%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% ws-p8-50x6-00.tex : 20-11-97 %% This Latex2e file rewritten from various sources for use in the %% preparation of the (smaller [8.50''x6.00'']) single-column proceedings %% Volume, latest version by R. Sankaran with acknowledgements to Susan %% Hezlet and Lukas Nellen. Please comments to:rsanka@wspc.com.sg %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Transition on Grothendieck polynomials Alain Lascoux oct 2000 % last corrections November 20, 2000 \documentclass{ws-p8-50x6-00} \usepackage{amsmath,amsfonts,amsthm} %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% Macros personnelles %%%%%%%%%%%%%% \font\bb=msbm10 \def\s{\scriptstyle } \def\ff{ f^{\l,n}} \def\ss{\sigma} \def\a{\alpha} \def\b{\beta} \def\c{\gamma} \def\g{\gamma} \def\l{\lambda} \def\L{\Lambda} \def\e{\epsilon} \def\d{\partial} \def\tF{{\widetilde{F}}} \def\bp{\widehat{\pi}} \def\N{{\mathbb N}} \def\Z{{\mathbb Z}} \def\A{{\mathbb A}} \def\B{{\mathbb B}} \def\C{{\mathbb C}} \def\K{{\mathbb K}} \def\cF{{\mathcal F}} \def\cH{{\mathcal H}} \def\S{{\mathfrak S}} \def\Sym{{\mathfrak Sym}} \def\GP{Grothendieck polynomials } \def\moins{\raise 1pt\hbox{{$\scriptstyle -$}}} \def\plus{\raise 1pt\hbox{{$\scriptstyle +$}} } %%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} %%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{Transition on Grothendieck Polynomials} \author{A. Lascoux} \address{CNRS, Institut Gaspard Monge, Universit\'e de Marne-la-Vall\'ee\\ \small 77454 Marne-la-Vall\'ee Cedex, France\\ E-mail:Alain.Lascoux@univ-mlv.fr} \maketitle \abstracts{ Grothendieck polynomials represent Schubert varieties in the Grothendieck ring of the flag manifold (for $Gl(n,\C)$). We describe how general \GP are related to those for Grasmann manifolds, which themselves are deformations of Schur functions.} \section{Introduction} Given a vector bundle $V$ of rank $n$, the Grothendieck ring of classes of vector bundles of the relative flag manifold $\cF(V)$ is generated by the classes $a_1,\ldots, a_n$ of the so-called {\it tautological line bundles} on $\cF(V)$. The structure sheaf of a Schubert variety in $\cF(V)$, having a finite resolution by vector bundles, can be expressed as a (Laurent) polynomial in the $a_i,1/a_i$. Explicit representatives $G_\ss$, $\ss\in \S_n$ were defined in \cite{LS2},\cite{LS3} under the name ``\GP'' (we recall their definition in Eq.~\ref{eq:groth}). We shall also use {\it codes} of permutations rather than permutations, and in that case, note $G[J]$, $J\in \N^n$ (codes are defined in section 2). I gave in \cite{La1},\cite{La2} two different expressions of the $G[J]$, $J$ partition, that I recall in the last section. As usual in the theory of symmetric functions, it is appropriate to let the number of indeterminates be infinite and to obtain the finite case by specializing $x_{n+1}=0=x_{n+2}=\cdots$. When using these conventions, we shall write partitions decreasingly and write $g_\l$ for the symmetric function in an infinite number of variables obtained from the symmetric function $G[J]$ (in $x_1,\ldots, x_n$), with $J=[0\leq j_1\leq \cdots \leq j_n]$, $\l=[j_n,\ldots, j_1]$. In general, introducing some equivalence (taking the ``stable part'') that can be defined in four different manners which will be described in section 5, and that we shall note $\stackrel{ stab}{\sim}$, one can decompose any Grothendieck polynomial into a sum of Grasmannian ones \begin{equation} G_\ss \stackrel{ stab}{\sim} \sum c_\l g_\l \label{eq:stab} \end{equation} The similar decomposition for cohomology classes (represented by Schubert polynomials) embeds into the tradition of \lq\lq Schubert calculus" (for example, intersection multiplicities of Schubert varieties, generalizing the so-called {\it Littlewood-Richardson rule}). Buch \cite{Bu} studies the algebra generated by the $g_\l$, and the decomposition (\ref{eq:stab}) that he explicits for certain permutations. One motivation to write this text was to answer his question about positivity of the coefficients in (\ref{eq:stab}) (once taking the correct normalization), and I thank him for having sent me his preprint. Adapting what has been done for Schubert polynomials $X_\ss$, $\ss\in \S_n$, in \cite{LS1} and \cite{LS4}, we shall give in Eq.~(\ref{eq:trans4}) a formula which can be translated into a tree which describes the multiplicities in (\ref{eq:stab}). Moreover, this tree, as in the case of cohomology, gives a way to explicit the multiplicative structure of the Grothendieck ring of the Grassmannian, which is the problem considered by Buch\cite{Bu}. To generate the tree, one needs only to describe the multiplication of \GP by $1- 1/a_i$, $i=1,\ldots,n$, i.e. to describe the class of hyperplane sections of Schubert varieties. The similar rule for the cohomology ring is due to Monk (it was previously obtained by Chevalley) and leads to the explicit expression of Schubert polynomials as (positive) polynomials in the Chern classes of the tautological line bundles, as well as their stable part. The cohomological computations can easily be extended to two sets of indeterminates, due to the extension of Monk's rule (cf.\cite{KV}), or translated into planar combinatorial constructions. The same is true for \GP, but for simplicity, we shall restrict here to only one set of indeterminates. \GP are obtained from the action of the 0-Hecke algebra on the ring of polynomials. Let us emphasize that this action is not unique and depends upon several arbitrary parameters that we shall have to put into use (the action of the Iwahori-Hecke algebra of the symmetric group also depends upon several parameters, cf.\cite{LS5}). There exists quantum analogues of \GP and we refer to the work of Kirillov\cite{Ki} for their definition and properties. The geometry of Grassmann varieties is well understood thanks to Schubert, Giambelli and their successors\cite{KL}. It is hoped that relating the Schubert subvarieties of a flag manifold to those of a Grassmannian will be of some help to understand those varieties, the singularities of which we still do not know how to describe \cite{De1}. In the present article, we do not consider Schubert varieties, but only the classes of their structure sheaves. Much of the combinatorics of the symmetric group that appear in my work with M.P. Sch\"utzenberger finds application to Schubert calculus, and I mentionned in this text the connections with \cite{LS1}--\cite{LS8}. \section{0-Hecke algebra} The 0-Hecke algebra ${\mathcal H}$ of the symmetric group $\S_n$ is the algebra generated by $T_1,\ldots, T_{n-1}$ satisfying the braid relations $$ T_i T_{i+1}T_i= T_{i+1}T_iT_{i+1} \quad \& \quad T_i T_j=T_jT_i \ , \ |i-j|\neq 1 $$ together with $T_i^2=T_i$. Given arbitrary parameters $\a,\b,\c,\delta \in \C$, such that $\a\delta -\b\c=1$, one gets an action of ${\mathcal H}$ on the ring $\C[a_1^\pm,\ldots , a_n^\pm]$ by setting $$T_i : f \mapsto f\, T_i:= {(\a a_i+\b)(\c a_{i+1}+\delta)f - (\a a_{i+1}+\b)(\c a_i+\delta) f^{s_i} \over a_i -a_{i+1} } $$ where $s_i$ exchanges $x_i,x_{i+1}$ (cf.\cite{LS2},\cite{LS5} for a more general statement; operators on polynomials will be denoted {\bf on the right}). Moreover, the elements $1-T_i$ also satisfy the braid relations together with $(1-T_i)^2= 1-T_i$. The operators $$ f \mapsto f\d_i := {f -f^{s_i} \over a_i - a_{i+1}} $$ are {\it Newton's divided differences}. The operators $$ f \mapsto f\pi_i := f\, a_i\, \d_i = {a_i f - a_{i+1} f^{s_i} \over a_i - a_{i+1}} $$ and the operators $$ f \mapsto f\bp_i := f\d_i\, a_{i+1} = {a_{i+1} f - a_{i+1} f^{s_i} \over a_i - a_{i+1}} $$ are due to Demazure \cite{De2} and called {\it isobaric divided differences} in \cite{LS2}. Let us also use the variables $x_i :=1-1/a_i$, and the corresponding operators that we shall distinguish by superscripts (for example, $\d_1^x$ sends $x_1$ onto 1, and $a_1= 1/(1-x_1)$ onto $a_1a_2$). Since braid relations are satisfied, to any permutation $\ss$ are associated operators $T_\ss, \pi^a_\ss,\pi^x_\ss, \d^a_\ss, \d^x_\ss, \bp^a_\ss, \bp^x_\ss$ which can be obtained by taking any reduced decomposition of $\ss$ and evaluating the associated product of elementary operators. In particular, the operators associated to the {\it maximal permutation} $\omega\in \S_n,\, \omega:=[n,\ldots,1]$ play a fundamental r\^ole. We shall also need the isobaric divided differences corresponding to the shifted variables $a_1\!-\!1,a_2\!-\!1,\ldots,a_n\!-\!1$ that we shall denote $D_1,\ldots, D_{n-1}$ to avoid any confusion with the preceding ones. All these operators can be expressed in terms of Newton's divided differences only, but it greatly simplify computations, and comprehension, to introduce them instead of restricting to divided differences. Most of their properties directly arise from the following relations, which can be easily checked from their concrete action on polynomials or using the braid relations together with Leibniz formula: $ f\, g\, \d_i = f\, (g\d_i) + (f\,\d_i)\, g^{s_i}$. For example, Eq.~(\ref{eq:dd1})--(\ref{eq:dd8}) are statements involving only two indices $\{i,i\plus 1\}$. The operators commuting with multiplication by a function invariant under $s_i$, it is sufficient to check their action on the two polynomials $1$ and $x_i$. \begin{lemma} Divided differences satisfy the following relations~: \bea \pi_i^a =(1-x_{i+1})\, \d_i^x & ;& \bp_i^a = \d_i^x\, (1-x_i) \label{eq:dd1} \\[4pt] \d_i^a= (1-x_i)(1-x_{i+1})\, \d_i^x &=& \d_i^x\, (1-x_i)(1-x_{i+1}) \\[4pt] \pi_i^x=(a_i-1)a_{i+1}\, \d_i^a & ; & \bp_i^x = \d_i^a a_i (a_{i+1}-1) \label{eq:dd3} \\[4pt] \d_i^x = a_ia_{i+1}\, \d_i^a = a_{i+1}\, \pi_i^a &= & \bp_i^a\, a_i = \d_i^a a_i a_{i+1} \label{eq:dd4} \\[4pt] D_i = x_i(1-x_{i+1})\, \d_i^x & = & (a_i-1)\, \d_i^a \label{eq:dd5} \\[4pt] x_i\, \pi_i^a = \pi_i^a\, x_{i+1}+(1-x_{i+1}) & ;& (x_i-1)\, \pi_i^a = \bp_i^a\, (x_{i+1}-1)\label{eq:dd8} \\[4pt] \pi_i^a\, D_{n-1}\cdots D_1 & = & D_{n-1}\cdots D_1\, \pi_{i+1}^a \ , \ ii,\ss^k<\ss^i\}$. We shall write $G_\ss$ or $G[J]$ indifferently. \GP are stable with respect to the embeddings $\S_n = \S_n\times \S_1 \hookrightarrow \S_{n+1}$, and accordingly, one can freely add or suppress terminal 0's to codes without changing the polynomials : $G[J]= G[J0\ldots 0]$. Instead of starting from $G[\rho]=x^\rho$ for a given $n$, and letting $n$ vary, one can as well rewrite the preceding definition as follows. Say that a vector $\l=[\l_1,\ldots,\l_k]$ is {\it dominant} if $\l_1\geq \cdots \geq \l_k\geq 0$. Then \begin{equation} \left\{\ \begin{matrix} G[\l] &=& x^\l &\l\ {\rm dominant}\cr G[\ldots, j_k, j_{k+1},\ldots]\, \pi_k^a &=& G[\ldots, j_{k+1}\!-\! 1,j_k,\ldots] & j_j>j_{k+1} \cr \end{matrix} \right. \end{equation} This convention allows to have all $n$'s at the same time. \section{Symmetric Grothendieck Polynomials} On the Grothendieck ring of classes of vector bundles over a manifold act operators induced from the exterior powers and symmetric powers of vector bundles, that we shall denote $\L^i$ and $S^i$. Explicitely, taking an extra variable $z$, the generating function of the $S^i$ is such that \begin{equation} \sum_{i=0}^\infty z^i\, S^i(\B-\B')= \prod (1-zb')/\prod(1-zb) \end{equation} if $\B=\sum b$, $\B'=\sum b'$ are finite sums of classes of line bundles. More generally, given any partition $K=[k_1,\ldots, k_n]$, $0\leq k_1\leq \cdots \leq k_n$, one has a {\it Schur functor} $S_K$, the action of which on sums $\B=\sum_{i=1}^n b_i$ of line bundles is \begin{equation} S_K(\B) = \det\left|S^{k_j+j-i}(\B) \right|_{1\leq i,j\leq n} = \det\left|b_i^{k_j+n-j} \right|_{1\leq i,j\leq n}\, /\, \det\left| b_i^{n-j} \right|_{1\leq i,j\leq n} \end{equation} Still a little more generally again \cite{La2}, Schur functors indexed by partitions in $\N^n$ can be considered as morphisms from the $n$-th tensor power of a $\l$-ring into itself : \begin{equation} S_K(\B^1, \ldots, \B^n) := \det\left|S^{k_j+j-i}(\B^j) \right|_{1\leq i,j\leq n} \end{equation} where $\B^1,\ldots,\B^n$ are arbitrary elements of the $\l$-ring. \section{ Stable part by restriction or symmetrization.} Let $k,n\in \N$, $k\geq n$, $I\in \N^k$. Then $G[0^{k-1}I]$ is symmetrical in $x_1,\ldots, x_k$ and therefore the restriction $$G[0^{k-1}I]\big|_{x_{k+1}=0=x_{k+2}=\cdots}$$ is a symmetric function $F[I]$ of $k$ variables. Similarly to the case of Schubert polynomials (cf.\cite{LS1},\cite{Ma}) we shall adopt the common terminology and call $F[I]$ the {\it stable part} of $G[I]$, writing also $F_\ss$ if $\ss$ is any permutation of code $[I0\,\ldots 0]$. One needs to check some stability with respect to $k$, since one has only required $k\geq n$. This will be clarified by the following property. \vfill\eject \begin{theorem} 1) Let $I\in \N^n$. Then \begin{equation} G[I] D_n\cdots D_1 = G[0I] \label{eq:stab1} \end{equation} 2) Let $\omega$ be the maximal permutation of $\S_n$. Then \begin{equation} G[I] D_\omega= F[I] \in \Sym(x_1,\ldots, x_n) \label{eq:stab2} \end{equation} \end{theorem} \noindent {\it Proof.} $G[I]$ is the image of some $G[J]$, $J\in \N^n$ dominant, under some product of $\pi_i^a$, $i\leq n-1$. However, $G[J]$ can be written as a determinant (with $\A_k:=:a_1+\cdots+a_k$)~: $$G[J]=G[J0]= (a_1\cdots a_{n+1})^{-n}\, S_{n^{n+1}} (\A_{n+1}\moins 0,\, \A_n\moins j_n, \ldots,\, \A_1\moins j_1)\ .$$ Since $\forall i,j,k\in \N$, one has $S^k(\A_i-j)\, D_i= S^k(\A_{i+1}-j\moins 1)$ ($D_i$ being the isobaric divided difference with respect to the pair $(a_i\moins 1, a_{i+1}\moins 1)$), each $D_i$ in the product $D_n\cdots D_1$ operates in turn on a single column of the determinant, the others being symmetrical in $(a_i-1, a_{i+1}-1)$. Therefore the image of $G[J0]$ under $D_n\cdots D_1$ is $$ (a_1\cdots a_{n+1})^{-n} \, S_{n^{n+1}} (\A_{n+1}-0,\, \A_{n+1}-j_n\moins 1, \ldots,\, \A_{n+1}-j_1\moins 1) $$ i.e. is equal to the image of $G[j_1+1,\ldots, j_n+1,0]$ under $\pi_n^a\cdots \pi_1^a$ which by definition is $G[0J]$. Now, if $G[I]=G[J] \pi_i^a\pi_j^a\cdots$ , then the commutations (\ref{eq:dd6}) show that $$G[0I]=G[0J]\, \pi_{i+1}^a\pi_{j+1}^a\cdots = G[J]D_n\cdots D_1 \pi_{i+1}^a\pi_{j+1}^a\cdots = G[I]D_n\cdots D_1$$ and this proves Eq.~(\ref{eq:stab1}). To simplify indices, let us take $k=n=4$ to attack Eq.~(\ref{eq:stab2}) The morphism $G[I] \mapsto G[000I]$, $I\in \N^4$ is $D_\nu:= (D_4\cdots D_1)\, (D_5\cdots D_1)\,(D_6\cdots D_1)$. The specialization $x_5=x_6=x_7$ of $G[000I]$ is the same as the specialization of $G[000I]D_\eta$, with $\eta= s_6 s_5s_4 s_6 s_5 s_6$. However, $\nu\eta= \omega_7$ (= the maximal element of $\S_7$), and thus is equal to $\omega_4\, (\omega_4 \omega_7)$. Now, given any symmetric function of $x_1,\ldots, x_4$, applying to it any sequence of $D_i$ and specializing $x_5=0=x_6= \cdots$ does not change the function. Therefore $G[I]\, D_{\omega_4}$, which indeed is symmetrical in $x_1,\ldots, x_4$, coincides with $G[I] D_{\omega_7}\big|_{x_5=0=x_6= \cdots}$ \qed \smallskip Notice that if $I$ is the code of a vexillary permutation, then $G[I 0^{n-1}]$ has a determinantal expression in terms of the $S^k(\A_i -j)$, and the morphism $G[I0^{n-1}]\mapsto G[0^{n-1}I]$ consists in replacing in the determinantal expression of $G[I0^{n-1}]$ each term $S^k(\A_i -j)$ by $S^k(\A_{i+n-1} -j\moins n\plus 1)$. Specializing $x_{n+1},x_{n+2},\ldots$ to 0 , that is, $a_{n+1}, a_{n+2}, \ldots$ to 1, is obtained by putting $\A_{i+n}=\A_n+i$, and thus the stable part of $G[I]$ is a determinant in the $S^k(\A_n\moins j)$ (which is recognized to be the one expressing $G[J]$, with $J$= increasing reordering of $I$). In other words, if $I\in \N^n$ is the code of a vexillary permutation, then $G[I]\, D_{\omega} = G[J]$. This leads to define, for a general $I\in \N^n$, a {\it G-key polynomial} $KG[I]$ by \begin{equation} \left\{\ \begin{matrix} KG[I] &=& G[I] &I\ {\rm dominant}\cr KG[\ldots, j_k, j_{k+1},\ldots]\, D_k &=& KG[\ldots, j_{k+1},j_k,\ldots] & j_k>j_{k+1} \cr \end{matrix} \right. \end{equation} Using that $D_i D_\omega = D_\omega$, $ir$, let $${\mathcal A}_{n,r}(x):= (1+xT_{n-1})\cdots (1+xT_r) $$ \smallskip\noindent {\bf Definition\cite{FK}.} \quad For any $n$, and any $\ss\in\S_n$, the {\it stable \GP} $\tF_\ss$ are the coefficients of the generating function \begin{equation} {\mathcal A}_{n,1}(x_1)\cdots {\mathcal A}_{n,1}(x_n) = \sum\nolimits_{\ss\in \S_n}\, \tF_\ss\, T_\ss \ . \end{equation} On the other hand, the generating function\cite{FK} of \GP $G_\ss$, $\ss\in\S_{2n-1}$, is \begin{equation} {\mathcal G}_{2n-1} := {\mathcal A}_{2n-1,1}(x_1) \, {\mathcal A}_{2n-1,2}(x_2) \cdots {\mathcal A}_{2n-1,2n-1}(x_{2n-1}) \ . \end{equation} Now, if $T_iT_j\cdots T_k$ is equal to $T_\ss$, $\ss\in \S_n$, $\ss$ having code $I$, then $T_{i+n-1}T_{j+n-1} \cdots T_{k+n-1}$ is equal to $T_{\ss'}$, $\ss'$ having code $[0^{n-1}I]$. Therefore, the coefficient of $T_{\ss'}$ in the specialization $T_1=0=\cdots =T_{n-1}$; $x_{n+1}=0=\cdots= x_{2n-1}$ of ${\mathcal G}_{2n-1}$ is the same as the coefficient of $T_\ss$ in ${\mathcal A}_{n,1}(x_1)\cdots {\mathcal A}_{n,1}(x_n)$, that is, one has the identity \begin{equation} \tF_\ss = G[0^{n-1}I]\,\big|_{x_{n+1}=0=\cdots= x_{2n-1}} = F_\ss \ . \end{equation} \smallskip In fact, \GP already appear at the level of the ``twisted'' group algebra of the symmetric group (i.e. the algebra generated by the $s_i$'s, $x_j$'s), as coefficients in the expansion of isobaric divided differences $\pi_\ss$ $\pi_\ss$ in the basis of permutations (cf.\cite{LS8}). Yang-Baxter equation in the 0-Hecke algebra also lead to similar expansions (\cite{FK}, \cite{LLT}). \section{Transition on \GP} Let $\ss$ be a permutation $J$ its code, and $r$ be the length of $J$ (i.e. $j_r>0$, $j_i=0\ \forall i>r$). Let $\nu$ be the permutation of code $[j_1,\ldots,j_{r-1},j_r-1]$. Recall that for what concerns Schubert polynomials in $x_1,x_2,\ldots$, one has the equation \begin{equation} X_\ss= x_r X_\nu + \sum\nolimits_{i\in {\mathcal I}(\ss)} X_{\tau_{ji}\nu}\ , \label{eq:trans1} \end{equation} sum over all transpositions $\tau_{ji} :j:=\nu_r , i\in {\mathcal I}(\ss)$ (i.e. $i$ such that such that $\ell(\tau_{ji}\nu) = \ell(\nu)+1$). These equations, called {\it transition} in \cite{LS1}, refine the Ehresmann-Bruhat order on the symmetric group and give the expression of Schubert polynomials as a positive sum of monomials; they also furnish the stable part of them as a sum of Schur functions, if one cancels the successive terms $x_r X_\nu$, when iterating Eq.~(\ref{eq:trans1}). One has similar transitions for \GP. But first, one needs to explicit products of $G_\ss$ by $x_r$'s. In \cite{La3}, I described the $G_\ss /(a_1\cdots a_r)$. It requires a small adaptation to get products by $1/a_r=1-x_r$ instead. On the other hand, I cannot relate products $G_\ss (1-x_r)$ to the products $a^\l G_\ss$, $\l$ dominant, considered in \cite{FL} under the name {\it Pieri formula}. Geometrically these last products are classes of restriction of ample line bundles over Schubert subvarieties, and not intersections as in our case. Let us keep the same hyptheses as in Eq.~(\ref{eq:trans1}), ordering the values $i_1\cdots \ss_{r-1}$. If $j_{r-1}\neq 0$, then $G[j_1,\ldots ,j_r]= x_1\cdots x_r G[j_1-1,\ldots, j_r-1]$ and one is reduced to study a polynomial of smaller degree. The fundamental case to study is $J=[j_1,\ldots,j_{r-2},0,j_r]$, $j_{r-2}>0$. By induction on $r$, we know that \bea G[j_1,\ldots,j_{r-2},j_r+1] &=& G[j_1,\ldots,j_{r-2},j_r] + \nonumber \\[2pt] && (x_{r-1}-1) (1-\tau_{j i_k})\star \cdots\star (1-\tau_{j i_2}) \star G[j_1,\ldots,j_{r-2},j_r] \nonumber \\ && \label{eq:trans3} \eea \smallskip\noindent the transpositions appearing in this formula being the same as in Eq.~(\ref{eq:trans2}), but for $\tau_{j 1}$ which is missing ( the hypothesis $j_{r-2}>0$ implies $i_1=1$). Taking the image of (\ref{eq:trans3}) under $\pi_{r-1}^a$ with the help of (\ref{eq:dd8}), one gets \noindent\medskip $G[j_1,\ldots,j_{r-2},0,j_r]= $\\ \hbox{$G[j_1,\ldots,j_{r-2},0,j_r\moins 1] + (1\moins \tau_{j i_k})\star \cdots\star (1\moins \tau_{j i_2}) \star G[j_1,\ldots,j_{r-2},j_r,0] \bp_{r\moins 1}^a (x_r\moins 1) $}\\ $ = G[j_1,\ldots,j_{r-2},0,j_r-1] + \cdots\star (1-\tau_{j i_2})$\\ \hspace*{130pt} \hbox{$\star \bigl( G[j_1,\ldots,j_{r-2},0,j_r-1]- G[j_1,\ldots,j_{r-2},j_r,0] \bigr)$}\\ $= G[j_1,\ldots,j_{r-2},0,j_r-1] + (1-\tau_{j i_k}) \cdots\star (1-\tau_{j i_2}) \star (1-\tau_{j1})$\\ \hspace*{165pt}\hbox{$\star G[j_1,\ldots,j_{r-2},0,j_r-1]\, (x_r-1) $}\\ which is the required expression for those special $J$. \smallskip A general $G[J']$, $J'\in \N^r$, will be obtained from a $G[J]$, $j_1\geq \cdots \geq j_{r-1}$ by repeated use of the $\pi_k^a$, $i\leq r-2$. Suppose that Eq.~(\ref{eq:trans2}) is true for a permutation $\ss$. Let $p\leq r-2$ be an integer such that $\nu_p>\nu_{p+1}$. Then, except when both $\nu_p$ and $\nu_{p+1}$ belong to ${\mathcal I}(\ss)$, the image of (\ref{eq:trans2}) under $\pi_p^a$ is a decomposition $$ G_{\ss s_p}= G_{\nu s_p} + (x_r-1)\, (1-\tau_{j i_k})\star \cdots\star (1-\tau_{j i_1}) \star G_{\nu s_p} $$ of the type required by the theorem. On the other hand, if $\nu_p=\b, \nu_{p+1}=\a\in {\mathcal I}(\ss)$, one writes the RHS of (\ref{eq:trans2}) as a summation $$ (x_r-1) \sum \eta' \star (1-\tau_{j\b})(1-\tau_{j\a}) \star \eta'' \star G_\nu \ , $$ sum over certain pairs of permutations $\eta', \eta''$. Thus terms occur in four-tuples $$ (x_r-1) \bigl( G_{\cdots \b\a\cdots j\cdots} - G_{\cdots j \a\cdots \b\cdots} - G_{\cdots \b j\cdots \a \cdots} + G_{\cdots j\b\cdots \a \cdots} \bigr)\ ,$$ writing only the indices at places $p,p+1,r$. Now, the image of such a term under $\pi_p^a$ is $$ (x_r-1) \bigl( G_{\cdots \a\b\cdots j\cdots} - G_{\cdots \a j \cdots \b\cdots} \bigr) $$ and therefore the decomposition of $G_{\ss s_p}$ is obtained from the one of $G_\ss$ by replacing the double factor $(1-\tau_{j\b})(1-\tau_{j\a})$ by the single one $(1-\tau_{j\b})$, and exchanging $\nu$ for $\nu s_p$. \qed Specializing $x_r$ to 0, one finally gets \begin{theorem} 1) Let $\ss$ be a permutation, $\ss_1=1$. Using conventions (\ref{eq:trans1}), the stable part of $G_\ss$ is equal to that of \begin{equation} \Bigl(1- (1-\tau_{j i_k})\star \cdots\star (1-\tau_{j i_1}) \Bigr) \star G_\nu \ . \label{eq:trans4} \end{equation} 2) There exists positive integers $m_J^\ss$ such that \begin{equation} G_\ss \stackrel{ stab}{\sim}\ \sum\nolimits_J (-1)^{\ell(\ss)-|J|} \ m_J^\ss\, G[J] \ , \label{eq:trans5} \end{equation} sum over all partitions $J$. \end{theorem} \noindent {\it Proof.}\quad One gets 1) from a transition by specializing $x_r$ to 0. Instead of having to embed $\S_n$ into $\S_1\times \S_n$, we have choosen the hypothesis $\ss_1=1$. From the theory of Schubert polynomials \cite{LS4}, one knows that all the permutations appearing when iterating sufficiently many times a Schubert-transition are vexillary. Now, in the case where $\ss$ is vexillary, the set ${\mathcal I}(\ss)$ is of cardinality 1, and every long sequence of transitions starting from $\ss$ encounter a grassmannian permutation (with code obtained by reordering increasingly the code of $\ss$). There are more permutations appearing in Eq.~(\ref{eq:trans2}) than Eq.~(\ref{eq:trans1}). To adapt the argument of \cite{LS4}, one has to modify the weight function $(m-r)\ell(\ss)$, where $m$ is the first index such $\ss_m>\ss_{m+1}$. Buch proposes the function $ \sum_{i\geq 1} 2^i j_{i+m}$, where $[j_1,j_2,..]$ is the code of $\ss$. This function vanishes for grasmannian permutations, and strictly decreases in a transition $\ss \mapsto \eta$, $\eta$ being any permutation appearing on the RHS of (\ref{eq:trans2}). In summary, the stable part of $G_\ss$ belongs to the space generated by the stable parts of the $G_\c$, $\c$ grassmannian. However, each product by a transposition appearing in a transition increases length (of permutations) by 1, and signs are as stated, since $\ell(\ss)= |code(\ss)|$ for any permutation $\ss$ (this positivity was conjectured by Buch \cite{Bu}, as we already mentionned in the introduction). \qed As an example of decomposition given by theorem 4, let us take $\ss= [1, 5, 2, 4, 3, 9, 6, 7, 8]$; this implies $J=[0,3,0,1,0,3]$, $r=6$. One has $$ X_{152439678}= x_6 X_{152438679} + X_{15248367}+ X_{15283467}+X_{18243567} \ ,$$ and therefore ${\mathcal I}= \{3,4,5\}$, $$ G_{152439678} = G_{152438679} + (x_6-1) (1-\tau_{85})\star (1-\tau_{84})\star (1-\tau_{83})\star G_{152438679} \ .$$ Expression (32) develops in this case into $(1-\tau_{85})\star (1-\tau_{84})\star (1-\tau_{83})\star G_{152438679}= (1-\tau_{85})\star (1-\tau_{84}) \bigl(G_{152438679}-G_{152483679} \bigr)= (1-\tau_{85})\star \bigl(G_{152438679}-G_{152483679} -G_{152834679}+G_{152843679} \bigr) = G_{152438679}-G_{152483679} -G_{152834679}+G_{152843679} -G_{182435679}+G_{182453679} +G_{182835679}- G_{182543679}$. \smallskip Iterating transitions, and using codes now, one finally gets a sum \noindent\smallskip $ G[030103] \stackrel{ stab}{\sim} G[0601] + G[0502] + G[034] - G[053] - G[0602] + G[05011] + G[0412] + G[0331] - G[0431] - G[05021] - G[06011] - G[0512] - G[0341] + G[0531] + G[06021] $ \noindent of vexillary \GP, which implies, by sorting indices, \noindent $ G[030103] \stackrel{ stab}{\sim} g_{61}+g_{52}+g_{43}-g_{53}-g_{62}+g_{511}+g_{421} +g_{331}-2g_{431}-2g_{521}-g_{611}+ g_{531}+g_{621}$. \medskip One has a similar decomposition of the stable part of Schubert polynomials in terms of Schur functions (cf.\cite{LS3},\cite{EG}). This decomposition is a direct by-product (just sorting increasingly all indices) of the decomposition of Schubert polynomials in terms of key polynomials (cf.\cite{LS6},\cite{RS}), which involves finding all Young tableaux (in the alphabet $s_1,s_2,\ldots$) which are reduced deompositions of a given permutation. Fomin and Greene\cite{FG} obtain a decomposition of the stable part of $G_\ss$ in terms of Schur functions, by enumerating tableaux in the $T_i$'s which are decompositions, in the 0-Hecke algebra, of a given $T_\ss$. Reconstructing Eq.~(\ref{eq:trans5}) from their result, however, is not possible since many cancelations occur : Schur functions are not the appropriate symmetric functions for what concerns the Grothendieck ring. We shall give elsewhere a decomposition of \GP in terms of G-key polynomials. Let us mention that \GP are given by a simple statistics on alternating sign matrices. This also will be written elsewhere. \section{ Variations among determinantal expressions} There are many determinantal expressions of classes of Schubert subvarieties of Grassmannians, that is, of $g_\l$, $\l$ partition, or of $G[J]$, $J$ increasing. Let us illustrate them on a concrete example, the case of a determinantal variety defined by the vanishing of minors of order 2 of a matrix of order 4. With the notations used in this text, the Grothendieck polynomial which is to be explicited is $G_{3412}= G[0022]$. By definition, it is \begin{equation} (1-1/a_1)^5 (1-1/a_2)^4 (1-1/a_3)^1\, \pi_{4321}^a \label{eq:gr1} \end{equation} \begin{equation} = (1-1/a_1)^4 (1-1/a_2)^4\, \pi_{4321}^a \label{eq:gr2} \end{equation} \begin{equation} = (a_1-1)^4 (a_2-1)^4 a_3^4 a_4^4\, \pi_{4321}^a\, {1\over (a_1a_2a_3a_4)^4} \label{eq:gr3} \end{equation} \begin{equation} = (a_1-1)^4 (a_2-1)^4 a_3^4 a_4^4 \, a_1^3a_2^2a_3^1 \, \d_{4321}^a\, {1\over (a_1a_2a_3a_4)^4} \label{eq:gr4} \end{equation} \begin{equation} det \left|\, a_i^3(a_1-1)^4 \, ,\, a_i^2(a_1-1)^4 \, ,\, a_i^5 \, ,\, a_i^4\, \right|\, { (a_1a_2a_3a_4)^{-4}\over \Delta(\A) } \label{eq:gr5} \end{equation} with $\Delta(\A):= \prod_{i