarly,
$$M(X,\zz)_{\mu\nu} = |M(\zz ,X)_{\nu\omega,\mu\omega}| \ , $$
where $|\, \cdot\, |$ denotes the absolute value.
\end{theorem}

\Proof The first matrix corresponds to the expansions
$$
Z_\mu = \sum_\nu \<Z_\mu\, ,\, X_{\omega\nu}\> X_\nu \ .
$$
The inverse formulas are, according to Lemma \ref{ADJ2},
$$
X_\nu = \sum_{\mu} \<\bz_{\omega\mu}\, ,\, X_\nu\> Z_\mu \ .
$$
But now, $\<\bz_{\omega\mu}\, ,\, X_\nu\> = |\<Z_{\omega\mu}\, ,\, X_\nu\> |$,
whence the first part of the theorem. The second part is similar. \cqfd

Thus, the inverse of the matrix $M(Z,X)$ is obtained from $M(Z,X)$
by reflection through the antidiagonal $(\mu,\nu)\longrightarrow
(\omega\nu,\omega\mu)$ and supression of the signs. 

\begin{corollary}
For any pair of permutations,
$$
(X_\mu\, ,\, \zz_\eta) =| \< X_{\omega\mu\omega}\, ,\, Z_{\omega\eta\omega}\>|
$$
and
$$
(X_\mu\, ,\, Z_\eta) =| \< X_{\omega\mu\omega}\, ,\, \zz_{\omega\eta\omega}\>|
\ .
$$
\end{corollary}

Indeed, 
$$
X_\mu = \sum_\eta (X_\mu\, ,\, \zz_\eta ) Z_{\omega\eta}
$$
but we have just seen that the coefficients of the expansion of $X_\mu$
in the basis $Z_{\omega\eta}$ are the $\<\bz_\eta\, ,\, X_\mu\>$.
\cqfd





\section{Schubert expansions of Chern classes }

Let $|I|$ be a composition of $n$, \ie $I\in\N^r$  with
$|I|:=i_1+\ldots +i_r=n$, and let $J_1,\ldots,J_r$ be the associated
decomposition of the interval $[1,n]$, that is
$$
J_1=[1,i_1]\ , \ \ J_2=[i_1+1,i_1+i_2]\ , \ \ \ldots \ , \ \
J_r=[i_1+\cdots +i_{r-1}+1, n] \ .
$$
Let $\F_I$ be the variety of  flags
$$
V_0=\{0\} \subset V_1 \subset V_2 \subset \ldots \subset V_r = V
$$
such that $\dim V_k = i_1+\cdots +i_k$.
Let also $\S_I$ denote the Young subgroup
$$
\S(J_1)\times\S(J_2)\times\cdots\times\S(J_r) \ \subset  \S_n 
$$
associated to the composition $I$. 
The Chern class $\theta^I$ of the tangent bundle of $\F_I$ is the  maximal 
$\S_I$-invariant factor
of $\theta$:  
$$
\theta^I = \theta / (\theta_{J_1}\theta_{J_2}\cdots \theta_{J_r})
$$
where
$$
\theta_{J_k} = \prod_{\scriptstyle {i < j \atop i,j\in J_k}}(1+x_i-x_j) \ .
$$

A basis of the cohomology ring $H^*(\F_I)$ is the set of Schubert
polynomials $X_\mu$ with $\mu$ minimal in its right coset $\mu\S_I$.
In other words, one restricts the Schubert basis $(X_\mu)$ to those
$\mu$ such that $\mu_1<\ldots <\mu_{i_1}$, $\mu_{i_1+1}<\ldots <
\mu_{i_1+i_2}$, $\ldots$, $\mu_{i_1+\cdots+i_{r-1}+1} < \ldots < \mu_n$.
Since $\mu_i < \mu_{i+1}$ iff $X_\mu$ is symmetrical in $x_i$ and $x_{i+1}$,
the Schubert basis of $H^*(\F_I)$ consists of those Schubert polynomials
which are invariant under $\S_I$.

Define the  {\it Chern coefficient} $c_\mu[\F_I]$  of the variety $\F_I$
as the coefficient of (the class) of $X_\mu$ in the expansion of
$\theta^I$ on the Schubert basis of $H^*(\F_I)$. In other words,
\begin{equation}
c_\mu[\F_I] = \<\theta^I\, ,\, X_{\omega\mu}\> \ .
\end{equation}
As in the case of the full flag variety $\F$, these scalar products can be
computed with the help of the scalar product $(\ , \ )$.

Let $\omega_I$ be the maximal element of $\S_I$, and $\zeta_I:=\omega_I\omega$.
We have seen that
$$
\carre_\omega = \d_\omega\theta =
\d_{\zeta_I}\d_{\omega_I} \theta^I \theta_{J_1}\theta_{J_2}\cdots\theta_{J_r}
\ \ .
$$
The operator $\d_\omega$ factorizes
$$
\d_\omega = \d_{\zeta_i}\d_{\omega_I}
$$
so that
$$
c_\mu[\F_I] = \d_\omega\left( \theta^I X_{\omega\mu} \right) 
$$
\begin{equation}\label{E1}
=\d_{\zeta_i}\d_{\omega_i} \left( \theta^I X_{\omega\mu} \right)
= \d_{\zeta_i}\left( \theta^I\d_{\omega_i} X_{\omega\mu} \right)
\end{equation}
\begin{equation}\label{E2}
= \d_{\zeta_I} \left( \theta^I X_{\omega\mu\omega_I} \right)
=\d_{\zeta_i} \left( \theta^I X_{\omega\mu\omega_I} \right)
 \left( \d_{\omega_I} \theta_{J_1}\theta_{J_2}\cdots \theta_{J_r}/I! \right)
\end{equation}
\begin{equation}\label{E3}
= {1\over I!} \d_\omega \left ( \theta\, X_{\omega\mu\omega_I} \right)
\end{equation}
$$
= {1\over I!} \left( 1\, ,\, X_{\omega\mu\omega_I} \right)
$$
Equality (\ref{E1}) follows from the fact that $\theta^I$
is invariant under $\S_I$ and thus commutes with $\d_{\omega_I}$. Now,
$\theta$ is of degree ${n \choose 2}$, and $\d_\omega$ decreases
degrees by $\ell(\omega)={n\choose 2}$. Thus $\d_\omega(\theta)$ is
a scalar which is checked to be $n!$. More generally, by direct product,
one has for the maximal element of the Young subgroup $\S_I$
$$
\d_{\omega_I} \theta_{J_1}\cdots \theta_{J_r} = I! := i_1!\cdots i_r!
$$
and equality (\ref{E2}) follows from this identity. 

Since $\theta^I$ as well as $X_{\omega\mu\omega_I}$ are invariant
under $\S_I$, they commute with $\d_{\omega_I}$,
which is step (\ref{E3}).

Summarizing, we have the following expression for the components of the Chern 
class
of $\F_I$ on the Schubert basis.

\begin{theorem}
Let $I=(i_1,\ldots,i_r)$ be a composition of $n$, $\S_I$ and
$\F_I$ the corresponding Young subgroup and flag variety.
Let $\mu$ be a permutation which is minimum in its coset $\mu\S_I$.
Then, the Chern coefficient $c_\mu[\F_I]$ is given by
$$
c_\mu[\F_I]= (1\, ,\, X_{\omega\mu\omega_I})/I! \ .
$$
\end{theorem}

In particular, for the full flag variety (case $I=(1,1,\ldots,1)$), one has
\begin{equation}
c_\mu[\F]= (1\, ,\, X_{\omega\mu}) = \carre_\omega (X_{\omega\mu})
\end{equation}
and these numbers constitute the first column of the matrix $M(X,\zz)$.
Equivalently, they are equal to the absolute values of the entries of the
last row of $M(\zz,X)$.

In the case of a Grassmann manifold $G(p,p+q)=\F_{(p,q)}$, the basis of
$H^*(\F_{(p,q)})$ consists of those $X_\mu$ for which
$\mu_1<\ldots <\mu_p$ and $\mu_{p+1}<\ldots <\mu_{p+q}$ (Grassmannian
permutations). In fact, for such a permutation, $X_\mu$ is equal
to the Schur function indexed by the partition $(\mu_1-1,\mu_2-2,\ldots,\mu_p-p)$
on the set of variables $\{x_1,\ldots,x_p\}$. Thus, the Chern coefficient 
$c_\mu[\F_{(p,q)}]$ is 
\begin{equation}
c_\mu[\F_{(p,q)}] =
\carre_\omega \left(
X_{(n+1-\mu_p,\ldots,n+1-\mu_1,n+1-\mu_n,\ldots,n+1-\mu_{p+1}) } 
\right) \ .
\end{equation}
For example, up to a factor $(2!)^2$, the Chern coefficients of
$\F_{(2,2)}$ are 
4,\,16,\,28,\,28,\,48,\,24. They are 
given by the absolute values of the six entries of
the bottom row of the matrix $M(\zz,X)$ corresponding to columns
indexed by permutations $\omega\mu$ where $\mu$ is Grassmannian.


















\newpage
\section{Tables for $n=4$}

{\parindent=0mm

\subsection{\bf Affine Schubert polynomials}

\vskip 15mm
$Z_{4321}$ = \ $ \ x_1^{3}x_2^{2}x_3\ $ \ 

$Z_{4312}$ = \ $ \ x_1^{3}x_2^{2}\ 
$ \ 

$Z_{4231}$ = \ $ \ x_1^{3}x_2x_3\ $ \ 

$Z_{4213}$ = \ $ \ x_1^{3}x_2\  -1  /  2 \ x_1^{3}x_2x_3\ 
 - \ x_1^{3}x_2^{2}\ $ \ 

$Z_{4132}$ = \ $ \ x_1^{3}x_3\  + \ x_1^{3}x_2\ 
 - \ x_1^{3}x_2x_3\  -1  /  2 \ x_1^{3}x_2^{2}\ $ \ 

$Z_{4123}$ = \ $ \ x_1^{3}\ 
$ \ 

$Z_{3421}$ = \ $ \ x_1^{2}x_2^{2}x_3\ $ \ 

$Z_{3412}$ = \ $ \ x_1^{2}x_2^{2}\ 
$ \ 

$Z_{3241}$ = \ $ \ x_1^{2}x_2x_3\  -1  /  2 \ x_1^{2}x_2^{2}x_3\ 
 - \ x_1^{3}x_2x_3\ $ \ 

$Z_{3214}$ = \ $ \ x_1^{2}x_2\  -2  /  3 \ x_1^{2}x_2x_3\ 
 -3  /  2 \ x_1^{2}x_2^{2}\  +  1  /  3 \ x_1^{2}x_2^{2}x_3\ 
 -2 \ x_1^{3}x_2\  +  2  /  3 \ x_1^{3}x_2x_3\  + \ x_1^{3}x_2^{2}\ 
$ \ 

$Z_{3142}$ = \ $ \ x_1^{2}x_3\  + \ x_1^{2}x_2\  - \ x_1^{2}x_2x_3\ 
 -2  /  3 \ x_1^{2}x_2^{2}\  - \ x_1^{3}x_3\  - \ x_1^{3}x_2\ 
$ \ 

$Z_{3124}$ = \ $ \ x_1^{2}\  -1  /  2 \ x_1^{2}x_3\  -1  /  2 \ x_1^{2}x_2\ 
 +  1  /  2 \ x_1^{2}x_2x_3\  -2 \ x_1^{3}\  +  1  /  2 \ x_1^{3}x_3\ 
 +  1  /  2 \ x_1^{3}x_2\ $ \ 

$Z_{2431}$ = \ $ \ x_1x_2^{2}x_3\  + \ x_1^{2}x_2x_3\ 
 - \ x_1^{2}x_2^{2}x_3\  -1  /  2 \ x_1^{3}x_2x_3\ $ \ 

$Z_{2413}$ = \ $ \ x_1x_2^{2}\ 
 -1  /  2 \ x_1x_2^{2}x_3\  + \ x_1^{2}x_2\  -1  /  2 \ x_1^{2}x_2x_3\ 
 -2 \ x_1^{2}x_2^{2}\  +  1  /  2 \ x_1^{2}x_2^{2}x_3\ 
 -1  /  2 \ x_1^{3}x_2\  +  1  /  4 \ x_1^{3}x_2x_3\  +  1  /  2 \ x_1^{3}x_2^{2}\ 
$ \ 

$Z_{2341}$ = \ $ \ x_1x_2x_3\ $ \ 

$Z_{2314}$ = \ $ \ x_1x_2\  -2  /  3 \ x_1x_2x_3\  - \ x_1x_2^{2}\ 
 - \ x_1^{2}x_2\ $ \ 

$Z_{2143}$ = \ $ \ x_1x_3\  + \ x_1x_2\  -3  /  2 \ x_1x_2x_3\ 
 -2  /  3 \ x_1x_2^{2}\  +  1  /  3 \ x_1x_2^{2}x_3\  + \ x_1^{2}\ 
 -2 \ x_1^{2}x_3\  -8  /  3 \ x_1^{2}x_2\  +  7  /  3 \ x_1^{2}x_2x_3\ 
 +  4  /  3 \ x_1^{2}x_2^{2}\  -1  /  3 \ x_1^{2}x_2^{2}x_3\ 
 -3  /  2 \ x_1^{3}\  +  2 \ x_1^{3}x_3\  +  7  /  3 \ x_1^{3}x_2\ 
 -2  /  3 \ x_1^{3}x_2x_3\  -1  /  3 \ x_1^{3}x_2^{2}\ 
$ \ 

$Z_{2134}$ = \ $ \ x_1\  -1  /  2 \ x_1x_2\  +  1  /  2 \ x_1x_2^{2}\  - \ x_1^{2}\ 
 +  1  /  2 \ x_1^{2}x_2\  + \ x_1^{3}\ $ \ 

$Z_{1432}$ = \ $ \ x_2^{2}x_3\ 
 + \ x_1x_2x_3\  + \ x_1x_2^{2}\  -2 \ x_1x_2^{2}x_3\  + \ x_1^{2}x_3\ 
 + \ x_1^{2}x_2\  -2 \ x_1^{2}x_2x_3\  -3  /  2 \ x_1^{2}x_2^{2}\ 
 + \ x_1^{2}x_2^{2}x_3\  -2  /  3 \ x_1^{3}x_3\  -2  /  3 \ x_1^{3}x_2\ 
 +  2  /  3 \ x_1^{3}x_2x_3\  +  1  /  3 \ x_1^{3}x_2^{2}\ 
$ \ 

$Z_{1423}$ = \ $ \ x_2^{2}\  + \ x_1x_2\  - \ x_1x_2^{2}\  + \ x_1^{2}\ 
 - \ x_1^{2}x_2\  -2  /  3 \ x_1^{3}\ $ \ 

$Z_{1342}$ = \ $ \ x_2x_3\  + \ x_1x_3\ 
 + \ x_1x_2\  -2 \ x_1x_2x_3\  -1  /  2 \ x_1^{2}x_3\  -1  /  2 \ x_1^{2}x_2\ 
 +  1  /  2 \ x_1^{2}x_2x_3\  +  1  /  2 \ x_1^{3}x_3\ 
 +  1  /  2 \ x_1^{3}x_2\ $ \ 

$Z_{1324}$ = \ $ \ x_2\  -1  /  2 \ x_2x_3\  - \ x_2^{2}\ 
 + \ x_1\  -1  /  2 \ x_1x_3\  -5  /  2 \ x_1x_2\  + \ x_1x_2x_3\  +  2 \ x_1x_2^{2}\ 
 -3  /  2 \ x_1^{2}\  +  1  /  4 \ x_1^{2}x_3\  +  9  /  4 \ x_1^{2}x_2\ 
 -1  /  4 \ x_1^{2}x_2x_3\  -1  /  2 \ x_1^{2}x_2^{2}\ 
 + \ x_1^{3}\  -1  /  4 \ x_1^{3}x_3\  -1  /  4 \ x_1^{3}x_2\ 
$ \ 

$Z_{1243}$ = \ $ \ x_3\  + \ x_2\  - \ x_2x_3\  -1  /  2 \ x_2^{2}\  + \ x_1\ 
 - \ x_1x_3\  -3  /  2 \ x_1x_2\  + \ x_1x_2x_3\  +  1  /  2 \ x_1x_2^{2}\ 
 -1  /  2 \ x_1^{2}\  +  1  /  2 \ x_1^{2}x_2\ $ \ 

$Z_{1234}$ = \ $ \ 1\ 
$ \ 


\newpage
\subsection{Adjoint polynomials }

\vskip 5mm

$Z^\vee_{4321}$ = \ $ \ x_1^{3}x_2^{2}x_3\ $ \ 

$Z^\vee_{4312}$ = \ $ \ x_1^{3}x_2^{2}\ 
 -2 \ x_1^{3}x_2^{2}x_3\ $ \ 

$Z^\vee_{4231}$ = \ $ \ x_1^{3}x_2x_3\  -2 \ x_1^{3}x_2^{2}x_3\ 
$ \ 

$Z^\vee_{4213}$ = \ $ \ x_1^{3}x_2\  -3  /  2 \ x_1^{3}x_2x_3\  -3 \ x_1^{3}x_2^{2}\ 
 +  3 \ x_1^{3}x_2^{2}x_3\ $ \ 

$Z^\vee_{4132}$ = \ $ \ x_1^{3}x_3\  + \ x_1^{3}x_2\ 
 -3 \ x_1^{3}x_2x_3\  -3  /  2 \ x_1^{3}x_2^{2}\  +  3 \ x_1^{3}x_2^{2}x_3\ 
$ \ 

$Z^\vee_{4123}$ = \ $ \ x_1^{3}\  -2 \ x_1^{3}x_3\  -4 \ x_1^{3}x_2\ 
 +  6 \ x_1^{3}x_2x_3\  +  6 \ x_1^{3}x_2^{2}\  -6 \ x_1^{3}x_2^{2}x_3\ 
$ \ 

$Z^\vee_{3421}$ = \ $ \ x_1^{2}x_2^{2}x_3\  -2 \ x_1^{3}x_2^{2}x_3\ 
$ \ 

$Z^\vee_{3412}$ = \ $ \ x_1^{2}x_2^{2}\  -2 \ x_1^{2}x_2^{2}x_3\ 
 -2 \ x_1^{3}x_2^{2}\  +  4 \ x_1^{3}x_2^{2}x_3\ 
$ \ 

$Z^\vee_{3241}$ = \ $ \ x_1^{2}x_2x_3\  -3  /  2 \ x_1^{2}x_2^{2}x_3\ 
 -3 \ x_1^{3}x_2x_3\  +  3 \ x_1^{3}x_2^{2}x_3\ $ \ 

$Z^\vee_{3214}$ = \ $ \ x_1^{2}x_2\ 
 -4  /  3 \ x_1^{2}x_2x_3\  -5  /  2 \ x_1^{2}x_2^{2}\ 
 +  2 \ x_1^{2}x_2^{2}x_3\  -4 \ x_1^{3}x_2\  +  4 \ x_1^{3}x_2x_3\ 
 +  6 \ x_1^{3}x_2^{2}\  -4 \ x_1^{3}x_2^{2}x_3\ 
$ \ 

$Z^\vee_{3142}$ = \ $ \ x_1^{2}x_3\  + \ x_1^{2}x_2\  -3 \ x_1^{2}x_2x_3\ 
 -4  /  3 \ x_1^{2}x_2^{2}\  +  8  /  3 \ x_1^{2}x_2^{2}x_3\ 
 -3 \ x_1^{3}x_3\  -3 \ x_1^{3}x_2\  +  8 \ x_1^{3}x_2x_3\ 
 +  8  /  3 \ x_1^{3}x_2^{2}\  -16  /  3 \ x_1^{3}x_2^{2}x_3\ 
$ \ 

$Z^\vee_{3124}$ = \ $ \ x_1^{2}\  -3  /  2 \ x_1^{2}x_3\  -7  /  2 \ x_1^{2}x_2\ 
 +  9  /  2 \ x_1^{2}x_2x_3\  +  5 \ x_1^{2}x_2^{2}\ 
 -4 \ x_1^{2}x_2^{2}x_3\  -4 \ x_1^{3}\  +  9  /  2 \ x_1^{3}x_3\ 
 +  25  /  2 \ x_1^{3}x_2\  -12 \ x_1^{3}x_2x_3\  -12 \ x_1^{3}x_2^{2}\ 
 +  8 \ x_1^{3}x_2^{2}x_3\ $ \ 

$Z^\vee_{2431}$ = \ $ \ x_1x_2^{2}x_3\  + \ x_1^{2}x_2x_3\ 
 -3 \ x_1^{2}x_2^{2}x_3\  -3  /  2 \ x_1^{3}x_2x_3\  +  3 \ x_1^{3}x_2^{2}x_3\ 
$ \ 

$Z^\vee_{2413}$ = \ $ \ x_1x_2^{2}\  -3  /  2 \ x_1x_2^{2}x_3\  + \ x_1^{2}x_2\ 
 -3  /  2 \ x_1^{2}x_2x_3\  -4 \ x_1^{2}x_2^{2}\  +  9  /  2 \ x_1^{2}x_2^{2}x_3\ 
 -3  /  2 \ x_1^{3}x_2\  +  9  /  4 \ x_1^{3}x_2x_3\  +  9  /  2 \ x_1^{3}x_2^{2}\ 
 -9  /  2 \ x_1^{3}x_2^{2}x_3\ $ \ 

$Z^\vee_{2341}$ = \ $ \ x_1x_2x_3\  -2 \ x_1x_2^{2}x_3\ 
 -4 \ x_1^{2}x_2x_3\  +  6 \ x_1^{2}x_2^{2}x_3\  +  6 \ x_1^{3}x_2x_3\ 
 -6 \ x_1^{3}x_2^{2}x_3\ $ \ 

$Z^\vee_{2314}$ = \ $ \ x_1x_2\  -4  /  3 \ x_1x_2x_3\ 
 -3 \ x_1x_2^{2}\  +  8  /  3 \ x_1x_2^{2}x_3\  -5 \ x_1^{2}x_2\ 
 +  16  /  3 \ x_1^{2}x_2x_3\  +  10 \ x_1^{2}x_2^{2}\ 
 -8 \ x_1^{2}x_2^{2}x_3\  +  8 \ x_1^{3}x_2\  -8 \ x_1^{3}x_2x_3\ 
 -12 \ x_1^{3}x_2^{2}\  +  8 \ x_1^{3}x_2^{2}x_3\ 
$ \ 

$Z^\vee_{2143}$ = \ $ \ x_1x_3\  + \ x_1x_2\  -5  /  2 \ x_1x_2x_3\  -4  /  3 \ x_1x_2^{2}\ 
 +  2 \ x_1x_2^{2}x_3\  + \ x_1^{2}\  -4 \ x_1^{2}x_3\ 
 -16  /  3 \ x_1^{2}x_2\  +  9 \ x_1^{2}x_2x_3\  +  16  /  3 \ x_1^{2}x_2^{2}\ 
 -6 \ x_1^{2}x_2^{2}x_3\  -5  /  2 \ x_1^{3}\  +  7 \ x_1^{3}x_3\ 
 +  9 \ x_1^{3}x_2\  -12 \ x_1^{3}x_2x_3\  -6 \ x_1^{3}x_2^{2}\ 
 +  6 \ x_1^{3}x_2^{2}x_3\ $ \ 

$Z^\vee_{2134}$ = \ $ \ x_1\  -2 \ x_1x_3\  -7  /  2 \ x_1x_2\ 
 +  5 \ x_1x_2x_3\  +  9  /  2 \ x_1x_2^{2}\  -4 \ x_1x_2^{2}x_3\ 
 -5 \ x_1^{2}\  +  8 \ x_1^{2}x_3\  +  31  /  2 \ x_1^{2}x_2\ 
 -18 \ x_1^{2}x_2x_3\  -15 \ x_1^{2}x_2^{2}\  +  12 \ x_1^{2}x_2^{2}x_3\ 
 +  11 \ x_1^{3}\  -14 \ x_1^{3}x_3\  -26 \ x_1^{3}x_2\ 
 +  24 \ x_1^{3}x_2x_3\  +  18 \ x_1^{3}x_2^{2}\ 
 -12 \ x_1^{3}x_2^{2}x_3\ $ \ 

$Z^\vee_{1432}$ = \ $ \ x_2^{2}x_3\  + \ x_1x_2x_3\ 
 + \ x_1x_2^{2}\  -4 \ x_1x_2^{2}x_3\  + \ x_1^{2}x_3\ 
 + \ x_1^{2}x_2\  -4 \ x_1^{2}x_2x_3\  -5  /  2 \ x_1^{2}x_2^{2}\ 
 +  6 \ x_1^{2}x_2^{2}x_3\  -4  /  3 \ x_1^{3}x_3\ 
 -4  /  3 \ x_1^{3}x_2\  +  4 \ x_1^{3}x_2x_3\  +  2 \ x_1^{3}x_2^{2}\ 
 -4 \ x_1^{3}x_2^{2}x_3\ $ \ 

$Z^\vee_{1423}$ = \ $ \ x_2^{2}\  -2 \ x_2^{2}x_3\ 
 + \ x_1x_2\  -2 \ x_1x_2x_3\  -5 \ x_1x_2^{2}\  +  8 \ x_1x_2^{2}x_3\ 
 + \ x_1^{2}\  -2 \ x_1^{2}x_3\  -5 \ x_1^{2}x_2\ 
 +  8 \ x_1^{2}x_2x_3\  +  10 \ x_1^{2}x_2^{2}\  -12 \ x_1^{2}x_2^{2}x_3\ 
 -4  /  3 \ x_1^{3}\  +  8  /  3 \ x_1^{3}x_3\  +  16  /  3 \ x_1^{3}x_2\ 
 -8 \ x_1^{3}x_2x_3\  -8 \ x_1^{3}x_2^{2}\  +  8 \ x_1^{3}x_2^{2}x_3\ 
$ \ 

$Z^\vee_{1342}$ = \ $ \ x_2x_3\  -2 \ x_2^{2}x_3\  + \ x_1x_3\  + \ x_1x_2\  -6 \ x_1x_2x_3\ 
 -2 \ x_1x_2^{2}\  +  8 \ x_1x_2^{2}x_3\  -7  /  2 \ x_1^{2}x_3\ 
 -7  /  2 \ x_1^{2}x_2\  +  25  /  2 \ x_1^{2}x_2x_3\  +  5 \ x_1^{2}x_2^{2}\ 
 -12 \ x_1^{2}x_2^{2}x_3\  +  9  /  2 \ x_1^{3}x_3\ 
 +  9  /  2 \ x_1^{3}x_2\  -12 \ x_1^{3}x_2x_3\  -4 \ x_1^{3}x_2^{2}\ 
 +  8 \ x_1^{3}x_2^{2}x_3\ $ \ 

$Z^\vee_{1324}$ = \ $ \ x_2\  -3  /  2 \ x_2x_3\ 
 -3 \ x_2^{2}\  +  3 \ x_2^{2}x_3\  + \ x_1\  -3  /  2 \ x_1x_3\ 
 -15  /  2 \ x_1x_2\  +  9 \ x_1x_2x_3\  +  13 \ x_1x_2^{2}\  -12 \ x_1x_2^{2}x_3\ 
 -9  /  2 \ x_1^{2}\  +  21  /  4 \ x_1^{2}x_3\  +  73  /  4 \ x_1^{2}x_2\ 
 -75  /  4 \ x_1^{2}x_2x_3\  -22 \ x_1^{2}x_2^{2}\  +  18 \ x_1^{2}x_2^{2}x_3\ 
 +  6 \ x_1^{3}\  -27  /  4 \ x_1^{3}x_3\  -75  /  4 \ x_1^{3}x_2\ 
 +  18 \ x_1^{3}x_2x_3\  +  18 \ x_1^{3}x_2^{2}\ 
 -12 \ x_1^{3}x_2^{2}x_3\ $ \ 

$Z^\vee_{1243}$ = \ $ \ x_3\  + \ x_2\  -3 \ x_2x_3\ 
 -3  /  2 \ x_2^{2}\  +  3 \ x_2^{2}x_3\  + \ x_1\ 
 -5 \ x_1x_3\  -13  /  2 \ x_1x_2\  +  14 \ x_1x_2x_3\  +  15  /  2 \ x_1x_2^{2}\ 
 -12 \ x_1x_2^{2}x_3\  -7  /  2 \ x_1^{2}\  +  11 \ x_1^{2}x_3\ 
 +  31  /  2 \ x_1^{2}x_2\  -26 \ x_1^{2}x_2x_3\  -15 \ x_1^{2}x_2^{2}\ 
 +  18 \ x_1^{2}x_2^{2}x_3\  +  5 \ x_1^{3}\  -14 \ x_1^{3}x_3\ 
 -18 \ x_1^{3}x_2\  +  24 \ x_1^{3}x_2x_3\  +  12 \ x_1^{3}x_2^{2}\ 
 -12 \ x_1^{3}x_2^{2}x_3\ $ \ 

$Z^\vee_{1234}$ = \ $ \ 1\  -2 \ x_3\  -4 \ x_2\ 
 +  6 \ x_2x_3\  +  6 \ x_2^{2}\  -6 \ x_2^{2}x_3\  -6 \ x_1\ 
 +  10 \ x_1x_3\  +  22 \ x_1x_2\  -28 \ x_1x_2x_3\  -26 \ x_1x_2^{2}\ 
 +  24 \ x_1x_2^{2}x_3\  +  16 \ x_1^{2}\  -22 \ x_1^{2}x_3\ 
 -48 \ x_1^{2}x_2\  +  52 \ x_1^{2}x_2x_3\  +  44 \ x_1^{2}x_2^{2}\ 
 -36 \ x_1^{2}x_2^{2}x_3\  -22 \ x_1^{3}\  +  28 \ x_1^{3}x_3\ 
 +  52 \ x_1^{3}x_2\  -48 \ x_1^{3}x_2x_3\  -36 \ x_1^{3}x_2^{2}\ 
 +  24 \ x_1^{3}x_2^{2}x_3\ $ \ 
















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

\newpage
\subsection{Transition matrices with Schubert polynomials}

The following matrices give the decompositions of the polynomials
$Z_\mu$ and $\zz_\mu$  in the basis
of Schubert polynomials.  Rows
and columns are indexed by permutations in reverse lexicographic order:
$$[4321,\,4312,\,4231,\,4213,\,4132,\,4123,\,3421,\,3412,\,3241,\,3214,\,3142,\,3124,$$
$$2431,\,2413,\,2341,\,2314,\,2143,\,2134,\,1432,\,1423,\,1342,\,1324,\,1243,\,1234]$$
The bar over a number is to be interpreted as a minus sign.

\subsubsection{$M(Z,X)$}

The entry in row $\mu$ and column $\nu$ of the following
matrix is equal to the coefficient of $X_\nu$ in 
$Z_\mu$.  This number is also the coefficient of
$\zz_{\omega\mu}$ in $X_{\omega\nu}$. 

\vskip 15mm


  $$\vbox{ \halign{$#$&&$\kern 5pt #$\cr
 1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  \overline 1  &  \overline 1  /  2  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  \overline 1  /  2  &  \overline 1  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  0  &  0  &  0  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  0  &  0  &  0  &  0  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  0  &  0  &  0  &  0  &  0  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  0  &  \overline 1  &  0  &  0  &  0  &  \overline 1  /  2  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  1  &  2  /  3  &  \overline 2  &  0  &  0  &  1  /  3  &  \overline 3  /  2  &  \overline 2  /  3  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  0  &  0  &  0  &  \overline 1  &  0  &  0  &  \overline 2  /  3  &  \overline 1  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  0  &  0  &  0  &  1  /  2  &  \overline 2  &  0  &  0  &  1  /  2  &  0  &  \overline 1  /  2  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  0  &  \overline 1  /  2  &  0  &  0  &  0  &  \overline 1  &  0  &  0  &  0  &  0  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  1  /  2  &  1  /  4  &  \overline 1  /  2  &  0  &  0  &  1  /  2  &  \overline 2  &  0  &  0  &  0  &  0  &  \overline 1  /  2  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \overline 1  &  \overline 2  /  3  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  \overline 1  /  3  &  \overline 2  /  3  &  1  /  3  &  2  &  \overline 3  /  2  &  \overline 1  /  3  &  4  /  3  &  2  &  0  &  \overline 2  &  0  &  1  /  3  &  \overline 2  /  3  &  \overline 3  /  2  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  0  &  0  &  0  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  \overline 1  &  0  &  1  /  2  &  0  &  \overline 1  /  2  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  1  /  3  &  2  /  3  &  0  &  \overline 2  /  3  &  0  &  1  &  \overline 3  /  2  &  0  &  0  &  0  &  0  &  \overline 2  &  0  &  0  &  0  &  0  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  \cr
 0  &  0  &  0  &  0  &  0  &  \overline 2  /  3  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \overline 1  &  0  &  0  &  0  &  0  &  0  &  1  &  0  &  0  &  0  &  0  &  \cr
 0  &  0  &  0  &  0  &  1  /  2  &  0  &  0  &  0  &  1  /  2  &  0  &  \overline 1  /  2  &  0  &  0  &  0  &  \overline 2  &  0  &  0  &  0  &  0  &  0  &  1  &  0  &  0  &  0  &  \cr
 0  &  0  &  0  &  0  &  \overline 1  /  4  &  1  &  0  &  \overline 1  /  2  &  \overline 1  /  4  &  0  &  1  /  4  &  \overline 1  /  2  &  0  &  2  &  1  &  \overline 1  &  0  &  0  &  0  &  \overline 1  &  \overline 1  /  2  &  1  &  0  &  0  &  \cr
 0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  1  /  2  &  1  &  0  &  0  &  0  &  0  &  \overline 1  /  2  &  \overline 1  &  0  &  1  &  0  &  \cr
 0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  1  &  \cr
 }  }$$ 

\newpage
\subsubsection{$M(\zz,X)$}

The entry in row $\mu$ and column $\nu$ of the following
matrix is equal to the coefficient of $X_\nu$ in 
$\zz_\mu$. This number is also the coefficient of $Z_{\omega\mu}$
in $X_{\omega\nu}$.



\vskip 25mm

$$\vbox{ \halign{$#$&&$\kern 5pt #$\cr
 1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 \overline 2  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 \overline 2  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 3  &  \overline 3  &  \overline 3  /  2  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 3  &  \overline 3  /  2  &  \overline 3  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 \overline 6  &  6  &  6  &  \overline 2  &  \overline 2  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 \overline 2  &  0  &  0  &  0  &  0  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 4  &  \overline 2  &  0  &  0  &  0  &  0  &  \overline 2  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 3  &  0  &  \overline 3  &  0  &  0  &  0  &  \overline 3  /  2  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 \overline 4  &  6  &  4  &  \overline 4  &  0  &  0  &  2  &  \overline 5  /  2  &  \overline 4  /  3  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 \overline {16}  /  3  &  8  /  3  &  8  &  0  &  \overline 3  &  0  &  8  /  3  &  \overline 4  /  3  &  \overline 3  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 8  &  \overline {12}  &  \overline {12}  &  8  &  9  /  2  &  \overline 4  &  \overline 4  &  5  &  9  /  2  &  \overline 2  &  \overline 3  /  2  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 3  &  0  &  \overline 3  /  2  &  0  &  0  &  0  &  \overline 3  &  0  &  0  &  0  &  0  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 \overline 9  /  2  &  9  /  2  &  9  /  4  &  \overline 3  /  2  &  0  &  0  &  9  /  2  &  \overline 4  &  0  &  0  &  0  &  0  &  \overline 3  /  2  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 \overline 6  &  0  &  6  &  0  &  0  &  0  &  6  &  0  &  \overline 2  &  0  &  0  &  0  &  \overline 2  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 8  &  \overline {12}  &  \overline 8  &  8  &  0  &  0  &  \overline 8  &  10  &  8  /  3  &  \overline 2  &  0  &  0  &  8  /  3  &  \overline 3  &  \overline 4  /  3  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 6  &  \overline 6  &  \overline {12}  &  2  &  7  &  \overline 5  /  2  &  \overline 6  &  {16}  /  3  &  7  &  0  &  \overline 4  &  0  &  2  &  \overline 4  /  3  &  \overline 5  /  2  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 \overline {12}  &  18  &  24  &  \overline {12}  &  \overline {14}  &  11  &  {12}  &  \overline {15}  &  \overline {14}  &  3  &  8  &  \overline 3  &  \overline 4  &  9  /  2  &  5  &  \overline 3  /  2  &  \overline 2  &  1  &  0  &  0  &  0  &  0  &  0  &  0  &  \cr
 \overline 4  &  2  &  4  &  0  &  \overline 4  /  3  &  0  &  6  &  \overline 5  /  2  &  0  &  0  &  0  &  0  &  \overline 4  &  0  &  0  &  0  &  0  &  0  &  1  &  0  &  0  &  0  &  0  &  0  &  \cr
 8  &  \overline 8  &  \overline 8  &  8  /  3  &  8  /  3  &  \overline 4  /  3  &  \overline {12}  &  10  &  0  &  0  &  0  &  0  &  8  &  \overline 3  &  0  &  0  &  0  &  0  &  \overline 2  &  1  &  0  &  0  &  0  &  0  &  \cr
 8  &  \overline 4  &  \overline {12}  &  0  &  9  /  2  &  0  &  \overline {12}  &  5  &  9  /  2  &  0  &  \overline 3  /  2  &  0  &  8  &  0  &  \overline 4  &  0  &  0  &  0  &  \overline 2  &  0  &  1  &  0  &  0  &  0  &  \cr
 \overline {12}  &  18  &  18  &  \overline {12}  &  \overline {27}  /  4  &  6  &  18  &  \overline {22}  &  \overline {27}  /  4  &  3  &  9  /  4  &  \overline 3  /  2  &  \overline {12}  &  10  &  6  &  \overline 3  &  0  &  0  &  3  &  \overline 3  &  \overline 3  /  2  &  1  &  0  &  0  &  \cr
 \overline {12}  &  {12}  &  24  &  \overline 4  &  \overline {14}  &  5  &  18  &  \overline {15}  &  \overline {14}  &  0  &  8  &  0  &  \overline {12}  &  9  /  2  &  11  &  0  &  \overline 2  &  0  &  3  &  \overline 3  /  2  &  \overline 3  &  0  &  1  &  0  &  \cr
 24  &  \overline {36}  &  \overline {48}  &  24  &  28  &  \overline {22}  &  \overline {36}  &  44  &  28  &  \overline 6  &  \overline {16}  &  6  &  24  &  \overline {20}  &  \overline {22}  &  6  &  4  &  \overline 2  &  \overline 6  &  6  &  6  &  \overline 2  &  \overline 2  &  1  &  \cr
 }  }$$












% ***************************************************************************
%                      BIBLIOGRAPHY
% ***************************************************************************
\newpage\footnotesize
\begin{thebibliography}{Lod11A}

\bibitem{BGG} \sc I.N. Bernstein, I.M. Gelfand {\rm and} S.I. Gelfand, \it
Schubert cells and the cohomology of the spaces $G/P$\rm, Russian Math.
Surveys {\bf 28} (1973), 1-26.

\bibitem{Che1} \sc I.V. Cherednik, \it On $R$-matrix quantization
of formal loop groups\rm, {\it in} Group theoretical
methods in physics, Proc. 3rd Semin. Yurmala (USSR), 1985,
vol. 2, 161-180 (1986).

\bibitem{Che2} \sc I.V. Cherednik, \it Quantum groups as hidden symmetries
of classic representation theory\rm, {\it in} Differential geometric
methods in theoretical physics (A.I. Solomon ed.), World Scientific,
Singapore, 1989, 47-54.

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