%paper for the special volume of MMJ, dedicated to W. Fulton %version: 23.06.2000 %AMSTEX \input amstex \documentstyle {amsppt} \def\S{\frak S} \def\Sym{Sym} \def\ss{s_{\heartsuit}} \def\vv{\upsilon} \def\d{\partial} \def\dd{\partial_{\heartsuit}} \def\l{\lambda} \def\t{\tau} \def\Pol{{\Cal P}ol} \def\X{{\Cal X}} \def\Y{{\Cal Y}} \def\Z{\Bbb Z} \def\N{\Bbb N} \def\Q{\widetilde{Q}} \def\P{\widetilde{P}} \def\ov{\overline } \def \card {\operatorname{card}} \def \Im {\operatorname{Im}} \def \strict {\operatorname{strict}} \def \deg {\operatorname{deg}} \let \hoto =\hookrightarrow \let \loto =\longrightarrow \def\semi{\ltimes} \def\trait{\vskip 5mm \hrule \vskip 5mm} \def\sv{\sevenrm } \def\s{\scriptstyle } \catcode`@=11 \def\m@th{\mathsurround=\z@} \def\smatrix#1{\null\,\vcenter{\baselineskip=0pt\m@th \ialign{\hfil$\s ##$\hfil&&\ \hfil$\s ##$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,} \def\limproj{\mathop{lim\,proj}} \def\limproj{\mathop{\oalign{\hfil$lim$\hfil\cr $\longleftarrow$\cr}}} \magnification 1200 \document \topmatter \title{Orthogonal divided differences and Schubert polynomials, $\P$-functions, and vertex operators} \endtitle \rightheadtext{Lascoux and Pragacz} \leftheadtext{Orthogonal divided differences and vertex operators} \author Alain Lascoux \ and \ Piotr Pragacz \endauthor \thanks This research was supported by grant No. 5031 of French-Polish cooperation C.N.R.S.--P.A.N. and KBN grant No. 2P03A 05112. \endthanks \endtopmatter \centerline{\it Dedicated to Bill Fulton on his 60th birthday} \vskip20pt \leftline{\bf Abstract} \noindent {\sv Using divided differences associated with the orthogonal groups, we investigate the structure of the polynomial rings over the rings of invariants of the corresponding Weyl groups. We study in more detail the action of orthogonal divided differences on some distinguished symmetric polynomials ($\P$-polynomials) and relate it to vertex operators. Relevant families of orthogonal Schubert polynomials, generalizing $\P$-polynomials, and well-suited to intersection theory computations, are also studied.} \vskip20pt \leftline{\bf Contents:} \noindent Introduction. \ \ 1.Divided differences. \ \ 2.Bases of polynomial rings. \ \ 3.Vertex operators. \ \ 4.Applications to $\P$-polynomials and orthogonal Schubert polynomials. \ \ Appendix: results in type B. \ \ References. \vskip30pt \head {\bf Introduction} \endhead Divided differences were introduced by Newton in his famous interpolation formula (cf. [N, pp.481--483], and [L] for some historical comments). Their importance in geometry was shown by Bernstein-Gel'fand-Gel'fand [BGG] and Demazure [D1,D2] in the context of {\it Schubert calculus} for generalized flag varieties associated with semisimple algebraic groups in the early 1970's. More recently, simple divided differences, interpreted as {\it correspondences} in flag bundles, were extensively used in the sequence of papers [F1,F2,F3] by Fulton in the context of degeneracy loci associated with classical groups. Still another interpretation of divided differences, as {\it Gysin morphisms} in the cohomology of flag bundles associated with semisimple algebraic groups, was discussed in [P2, Sect.4] and [PR, Sect.5]. We refer to the lecture notes [FP] for an introduction. The case of $SL(n)$ has been developed by the first author and Sch\"utzenberger (see e.g. \ [LS1,LS2,LS3] and [M2]). For other classical groups, parallel studies were done by Billey-Haiman [BH], Fomin-Kirillov [FK], Ratajski and the second author [PR], and by the authors [LP1]. The present paper is a continuation of [LP1]. Here we study divided differences associated with the orthogonal groups $SO(2n)$ and $SO(2n+1)$ (i.e. for types D and B). The results for type B are an immediate adaptation of the results for type C given in [LP1]. We summarize them in the appendix. We add, however, a certain new result (Theorem 9) for type B, whose C-analogue was not needed in our former paper [LP1]. Our results for type D require some new computations with vertex operators, that are furnished in Section 3 of the present paper and summarized in our main Theorem 11. In order to simplify the computations with divided differences, we display them as planar arrays, which allows us to perform some kind of `` jeu de taquin". This offers a certain technical novelty w.r.t. our former paper [LP1]. In type $C_n$ (or $B_n$), the key r\^ole was/is played by the divided differences of the form $$ (\d_0 \d_1 \cdots \d_{n-1}) \cdots (\d_0 \d_1 \cdots \d_{n-k})\,, \eqno(*) $$ where $k\le n$. It appears that in type $D_n$ a similar r\^ole is played by the divided differences of the form $$ (\dd \d_2\cdots \d_{n-1} \d_1 \d_2 \cdots \d_{n-2}) \cdots (\dd \d_2\cdots \d_{n-2k+1} \d_1 \d_2 \cdots \d_{n-2k})\,, \eqno(**) $$ where $k\le n/2$. \ Here $\d_i$, for $i>0$, are Newton's (simple) divided differences: $$ f\, \d_i := {{f-f(\ldots,x_{i+1},x_i,\ldots)}\over{x_i-x_{i+1}}}\,, $$ and moreover we set $$ f\, \d_0 := {{f-f(-x_1,x_2,\ldots)}\over{-x_1}}\,, $$ $$ f\, \dd := {{f-f(-x_2,-x_1,x_3,\ldots)}\over{-x_1 - x_2}}\,. $$ We compose the simple orthogonal divided differences in (*) and (**) {\it from left to right}. As the the Weyl group of type $D$ is naturally embedded in the Weyl group of type $B$, the divided difference (**) can be expressed in terms of (*). Such basic relations are given in Proposition 6 and Corollary 8. The symmetric functions which are most adapted to orthogonal divided differences are $\P$-polynomials [PR], which are a variant of Schur $P$-polynomials. Our paper is of an algebro-combinatorial nature but its motivation comes from geometry. The algebro-combinatorial properties studied here should be useful in Schubert calculus associated with orthogonal groups and the related degeneracy loci. The computations of this paper are closely related to the ones in [LLT1]; we plan to develop this link in some future publication. The algebro-combinatorial techniques, used in the present paper, are chosen to be as elementary as possible. This should help the reader, with more geometric and less algebro-combinatorial background, to read the paper. We mention however, that several results, used in the proof of Theorem 9, in the appendix, are particular instances of more general properties of Hall--Littlewood polynomials (see [LLT2] and [LP2]). Let us remark that there is also an interesting algebra and combinatorics of ``isobaric divided differences", with associated {\it Grothendieck polynomials} (cf. [FL]). \smallskip This work has extensively used ACE ([V]) for explicit computations. \smallskip It is our pleasure and honor to dedicate the present article to the mathematician whose recent work has illuminated important connections between geometry and combinatorics. \bigskip\smallskip \noindent {NOTATION AND CONVENTIONS :} \smallskip A {\it vector} (of {\it length} $m$) is a sequence $[v_1,\ldots,v_m] \in \Bbb Z^m$. \hfill\break We will compare vectors of the same lengths, writing $$ [v_1,\ldots,v_m] \subseteq [u_1,\ldots,u_m] $$ if $v_i \le u_i$ for all $i=1,\ldots,m$. Given a vector $\alpha=[\alpha_1,\ldots,\alpha_m]$, we will write $|\alpha|$ for the sum of its components. A {\it partition} is an equivalence class of sequences $[i_1 \ge \cdots \ge i_m] \in \N^m$, where we identify the sequences $[i_1,\ldots, i_m]$ with $[i_1,\ldots,i_m,0]$. We denote the corresponding partition by $I=(i_1,\ldots,i_m)$, by taking any representative sequence. A {\it part} of a partition $I$, is a nonzero component of any sequence that represents $I$. The {\it length} of a partition $I$ is the number of its nonzero parts, denoted $\ell(I)$. We call a partition {\it strict} if all its parts are different. We write $I\subseteq J$ for two partitions $I$ and $J$ (of possibly different lengths) if the same relation holds for any pair of the same length representing them. All operators act, in this paper, on their {\it left}. Polynomials are usually treated as operators acting by multiplication. \bigskip\medskip \head {\bf 1. Divided differences} \endhead Let $n$ be a fixed (throughout the paper) positive integer. The symmetric group (i.e. the Weyl group of type A) $\frak S_n$ is the group with generators $s_1,\ldots,\, s_{n-1}$ subject to the relations $$ s_i^2=1, \quad \quad s_{i-1}\, s_i\, s_{i-1} =s_i s_{i-1}\, s_i \quad ,\quad s_is_j =s_js_i \quad \forall i,j: |i-j|>1 \ . \eqno (1.1) $$ We shall call $s_1,\ldots,\, s_{n-1}$ \ {\it simple transpositions} of $\S_n$. The hyperoctahedral group (i.e. the Weyl group of type B) $\frak B_n$ is an extension of $\S_n$ by an element $s_0$ such that $$ s_0^2=1, \quad s_0\, s_1\, s_0\, s_1=s_1\, s_0\, s_1\, s_0, \quad s_0\, s_i=s_i\, s_0 \quad \hbox{for} \quad i\ge 2. \eqno (1.2) $$ The Weyl group $\frak D_n$ of type $D$ is the extension of $\S_n$ by an element $\ss$ such that $$ \ss^2=1, \quad s_1 \ss\ =\ss\, s_1, \quad \ss\, s_2\, \ss \, = s_2 \ss\, s_2, \quad \ss\, s_i=s_i\, \ss \quad \hbox{for} \quad i> 2. \eqno (1.3) $$ The group $\frak D_n$ can be thought as a subgroup of $\frak B_n$ by sending $\ss$ to $s_0\, s_1\, s_0$. \medskip The above three groups act on vectors of length $n$ by $$ [v_1,\ldots, v_n]\, s_i := [v_1,\ldots, v_{i-1},v_{i+1},v_i, v_{i+2},\ldots,v_n] $$ $$ [v_1,\ldots, v_n]\, s_0 := [-v_1, v_2,\ldots,v_n] $$ $$ [v_1,\ldots, v_n]\, \ss := [-v_2,-v_1,\, v_3,\ldots,v_n]$$ The orbit of the vector $v=[1,\ldots,\, n]$ under $W=\S_n,\, \frak B_n$ or $\frak D_n$, is in bijection with the elements of $W$, and we shall code each $w\in W$ by the vector $[1,\ldots,n]\, w$, writing $\ov \imath$ instead of $-i$. \medskip The three groups $W= \S_n,\, \frak B_n,\, \frak D_n$ also act on the ring of polynomials in $n$ indeterminates $\Cal X=\{ x_1,\ldots, x_n\} $: the simple transposition $s_i$, $i\geq 1$, exchanges $x_i$ and $x_{i+1}$, $s_0$ sends $x_1$ to $-x_1$, $\ss$ sends $x_1$ to $-x_2$ and $x_2$ to $-x_1$, the action being trivial in the non-listed cases. We shall denote by $f^w$ the image of a polynomial $f\in \Z[\Cal X]$ under $w\in W$, and write $x^\alpha$, with $\alpha=[\alpha_1,\ldots,\alpha_n]\in \N^n$, for the monomial $x_1^{\alpha_1}\cdots x_n^{\alpha_n}$. \medskip Let $\Pol$ be the ring of polynomials in $\X$ with coefficients in $\Z[{1\over 2}]$ (we need only division by 2). For any $m\leq n$, let $\Sym(m|n-m)$ denote the subring of $\Pol$ consisting of polynomials invariant under all $s_i$, $1\leq i\leq n-1$, $i\neq m$, and let $\Sym(n)=\Sym(n|0)=\Sym(0|n)$ be the ring of symmetric polynomials. It contains as subrings $\Sym^B(n)$, the ring of polynomials invariant under $\frak B_n$, and $\Sym^D(n)$, the invariants of $\frak D_n$. It is easy to see that $\Pol$ is a free module over these different rings, generated by $x^\alpha$, $\alpha\subseteq [n-1,\ldots,0]$ (or $\alpha\subseteq [0,\ldots,n-1]$) over $\Sym(n)$, by $x^\alpha$, $\alpha\subseteq [2n-1,2n-3,\ldots,1]$ (or $\alpha\subseteq [1, \ldots, 2n-3,2n-1]$) over $\Sym^B(n)$, and by $x^\alpha$, $\alpha\subseteq [2n-2,2n-4,\ldots,2,0]$ (or $\alpha\subseteq [0,2,\ldots, 2n-2]$) over $\Sym^D(n)$. \medskip The respective elements of maximal length in each of the groups are $$\omega := [n,\ldots,\, 1] \ \hbox{for}\ \S_n \ ,$$ $$ w_0^B := [\ov 1,\ldots,\, \ov n] \ \hbox{for}\ \frak B_n\ ,$$ $$ w_0^D := \left\{ \matrix [\ov 1,&\ldots,&\ov n] & n & \hbox{even} \cr [1,&\ov 2,&\ldots,&\ov n]&n &\hbox{odd} \cr \endmatrix \right. $$ for $\frak D_n$. We shall also need the following element of $\frak D_n$: $$ \vv := \omega w_0^D = \left\{ \matrix [\ov n,&\ldots,&\ov 1] & n & \hbox{even} \cr [n,&\ov{n-1},&\ldots,&\ov1] & n & \hbox {odd\, .} \cr \endmatrix \right. $$ \smallskip Relations between reduced decompositions in $W$ can be represented planarly. By definition, a planar display will be identified with its reading from left to right and top to bottom ({\it row--reading}). We shall also use {\it column--reading}, that is, reading successive columns downwards, from left to right. For example, we will write $$\matrix 2 \cr 1 & 2 \cr \endmatrix \equiv\ \matrix 1 & 2 \cr & 1 \cr \endmatrix $$ for the following equality for simple transpositions: $$ s_2\, s_1\, s_1 = s_1\, s_2\, s_1 \ . $$ Suppose that a rectangle is filled row-wise from left to right, and column-wise from bottom to top with consecutive numbers from $\{1,\ldots, n-1\}$. Then one easily checks that its row--reading and column--reading produce two words which, interpreted as words in the $s_i$, are congruent modulo the Coxeter relations. Here is an example of such a congruence : $$ \matrix 3456 \cr &2345 \cr &&1234 \cr \endmatrix \equiv \matrix 3 \cr 2 \cr 1 \cr \endmatrix \cdot \matrix 4 \cr 3 \cr 2 \endmatrix \cdot \matrix 5 \cr 4 \cr3 \endmatrix \cdot \matrix 6 \cr 5 \cr 4 \cr \endmatrix $$ the congruence class being conveniently denoted by the rectangle $ \smatrix{3 & 4 & 5 & 6 \cr 2 & 3 & 4 & 5 \cr 1 & 2 & 3 & 4 \cr}$. More generally, the planar arrays that we shall write, will have the property that their row--reading and column--reading are congruent modulo Coxeter relations \ (cf. [LS2,LS3] or [EG] for a ``jeu de taquin" on reduced decompositions). In this notation, one has, for any integers $a,b,c,d,k$ : $1\leq a1$. The elements $\Omega^W$, as operators on the ring of polynomials $\Pol$, can be obtained from the cases of $\S_2$, $\frak B_1$, $\frak D_2$. To see this, we first need to define {\it simple divided differences}~: $$ \Pol \ni f \mapsto f\, \d_i := (f-f^{s_i})/(x_i-x_{i+1}), \quad i\ge 1 \,, \eqno(1.8) $$ $$ \Pol \ni f \mapsto f\, \d_0 := (f-f^{s_0})/{(-x_1)}\ , \eqno(1.9) $$ $$ \Pol \ni f \mapsto f\, \dd := (f-f^{\ss} )/{(-x_1 - x_2)} \ . \eqno(1.10) $$ The $\d_i,\, \d_0,\, \dd$ satisfy the Coxeter relations (1.1)--(1.3), together with the relations $$ \dd^2= 0 = \d_i^2 \ \ \ \hbox{for} \ \ \ 0\leq i j\ge 1} (x_i^2 -x_j^2) = {1\over 2^{n-1}} x^{[1,3,\ldots,2n-1]} \, \Omega^{\frak B_n} \ , \eqno(1.13) $$ $$ \hbox{and} \ \ \Delta^{D} := \prod_{n\ge i>j \ge 1} (x_i^2 -x_j^2) = {1\over 2^{n-1}}x^{[2,4,\ldots,2n-2]} \, \Omega^{\frak D_n} \ . \eqno(1.14) $$ The Weyl character formula for types A, B, and D can be written as \proclaim{Lemma 1} \ For each of the groups $W=\S_n,\, \frak B_n$ or $\frak D_n$, the alternating sum $\Omega^W$, as an operator on the ring of polynomials $\Pol$, is related to the maximal divided difference by $$ \Omega^{\S_n} {1\over \Delta} = \d_\omega\,, \ \ \ \ \ \Omega^{\frak B_n} {1\over \Delta^B} = (-1)^{n\choose 2}\, \d_{w_0^B}\,, \ \ \ \hbox{and} \ \ \ \Omega^{\frak D_n} {1\over \Delta^D} = (-1)^{n\choose 2} \, \d_{w_0^D} . $$ \endproclaim Indeed, all the operators in Lemma 1 commute with multiplication by polynomials which are invariant under $W$. Moreover, they decrease degree by the length of the maximal element of the group. Since $\Pol$ is a module over $\Sym^W(n)$ with a basis of monomials of degree strictly less than this length, except for a single monomial, it remains only to check that the actions of $\Omega$'s and $\partial$'s agree on this monomial, which offers no difficulty. \bigskip\medskip \head {\bf 2. Bases of polynomial rings} \endhead The monomials mentioned in the previous section are not an appropriate basis, when interpreted in terms of cohomology classes for the flag variety. Define, for the rest of this paper, the vector $$ \rho:=[n-1,\ldots, 1,0]\, . \eqno(2.1) $$ Motivated by geometry, one defines recursively {\it Schubert polynomials} $Y_\alpha$, for any sequence $\alpha\in \N^n$, with $\alpha\subseteq \rho$, by $$ Y_\alpha\, \d_i = Y_{\beta} \ , \ \ \ \hbox{if} \ \ \alpha_i >\alpha_{i+1} \ \ , \eqno (2.2) $$ where $$ \beta = [\alpha_1,\ldots,\alpha_{i-1}, \alpha_{i+1}, \alpha_i-1, \alpha_{i+2},\ldots, \alpha_n]\,, $$ starting from $Y_\rho= x^\rho$ (cf. [LS1], [M2]). In particular, if $\alpha\in \N^n$ is weakly decreasing, then $Y_\alpha$ is equal to the {\it monomial} \ $x^\alpha$\,. If, on the contrary, $\alpha_1\leq \cdots \leq \alpha_k $ and $\alpha_{k+1}=\cdots =\alpha_n=0$, for some $k\le n$, then $Y_\alpha$ coincides with the {\it Schur polynomial} $s_\l(x_1,\ldots,x_k)$, where $\l = (\alpha_k,\ldots, \alpha_1)$. \medskip \noindent CONVENTION: \ Let $\alpha \in \Bbb N^k$. Then we shall write $Y_{\alpha}$ for $Y_{\alpha,0,\ldots,0}$. \smallskip We also record, for later use, the following equality: for $\alpha = [\alpha_1, \ldots ,\alpha_k] \in \Bbb N^k$, $$ Y_{\alpha} \, x_1 \cdots x_k = Y_{[\alpha_1+1,\ldots, \alpha_k+1]}. \eqno(2.3) $$ \smallskip On $\Pol$ there is a scalar product: $$ ( \ , \ ): \Pol \times \Pol \to \Sym(n), $$ defined for $f, g\in \Pol$ by $$ (f,g): = f g \, \d_{\omega}. \eqno(2.4) $$ There exists an involution \footnote {This involution is: \ $\hbox {code}(w)= \alpha \mapsto \alpha'=\hbox{code} (w \omega)$ \ (cf. [M2]).} \ $\alpha \mapsto \alpha'$ \ such that $$ \bigl(Y_{\alpha}^\omega , Y_{\beta'}\bigr) = (-1)^{|\alpha|} \delta_{\alpha \beta}\,. \eqno(2.5) $$ Moreover, when $\alpha,\beta \subseteq \rho$ are such that $|\alpha| + |\beta| = |\rho|$, then one has $$ \bigl(Y_{\alpha}, Y_{[n-1-\beta_1, n-2-\beta_2,\ldots,0-\beta_n]}\bigr)= \delta_{\alpha \beta}. \eqno(2.6) $$ We also will need \ $\Q$-polynomials \ of [PR]. We set $\Q_i:= e_i=e_i(\X)$, the $i$-th elementary symmetric polynomial in $\X$\,. Given two nonnegative integers $i \ge j$, we adapt Schur's definition of his $Q$-functions by putting $$ \Q_{i,j}:=\Q_i \Q_j +2\sum\limits^j_{p=1}(-1)^p\Q_{i+p} \Q_{j-p}\,. \eqno(2.7) $$ Given any partition $I=(i_1,\ldots,i_k)$, where we can assume $k$ to be even, we set $$ \Q_I:= \hbox{Pfaffian} (M)\,, \eqno(2.8) $$ where $M=(m_{p,q})$ is the $k\times k$ skew-symmetric matrix with $m_{p,q}= \Q_{i_p,i_q}$ for $1 \le p < q \le k$. \smallskip Equivalently, for any partition $I=(i_1\ge i_2\ge\ldots\ge i_{\ell}>0)$\,, the polynomial $\Q_I=\Q_I(X)$ is defined recurrently on $\ell$ by putting for odd $\ell$\,, $$ \Q_I:=\sum\limits^{\ell}_{j=1}(-1)^{j-1} \Q_{i_j} \Q_{(i_1,\ldots,i_{j-1},i_{j+1},\ldots,i_{\ell})} \eqno(2.9) $$ and for even $\ell$\,, $$ \Q_I:=\sum\limits^{\ell}_{j=2}(-1)^j \Q_{i_1,i_j} \Q_{(i_2,\ldots,i_{j-1},i_{j+1},\ldots, i_{\ell})}\,. \eqno(2.10) $$ For any positive integer $k$, let $\rho(k)$ denote the partition $$ \rho(k):=(k,k-1,\ldots, 1)\,. \eqno(2.11) $$ The ring $\Sym(n)$ is a free module over the ring of polynomials symmetric in $x_1^2,\ldots,x_n^2$, with a basis provided by the $\Q_I(\X)$, where $I\subseteq \rho(n)$ ranges over strict partitions. As functions of $x_1,\ldots, x_m$, the $\Q$-polynomials can also be defined recursively by induction on $m$, involving now all partitions without restriction, as for Hall-Littlewood polynomials: for any strict partition $I$, one has $$ \Q_I(x_1,\ldots,x_m) = \sum_{j=0}^{\ell(I)} x_m^j \bigr( \sum_{|I|-|J|=j} \Q_J(x_1,\ldots,x_{m-1})\bigl), \eqno(2.12) $$ where the sum is over all (i.e. not necessarily strict) partitions $J\subseteq I$ such that $I/J$ has at most one box in every row (cf. [PR, Prop. 4.1]). Moreover, given a partition $I'=(\ldots,i,j,j,k,\ldots)$ and denoting $I=(\ldots,i,k,\ldots )$, one has the {\it factorization property} $$ \Q_{I'}=\Q_{j,j} \Q_I. \eqno(2.13) $$ We define, for a strict partition $I$, $$ \P_I:= 2^{-\ell(I)}\, \Q_I\,. \eqno(2.14) $$ The ring $\Sym(n)$ is a free module over $\Sym^D(n)$ with a basis provided by the $\P_I$, where $I$ ranges over strict partitions contained in $\rho(n-1)$. Now we will need the following divided difference: $$ \d_v =(\dd \d_2\cdots \d_{n-1}\, \d_1\cdots \d_{n-2}) \cdots (\dd \d_2\d_3\, \d_1\d_2)\, \dd \ \ \ \ \hbox{$n$ even} \eqno(2.15) $$ and $$ \d_v =(\dd \d_2\cdots \d_{n-1}\, \d_1\cdots \d_{n-2}) \cdots (\dd \d_2\d_3\d_4\, \d_1\d_2\d_3)\, (\dd \d_2\d_1) \ \ \hbox{ $n$ odd} \eqno(2.16) $$ Denote by $$ \langle \ ,\ \rangle : \Sym(n) \times \Sym(n) \to Sym^D(n) $$ the scalar product defined for $f, g\in Sym(n)$ by $$ \langle f,g \rangle := f g \, \d_v . \eqno(2.17) $$ For strict partitions $I,J\subseteq \rho(n-1)$, one has $$ \langle \P_I,\P_{\rho(n-1) \smallsetminus J} \rangle = (-1)^{n\choose 2}\, \delta_{I J}, \eqno(2.18) $$ where $\rho(n-1) \smallsetminus I$ is the strict partition whose parts complement the parts of $I$ in $\{n-1,n-2,\ldots, 1\}$ \ (cf. [PR]). Consequently, the polynomial ring $\Pol= \Bbb Z [{1 \over 2}][x_1,\ldots, x_n]$ is a free $\Sym^D(n)$-module with a basis $Y_{\alpha}\, \P_I$, where $\alpha$ ranges over subsequences contained in $\rho$ and $I$ runs over all strict partitions contained in $\rho(n-1)$. Note that the element of maximal degree of this basis is $x^{\rho} \P_{\rho(n-1)}$. Let $$ [ \ , \ ]: \Pol \times \Pol \to \Sym^D(n) $$ be a scalar product, defined for $f,g \in \Pol$ by $$ [f,g]:= f g \,\d_{w_0^D}. \eqno(2.19) $$ One has, for $\alpha, \beta \subseteq \rho$ and strict partitions $I,J\subset \rho(n-1)$, $$ \bigl[Y_{\alpha}^{\omega}\, \P_I , Y_{\beta'}\, \P_{\rho(n-1)\smallsetminus J} \bigr] = (-1)^{|\alpha|+{n \choose 2}}\, \delta_{\alpha \beta}\, \delta_{I J}\,. \eqno(2.20) $$ (See (2.5).) \medskip Let $\Y=\{ y_1,\ldots,y_n\}$ be a second set of indeterminates of cardinality $n$. The symbol $\equiv$ will mean: \ ``congruent modulo the ideal generated by the relations \hfill\break $ f(x_1^2,\ldots, x_n^2) = f(y_1^2,\ldots, y_n^2)$, $f\in Sym(n)$, together with $x_1\cdots x_n = y_1\cdots y_n$". \medskip Following Fulton [F2,F3], define $$ \aligned F(\X,\Y) &:=\ |\P_{n+j-2i}(\X)+\P_{n+j-2i}(\Y)|_{1\le i,j\le n-1} \\ & \\ &=\ {\left|\ \CD \P_{n-1}(\X)+\P_{n-1}(\Y) & 0 &.\;.\;. & \\ \P_{n-3}(\X)+\P_{n-3}(\Y) &\ \P_{n-2}(\X)+\P_{n-2}(\Y) &\ .\;.\;.\ & \\ \vbox{\offinterlineskip\hbox{.}\vskip3pt\hbox{.}\vskip3pt\hbox{.}} &\vbox{\offinterlineskip\hbox{.}\vskip3pt\hbox{.}\vskip3pt\hbox{.}} &\vbox{\offinterlineskip\hbox{.}\vskip3pt\hbox{\ \ .}\vskip3pt \hbox{\ \ \ \ .}} & \\ && 1 && \P_1(\X)+\P_1(\Y) \\ \endCD\ \right|}\,. \\ \endaligned \eqno(2.21) $$ Following [PR], define $$ \P(\X, \Y):= \sum \P_I(\X)\, \P_{\rho(n-1) \smallsetminus I}(\Y) \ , \eqno(2.22) $$ where the summation is over all strict partitions $I\subseteq \rho(n-1)$. The reasoning in [LP1, Sect.2] made for case $C_n$ adapts to case $D_n$ and furnishes: \proclaim {Proposition 2} \ We have \noindent (i) \ $$F(\X,\Y) \equiv \P(\X,\Y)\,. \eqno(2.23)$$ \noindent (ii) \ For every $w\in \frak D_n\smallsetminus \S_n$, $$\P(\X^w, \X)=0\,, \eqno(2.24) $$ and for every $w\in \S_n$, $$ \P(\X^w, \X)= \P(\X, \X)= s_{\rho(n-1)}(\X)\,. \eqno(2.25) $$ \noindent (iii) \ For every $f\in \Sym(n)$, $$ \langle f(\X)\,, F(\X,\Y) \rangle \equiv (-1)^{n\choose 2}\, f(\Y) \ . \eqno(2.26) $$ \noindent (iv) \ For every $f\in \Pol$, $$ \bigl[ f(\X)\,, \prod_{n \ge i>j \ge 1}(x_i-y_j)\, F(\X,\Y) \,\bigr] \equiv f(\Y)\,. \eqno(2.27) $$ \endproclaim In other words, $F(\X,\Y)$ is a reproducing kernel for the scalar product $\langle \ , \ \rangle$, and $\prod_{i>j}(x_i-y_j)\, F(\X,\Y)$ is a reproducing kernel for $[ \ , \ ]$. One can show that the ``vanishing property" {\it (ii)} characterizes $\P(\X,\Y)$ up to $\equiv$. The congruence {\it (i)} can be also derived from geometry by comparing the classes of diagonals in flag bundles associated with $SO(2n)$ given in [F2,F3] and [PR] (see also [G]). \bigskip\medskip \head {\bf 3. Vertex Operators} \endhead In this section we shall mainly make computations using the following two divided differences: \proclaim {Definition 3} \ For $k\le n$, we set $$ \nabla_k^B(n):= (\d_0 \d_1 \cdots \d_{n-1}) \cdots (\d_0 \d_1 \cdots \d_{n-k}) \,. \eqno(3.1) $$ For $k\le n/2$, we put $$ \nabla_k^D(n) :=(\dd \d_2\cdots \d_{n-1} \d_1 \d_2 \cdots \d_{n-2}) \cdots (\dd \d_2\cdots \d_{n-2k+1} \d_1 \d_2 \cdots \d_{n-2k}). \eqno(3.2) $$ \endproclaim We shall need the following fact from [LP1], quoted in the appendix: \proclaim{Fact 4} \ Let $k\leq n$ and let $\alpha =[\alpha_1 \leq \cdots \leq \alpha_k] \in \N^{k}$ with $\alpha_k \le n-k$. Suppose that $I\subseteq \rho(n)$ is a strict partition. Then the image of $\Q_I \, Y_\alpha$ under $\nabla_k^B(n)$ is 0 unless $n-0-\alpha_1,\ldots, n-(k-1) -\alpha_k$ are parts of $I$. In this case, the image is $(-1)^{k(n-1)+s} 2^k\Q_J$, where $J$ is the strict partition with parts $$\{i_1,\ldots, i_{\ell(I)} \} \smallsetminus \{ n-0-\alpha_1,\ldots, n-(k-1)-\alpha_k \}, $$ and $s$ is the sum of positions of the parts erased in $I$. \endproclaim \bigskip \noindent {\bf Example 5.} \ For $n=7$ and $k=2$\,, we have $$ \aligned &\Q_{(5,4,3,2,1)} \, Y_{[2,5]} \, \nabla_2^B(7) \cr &=\Q_{(5,4,3,2,1)} \, Y_{[2,5]}\, (\d_0 \d_1 \d_2 \d_3 \d_4 \d_5 \d_6) (\d_0 \d_1 \d_2 \d_3 \d_4\d_5)= 4 \Q_{(4,3,2)}\,, \cr \endaligned $$ and for $k=3$\,, we have $$ \Q_{(7,5,4,3,1)} \, Y_{[2,3,4]} \nabla_3^B(7) = - 8 \Q_{(7,4)}\,. $$ \bigskip The following result establishes a basic relation between the $\nabla^D$'s and $\nabla^B$'s: \proclaim{Proposition 6} \quad Let $k$ be a positive integer. As operators on $Sym(2k)$, $$ \nabla_k^D(2k) = x_1\cdots x_{2k} \, \nabla_{2k}^B(2k) +x_1\cdots x_{2k-1} \, \nabla_{2k-1}^B(2k) \,. \eqno(3.3) $$ \endproclaim Before proving (3.3), we illustrate it by the following examples: \bigskip \noindent {\bf Example 7.} As operators on $\Sym(2)$, $$ \nabla_1^D(2) = \dd = x_1x_2 \d_0\d_1\d_0 + x_1 \d_0\d_1. $$ As operators on $\Sym(4)$, $$ \aligned \nabla_2^D(4) &= (\dd\d_2\d_3\d_1\d_2)\dd= \cr &x_1x_2x_3x_4\, (\d_0\d_1\d_2\d_3)(\d_0\d_1\d_2)(\d_0\d_1)\d_0 +x_1x_2x_3\, (\d_0\d_1\d_2\d_3)(\d_0\d_1\d_2)(\d_0\d_1). \endaligned $$ \medskip \noindent The RHS of the last equation is depicted planarly as $$ x_1 x_2 x_3 x_4 \matrix \d_0 & \d_1 & \d_2 & \d_3 \cr & \d_0 & \d_1 & \d_2 \cr & & \d_0 & \d_1 \cr & & & \d_0 \cr \endmatrix +x_1 x_2 x_3 \matrix \d_0 & \d_1 & \d_2 & \d_3 \cr & \d_0 & \d_1 & \d_2 \cr & & \d_0 & \d_1 \cr \endmatrix $$ \bigskip \demo {Proof of the proposition} \ In this proof, let $\X:=\{x_1,\ldots,x_{2k} \}$. Both sides of (3.3) are $Sym^B(2k)$-linear. The operator $\nabla_k^D(2k)$ sends all $\Q_I(\X)$, $I\subseteq \rho(2k-1)$ to 0, except for $\Q_{\rho(2k-1)}$ which is sent to $(-1)^k 2^{2k-1}$ (cf.(2.18)). Thus $\nabla_k^D(2k)$ annihilates all $\Q_I(\X)$, $I\subseteq \rho(2k)$, except for $I=\rho(2k-1)$ which is sent to $(-1)^k 2^{2k-1}$ and $I=\rho(2k)$ which is sent to $(-1)^k 2^{2k-1} x_1 \cdots x_{2k}$. The action of $$ x_1\cdots x_{2k} \nabla_{2k}^B(2k) $$ is given by Fact 4. Only $\Q_{\rho(2k)}(\X)$ survives and is sent to $(-1)^k 2^{2k} x_1\cdots x_{2k}$. We will now calculate the action of $$ x_1\cdots x_{2k-1} \, \nabla_{2k-1}^B(2k) $$ on the $\Q_I(\X)$, where $I\subseteq \rho(2k)$ is a strict partition. We set, temporarily in this proof, $$ \nabla:=\nabla_{2k-1}^B(2k) \ \ \ \ \hbox{and} \ \ \ \ \nabla':=\nabla_{2k-1}^B(2k-1)\,, $$ so that $$ \nabla=\nabla' \d_{2k-1} \cdots \d_1. $$ Let $\X':=\{x_1,\ldots,x_{2k-1}\}$. We decompose $\Q_I(\X)$ as a sum of products of powers of $x_{2k}$ times some $\Q_J(\X')$, according to the formula (2.12): $$ \Q_I(\X)= \sum \Q_J(\X')\, x_{2k}^{m_J} \ . $$ Let $\overline J$ be the strict partition obtained from a partition $J$ by subtracting all the pairs of equal parts. We have three cases to examine: \smallskip \noindent 1. Let $i_1\le 2k-2$. Then for each $J$, $|\overline J| + m_J + 2k-1 < \hbox {deg} \nabla$, and hence $$ x_1\cdots x_{2k-1} \Q_I(\X) \nabla = 0. $$ \smallskip \noindent 2. Let $i_1=2k-1$. For degree reasons, $\Q_I(\X)x_1\cdots x_{2k-1} \nabla \ne 0$ is possible only if $I=\rho(2k-1)$ ($I$ being a strict partition). \noindent \proclaim {Claim} \ We have $$ x_1\cdots x_{2k-1} \Q_J(\X') \nabla' \ne 0 \eqno(3.4) $$ only if $J = \overline J = \rho(2k-2)$. \endproclaim Indeed, suppose first that $j_1=2k-1$. Then $$ x_1\cdots x_{2k-1} \Q_J(\X') = \Cal P \cdot \Q_H(\X'), $$ where $\Cal P$ is a polynomial symmetric in $x_1^2,\ldots,x_{2k-1}^2$, and a strict partition $H$ has no part equal to $2k-1$. Since this expression is annihilated by $\nabla'$, we cannot have (3.4). So, for degree reasons, (3.4) holds only if $j_1=2k-2$. Suppose now that $j_2=2k-2$. We get $$ x_1\cdots x_{2k-1} \Q_J(\X') = \Cal P \cdot \Q_H(\X'), $$ where $\Cal P$ is a polynomial symmetric in $x_1^2,\ldots,x_{2k-1}^2$, and a strict partition $H$ has no part equal to $2k-2$. Since this expression is annihilated by $\nabla'$, we cannot have (3.4). So, for degree reason, (3.4) holds only if $j_2=2k-3$. Continuing this way, we get the claim. \smallskip For $J=\rho(2k-2)$, we compute $$ x_1\cdots x_{2k-1} \Q_J(\X') x_{2k}^{2k-1} \nabla =\Q_{\rho(2k-1)}(\Cal X')\nabla' x_{2k}^{2k-1}\, \d_{2k-1} \cdots \d_1 = (-1)^{k-1} 2^{2k-1}. $$ \smallskip \noindent 3. Let $i_1=2k$. Then $(i_2,\ldots ) \subseteq \rho(2k-1)$. We have $$ \aligned & x_1\cdots x_{2k-1} \Q_I(\X) \nabla = x_1^2 \cdots x_{2k-1}^2 x_{2k} \Q_{(i_2,\ldots)}(\X) \nabla \cr =& \Q_{(i_2,\ldots)}(\X) \nabla' x_1^2 \cdots x_{2k-1}^2 x_{2k}\, \d_{2k-1} \cdots \d_1 \ . \endaligned $$ Now for $H\subseteq (i_2,\ldots) \subseteq \rho(2k-1)$, $\Q_H(\X')\nabla' \ne 0$ iff $H=\rho(2k-1)$, so iff $(i_2,\ldots)=\rho(2k-1)$. We have $$ \Q_{\rho(2k-1)}(\Cal X) \nabla' =\Q_{\rho(2k-1)}(\Cal X')\nabla' = (-1)^k 2^{2k-1} $$ and $$ x_1^2 \cdots x_{2k-1}^2 x_{2k}\, \d_{2k-1} \cdots \d_1 = - x_1 \cdots x_{2k}. $$ Summarizing, $$ \Q_I(\Cal X) x_1 \cdots x_{2k-1} \nabla \ne 0 $$ only if $I=\rho(2k-1)$, when we get $(-1)^{k-1} 2^{2k-1}$; or $I=\rho(2k)$, when we get $(-1)^{k-1} 2^{2k-1} x_1\cdots x_{2k}$. Finally, comparing the computed values of the $\Q_I(\X)$ under the operators: $$ \nabla_k^D(2k)\,, \ \ \ \ \ x_1\cdots x_{2k}\, \nabla_{2k}^B(2k), \ \ \hbox {and} \ \ \ \ \ x_1\cdots x_{2k-1} \, \nabla_{2k-1}^B(2k), $$ that are possibly nonzero only for $I=\rho(2k)$ and $\rho(2k-1)$, we get the desired formula (3.3). (Note that we have also used the equality $2^{p-1}=2^p - 2^{p-1}$.) \qed \enddemo \medskip \proclaim{Corollary 8} \ Let $k$ be a positive integer such that $k \leq n/2$. As operators on the ring \ $\Sym(2k\, |\, n-2k)$, $$ \nabla_k^D(n) = x_1\cdots x_{2k} \, \nabla_{2k}^B(n) +x_1\cdots x_{2k-1} \, \nabla_{2k-1}^B(n) \, \d_1 \cdots \d_{n-2k}\,. \eqno(3.5) $$ \endproclaim This property is obtained from Proposition 6 by composing the expression for the operator $\nabla_k^D(2k)$ with the divided difference $$ (\d_{2k} \cdots \d_{n-1}) \cdots (\d_2 \cdots \d_{n-2k+1}) (\d_1\cdots \d_{n-2k}) = \matrix \d_{2k} & \cdots & \d_{n-1} \cr \vdots & & \vdots \cr \d_2 & \cdots & \d_{n-2k+1} \cr \d_1 & \cdots & \d_{n-2k} \cr \endmatrix $$ \bigskip In the proof of Theorem 11, we will need the following supplement to Fact 4: \proclaim{Theorem 9} \ Let $k\leq n$ and let $\alpha =[\alpha_1 \leq \cdots \leq \alpha_k] \in \N^{k}$ with $\alpha_k=n-k+1$. Suppose that $I\subseteq \rho(n)$ is a strict partition. Then the image of \ $\Q_I \, Y_\alpha$ \ under $\nabla_k^B(n)$ is 0 unless $\ell(I)\not\equiv n$ (mod $2$) and $n-0-\alpha_1,\ldots, n-(k-2) -\alpha_{k-1}$ are parts of $I$. In this case, the image is $(-1)^{(k-1)(n-1)+1+s} 2^k\Q_J$, where $J$ is the strict partition with parts $$\{i_1,\ldots, i_{\ell(I)} \} \smallsetminus \{ n-0-\alpha_1,\ldots, n-(k-2)-\alpha_{k-1} \} \,, $$ and $s$ is the sum of positions of the parts erased in $I$. \endproclaim The proof of this theorem will be given in the appendix. \bigskip \noindent {\bf Example 10.} (i) For $n=5$ and $k=1$, we have $$ x_1^5 \Q_{(5,3,2,1)} \d_0 \d_1 \d_2 \d_3 \d_4 = - 2 \Q_{(5,3,2,1)} \ \ \ \hbox{and} \ \ \ x_1^5 \Q_{(5,2,1)} \d_0 \d_1 \d_2 \d_3 \d_4 = 0. $$ \noindent (ii) For $n=7$ and $k=2$, we have $$ \Q_{(7,6,4,1)}\, Y_{[1,6]}\, \nabla_2^B(7) = - 4 \Q_{(7,4,1)} $$ and $$ \Q_{(7,6,4,3,1)}\, Y_{[1,6]}\, \nabla_2^B(7) = 0. $$ \noindent (iii) For $n=7$ and $k=4$, we have $$ \Q_{(7,6,4,3,2,1)}\, Y_{[1,2,2,4]}\, \nabla_{4}^B(7) =16 \Q_{(7,2,1)} $$ and $$ \Q_{(7,6,4,3,2,1)}\, Y_{[1,3,4,4]}\, \nabla_{4}^B(7) =-16 \Q_{(7,4,2)}. $$ \bigskip The following theorem is the main result of the present paper: \proclaim{Theorem 11} \quad Let $k$ be a positive integer such that $k \leq n/2$. Suppose that $I\subseteq \rho(n-1)$ is a strict partition. Let $\alpha = [\alpha_1 \leq \alpha_2\leq \cdots \leq \alpha_{2k}] \in \N^{2k}$ with $\alpha_{2k} \leq n-2k$. Then the image of $\P_I \, Y_{\alpha}$ under $\nabla_k^D(n)$ is $0$ unless all the integers $n-1 -\alpha_1,\ldots,\, n-2k -\alpha_{2k}$ belong to $\{ i_1,\ldots,i_{\ell(I)},0 \}$. In this case, the image is $(-1)^s \P_J$, where $J$ is the strict partition with parts $$ \{ i_1,\ldots,i_{\ell(I)} \} \smallsetminus \{ n-1-\alpha_1,\ldots, n-2k -\alpha_{2k} \} \, . $$ Moreover, let $s'$ be the sum of positions of the parts erased in $I$, and $s'':=\ell(I)+1$. Then $s=s'$ if $\alpha_{2k} < n-2k$, and $s=s' + s''$ if $\alpha_{2k} = n-2k$. \footnote{It is convenient to treat here $0=n-2k-\alpha_{2k}$ as an ``extra part" of $I$, and take $s$ to be the sum of positions of all the parts erased in $I$, including the extra part.} \endproclaim \bigskip \noindent {\bf Example 12.} \ (i) For $n=7$ and $k=1$, we have $$ \P_{(5,4,3,2,1,0)}\, Y_{[1,3]} \nabla_1^D(7)= \P_{(5,4,3,2,1,0)}\, Y_{[1,3]}\, \dd \d_2 \d_3 \d_4 \d_5 \d_6 \d_1 \d_2 \d_3 \d_4 \d_5 = - \P_{(4,3,1)} $$ and $$ \P_{(6,4,3,2,1,0)}\, Y_{[2,5]} \, \nabla_1^D(7)= \P_{(6,3,2,1)}. $$ \smallskip \noindent (ii) For $n=7$ and $k=2$, we have $$ \P_{(6,5,4,3,2,1,0)} \, Y_{[1,1,1,2]}\, \nabla_2^D(7) = - \P_{(6,2)} $$ and $$ \P_{(6,5,4,3,2,1,0)} \, Y_{[1,1,1,3]}\, \nabla_2^D(7) = \P_{(6,2,1)}\,. $$ \bigskip \demo{Proof of the theorem} \ To compute the action of $\nabla_k^D(n)$, one uses its decomposition into a sum of two operators, given in Eq.(3.5). The image of \ $\Q_I(\X)\, Y_\alpha$ \ under the first operator $$ \Omega_1 := x_1\cdots x_{2k} \, \nabla_{2k}^B(n) $$ is given by Fact 4 combined with Eq.(2.3) if \ $\alpha_{2k}0)\subseteq \rho(n-1)$ be a strict partition. If $n$ and $\ell$ are of the same parity, we set $h:=n-\ell$, and $\{ j_1< \cdots < j_h \}: = \{ 1,\ldots,n \} \smallsetminus \{ i_1+1,\ldots, i_{\ell}+1 \}$. If $n$ and $\ell$ are of different parity, we set $h:=n-\ell-1$, and $\{ j_1< \cdots < j_h \}: = \{ 1,\ldots,n \} \smallsetminus \{ i_1+1,\ldots, i_{\ell}+1,1 \}$. \smallskip Then for $\alpha(I): = [n-j_1,\ldots,n-j_h,0,\ldots,0]$ and $k:=h/2$, $$ x^{\alpha(I)} \, \P_{\rho(n-1)}\,\partial_{[2k,\ldots,1]} \nabla_k^D(n) = (-1)^s\, \P_I\,, \eqno(4.1) $$ where $s$ is the number of positions of the parts erased in $\rho(n-1)$ in order to get the partition $I$. \endproclaim The assertion of the lemma is a direct consequence of Theorem 11 and the definition of a Schur $S$-polynomial via the Jacobi symmetrizer. \medskip Now, with every strict partition $I=(i_1,\ldots,i_{\ell}>0)$\,, we associate the following element $v(I)\in \frak D_n$\,. If $n-\ell$ is even, we set $$ v(I):=[i_1+1, i_2+1, \ldots, i_{\ell}+1, \overline{j_1}, \ldots, \overline{j_h}]\,, \eqno(4.2) $$ and if $n-\ell$ is odd, $$ v(I):=[i_1+1, i_2+1, \ldots, i_{\ell}+1,1, \overline{j_1}, \ldots ,\overline{j_h}]\,. \eqno(4.3) $$ (The notation is the same as in Lemma 14.) \proclaim{Theorem 15} \ For a strict partition $I\subseteq \rho(n-1)$, $$ x^{\rho} \P_{\rho(n-1)} \, \partial_{v(I)} = (-1)^{|I|+{n\choose 2}} \P_I. \eqno(4.4) $$ \endproclaim The proof of this result is analogous to the proof of [LP1, Thm. A.6]. Using the notation of Lemma 14, we have $$ \d_{v(I)} = \d_{\sigma}\, \d_{[k,k-1,\ldots,1]}\, \nabla_k^D(n)\,, \eqno(4.5) $$ where $$ \sigma= \left\{ \matrix [j_1,\ldots,j_h,i_1+1,\ldots,i_{\ell}+1], & n & \hbox{and} & \ell & \hbox{of the same parity} \cr [j_1,\ldots,j_h,i_1+1,\ldots,i_{\ell}+1,1], & n & \hbox{and} & \ell & \hbox{of different parity}\,. \cr \endmatrix \right. $$ Note that $x^{\rho}\, \d_{\sigma}= x^{\alpha(I)}$, and hence the assertion follows by Lemma 14. \smallskip This result leads to the following characterization of $\P$-polynomials via orthogonal divided differences: \proclaim{Corollary 16} For a strict partition $I\subseteq \rho(n-1)$, we set $w(I):= v(I)^{-1} w_0^D$. More explicitly, for even $\ell$, $w(I)=[\overline{i_1+1},\ldots,\overline{i_{\ell}+1}, j_1,\ldots,j_h]^{-1}$, and for odd $\ell$, $w(I)=[\overline{i_1+1},\ldots,\overline{i_{\ell}+1},\overline 1,j_1, \ldots,j_h]^{-1}$. Then $w=w(I)$ is the unique element of $\frak D_n$ such that $\ell(w)=|I|$ and $\P_I \, \partial_w \ne 0$. In fact, \ $\P_I \, \partial_{w(I)}= (-1)^{|I|}$. \endproclaim This can be also seen by geometric considerations (see [P1] and [LP1]), with the help of the characteristic map ([B], [D1,D2]). \medskip More generally, consider, for any $w\in \frak D_n$, the {\it orthogonal Schubert polynomial} $$ X_w^D=X_w^D(n)= x^{\rho} \P_{\rho(n-1)} \partial_{w_0^D w} \eqno(4.6) $$ of degree $\ell(w)$. Arguing in the same way as in [LP1, pp.33--36], one shows that these Schubert polynomials have the {\it stability property} in the sense that for $w\in \frak D_n \subset \frak D_{n+1}$, $$ X_w^D(n+1)|_{x_{n+1}=0} = X_w^D(n). \eqno(4.7) $$ Together with the ``maximal Grassmannian property" from Theorem 15, asserting that, for even $\ell$, $$ X_{[\overline{i_1+1},\ldots,\overline{i_{\ell}+1},j_1,\ldots,j_h]}^D= (-1)^{|I|+{{n}\choose 2}} \P_I\,, \eqno(4.8) $$ and for odd $\ell$, $$ X_{[\overline{i_1+1},\ldots,\overline{i_{\ell}+1},\overline 1, j_1,\ldots,j_h]}^D= (-1)^{|I|+{{n}\choose 2}} \P_I\,, \eqno(4.9) $$ this shows that they provide a natural tool for the cohomological study of Schubert varieties for the orthogonal group $SO(2n)$ and the related degeneracy loci. \medskip We also record \proclaim{Proposition 17} \ For a strict partition $I=(i_1,i_2,i_3,i_4, \ldots) \subseteq \rho(n-1)$, $$ \P_I \, \dd \d_2 \cdots \d_{i_1} \d_1 \d_2 \cdots \d_{i_2} = (-1)^{i_1+i_2} \P_{(i_3, i_4, \ldots)}. \eqno(4.10) $$ \endproclaim To see this, we argue in a manner similar to the proof of [LP1, Prop. 5.12]. For $J=(i_3,i_4,\ldots)$, we choose the presentation from Lemma 14: $$ \pm \P_I = x^{\alpha(I)}\, \P_{\rho(n-1)} \d_u \ \ , \ \ \ \ \pm \P_J = x^{\alpha(J)}\, \P_{\rho(n-1)} \d_v $$ for appropriate $u, v \in \frak D_n$. Let $\sigma \in \frak S_n$ be the permutation such that $$ v=\sigma\, u\, \ss s_2 \cdots s_{i_1} s_1 s_2 \cdots s_{i_2}\,. $$ The assertion now follows from \ $x^{\alpha(I)} \d_{\sigma}= x^{\alpha(J)}$. \bigskip\medskip \head {\bf Appendix: results in type B} \endhead In this appendix we give a summary of the results for type $B_n$. They are obtained directly from the results for type $C_n$ in [LP1], by changing $\d_0$ in loc.cit. to $-2 \d_0$, and read as follows: writing $\nabla:= \nabla_n^B(n)$, we have \proclaim {Theorem 18} \ (i) \ For $\alpha\in \N^n,\, \alpha \subseteq \rho$, $$ \ Y_{\alpha}\, \P_{\rho(n)} \, \nabla = (-1)^{|\alpha|+{{n+1}\choose 2}} Y_{\alpha}^{\omega}. \eqno(5.1) $$ \medskip \noindent (ii) \ For strict $I\varsubsetneq \rho(n)$, and $\alpha \subseteq \rho$, $$ \ Y_{\alpha} \, \P_I \, \nabla = 0 \ . \eqno(5.2) $$ \endproclaim \bigskip Denote by $$ \langle \ ,\ \rangle : \Sym(n) \times \Sym(n) \to Sym^D(n) $$ the scalar product defined for $f, g\in Sym(n)$ by $$ \langle f,g \rangle := f g \, \nabla\,. \eqno(5.3) $$ For strict partitions $I,J\subseteq \rho(n)$, one has $$ \langle \P_I,\P_{\rho(n) \smallsetminus J} \rangle = (-1)^{{{n+1}\choose 2}}\, \delta_{I J}, \eqno(5.4) $$ where $\rho(n) \smallsetminus J$ is the strict partition whose parts complement the parts of $J$ in $\{n,n-1,\ldots, 1\}$ \ (cf. [PR]). Consequently, the polynomial ring $\Pol= \Bbb Z [{1 \over 2}][x_1,\ldots, x_n]$ is a free $\Sym^B(n)$-module with basis $Y_{\alpha}\, \P_I$, where $\alpha$ ranges over subsequences contained in $\rho$ and $I$ runs over all strict partitions contained in $\rho(n)$. Note that the element of the maximal degree of this basis is $x^{\rho} \P_{\rho(n)}$. Let $$ [ \ , \ ]: \Pol \times \Pol \to \Sym^B(n) $$ be a scalar product, defined for $f,g \in \Pol$ by $$ [f,g]:= f g \,\d_{w_0^B}. \eqno(5.5) $$ One has, for $\alpha, \beta \subseteq \rho$ and strict partitions $I,J\subset \rho(n)$, $$ \bigl[Y_{\alpha}^{\omega}\, \P_I , Y_{\beta'}\, \P_{\rho(n)\smallsetminus J} \bigr] = (-1)^{|\alpha| + {{n+1}\choose 2}}\, \delta_{\alpha \beta}\, \delta_{I J}\,. \eqno(5.6) $$ (See (2.5).) \medskip Let $\Y=\{ y_1,\ldots,y_n\}$ be a second set of indeterminates of cardinality $n$. The symbol $\equiv$ will mean: \ ``congruent modulo the ideal generated by the relations \hfill\break $ f(x_1^2,\ldots, x_n^2) = f(y_1^2,\ldots, y_n^2)$, where $f\in Sym(n)$." \medskip Following Fulton [F2,F3], define $$ \aligned F(\X,\Y) &:=\ |\P_{n+1+j-2i}(\X)+\P_{n+1+j-2i}(\Y)|_{1\le i,j\le n} \\ & \\ &=\ {\left|\ \CD \P_{n}(\X)+\P_{n}(\Y) & 0 &.\;.\;. & \\ \P_{n-2}(\X)+\P_{n-2}(\Y) &\ \P_{n-1}(\X)+\P_{n-1}(\Y) &\ .\;.\;.\ & \\ \vbox{\offinterlineskip\hbox{.}\vskip3pt\hbox{.}\vskip3pt\hbox{.}} &\vbox{\offinterlineskip\hbox{.}\vskip3pt\hbox{.}\vskip3pt\hbox{.}} &\vbox{\offinterlineskip\hbox{.}\vskip3pt\hbox{\ \ .}\vskip3pt \hbox{\ \ \ \ .}} & \\ && 1 && \P_1(\X)+\P_1(\Y) \\ \endCD\ \right|}\,. \\ \endaligned \eqno(5.7) $$ Following [PR], define $$ \P(\X, \Y):= \sum \P_I(\X)\, \P_{\rho(n) \smallsetminus I}(\Y) \ , \eqno(5.8) $$ where the summation is over all strict partitions $I\subseteq \rho(n)$. The reasoning in [LP1, Sect.2] made for case $C_n$ adapts to case $B_n$ and furnishes: \proclaim {Proposition 19} \ We have \noindent (i) \ $$F(\X,\Y) \equiv \P(\X,\Y). \eqno(5.9)$$ \noindent (ii) \ For every $w\in \frak B_n\smallsetminus \S_n$, $$ \P(\X^w, \X)=0\,, \eqno(5.10) $$ and for every $w\in \S_n$, $$ \P(\X^w, \X)= \P(\X, \X)= s_{\rho(n)}(\X)\,. \eqno(5.11) $$ \noindent (iii) \ For every $f\in \Sym(n)$, $$ \langle f(\X)\,, F(\X,\Y) \rangle \equiv (-1)^{{{n+1} \choose 2}}\, f(\Y) \ . \eqno(5.12) $$ \noindent (iv) \ For every $f\in \Pol$, $$ \bigl[ f(\X)\,, \prod_{n \ge i>j \ge 1}(x_i-y_j)\, F(\X,\Y) \,\bigr] \equiv f(\Y)\,. \eqno(5.13) $$ \endproclaim In other words, $F(\X,\Y)$ is a reproducing kernel for the scalar product $\langle \ , \ \rangle$, and $\prod_{i>j}(x_i-y_j)\, F(\X,\Y)$ is a reproducing kernel for $[ \ , \ ]$. One can show that the ``vanishing property" {\it (ii)} characterizes $\P(\X,\Y)$ up to $\equiv$. The congruence {\it (i)} can be also derived from geometry by comparing the classes of diagonals in flag bundles associated with $SO(2n+1)$ given in [F2,F3] and [PR] (see also [G]). \medskip \proclaim{Proposition 20} Suppose $n\ge p>0$. Let $IpJ\subseteq \rho(n)$ be a strict partition and $H\subseteq \rho(n)$ a strict partition not containing $p$. Then $$ x_1^{n-p}\P_{IpJ} \, \d_0 \d_1 \cdots \d_{n-1} =(-1)^{\ell(I)+n} \P_{IJ} \eqno(5.14) $$ and $$ x_1^{n-p}\P_H \, \d_0 \d_1 \cdots \d_{n-1} = 0 \,. \eqno(5.15) $$ \endproclaim More generally, \proclaim{Theorem 21} \quad Let $00)$ the following element of $\frak B_n$: $$ v(I):=[i_1,\ldots,i_{\ell},\overline{j_1}, \ldots , \overline{j_h}]\,, \eqno(5.17) $$ where $j_1 < \cdots < j_h$\,. \smallskip \proclaim{Theorem 23} \ For every strict partition $I\subseteq \rho(n)$, $$ x^{\rho} \P_{\rho(n)} \, \partial_{v(I)} = (-1)^{|I|+{{n+1}\choose 2}} \P_I. \eqno(5.18) $$ \endproclaim This leads to the following characterization of $\P$-polynomials via divided differences: \proclaim{Corollary 24} \ For any strict partition $I$, let $w(I):= v(I)^{-1} w_0^B$, that is $w(I)= [\overline{i_1},\ldots,\overline{i_{\ell}},j_1,\ldots,j_h]^{-1}$\,. Then $w=w(I)$ is the unique element of $\frak B_n$ such that $\ell(w)=|I|$ and $\P_I \, \partial_w \ne 0$. In fact, $\P_I \, \partial_{w(I)}=(-1)^{|I|}$. \endproclaim This can be also seen by geometric considerations (see [P1] and [LP1]), with the help of the characteristic map ([B], [D1,D2]). \smallskip More generally, consider, for any $w\in \frak B_n$, the {\it orthogonal Schubert polynomial} $$ X_w^B= X_w^B(n) = x^{\rho} \P_{\rho(n)} \partial_{w_0^B w} \eqno(5.19) $$ of degree $\ell(w)$. Arguing in the same way as in [LP1, pp.33--36], one shows that these Schubert polynomials have the {\it stability property} in the sense that for $w\in \frak B_n \subset \frak B_{n+1}$, $$ X_w^B(n+1)|_{x_{n+1}=0} = X_w^B(n). \eqno(5.20) $$ Together with the ``maximal Grassmannian property" from Theorem 23, asserting that $$ X_{[\overline{i_1},\ldots,\overline{i_{\ell}},j_1,\ldots,j_h]}^B= (-1)^{|I|+{{n+1}\choose 2}} \P_I\,, \eqno(5.21) $$ this shows that they provide a natural tool for the cohomological study of Schubert varieties for the orthogonal group $SO(2n+1)$ and the related degeneracy loci. \bigskip We now give \smallskip \noindent {\bf Proof of Theorem 9} \smallskip Given a symmetric function $f$, let $D_f$ be the Foulkes derivative i.e. the adjoint operator to the multiplication by $f$ w.r.t. the standard scalar product on the ring $\Sym$ of symmetric functions in a countable number of variables (cf. [M1]). We use the following vertex operators on $\Sym$: $$ U^s := 1 - D_{P_1} s_1 + D_{P_2} s_2 - \cdots \,, \eqno(5.22) $$ $$ U^e := 1 - D_{P_1} e_1 + D_{P_2} e_2 - \cdots \,, \eqno(5.23) $$ and $$ V^e:=1 - D_{e_1} P_1+ D_{e_2} P_2 - \cdots \,. \eqno(5.24) $$ We refer to [LP1, p.24] for the definitions of Schur $P$-functions $P_I$ [S]. In loc. cit. the reader can also find a definition of $Q'$-functions $Q'_I$ [LLT2], used in the following proposition: \proclaim {\bf Proposition 25} \ Let $I$ be a strict partition. We have the following identities of symmetric functions in $\Sym$: $$ \Q_I \, U^s = \left\{ \matrix \Q_I, & \ell(I) & \hbox{even} \cr 0, &\ell(I) &\hbox{odd}\,; \cr \endmatrix \right. \eqno(5.25) $$ $$ Q_I' \, U^e = \left\{ \matrix Q_I', & \ell(I) & \hbox{even} \cr 0, &\ell(I) &\hbox{odd}\,; \cr \endmatrix \right. \eqno(5.26) $$ and $$ P_I \, V^e = \left\{ \matrix P_I, & \ell(I) & \hbox{even} \cr 0, &\ell(I) &\hbox{odd}\,. \cr \endmatrix \right. \eqno(5.27) $$ \endproclaim \demo{Proof} First of all, arguing as in [LP1, pp.24--27], with the help of the operators $V^e$, $U^s$, and $U^e$ instead of $V^e_k$, $U^s_k$, and $U^e_k$ in loc. cit., we note that the equalities (5.25), (5.26), and (5.27) are equivalent. \smallskip We show (5.27). It suffices to prove the statement when the set of indeterminates $\{x_1,\ldots, x_n\}$ is of finite cardinal $n>|I|$\,. Besides the well-known equality: for $k>0$, $$ x_1^k \prod_{2\le i \le n} (x_1+x_i)\, \d_1 \d_2 \cdots \d_{n-1} = P_k(x_1,\ldots,x_n)\,, \eqno(5.28) $$ we also need the following formula from [P2]: \proclaim {Fact 26} \ For a strict partition $I$, $$\aligned P_I(x_2,\ldots,x_n)& \, \prod_{2\le i \le n}(x_1 + x_i)\, \d_1 \d_2 \cdots \d_{n-1} \cr &= \left\{ \matrix (-1)^{n-1} P_I(x_1,\ldots,x_n), & n-\ell(I) & \hbox{odd} \cr 0, &n-\ell(I) &\hbox{even}. \cr \endmatrix \right. \endaligned \eqno(5.29) $$ \endproclaim \noindent (More precisely, Eq.(5.29) is a special case of the following formula given in [P2, Prop.1.3(ii)]: \ Let $q$, $r$, $k$, and $h$ be integers such that $0