%==================================================================== % % Quantum double Schubert polynomials, quantum Schubert polynomials % and Vafa-Intriligator formula % % Anatol N. Kirillov and Toshiaki Maeno, October 1996 % % LATEX %==================================================================== \documentstyle[12pt]{article} %\documentclass[12pt]{amsart} %\hsize=5.0truein %\vsize=7.8truein %\parindent=15pt \def \qed{\hfill $\vrule height 2.5mm width 2.5mm depth 0mm $} \newtheorem{lem}{Lemma} \newtheorem{th}{Theorem} \newtheorem{pr}{Proposition} \newtheorem{col}{Corollary} \newtheorem{de}{Definition} \newtheorem{con}{Conjecture} \newtheorem{cla}{Claim} \def\al{\alpha} \def\Res{\hbox{\rm Res}} \def\mod{\hbox{\rm mod~}} \def\Fl{\hbox{$Fl_n$}} \def\ds{\displaystyle} \def\wh{\widehat} \def\wt{\widetilde} \def\ld{\lambda} \def\covec{\buildrel\leftarrow\over } %\font\germ=eufm10 \font\germ=eufm10 \def\g{\hbox{\germ g}} \def\s{\hbox{\germ S}} \def\h{\hbox{\germ h}} \def\d{\hbox{\germ D}} \def\a{\hbox{\germ A}} \def\t{\hbox{\germ T}} \begin{document} %\rightline{q-alg/9610022} \title{Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa--Intriligator formula.} \vskip 0.4cm \author{\Large {Anatol N. Kirillov}\thanks{On leave from {\it Steklov Mathematical Institute, Fontanka 27, St.Petersburg, 191011, Russia}}~ \Large{and Toshiaki Maeno}\thanks{Supported by JSPS Research Fellowships for Young Scientists} \medskip \\ {\small {\it Department of Mathematical Sciences, University of Tokyo,}}\\ {\small {\it Komaba, Meguro-ku, Tokyo 153, Japan}}} \date{} \maketitle %\vskip 0.2cm \begin{abstract} We study the algebraic aspects of equivariant quantum cohomology algebra of the flag manifold. We introduce and study the quantum double Schubert polynomials $\wt{\s}_w(x,y)$, which are the Lascoux--Sch\"utzenberger type representatives of the equivariant quantum cohomology classes. Our approach is based on the quantum Cauchy identity. We define also quantum Schubert polynomials $\wt{\s}_w(x)$ as the Gram--Schmidt orthogonalization of some set of monomials with respect to the scalar product, defined by the Grothendieck residue. Using quantum Cauchy identity, we prove that $\wt{\s}_w(x)=\wt{\s}_w(x,y)|_{y=0}$ and as corollary obtain a simple formula for the quantum Schubert polynomials $\wt{\s}_w(x)=\partial_{ww_0}^{(y)}\wt{\s}_{w_0}(x,y)|_{y=0}$\,. We also prove the higher genus analog of Vafa--Intriligator's formula for the flag manifolds and study the quantum residues generating function. We introduce the extended Ehresman--Bruhat order on the symmetric group and formulate the equivariant quantum Pieri rule. \end{abstract} \newpage %\vskip 0.5cm %{\bf $\S 0.$ \section{Introduction.} The structure constants of the quantum cohomology ring are given by the third derivatives of the Gromov-Witten potential $F.$ The Gromov-Witten potential $F$ is a generating function of the Gromov-Witten invariants. The axioms of the tree level Gromov-Witten invariants $$ \langle I_{0, m, \beta}^{V} \rangle : H^{\ast}(V,{\bf Q})^{\otimes m} \longrightarrow {\bf Q}, $$ $\beta \in H_2 (V,{\bf Z}),$ for a target space $V$ are given by Kontsevich and Manin [KM]. Let $X_1,\ldots ,X_m$ be cycles on $V$ and $X_1^{\ast},\ldots , X_m^{\ast}$ their dual classes. Then the invariant $\langle I_{0,m,\beta}^V \rangle (X_1^{\ast}\otimes \cdots \otimes X_m^{\ast} )$ can be considered as the virtual number of the stable maps $f$ from $m$-pointed rational curve $({\bf P}^1 ; p_1,\ldots ,p_m)$ to $V,$ such that the image of $f$ represents the homology class $\beta$ and $f(p_i)\in X_i.$ In case of flag variety $Fl_n:=SL_n/B$ of type $A_{n-1}$ the potential $F$ is given as follows. Let $\Omega _v$ be the dual class of the Schubert cycle $X_v$ corresponding to a permutation $v\in S_n.$ Then the potential $F_{\omega}((t_v)_{v\in S_n})$ is defined by $$ F_{\omega}((t_v)_{v\in S_n}) = \sum_{\beta} \sum_{m=\Sigma m_v \geq 3} \exp(-\int_{\beta}\omega) \frac{ \langle I_{0,m,\beta}^{V} \rangle (\ds\bigotimes _{v\in S_n}\Omega_v^{\otimes m_v})} {\ds\prod_{v\in S_n} m_v !}\prod_{v\in S_n}t^{m_v}, $$ where $\omega$ is a K\"ahler form. For each point $t\in H^{\ast} (Fl_n),$ the quantum multiplication law is given by $$ \Omega_u \ast \Omega_v = \sum_{w\in S_n} \frac{\partial ^3 F_{\omega}}{\partial t_u \partial t_v \partial t_w}(t) \Omega_{ww_0}, $$ where $w_0$ is the permutation of maximal length. The algebra with this multiplication law is called a quantum cohomology ring, which is denoted by $QH^{\ast}_t (Fl_n).$ The associativity of the quantum multiplication is equivalent to the Witten--Dijkgraaf--Verlinde--Verlinde (WDVV) equation \[ \sum_{v\in S_n} \frac{\partial ^3 F_{\omega}} {\partial t_{u_1} \partial t_{u_2} \partial t_{v}} \frac{\partial ^3 F_{\omega}} {\partial t_{vw_0} \partial t_{u_3} \partial t_{u_4}} = \sum_{v\in S_n} \frac{\partial ^3 F_{\omega}} {\partial t_{u_2} \partial t_{u_3} \partial t_{v}} \frac{\partial ^3 F_{\omega}} {\partial t_{vw_0} \partial t_{u_1} \partial t_{u_4}}, \] for any $u_1,u_2,u_3,u_4 \in S_n.$ >From [KM, Proposition 4.4], the potential $F_{\omega}$ satisfies \[ F_{\omega} (t) = F_{\omega - t^{(2)}}(t-t^{(2)}), \] where $t^{(2)} = \ds\sum_{l(v) = 1} t_v \Omega_v .$ Hence, we may assume $\omega = 0.$ The potential $F$ is decomposed as a sum of the classical part $f_{\rm cl}$ and the quantum correction $f,$ where \[ f_{\rm cl} = \frac{1}{6} t_{id} \left( \sum_{u,v}\delta_{u,vw_0}t_u t_v \right) , \] and \[f = \sum_{\beta}\sum_{m = \Sigma m_v \geq 3} \frac{\langle I_{0,m,\beta}^{Fl} \rangle (\ds\bigotimes _{l(v) \geq 1}\Omega _v^{\otimes m_v})} {\ds\prod_{l(v)\geq 1} m_v !}\prod_{l(v)\geq 1}t^{m_v}. \] >From the axioms of the Gromov-Witten invariants ([KM,(2.2.4)]), we have \[ \langle I_{0,m,\beta}^{Fl} \rangle (\bigotimes _{l(v)\geq 1} \Omega _{v}^{\otimes m_v}) = \langle I_{0,m-\Sigma_{l(u)=1}m_u,\beta}^{Fl} \rangle (\bigotimes _{l(v) > 1} \Omega _{v}^{\otimes m_v} ) \prod_{l(u)=1} \left( \int_{\beta}\Omega _u \right). \] Hence, the quantum correction $f$ is expressed as \[ f = \sum_{m_v, b_u} N((m_v)_{l(v)>1} \mid (b_u)_{l(u)= 1}) \exp(\sum_{l(u)=1}(b_u t_u))\prod_{l(v)>1}\frac{t_v^{m_v}} {m_v !}, \] where \[ N((m_v) \mid (b_u)) = \langle I_{0, \Sigma m_v, \Sigma b_u X_u}^{Fl} \rangle \left( \bigotimes _{l(v)>1} \Omega _v^{\otimes m_v} \right). \] If $n=3,$ the WDVV equations with the initial condition \[ N(0,0,1 \mid 1,0) = N(0,0,1 \mid 0,1) = 1 \] or \[ N(2,0,0 \mid 1,0) = N(0,2,0 \mid 0,1) = 1 \] determine all the coefficients $N(\lambda, \mu, \nu \mid a,b)$ uniquely. In fact, Di Francesco and Itzykson [FI] gave the coefficients $N(\lambda, \mu, \nu \mid a,b)$ for $a+b \leq 10.$ Let $x_1 = \Omega _{(1,2)}$ and $x_i = \Omega _{(i,i+1)} - \Omega _{(i-1,i)}$ for $2\leq i\leq n-1,$ where $(i,j)$ is the transposition that interchanges $i$ and $j.$ Since the classical cohomology ring is generated by $x_1,\ldots ,x_{n-1},$ the quantum cohomology ring $QH^{\ast}_t(Fl_n)$ is also generated by $x_1,\ldots ,x_{n-1}$ in the neighborhood of the origin $t=0.$ Moreover, we can choose a neighborhood of the origin on which $QH^{\ast}_t(Fl_n)$ is a complete intersection ring. Then, let ${\cal I}\subset {\bf C}[x_1,\ldots ,x_{n-1}]$ be the defining ideal of the quantum cohomology ring $QH^{\ast}_t(Fl_n).$ The Schubert class $\Omega _v$ is expressed by the Schubert polynomial ${\s}_v(x_1,\ldots , x_{n-1})$ in the classical cohomology ring. However, in the quantum cohomology ring, the class corresponding to the Schubert polynomial ${\s}_v$ is no longer the Schubert class $\Omega _v.$ Hence, the polynomial $\wt{\s}_{v} ^{t}(x_1,\ldots ,x_{n-1})$ expressing $\Omega _v$ gives a deformation of the Schubert polynomial. We call it a big quantum Schubert polynomial. We identify the residue pairing defined by ${\cal I}$ with the intersection form on the cohomology ring, so the big quantum Schubert polynomials are obtained by orthogonalization the basis consisting of the classical Schubert polynomials. It is difficult to describe the defining ideal $\cal I$ of the big quantum cohomology ring for generic $t,$ so the big quantum Schubert polynomials are complicated in general. However, in the case where parameters $t_v = 0$ for all permutations $v\in S_n$ such that $l(v) > 1,$ the defining relations of the quantum cohomology ring are known by the results of A. Givental and B. Kim [GK] and I. Ciocan-Fontanine [C]. We call it the small quantum cohomology ring. The structure constants of the small quantum cohomology ring are given by %\begin{eqnarray*} \[ \frac{\partial ^3 F}{\partial t_{v_1} \partial t_{v_2} \partial t_{v_3}} (t^{(2)})= \] \begin{eqnarray*} & = & \sum_{\beta}\sum_{m_u \geq 0} \frac{\langle I_{0,\Sigma m_u +3,\beta}^{Fl} \rangle (\Omega _{v_1} \otimes \Omega _{v_2} \otimes \Omega _{v_3} \otimes \ds(\bigotimes _{l(u) =1}\Omega_{u}^{\otimes m_u}))}{\ds\prod_{l(u)=1} m_u !} \prod_{l(u)=1}t_u^{m_u} \\ & = & \sum_{\beta} \langle I_{0,3,\beta}^{Fl} \rangle (\Omega_{v_1}\otimes \Omega_{v_2}\otimes \Omega_{v_2}) e^{\Sigma b_i t_{(i,i+1)}}, \end{eqnarray*} where the sum runs over \[ \beta = b_1 X_{(1,2)}+\cdots + b_{n-1}X_{(n-1,n)} \] with $b_i\in {\bf Z}_{\geq 0}.$ Hence the small quantum cohomology ring is determined by the invariants $\langle I_{0,3,\beta}^{Fl} \rangle .$ For a monomial $x_{i_1}\ast \cdots \ast x_{i_m},$ the $m$-point correlation function determined by the small quantum cohomology ring (the so-called small quantum cohomology ring correlation function) is defined to be \[ \langle x_{i_1} \cdots x_{i_m} \rangle = \int_{Fl_n}x_{i_1}\ast \cdots \ast x_{i_m}. \] If $m\geq 4,$ the small quantum cohomology ring correlation function can be expressed as follows: \[ \langle x_{i_1} \cdots x_{i_m} \rangle = \] \[ \sum_{\beta = \beta_1 + \cdots + \beta_{m-3}} \sum_{v_1,\ldots ,v_{m-3}} e^{ -\int_{\beta}\omega } \langle I_{0,3,\beta_1} \rangle (x_{i_1} \otimes x_{i_2} \otimes \Omega_{v_1} ) \langle I_{0,3,\beta_2} \rangle (\Omega_{v_1 w_0} \otimes x_{i_3} \otimes \Omega_{v_2} ) \] \[ \cdots \langle I_{0,3,\beta_{m-2}} \rangle (\Omega_{v_{m-3} w_0} \otimes x_{i_{m-1}} \otimes x_{i_m} ). \] Hence, if $m\ge 4$, the Gromov-Witten invariants $\langle I_{0,m,\beta} \rangle (x_{i_1} \otimes \cdots \otimes x_{i_m})$ do not appear as coefficients of small quantum cohomology ring correlation functions. Let us explain briefly the main results obtained in our paper. Follow to A.~Givental and B.~Kim [GK], and I.~Ciocan--Fontanine [C], we define the quantum elementary symmetric polynomials $\wt e_1,\ldots ,\wt e_n$ by the formula $$ \det\left(\begin{array}{ccccccc} x_1+t & q_1&0 &\ldots &\ldots &\ldots &0\\ -1 & x_2+t & q_2 &0 &\ldots &\ldots & 0\\ 0 & -1 &x_3+t & q_3 & 0 &\ldots & 0 \\ \vdots &\ddots &\ddots & \ddots &\ddots &\ddots &\vdots \\ 0&\ldots & 0 &-1&x_{n-2}+t &q_{n-2} & 0 \\ 0 &\ldots &\ldots & 0 &-1 & x_{n-1}+t & q_{n-1}\\ 0 & \ldots &\ldots &\ldots & 0 & -1 & x_n+t \end{array}\right) $$ $$= t^n + \wt e_1t^{n-1} + \wt e_2t^{n-2}+\cdots +\wt e_{n}, $$ where $q_i = e^{t_{(i,i+1)}}.$ The defining ideal $\wt I$ of the small quantum cohomology ring is generated by the quantum elementary symmetric polynomials, namely {\begin{equation}QH^*(Fl_n,{\bf Z}):=QH^{\ast}(Fl_n) = {\bf Z}[x_1,\ldots ,x_n;q_1, \ldots ,q_{n-1}]/(\wt e_1,\ldots ,\wt e_n). \label{211} \end{equation} In the classical case $q_1=\cdots =q_{n-1}=0$, on the quotient ring $$\a :={\bf Z}[ x_1,\ldots ,x_n]/(e_1(x),\ldots ,e_n(x))\simeq H^*(Fl_n,{\bf Z}) $$ there exists a natural pairing $\langle f,g\rangle =\eta (\partial_{w_0}(fg))$ which comes from the intersection pairing in the homology group $H_*(Fl_n,{\bf Z})$ of the flag variety. We can interpret the pairing $\langle ,\rangle$ as the Grothendieck residue pairing with respect to the ideal $I$ (see Subsection 2.5): $$\langle f,g\rangle =\Res_I(fg), $$ where $I=\wt I|_{q=0}$. Our first observation is that a natural residue pairing (we call it the quantum residue pairing) $$\langle f.g\rangle_Q=\Res_{\wt I}(fg) $$ on the quotient ring $\overline{\a} :={\bf Z}[x_1,\ldots ,x_n]/\wt I$ corresponds to the intersection pairing in quantum cohomology $QH^*(Fl_n,{\bf Z})$ under a natural isomorphism (\ref{211}). It is well-known (e.g. [LS2], [M]) that the classical Schubert polynomials form an orthonormal basis (with respect to the pairing $\langle ,\rangle$) in the cohomology ring of flag manifold and also give a linear basis in the quantum cohomology ring $QH^*(Fl_n,{\bf Z})$, [GK], [MS]. However, the classical Schubert polynomials do not orthogonal with respect to the quantum pairing any more. Thus, it is natural to ask: what kind of polynomials one can obtain applying the Gram--Schmidt orthogonalization to the classical Schubert polynomials with respect to the quantum pairing $\langle ,\rangle_Q$? Omiting some details with ordering (see Definition 5), the answer is: quantum Schubert polynomials. Our second observation is: to work with the equivariant quantum cohomology algebra ([GK], [K2]) is more convenient then with quantum cohomology ring itself. %In this case, we can describe the quantum Schubert polynomials %explicitly, so it is the small quantum cohomology ring that we study in %our paper. However, it turns out that to work with the equivariant %quantum cohomology is more convenient. The main reason is that one can find Lascoux--Sch\"utzenberger's type representative for any equivariant quantum cohomology class. In other words, each quantum double Schubert polynomial $\wt{\s}_w(x,y)$ can be obtained from the top one by using the divided difference operators acting on the $y$ variables. \medskip \\ {\bf Theorem-Definition A} \ {\it Let $x=(x_1,\ldots ,x_n)$, $y=(y_1,\ldots ,y_n)$ be two sets of variables, and $$\wt{\s}_{w_0}^{(q)}(x,y):=\prod_{i=1}^{n-1}\Delta_i(y_{n-i}~|~x_1,\ldots ,x_i), $$ where $\Delta_k(t~|~x_1,\ldots ,x_k):=\ds\sum_{j=0}^kt^{k-j}e_j(x_1,\ldots ,x_k~|~q_1,\ldots ,q_{k-1})$ is the generating function for the quantum elementary symmetric functions in $x_1,\ldots ,x_k$. Then \ $\wt{\s}_w^{(q)}(x,y)=\partial_{ww_0}^{(y)}\wt{\s}_{w_0}^{(q)}(x,y)$.} We define the quantum Schubert polynomials $\wt{\s}_w^{(q)}(x)$ as Gram--Schmidt's orthogonalization of the set of lexicographically ordered monomials\break $\{ x^I \mid I\subset (n-1,n-2,\ldots ,1,0) \}$ with respect to the quantum pairing $\langle ,\rangle_Q$, see Definition 5. One of our main results is the quantum analog of Cauchy's identity for (classical) Schubert polynomials, [M], (5.10). \medskip \\ {\bf Theorem B} \ {\it (Quantum Cauchy's identity) \begin{equation}\sum_{w\in S_n}\wt{\s}_w^{(q)}(x)\s_{ww_0}(y)= \wt{\s}_{w_0}^{(q)}(x,y). \end{equation}} We give a geometric proof of Theorem B in Section 7 using the arguments due to I.~Ciocan-Fontanine [C]; more particularly, we reduce directly a proof of Theorem B to that of the following geometric statement: \medskip \\ {\bf Lemma} \ {\it Let $I\subset\delta =(n-1,n-2,\ldots ,1,0)$ and $w\in S_n$ be a permutation, then \begin{equation} \langle\wt e_I(x),\wt{\s}_w(x)\rangle_Q=\langle e_I(x),\s_w(x)\rangle , \end{equation} where $e_I(x):=\ds\prod_{k=1}^{n-1}e_{i_k}(x_1,\ldots ,x_{n-k})$\\ (resp. $\wt e_I(x):=\ds\prod_{k=1}^{n-1}\wt e_{i_k}(x_1,\ldots ,x_{n-k}~|~q_1,\ldots ,q_{n-k-1})$)\\ is the elementary polynomial (resp. quantum elementary polynomial), see Section 5.2.} It is the formula (3) that we prove in Section 7 using the geometrical arguments from [C] and [K2]. By product, it follows from our proof that quantum Schubert polynomials $\hat{\s}_w(x)$ defined geometrically (see Section~6) coincide with those defined algebraically (see Definition 5): $$\hat{\s}_w(x) \equiv \wt{\s}_{w^{-1}}(x) \ (\mod \wt I). $$ It is interesting to note, that the intersection numbers $\langle e_I(x),\s_w(x)\rangle$ (which are nonnegative!) are precisely the coefficients of corresponding Schubert polynomial: $$\s_w(x)=\sum_{I\subset\delta}\langle e_I(x),\s_w(x)\rangle x^{\delta -I}. $$ The quantum Cauchy formula (2) plays the important role in our approach to the quantum Schubert polynomials. As a direct consequence of (2), we obtain the Lascoux--Sch\"utzenberger type formula for quantum Schubert polynomials (cf. Theorem-Definition A). \medskip \\ {\bf Theorem C} \ {\it Let $\wt{\s}_{w_0}(x,y)$ be as in Theorem-Definition A, then} $$\wt{\s}_{w}(x)=\partial_{ww_0}^{(y)}\wt{\s}_{w_0}(x,y)|_{y=0}. $$ In Section 5 we introduce a quantization map \[ P_n \rightarrow \overline{P}_n, \; \; f \mapsto \wt{f}. \] The quantization is a linear map which preserves the pairings, i.e., \[ \langle \wt{f},\wt{g} \rangle _{Q} = \langle f,g \rangle , \; \; f,g \in P_n. \] Using the quantum Cauchy formula (2), we prove that quantum double Schubert polynomials are the quantization of classical ones. Another class of polynomials having a nice quantization is the set of elementary polynomials \[ e_I(x) := \prod_{k=1}^{n-1}e_{i_k}(x_1,\ldots x_{n-k}), \; \; I=(i_1,\ldots ,i_{n-1})\subset \delta . \] It follows from Theorem B that quantization $\wt{e}_I(x)$ of elementary polynomial $e_I(x)$ is given by \[ \wt{e}_I(x) = \prod_{k=1}^{n-1} e_{i_k}(x_1,\ldots ,x_{n-k} \mid q_1,\ldots ,q_{n-k-1} ). \] More generally, we make a conjecture (''quantum Schur functions '') that quantization of the flagged Schur function (see [M], (3.1), (4.9) and (6.16)) \[ s_{\lambda / \mu }(X_1,\ldots ,X_n) = {\rm det}\left( h_{\lambda_i-\mu_j-i+j}(X_i)\right) _{1\leq i,j \leq n} \] is given by \[ \wt{s}_{\lambda / \mu }(X_1,\ldots ,X_n) = {\rm det}\left( \wt{h}_{\lambda_i-\mu_j-i+j}(X_i)\right) _{1\leq i,j \leq n}, \] where $\wt{h}_k(X)$ is the quantum complete homogeneous symmetric function of degree $k,$ and $X_1\subset \cdots \subset X_n$ are the flagged sets of variables (see Section 5). In Section 5.2 we consider a problem how to quantize monomials. It seems to be difficult to find an explicit determinantal formula for a quantum monomial $\wt{x}^I,$ i.e., to find a quantum analog of the Billey-Jockusch-Stanley formula for Schubert polynomials in terms of compatible sequences [BJS]. We prove the following formulae for quantum monomials \[ \wt{x}^I = \sum_{w\in S_n} \eta (\partial _{w}x^I ) \wt{\s}_w(x), \; \; I\subset \delta, \] \[ \wt{\s}_{w_0}(x,y) = \sum_{I\subset \delta}\wt{x}^I e_{\delta - I}(y). \] In section 8.1 we give a proof of the higher genus analog of the Vafa--\break Intriligator type formula for the flag manifold. In Section 8.3 we study a problem how to compute the quantum residues. This is important for computation of small quantum cohomology ring correlation functions (or correlation functions, for short) and the Gromov--Witten invariants, see Introduction and Theorem 11. We introduce the generating function \[ \Psi(t)= \langle \prod_{i=1}^{n-1}\frac{t_i}{t_i - x_i} \rangle \] for quantum residues and give a characterization of this function as the unique solution to some system of differential equations, see Proposition 14. In Appendix~B we calculate the generating function $\Psi(t)$ for the case $n=3$ explicitly. In Section 9 we introduce the extended Ehresman--Bruhat order and give a sketch of a proof of equivariant quantum Pieri rule. Details will appear elsewhere. In Appendix A one can find a list of explicit expressions for the quantum double Schubert polynomials for the symmetric group $S_4$. We would like to mention, that in the recent preprint ``Quantum Schubert polynomials'' by S.~Fomin, S.~Gelfand and A.~Postnikov, [FGP], developed a different approach to the theory of quantum Schubert polynomials, based on the remarkable family of commuting operators $X_i$ ([FGP], (3.2)). Among main results, obtained by S.~Fomin, S.~Gelfand and A.~Postnikov, are definitions, orthogonality, quantum Monk's formula and other properties of quantum Schubert polynomials; definition of quantization map and quantum multiplication. Besides some overlap with the preprint of S.~Fomin, S.~Gelfand and A.~Postnikov, our works were done independently and based on the different approaches, which allow to obtain the mutually complementary results. \section{Classical Schubert polynomials.} In this section we give a brief review of the theory of Schubert polynomials created by A.~Lascoux and M.-P.~Sch\"utzenberger. In exposition we follow to the I.~Macdonald book [M1] where proofs and more details can be found. \subsection{Divided differences.} Let $x_1,\ldots ,x_n,\ldots$ be independent variables, and let $$P_n:={\bf Z}[x_1,\ldots ,x_n] $$ for each $n\ge 1$, and \begin{equation} P_{\infty}:={\bf Z}[x_1,x_2,\ldots ]={\ds\bigcup_{n=1}^{\infty}P_n}. \end{equation} Let us denote by $\Lambda_n:={\bf Z}[x_1,\ldots ,x_n]^{S_n}\subset P_n$ the ring of symmetric polynomials in $x_1,\ldots ,x_n$, and by $H_n:=\{\ds\sum_{I=(i_1,\ldots ,i_n)}a_Ix^I~|~a_I\in{\bf Z}, \ 0\le i_k\le n-k, \forall k\}$ the additive subgroup of $P_n$ spanned by all monomials $x^I:=x_1^{i_1}x_2^{i_2}\ldots x_n^{i_n}$ with $I\subset\delta :=\delta_n=(n-1,n-2,\ldots ,1,0)$. For $1\le i\le n-1$ let us define a linear operator $\partial_i$ acting on $P_n$ \begin{equation} (\partial_i f)(x)= \frac{f(x_1,\ldots,x_i,x_{i+1},\ldots,x_n) -f(x_1,\ldots,x_{i+1},x_i,\ldots,x_n)} {x_i-x_{i+1}}. \end{equation} Divided difference operators $\partial_i$ satisfy the following relations \begin{eqnarray} \partial_i^2 & = & 0,\nonumber \\ \partial_i \partial_j & = & \partial_j \partial_i, \ \ {\rm if} \mid i-j \mid >1, \\ \partial_i \partial_{i+1} \partial_i & = & \partial_{i+1} \partial_i \partial_{i+1}, \nonumber \end{eqnarray} and the Leibnitz rule \begin{equation} \partial_i (fg) = \partial_i(f)g+s_i(f) \partial_i(g).\label{5} \end{equation} It follows from (\ref{5}) that $\partial_i$ is a $\Lambda_n$-linear operator. For any permutation $w\in S_n,$ let us denote by $R(w)$ the set of reduced words for $w,$ i.e. sequences $(a_1,\ldots,a_p)$ such that $w= s_{a_1}\cdots s_{a_p},$ where $p=l(w)$ is the length of permutation $w\in S_n$, and $s_i=(i,i+1)$ is the simple transposition that interchanges $i$ and $i+1.$ For any sequence ${\bf a}=(a_1,\ldots,a_p)$ of positive integers, we define \[ \partial_{\bf a}= \partial_{a_1}\cdots\partial_{a_p}. \] \begin{pr}\label{2} ([M1], (2.5),(2.6)) $\bullet$ If ${\bf a,b}\in R(w),$ then $\partial_{\bf a}=\partial_{\bf b}.$ \vskip 0.1cm $\bullet$ If ${\bf a}$ is not reduced, then $\partial_{\bf a}=0.$ \end{pr} >From Proposition \ref{2} it follows that an operator \[ \partial_w = \partial_{\bf a} \] is well-defined, where ${\bf a}$ is any reduced word for $w.$ By (\ref{5}), the operators $\partial_w$, $w\in S_n$, are $\Lambda_n$ linear, i.e. if $f\in\Lambda_n$, then $$\partial_w(fg)=f\partial_w(g). $$ \subsection{Schubert polynomials.} Let $\delta =\delta_n=(n-1,n-2,\ldots ,1,0)$, so that $x^{\delta}=x_1^{n-1}x_2^{n-2}\ldots x_{n-1}$. \begin{de} (Lascoux--Sch\"utzenberger [LS1]). For each permutation \hbox{$w\in S_n$} the Schubert polynomial ${\s}_w$ is defined to be $${\s}_w(x)=\partial_{w^{-1}w_0}(x^{\delta}), $$ where $w_0$ is the longest element of $S_n$. \end{de} \begin{pr} ([M1], (4.2),(4.5),(4.11),(4.15)).\\ $\bullet$ Let $v,w\in S_n$. Then $$\partial_v\s_w=\left\{\begin{array}{ll} \s_{wv^{-1}}, & \mbox{\rm if \ $l(wv^{-1})=l(w)-l(v)$,}\\ 0, & \mbox{\rm otherwise}. \end{array}\right. $$ $\bullet$ (Stability). Let $m>n$ and let $i:S_n\hookrightarrow S_m$ to be the natural embedding. Then $$\s_w=\s_{i(w)}. $$ $\bullet$ The Schubert polynomials $\s_w, w\in S_n$ form a ${\bf Z}$--basis of $H_n$.\\ $\bullet$ (Monk's formula). Let $f=\ds\sum_{i=1}^n\alpha_ix_i$, $w\in S_n$. Then \begin{eqnarray*}f\s_w &=&\sum (\alpha_i-\alpha_j)\s_{wt_{ij}},\\ \partial_w(fg) &=& w(f)\partial_wg+\sum (\al_i-\al_j)\partial_{wt_{ij}}g, \end{eqnarray*} where $t_{ij}$ is the transposition that interchanges $i$ and $j$, and both sums are over all pairs $i0$ and $I$ lexicographically smaller then $c(w)$. \end{pr} {\bf Remark 1} \ 1) (Definition of the code, [M1], p.9). \\ For a permutation $w\in S_n,$ we define \[ c_i = \sharp \{ j \mid i w(j) \} . \] The sequence $c(w)=(c_1,c_2,\ldots ,c_n)$ is called the code of $w.$ \smallskip \\ 2) Schubert polynomials are obtained as Gram-Schmidt's orthogonalization of the set of monomials ~$\{ x^I\}_{I\subset \delta}$~ ordered lexicographically. \subsection{Double Schubert polynomials.} Let $x=(x_1,\ldots ,x_n)$, $y=(y_1,\ldots ,y_n)$ be two sets of independent variables, and $$\s_{w_0}(x,y):=\prod_{i+j\le n}(x_i+y_j). $$ \begin{de} (Lascoux--Sch\"utzenberger [LS2]). For each permutation \hbox{$w\in S_n$,} the double Schubert polynomial $\s_w(x,y)$ is defined to be $$\s_w(x,y)=\partial_{w^{-1}w_0}^{(x)}\s_{w_0}(x,y), $$ where divided difference operator $\partial_{w^{-1}w_0}^{(x)}$ acts on the $x$ variables. \end{de} \begin{pr} ([M1], (6.3),(6.8)).\\ $\bullet$ $\s_w(x,y)=\ds\sum_u\s_u(x)\s_{uw^{-1}}(y)$, summed over all $u\in S_n$, such that\break $l(u)+l(uw^{-1})=l(w)$;\\ $\bullet$ (Interpolation formula). For all $f\in{\bf Z}[x_1,\ldots ,x_n]$ we have $$f(x)=\sum_w\s_w(x,-y)\partial_w^{(y)}f(y) $$ summed over all permutations $w\in S^{(n)}$. \end{pr} Double Schubert polynomials appear in algebra and geometry as cohomology classes related to degeneracy loci of flagged vector bundles. If $h:E\to F$ is a map of rank $n$ vector bundles on a smooth variety $X$, $$E_1\subset E_2\subset\cdots\subset E_n=E, \ F:=F_n\to F_{n-1}\to \cdots \to F_1 $$ are flags of subbundles and quotient bundles, then there is a degeneracy locus $\Omega_w(h)$ for each permutation $w$ in the symmetric group $S_n$, described by the conditions $$\Omega_w(h)=\{ x\in X~|~{\rm rank}(E_p(x)\to F_q(x))\le\#\{ i\le q, w_i\le p\} , \forall p,q\}. $$ For generic $h$,\ $\Omega_w(h)$ is irreducible, codim $\Omega_w(h)=l(w)$, and the class $[\Omega_w(h)]$ of this locus in the Chow ring of $X$ is equal to the double Schubert polynomial $\s_{w_0w}(x,-y)$, where \begin{eqnarray} x_i &=& c_1(\ker (F_i\to F_{i-1})),\nonumber\\ y_i &=& c_1(E_i/E_{i-1}), \ 1\le i\le n.\nonumber \end{eqnarray} It is well-known [F] that the Chow ring of flag variety $Fl_n$ admits the following description $$CH^*(Fl_n)\cong{\bf Z}[x_1,\ldots ,x_n,y_1,\ldots ,y_n]/J, $$ where $J$ is the ideal generated by $$e_i(x_1,\ldots ,x_n)-e_i(y_1,\ldots ,y_n), \ 1\le i\le n, $$ and $e_i(x)$ is the $i$-th elementary symmetric function in the variables $x_1,\ldots ,x_n$. $\bullet$ ([LS2], [KV]) The ring ${\bf Z}[x_1,\ldots ,x_n,y_1,\ldots ,y_n]/J$ is a free module of dimension $n!$ over the ring $R$, with basis either $\s_w(x)$, or $\s_w(x,y),\ w\in S_n$, where $$R:=\frac{{\bf Z}[x_1,\ldots ,x_n]\otimes{\bf Sym}[y_1,\ldots ,y_n]} {J}. $$ \subsection{Residue pairing.} Let $I$ be an ideal in $\bar P_n =R[x_1,\ldots ,x_n]$, $R\subset {\bf C}$, generated by a regular system of parameters $\varphi_1,\ldots ,\varphi_n$, and $\a :=\bar P_n /I$. \begin{pr} ([GH], [EL]). $\bullet$ $\dim_R\a <\infty$. $\bullet$ ${\cal H}:=\det \left(\ds\frac{\partial\varphi_i} {\partial x_j}\right)\not\in I$. \end{pr} Let $d_0:=\deg\cal H$, where we assume that $\deg x_i=1$ for all $1\le i\le n$. \begin{pr} ([EL]) $\bullet$ If $f\in\bar P_n$ and $\deg f=d_0$, then there exists a non zero $\al\in R$ such that $$f\equiv\frac{\al}{n!}{\cal H} \ (\mod I). $$ $\bullet$ If $f\in\bar P_n$, $f\ne 0$, and $\deg f>d_0$, then there exists $g\in\bar P_n$ such that $\deg g\le d_0$ and $g\equiv f \ (\mod I)$. \end{pr} \begin{de} (Grothendieck residue with respect to the ideal $I$). Let $f\in\bar P_n$ and $\deg fd_0$, then choose $g\in\bar P_n$ such that $g\equiv f \ (\mod I)$ and $\deg g\le d_0$, and define $$\Res_I(f):=\Res_I(g). $$ \end{de} We will use also notation $\langle f\rangle_I$ instead of $\Res_I(f)$. Finally, let us define a residue pairing $\langle , \rangle_I$ on $\overline P_n$ using the Grothendieck residue $$\langle f,g\rangle_I=\Res_I(f,g), \ \ f,g\in\overline P_n. $$ \begin{pr} ([GH]).\\ $\bullet$ If $f\in I$, then $\Res_I(f)=0$.\\ $\bullet$ The residue pairing $\langle , \rangle_I$ induces a non-degenerate pairing on $\a =\overline P /I$. \end{pr} We will use this general construction of residue pairing in the following two cases: i) $R={\bf Z}$, $I_n\subset P_n$ is an ideal generated by elementary symmetric polynomials $e_1(x), \ldots ,e_n(x)$. It is well-known that if $\Fl :=SL(n)/B$ is the flag variety of type $A_{n-1}$, then $$H^*(\Fl ,{\bf Z})\simeq P_n/I_n, $$ and residue pairing $\langle ,\rangle$ on $P_n/I_n$ coincides with the scalar product on $P_n/I_n$ induced by (\ref{100}). ii) $R={\bf Z}[q_1,\ldots ,q_{n-1}]$, $\wt I_n\subset\overline P_n$ is an ideal generated by the quantum elementary symmetric functions $\wt e_1(x),\ldots ,\wt e_n(x)$. It is a result of A.~Grivental and B.~Kim, and I.~Ciocan--Fontanine, that $$QH^*(\Fl )\simeq\overline P_n/\wt I_n, $$ and the residue pairing defined by $\wt I_n$ may be naturally identified with the intersection form on the quantum cohomology ring. We will call this residue pairing as quantum pairing on $\overline P_n/\wt I_n$ and denote it by $\langle ,\rangle_Q$. \section{Quantum double Schubert polynomials.} Quantum double Schubert polynomials are closely related with the equivariant quantum cohomology. Let us remind the result of A.~Givental and B.~Kim [GK] (see also [K2]) on the structure of the equivariant quantum cohomology algebra of the flag variety $Fl_n$: $$QH^*_{U_n}(Fl_n)\cong{\bf Z}[x_1,\ldots ,x_n,y_1,\ldots ,y_n,q_1,\ldots ,q_{n-1}]/\wt J, $$ where the ideal $\wt J$ generated by $$e_i(x_1,\ldots ,x_n~|~q_1,\ldots ,q_{n-1})-e_i(y_1,\ldots ,y_n), \ 1\le i\le n. $$ In classical case $q=0$, the double Schubert polynomials $\s_w(x,y)$ represent the equivariant cohomology classes [F]. Quantum double Schubert polynomials have to play the similar role for the quantum equivariant cohomology ring. Let us define at first the ``top'' quantum double Schubert polynomial $\wt{\s}_{w_0}(x,y)$. Let $x=(x_1,\ldots ,x_n)$, $y=(y_1,\ldots ,y_n)$ be two sets of variables, put $$\wt{\s}_{w_0}(x,y):=\wt{\s}_{w_0}^{(q)}(x,y)= \prod_{i=1}^{n-1}\Delta_i(y_{n-i}~|~x_1,\ldots ,x_i), $$ where $\Delta_k(t~|~x_1,\ldots ,x_k):=\ds\sum_{j=0}^kt^{k-j} e_j(x_1,\ldots ,x_k~|~q_1,\ldots ,q_{k-1})$ is the generating function for the quantum elementary symmetric polynomials in $x_1,\ldots ,x_k$, i.e. ~~ $ \Delta_k(t|x):=\ds\sum_{i=1}^ke_i(x|q)t^i=$ \begin{equation} \det\left(\begin{array}{ccccccc} x_1+t & q_1&0 &\ldots &\ldots &\ldots &0\\ -1 & x_2+t & q_2 &0 &\ldots &\ldots & 0\\ 0 & -1 &x_3+t & q_3 & 0 &\ldots & 0 \\ \vdots &\ddots &\ddots & \ddots &\ddots &\ddots &\vdots \\ 0&\ldots & 0 &-1&x_{k-2}+t &q_{k-2} & 0 \\ 0 &\ldots &\ldots & 0 &-1 & x_{k-1}+t & q_{k-1}\\ 0 & \ldots &\ldots &\ldots & 0 & -1 & x_k+t \end{array}\right) . \end{equation} \vskip 0.5cm %It is clear that %$$e_k(x|q)=\sum_l\sum_{\scriptsize\begin{array}{c} % 1\le j_1n$ and let $i:S_n\hookrightarrow S_m$ be the embedding. Then $$\wt{\s}_w(x,y)=\wt{\s}_{i(w)}(x,y). $$ $iii)$ One can check that the ring $${\bf Z}[x_1,\ldots ,x_n,y_1,\ldots ,y_n,q_1,\ldots ,q_{n-1}]/{\wt J} $$ is a free module of dimension $n!$ over the quotient ring $\wt R$ with basis either $\wt{\s}_w(x)$ or $\wt{\s}_w(x,y)$, $w\in S_n$, where $$\wt R:=\frac{{\bf Z}[x_1,\ldots ,x_n,q_1,\ldots ,q_{n-1}]\otimes {\bf Sym}[y_1,\ldots ,y_n]}{\wt J}. $$ \smallskip \\ {\bf Example.} Quantum double Schubert polynomials for $S_3$: \begin{eqnarray*}\wt{\s}_{s_1s_2s_1}(x,y) &=& (x_1+y_2)(x_1+y_1)(x_2+y_1)+q_1(x_1+y_2),\\ \wt{\s}_{s_2s_1}(x,y) &=& (x_1+y_1)(x_1+y_2)-q_1,\\ \wt{\s}_{s_1s_2}(x,y) &=& (x_1+y_1)(x_2+y_1)+q_1,\\ \wt{\s}_{s_1}(x,y) &=& x_1+y_1,\\ \wt{\s}_{s_2}(x,y) &=& x_1 +x_2+y_1+y_2,\\ \wt{\s}_{\rm id}(x,y) &=& 1. \end{eqnarray*} For the list of the quantum double Schubert polynomials corresponding to the symmetric group $S_4$, see appendix A. {\begin{th} Let $z=(z_1,\ldots ,z_n)$ be a third set of variables. Then \begin{equation} \langle\wt{\s}_{w_0}(x,y),\wt{\s}_{w_0}(x,z)\rangle_Q^{(x)}=C(y,z),\label{6} \end{equation} where the upper index $x$ means that the quantum pairing is taken in the $x$ variables, and $$C(x,y)=\sum_{w\in S_n}\s_w(x)\s_{w_0w}(y) $$ is the "canonical" element in the tensor product $H^*(\Fl )\otimes H^*(\Fl )$. \end{th} Theorem 1 plays the important role in our approach to the quantum Schubert polynomials. We will give a proof later, and now let us consider some applications of the formula (10). \section{Quantum Schubert polynomials.} \subsection{Definition.} Let us remind the result of A.~Givental and B.~Kim, and I.~Ciocan-Fontanine on the structure of the small quantum cohomology ring of flag variety $\Fl$ $$QH^*({\Fl})\cong{\bf Z} [ x_1,\ldots ,x_n,q_1,\ldots ,q_{n-1} ] /\wt I, $$ where the ideal $\wt I$ is generated by the quantum elementary symmetric polynomials $\wt e_i(x):= e_i(x_1,\ldots ,x_n|q_1,\ldots ,q_{n-1})$, $1\le i\le n$ with generating function $\Delta_n(t|x)$, see (9). %with generating function \ \ We define a pairing on the ring of polynomials ${\bf Z}[x;q]$ and the quantum cohomology ring $QH^*(\Fl )\simeq{\bf Z}[x;q]/\wt I$ using the Grothendieck residue $$\langle f,g\rangle_Q=\Res_{\wt I}(fg),~f,g\in{\bf Z}[x_1,\ldots ,x_n,q_1,\ldots ,q_{n-1}]. $$ Then 1) $\langle f,g\rangle_Q=0$ if $f\in \wt I$; \vskip 0.2cm 2) $\langle f,g\rangle_Q$ defines a nondegenerate pairing in $QH^*(\Fl )$. \begin{de}\label{102}~%\\ Define the quantum Schubert polynomials $\wt{\s}_w:=\wt{\s}_w(x)$ as Gram--Schmidt's orthogonalization of the set of lexicographically ordered monomials \hbox{$\{x^I~|~I\subset\delta\}$} with respect to the quantum residue pairing %classical Schubert polynomials $\s_u$ w.r.t. pairing $\langle f,g\rangle_Q$: 1) $\langle\wt{\s}_u,\wt{\s}_v\rangle_Q=\langle\s_u,\s_v\rangle =\left\{\begin{array}{ll} 1,& \mbox{\rm{if} $v=w_0 u$}\\ 0, & \mbox{\rm{otherwise}}\end{array}\right.$ 2) $\wt{\s}_w(x)=x^{c(w)}+\ds\sum_{In$ then $\wt h_k^m\in\wt I_n$. Hence, if $I\not\subset\delta$, then $h_I(x)\in I_n$ and $\wt h_I(x)\in \wt I_n$. It is not difficult to see that $P_n$ is a free $\Lambda_n$-module of rank $n!$ with basis $\{ h_I(x)~|~I\in \t\}$.\smallskip \\ {\bf Theorem 5'} {\it If $I\subset\delta$, then $\wt h_I(x)$ is the quantization of complete polynomial $h_I(x)$.} {\it Proof.} See Remark 10 in Section 7. \qed \subsection{Canonical involution $\omega$.} There exists an involution $\omega$ of the ring $\overline P_n[y]$ given by $\omega (x)={\covec x}$, $\omega (y)={\covec y}$, $\omega (q)= {\covec q}$, where for any sequences $z=(z_1,\ldots , z_m)$ we define $\covec z$ to be equal to $(z_m, z_{m-1},\ldots ,z_1):=\covec z$. It is clear from the definition of quantum elementary symmetric functions $e_i(x|q)$, see (9), Section 3, that $$\omega (e_i(x|q))=e_i(x|q) $$ and thus the involution $\omega$ preserves the ideal $\wt I_n$ (as well as the ideals $I_n$, $J_n$ and $\wt J_n$). \begin{pr} $$\omega (\wt{\s}_u^{(q)}(x,y))\equiv\epsilon (u) \wt{\s}_{w_0uw_0}^{(q)}(x,y)~\mod\wt J_n, $$ where $\epsilon (u)=(-1)^{l(u)}.$ \end{pr} {\it Proof.} First of all, if $u=w_0$, then \begin{equation} \omega (\wt{\s}_{w_0}^{(q)}(x,y))\equiv\epsilon (w_0) \wt{\s}_{w_0}^{(\covec q)}(\covec x,\covec y)~\mod \wt J_n.\label{21} \end{equation} But $\omega\partial_u=\epsilon (u)\partial_{w_0uw_0}\omega$ (see [M1], (2.12)). Thus, applying the divided difference operator $\epsilon (u) \partial_{w_0uw_0}^{(y)}$ to the both sides of (\ref{21}), we obtain $$\omega (\wt{\s}_{uw_0}^{(q)}(x,y))\equiv\epsilon (w_0) \wt{\s}_{w_0u}^{(\covec q)}(\covec x,\covec y)~\mod \wt J_n. $$ \qed Finally, let us describe the action of involution $\omega $ on the elementary polynomials. \begin{pr} $$\omega (\wt e_I(x))\equiv (-1)^{|I|} \wt h_{\covec I}(x)~\mod \wt I_n. $$ \end{pr}\bigskip {\bf Remark 9} \ To our knowledge, originaly, construction of the quantization map, using a remarkable family of commuting operators $X_i$, appeared in [FGP]. We use a different definition of quantization map, but it can be shown that two forms of quantizations are equivalent. For original proofs of Theorem 5 and 5', and Proposition 15, see Corollary 4.6, Corollary 7.16 and Proposition 7.13 in [FGP]. \section{Quantum cohomology ring of flag variety.} Quantum cohomology ring of the flag variety ${\Fl}$ is a deformed ring of the ordinary cohomology ring $H^*({\Fl} ,{\bf Z})$. The structure constants of the quantum cohomology ring are given by the Gromov--Witten invariants. Let $\Omega_{w_1},\ldots ,\Omega_{w_m}$ ($w_i\in S_n$) be Schubert cycles. We denote by $M_{\bar d}({\bf P}^1,Fl_n)$ the moduli space of morphisms from ${\bf P}^1$ to $\Fl$ of multidegree $\bar{d}=(d_1,\ldots ,d_{n-1})$. We consider the restriction of the universal map for $t\in{\bf P}^1$: $$ev_t~:~M_{\bar d}({\bf P}^1,\Fl )\times\{ t\}\hookrightarrow M_{\bar d}({\bf P}^1,\Fl )\times{\bf P}^1{\buildrel ev\over\rightarrow} \Fl ,~~(f,p)\mapsto f(p). $$ Let $\Omega_w(t)=ev_t^{-1}(\Omega_w)$. \begin{th} \mbox{(I.~Ciocan-Fontanine)}. If $\ds\sum_{i=1}^ml(w_i)=\frac{n(n-1)}{2} +2\sum d_i$ and\break $t_1,\ldots ,t_m\in{\bf P}^1$ are distinct, then for general translates of ~$\Omega_{w_i}$, the number of points in $\ds\bigcap_{i=1}^m\Omega_{w_i}(t_i)$ is finite and independent of $t_i$ and the translates of $\Omega_{w_i}$. \end{th} \begin{de} The Gromov--Witten invariant is defined as an intersection number $$\langle\Omega_{w_1}\ldots\Omega_{w_m}\rangle_{\bar d}=\left\{ \begin{array}{ll}\#\ds\bigcap_i\Omega_{w_i}(t_i), & \mbox{{\rm if} $\sum l(w_i)=\frac{n(n-1)}{2}+2\sum d_i$}\\ 0, & \mbox{\rm otherwise} \end{array}\right. . $$ \end{de} Now we can define the quantum multiplication as a linear map $$m_q~:~{\rm Sym~}(H^*(\Fl ,{\bf Z})[q_1,\ldots ,q_{n-1}])\to H^*(\Fl ,{\bf Z})[q_1,\ldots ,q_{n-1}] $$ given by $$m_q(\prod_{i=1}^m\Omega_{w_i})=\sum_{\bar d}q^{\bar d} \sum_w\langle\Omega_w\Omega_{w_1}\cdots\Omega_{w_m}\rangle_{\bar d} \Omega^*_w, $$ where $q^{\bar d}=q^{d_1}_1\cdots q_{n-1}^{d_{n-1}}$ and $(\Omega^*_w)$ is the dual basis of $(\Omega_w)$. Then the quantum cohomology ring $QH^*(\Fl )$ is a commutative and associative ${\bf Z}[q_1,\ldots ,q_{n-1}]$ -- algebra. Let $0=E_0\subset E_1\subset\cdots\subset E_n={\bf C}^n\otimes{\cal O}_F$ be the universal flag of subbundles on $\Fl$. \begin{th} \label{106}(A.~Givental and B.~Kim, I.~Ciocan--Fontanine).\\ The small quantum cohomology ring is generated by $x_i=c_1(E_{n-i+1}/E_{n-i})$, $i=1,\ldots ,n$, as a ${\bf Z}[q_1,\ldots ,q_{n-1}]$-algebra and $$QH^*(\Fl )\cong{\bf Z}[x_1,\ldots ,x_n,q_1,\ldots ,q_{n-1}]/(e_1(x|q),\ldots ,e_n(x|q)), $$ where $e_i(x|q)$ is given by the expansion of the following determinant $$\det\left(\begin{array}{ccccccc} x_1+t & q_1&0 &\ldots &\ldots &\ldots &0\\ -1 & x_2+t & q_2 &0 &\ldots &\ldots & 0\\ 0 & -1 &x_3+t & q_3 & 0 &\ldots & 0 \\ \vdots &\ddots &\ddots & \ddots &\ddots &\ddots &\vdots \\ 0&\ldots & 0 &-1&x_{n-2}+t &q_{n-2} & 0 \\ 0 &\ldots &\ldots & 0 &-1 & x_{n-1}+t & q_{n-1}\\ 0 & \ldots &\ldots &\ldots & 0 & -1 & x_n+t \end{array}\right) $$ $$=t^n+e_1(x|q)t^{n-1}+\cdots +e_n(x|q). $$ \end{th} It follows from Theorem \ref{106} that any Schubert cycle $\Omega_w$ may be expressed as a polynomial $\wh{\s}_w(x ,q)$ in $QH^*(\Fl )$. The polynomial $\wh{\s}_w(x ,q)$ is a deformation of the Schubert polynomial $\s_w(x )$ and $\wh{\s}_w(x ,0)=\s_w(x)$. Consider the correlation function $$\langle\Omega_{w_1}\ldots\Omega_{w_m}\rangle =\sum_{\bar d}q^{\bar d}\langle\Omega_{w_1}\ldots\Omega_{w_m}\rangle_{\bar d}. $$ Then $\wh{\s}_w(x ;q)$ is characterized by the condition $$\langle\Omega_w\Omega_{w_1}\ldots\Omega_{w_m}\rangle =\langle\wh{\s}_w(x ,q)\Omega_{w_1}\ldots\Omega_{w_m}\rangle $$ for any $w_1,\ldots ,w_m\in S_n$. $\wh{\s}_w(x ;q)$ is called a geometric quantum Schubert polynomial. By definition $\wh{\s}_w(x;q)\in QH^{2l(w)}(\Fl )$. %$$(\wt e_I,\wt e_J)_Q=(e_I,e_J). %$$ \section{Proofs of Theorem 3 and quantum Cauchy formula.} \begin{th} Let $I\in \t.$ Then \[ \langle \wt{e}_I(x), \wt{\s}_w (x) \rangle _Q = \langle e_I(x), \s_w(x) \rangle , \] for any permutation $w\in S_n.$ \end{th} {\it Proof.} The proof is based on the arguments due to I.~Ciocan-Fontanine [C]. To begin with, let us remind his results. We consider the hyper-quot scheme ${\cal HQ}_{\bar{d}}({\bf P}^1,Fl_n)$ associated to ${\bf P}^1$ with multidegree $\bar{d} = (d_1,\ldots,d_{n-1}).$ Let \[ {\bf C}^n \otimes {\cal O} \rightarrow {\cal T}_{n-1} \rightarrow \cdots \rightarrow {\cal T}_2 \rightarrow {\cal T}_1 \rightarrow 0 \] be the universal sequence of quotients on ${\bf P}^1 \times {\cal HQ}_{\bar{d}}({\bf P}^1,Fl_n)$ and $${\cal S}_i = {\rm Ker}\{ {\bf C}^n \otimes {\cal O} \rightarrow {\cal T}_{n-i} \} . $$ We also consider the dual sequence \[ {\bf C}^n \otimes {\cal O} \rightarrow {\cal S}_{n-1}^{\ast} \rightarrow \cdots \rightarrow {\cal S}_1^{\ast}. \] We fix a flag \[ 0=V_0 \subset V_1 \subset \cdots \subset V_n= {\bf C}^n \] and define the subschema $D_w^{p,q}$ of ${\bf P}^1 \times {\cal HQ}_{\bar{d}}({\bf P}^1,Fl_n)$ as the locus where ${\rm rank}(V_p \otimes {\cal O} \rightarrow {\cal S}_q^{\ast})\leq r_w(q,p),$ and $r_w(q,p):=\sharp \{ i \mid i\leq q, w_i\leq p \}.$ Let \[ D_w^{p,q}(t) = D_w^{p,q} \bigcap \{ \{t \} \times {\cal HQ}_{\bar{d}}({\bf P}^1,Fl_n) \} \] and \[ \overline{\Omega}_w(t) = \bigcap_{p,q=1}^{n-1} D_w^{p,q}(t). \] Then the class of $\bar{\Omega}_w(t)$ in the Chow ring $CH^{l(w)}({\cal HQ}_{\bar{d}}({\bf P}^1,Fl_n))$ is independent of $t\in {\bf P}^1$ and the flag $V_0\subset \cdots \subset V_n.$ The boundary ${\cal HQ}_{\bar{d}}({\bf P}^1,Fl_n) \setminus M_{\overline{d}}({\bf P}^1,Fl_n)$ consists of $n-1$ divisors ${\bf D}_1,\ldots,{\bf D}_{n-1},$ which are birational to $${\bf P}^1 \times {\cal HQ}_{\bar{d}_1}({\bf P}^1,Fl_n),\ldots , {\bf P}^1 \times {\cal HQ}_{\bar{d}_{n-1}}({\bf P}^1,Fl_n) $$ respectively, where $\bar{d}_{i}=(d_1,\ldots,d_i-1,\ldots,d_{n-1}).$ Let $x_i(t) = \overline{\Omega}_{s_i}(t) - \bar{\Omega}_{s_{i-1}}(t),$ then for any permutation $w\in S_n$ there exists an element $$G_w(t)\in CH_{\ast}(\bigcup_{i=1}^{n-2}{\bf D}_i) $$ such that \[ \bar{\Omega}_{w^{-1}}(t)-\s_w(x_1(t),\ldots ,x_{n-1}(t)) = j_{\ast}(G_{w^{-1}}(t)), \] where \[ j:\bigcup_{i=1}^{n-2}{\bf D}_i \rightarrow {\cal HQ}_{\bar{d}}({\bf P}^1,Fl_n) \] is the inclusion. Let $[m,k]\in S_n$ be the permutation \[ \left( \begin{array}{cccccccccccc} 1 & 2 & \ldots & m-k-1 & m-k & m-k+1 &\ldots & m & m+1 & \ldots & n \\ 1 & 2 & \ldots & m-k-1 & m & m-k &\ldots & m-1 & m+1 & \ldots & n \end{array} \right) . \] Then the geometric Schubert polynomial $\hat{\s}_{[m,k]^{-1}}(x)$ is the elementary symmetric function in $x_1,\ldots ,x_{m-1}$ of degree $k.$ Let $I=(i_1,\ldots,i_n).$ We have to calculate in $CH^{\ast}({\cal HQ}_{\bar{d}} ({\bf P}^1,Fl_n))$ \begin{equation} \left( \overline{ \bigcap_{\nu =1}^{n-1}\Omega_{[n-\nu+1,i_{\nu}]^{-1}}} \right) (t) \cdot \prod_{j=1}^{N}\overline{\Omega}_{w_j}(t_j) - \prod_{\nu =1}^{n-1}\s_{[n-\nu+1,i_{\nu}]}(x(t)) \cdot \prod_{j=1}^{N}\overline{\Omega}_{w_j}(t_j), \label{300} \end{equation} for distinct $t,t_1,\ldots, t_N$ and $w_1,\ldots ,w_N\in S_n$ such that $$\ds\sum_{\nu=1}^{n-1}i_{\nu} +\ds\sum_{j=1}^{N}l(w_j) = n(n-1)/2 + 2\ds\sum_{k=1}^{n-1} d_k, $$ where $\cap$ is the classical intersection product and \[ \left( \overline{ \bigcap_{\nu =1}^{n-1}\Omega_{[n-\nu+1,i_{\nu}]^{-1}} } \right) (t) \] is the corresponding degeneracy locus on ${\cal HQ}_{\bar{d}}({\bf P}^1,Fl_n).$ In order to calculate the expression (\ref{300}), we are going to prove at first that \[ \left( \overline{ \bigcap_{\nu =1}^{n-1} \Omega_{[n-\nu +1,i_{\nu}]^{-1}}} \right) (t) \cdot \prod_{j=1}^{N}\bar{\Omega}_{w_j}(t_j) - \prod_{\nu =1}^{n-1} \bar{\Omega}_{[n-\nu +1,i_{\nu}]^{-1}} (t) \cdot \prod_{j=1}^{N}\bar{\Omega}_{w_j} (t_j) = 0 .\] The LHS of the last expression can be computed as the number of points in \[ \prod_{\nu =1}^{n-1} \bar{\Omega}_{[n-\nu +1,i_{\nu}]^{-1}}(t) \cdot \prod_{j=1}^{N}\bar{\Omega}_{w_j} (t_j) \] suppoted on $\cup_{i=1}^{n-2}{\bf D}_i.$ Let $j_{\bar{d}_k}$ be the natural rational map \[ {\bf P}^1 \times {\cal HQ}_{\bar{d}_k} ({\bf P}^1,Fl_n) - \rightarrow {\cal HQ}_{\bar{d}}({\bf P}^1,Fl_n). \] >From Remark 3 in [C], \[ j_{\bar{d}_k}^{-1}(\bar{\Omega}_{[n-\nu+1,i_{\nu}]^{-1}}(t)) = \] \[\left\{ \begin{array}{cc} {\bf P}^1 \times \bar{\Omega}_{[n-\nu+1,i_{\nu}]^{-1}}(t) \cup \bigcap_{n-\nu-i_\nu+1\leq p \leq n-\nu-1} D_{[n-\nu+1,i_{\nu}]^{-1}}^{p,n-\nu-1}(t), & {\rm if} \; k=n-\nu, \\ {\bf P}^1 \times \bar{\Omega}_{[n-\nu+1,i_{\nu}]^{-1}}(t) & {\rm otherwise,} \end{array} \right. \] where \[ \bigcap_{n-\nu-i_\nu+1\leq p \leq n-\nu-1} D_{[n-\nu+1,i_{\nu}]^{-1}}^{p,n-\nu-1}(t) = \{ t\} \times \bar{\Omega}_{[n-\nu,i_{\nu}-1]^{-1}}(t) .\] Because, by assumption, \[\sum_{\nu =1}^{n-1}i_{\nu}+\sum_{\nu =1}^{n-1}l(w_j) = \frac{n(n-1)}{2} + 2 \sum_{k=1}^{n-1}d_k, \] we have \[ \bar{\Omega}_{[k,i_{n-k}-1]^{-1}}(t)\cdot \prod_{\nu \not= n-k} \bar{\Omega}_{[n-\nu +1,i_{\nu}]^{-1}}(t) \cdot \prod_{j=1}^{N}\bar{\Omega}_{w_j} (t_j) =0 \] on the hyper-quot scheme ${\cal HQ}_{\bar{d}_k}({\bf P}^1,Fl_n).$ Hence, we have equality \[ \left( \overline{ \bigcap_{\nu =1}^{n-1} \Omega_{[n-\nu +1,i_{\nu}]^{-1}}} \right) (t) \cdot \prod_{j=1}^{N}\bar{\Omega}_{w_j}(t_j) = \prod_{\nu =1}^{n-1} \bar{\Omega}_{[n-\nu +1,i_{\nu}]^{-1}} (t) \cdot \prod_{j=1}^{N}\bar{\Omega}_{w_j} (t_j) . \] Our next observation is that the intersection number in the RHS of the last equality is equal to $$\prod_{\nu =1}^{n-1}\overline\Omega_{[n-\nu +1,i_{\nu}]^{-1}}(s_{\nu}) \cdot\prod_{j=1}^N\overline\Omega_{w_j}(t_j), $$ where we can chose $s_1,\ldots ,s_{n-1}$, $t_1,\ldots ,t_N\in P^1$ to be the pairwise distinct points, since the class $[\Omega_w(t')]$ in the Chow ring $CH^*({\cal H}Q_{\overline d}(P^1,Fl_n))$ does not depends on the chose of $t'\in P^1$. Now we are going to use the following identity $$\prod_{k=1}^ma_k-\prod_{k=1}^mb_k=\sum_{k=1}^m\prod_{j=1}^{k-1}b_j (a_k-b_k)\prod_{j=k+1}^ma_j. $$ Let us take in the last equality $m=n-1$ \begin{eqnarray*} &&a_k:=\overline\Omega_{[n-k+1, i_k]^{-1}}(s_k),\\ &&b_k:=\s_{[n-k+1,i_k]}(x(s_k)). \end{eqnarray*} Then we obtain the following equality \begin{eqnarray*} & & \prod_{\nu =1}^{n-1}\overline\Omega_{[n-\nu +1,i_{\nu}]^{-1}}(s_{\nu}) \prod_{j=1}^N\overline\Omega_{w_j}(t_j)-\prod_{\nu =1}^{n-1} \s_{[n-\nu +1,i_{\nu}]}(x(s_{\nu}))\prod_{j=1}^N\overline\Omega_{w_j}(t_j)\\ &=&\left\{\sum_{k=1}^{n-1}\prod_{j=1}^{k-1}\overline\Omega_{[n-j+1,i_j]^{-1}} (s_j)\cdot j_*(G_{[n,i_k]^{-1}}(t))\prod_{j=k+1}^{n-1}\s_{[n-j+1,i_j]} (x(s_j))\right\}\cdot B, \end{eqnarray*} where $B:=\prod_{j=1}^N\overline\Omega_{w_j}(t_j)$. The contributions from $j_*((G_{[n,i_k]^{-1}}(t))$, $1\le k\le n-1$, can be computed by using the arguments in [C]. Inded, as in the proof of Theorem~4 in [C], the intersection number ($1\le k\le n-1$) $$\prod_{l=1}^{k-1}\overline\Omega_{[n-l+1,i_l]^{-1}} (s_l)\cdot j_*(G_{[n,i_k]^{-1}}(t))\prod_{l=k+1}^{n-1}\s_{[n-l+1,i_l]} (x(s_l))\cdot B $$ is the number of points in $$\prod_{l=1}^{k-1}\overline\Omega_{[n-l+1,i_l]^{-1}} (s_j)\prod_{l=k}^{n-1}\s_{[n-l+1,i_l]}(x(s_l))\cdot B $$ supported on $\bigcup_{j=1}^{n-2}D_i$. Hence, by induction, we have the following identity for correlation functions $$\langle\left(\bigcap_{\nu =1}^{n-1}\Omega_{[n-\nu +1,i_{\nu}]^{-1}}\right) \cdot\prod_{j=1}^N\overline\Omega_{w_j}\rangle =\langle\prod_{\nu =1}^{n-1} \wt{\s}_{[n-\nu +1,i_{\nu}]}\cdot\prod_{j=1}^N\overline\Omega_{w_j}\rangle , $$ where $\bigcap$ is the classical intersection product and $\cdot$ is the product in $${\rm Sym}~H^*(Fl_n,{\bf Z})[q_1,\ldots ,q_{n-1}]. $$ The last equality for correlation functions is equivalent to the following one \[ m_q(\Omega_{[n,i_1]^{-1}}\cap \Omega_{[n-1,i_2]^{-1}}, \cap\ldots\cap \Omega_{[l,i_{n-1}]^{-1}}, *) =m_q(\wt e_{i_1}\cdot\wt e_{i_2}\cdots \wt e_{i_{n-1}},*). \] This completes the proof. \qed \medskip \\ {\bf Remark 10} \ Using the similar geometrical arguments we can prove an analog of Theorem~9 for the quantum complete polynomials: \smallskip \\ {\bf Theorem 9'} {\it Let $I\subset\delta_n$. Then $$\langle\wt h_I(x),\wt{\s}_w(x)\rangle_Q=\langle h_I(x),\s_w(x)\rangle, $$ for any permutation $\omega\in S_n$.} It is easy to see that Theorem 5' is a corollary of Theorem 9'. \section{Correlation functions.} \subsection{Higher genus correlation function and the Vafa--Intriligator type formula.} Fix a Riemann surface $C$ of genus $g$. We denote by $M_d(C,F)$ the moduli space of morphism from $C$ to $\Fl $. One can define the higher genus Gromov--Witten invariants by method which is similar to that in the case of genus zero, [RT]. We have the following recursion relation for higher genus correlation function corresponding to the generating function for higher genus Gromov--Witten invariants $$\langle\Omega_{w_1}\ldots\Omega_{w_N}\rangle_g= \sum_{v\in S_n}\langle\Omega_{w_1}\ldots\Omega_{w_N}\Omega_v\Omega^*_v\rangle_{g-1} $$ (cf. Ruan--Tian [RT]). >From Corollary \ref{107} and Theorem~\ref{011} we can deduce the Vafa--Intriligator type formula for higher genus correlation functions, namely, let $\langle P(x_1,\ldots ,x_n)\rangle_g$ be the genus $g$ correlation function corresponding to a polynomial $P$, then \vskip 0.2cm \hskip 2.0cm $\langle P(x_1,\ldots ,x_n)\rangle_g=\Res_{\wt I}(P\Phi^g )=$ %\begin{eqnarray*} $$=\sum_{\wt e_1=\cdots =\wt e_n=0}P(x_1,\ldots ,x_n)\det\left (\frac{\partial\wt e_i}{\partial x_j}\right)^{-1}\left(\Phi (x_1,\ldots ,x_n)\right)^g, $$%\end{eqnarray*} where $\Phi (x)=\langle \wt{\s}_{w_0}(x,y),\wt{\s}_{w_0}(x,y)\rangle^{(y)}= \ds\sum_{w\in S_n}\wt{\s}_w(x)\wt{\s}_{w_0w}(x)=C^{(q,q)}(x,x)$. To simplify the formula above, we use the following observations $\bullet$ $\det\left(\ds\frac{\partial\wt e_i}{\partial x_j}\right)\equiv n!\:\wt{\s}_{w_0}(x)~(\mod\wt I);$ $\bullet$ $\Phi (x):=C^{(q,q)}(x,x)\equiv n!\:\wt{\s}_{w_0}(x)~(\mod\wt I).$ Hence, we obtain \begin{th} (Higher genus Vafa-Intriligator formula) $$\langle P(x_1,\ldots ,x_n)\rangle_g=\Res_{\wt I}(P\Phi^g)= \sum_{\wt e_1=\cdots =\wt e_n=0}P(x_1,\ldots ,x_n)\left(\wt{\s}_{w_0}(x) \right)^{g-1}, $$ where $\wt{\s}_{w_0}(x)=\wt e_{\delta}(x):=\wt e_1(x_1)\wt e_2(x_1,x_2) \ldots\wt e_{n-1}(x_1,\ldots ,x_{n-1})$, and $\wt e_i(z)$ is the quantum elementary polynomial of degree $i$ in the variable $z=(z_1,\ldots ,z_m)$, see Section 5.2. \end{th} %\smallskip \\ {\bf Remark 11} \ The polynomial $$C^{(q,q')}(x,y):=\sum_{w\in S_n}\wt{\s}_w^{(q)}(x) \wt{\s}_{w_0w}^{(q')}(y) $$ corresponds to the dual class of the diagonal in the quantum cohomology ring $QH^*(\Fl \times\Fl ,(q,q'))=QH^*(\Fl ,q)\otimes QH^*(\Fl ,q')$. \subsection{Witten--Dijkgraaf--Verlinde--Verlinde equations \break for symmetric group.} The Witten--Dijkgraaf--Verlinde--Verlinde equations (WDVV-equations) are equations on the correlation functions $\langle\wt{\s}_u\wt{\s}_v\wt{\s}_w\rangle\in{\bf Z}[q_1,\ldots ,q_{n-1}]$, where $u,v,w\in S_n$. The correlation functions satisfy the following conditions 1) Normalization: $$\langle 1\wt{\s}_v\wt{\s}_w\rangle =\langle \s_v,\s_w\rangle . $$ 2) Initial data: $$\langle \wt{\s}_{s_k}\wt{\s}_{s_k}\wt{\s}_{w_0}\rangle =q_k. $$ %where $\wt{\s}_k:=\wt{\s}_{s_k}$. 3) Degree conditions: $$\langle\wt{\s}_u\wt{\s}_v\wt{\s}_w\rangle =0 $$ if either $l(u)+l(v)+l(w)From the residue theorem, if $0\leq \nu_1+\cdots + \nu_n < n(n-1)/2,$ $0\leq \nu_i \leq n-i,$ then \[ \sum {\rm Res}_p \left( \frac{ x_1^{\nu_1}\cdots x_{n-1}^{\nu_{n-1}} dx_1 \wedge \cdots \wedge dx_n } {{\wt e}_1 \cdots {\wt e}_n} \right)=0. \] On the other hand, \[ \sum_{{\wt e}_i(p)=0} {\rm Res}_p \left( \frac{ x_1^{n-1}\cdots x_{n-1} dx_1 \wedge \cdots \wedge dx_n}{{\wt e}_1\cdots {\wt e}_n} \right) = -\sum_p{\rm Res}_p \omega , \] where $p$ runs over the common zeros of $\bar{e}^J_1, \ldots ,\bar{e}^J_n$ in $\ds\bigcup_{1\in J}B_J.$ Let $y_1 = 1/x_1,$ $z=(y_1,x_2,\ldots ,x_n)$ and \[ \bar{e}^{*}_1(z) = y_1(1+ y_1(x_2+ \cdots + x_n)). \] Then we have \ \ $ -\sum_p{\rm Res}_p \omega = $ \[ \sum_{\scriptsize{\begin{array}{cc}\bar{e}^{\ast}_1 = \bar{e}_2^{\{ 1\} } = \cdots =\bar{e}_n^{\{ 1\} } = 0 \\ {\rm in} \; \; {\rm the }\; \; {\rm locus} \; \; \{ y_1 =0 \}\end{array}} } {\rm Res}_p \left(\frac{ x_2^{n-2}\cdots x_{n-1} dy_1 \wedge \cdots \wedge dx_n } {\bar{e}^{\ast}_1(z)\cdot \bar{e}^{\{ 1\} }_2(z)\cdots \bar{e}^{\{ 1\} }_N(z)} \right) \] \[ = \sum_p{\rm Res}_p \left( \frac{ x_2^{n-2}\cdots x_{n-1} dx_2 \wedge \cdots \wedge dx_n } {{\wt e}_1(x_2,\ldots ,x_n) \cdots {\wt e}_{n-1}(x_2,\ldots ,x_n)} \right) = 1, \] by induction. \qed \bigskip >From the residue formula, the correlation function is given by the quantum residue ${\rm Res}_{\wt I},$ namely, \[ \langle P(x_1, \ldots , x_{n-1}) \rangle = {\rm Res}_{\wt I} P(t_1, \ldots ,t_{n-1}) . \] In order to relate the quantum residue with the classical one, we consider the quantum residue generating function \[ \Psi(t) = \langle \prod_{i=1}^{n-1} \frac{t_i} {t_i - x_i} \rangle = \sum_{\nu \in ({\bf Z}_{\geq 0})^{n-1}} \langle x^{\nu} \rangle t^{-\nu}.\] Then, we have \[ {\rm Res}_{\wt I} P(t_1,\ldots ,t_{n-1}) = {\rm Res}_I \left( P(x_1,\ldots ,x_{n-1}) \Psi(x) \right) . \] Hence, it is important to determine the generating function $\Psi(t).$ Let \[ f_i(t) = t^n + \sum_{j=0}^{n-1} \gamma_{n-j}^{(i)} t^j \] be the characteristic polynomial of the quantum multiplication by $x_i$ with respect to the basis consisting of the quantum Schubert polynomials. Let us consider the $(n!+1)\times (n!+1)$-matrix $C_n(t)$ such that \begin{eqnarray*}(C_n(t))_{1,j} &=& \frac{(-1)^{j-1} t^{n-j+2}}{(j-1)!}; \\ \\ (C_n(t))_{i,j} &=& \left\{ \begin{array}{ll} \ds\frac{(-1)^n}{(j-1)!} {i-2 \choose n-j+1} t^{j-2}, & {\rm if} \; \; i\geq 2, \; i+j \geq n+2, \\ \\ 0, & {\rm otherwise.} \end{array} \right. \end{eqnarray*} We define the differential operator $D_i$ by \[ D_i = (\gamma_{n!}^{(i)},\gamma_{n!-1}^{(i)},\ldots,1) \cdot C_n(t_i) \cdot ^{t}\! (1,\partial / \partial t_i,\ldots ,(\partial / \partial t_i )^{n!} ). \] \begin{pr} The generating function $\Psi(t)$ satisfies the system of differential equations \[ D_i \Psi(t) = 0, \; \; 1\leq i \leq n-1. \] Conversely, these differential equations and the initial values $\langle x_1^{\nu_1}\cdots x_{n-1}^{\nu_{n-1}} \rangle $ for $0\leq \nu_i \leq n!-1$ determine the generating function uniquely. \end{pr} {\it Proof.} Let $x$ be a variable. Since \[ \left( \begin{array}{c} 1 \\ x \\ x^2 \\ x^3 \\ \vdots \\ x^{n!} \end{array} \right) = C_n \cdot \left( \begin{array}{c} t(t-x)^{n!} \\ -x(t-x)^{n!-1} \\ 2! x(t-x)^{n!-2} \\ -3! x(t-x)^{n!-3} \\ \vdots \\ (-1)^{n!}(n!)!x \end{array} \right) , \] we have \[ D_i \Psi(t) = \langle \frac{f_i(x_i)}{(t_i - x_i)^{n!+1}} \prod_{j \not= i}\frac{t_j}{t_j - x_j} \rangle = 0. \] On the other hand, the recursive relations \[ \langle f_i(x_i) P(x_1,\ldots ,x_{n-1}) \rangle = 0 \] with the initial values $\langle x_1^{\nu_1}\cdots x_{n-1}^{\nu_{n-1}} \rangle $ for $0\leq \nu_i \leq n!-1$ determine the correlation function uniquely. \qed \smallskip \\ {\bf Remark 13} \ We can also consider another generating function \[ \langle \exp(x_1 t_1 + \cdots + x_n t_n) \rangle .\] This is the generating volume function in [GK]. This generating function satisfies \[ {\wt e}_i \left( \frac{\partial}{\partial t_1},\cdots , \frac{\partial}{\partial t_n}\right) \langle \exp (x_1 t_1 + \cdots + x_n t_n) \rangle = 0, \] for $1\leq i \leq n.$ \bigskip In the case of $n=3,$ we can calculate the generating function $\Psi (t)$ explicitly. The results of calculation one can find in the Appendix B. %{\bf $\S 3.$ \section{Extended Ehresman--Bruhat order and\break quantum Pieri rule.} Let us remind that the Ehresman--Bruhat order denoted by $\le$, is the partial order on $S_n$ that is the transitive closure of the relation $\rightarrow$. Relation $v\to w$ means that 1) $l(w)=l(v)+1$, 2) $w=v\cdot t$ where $t$ is a transposition. In other words, if $v$ and $w$ are permutations, $v\le w$ means that there exists $r\ge 0$ and $v_0,v_1,\ldots ,v_r$ in $S_n$ such that $$v=v_0\to v_1\to\cdots\to v_r=w. $$ Now let us define the extended Ehresman--Bruhat order $v\Leftarrow w$ on $S_n$. First of all, we define a relation $v\leftarrow w$ (see, also, [FGP]). Relation $v\leftarrow w$ means that 1) $w=v\cdot t$, where $t$ is a transposition, 2) $l(w)\ge l(v)+l(v^{-1}w)$. \smallskip \\ {\bf Remark 14} \ $i)$ It follows from [M], (1.10), that condition 2) is equivalent to the following one \smallskip \\ $2'$) $w(i)