% % A Pieri-type formula for isotropic flag manifolds % % Frank Sottile % Nantel Bergeron % % % \documentclass[12pt]{amsart} \usepackage{amssymb} \usepackage{epsf} \headheight=8pt \topmargin=0pt \textheight=624pt \textwidth=432pt \oddsidemargin=18pt \evensidemargin=18pt \newcommand{\DOT}{\setlength{\unitlength}{1pt}\begin{picture}(2.5,2)(1,1) \put(1,2){\circle*{2}}\end{picture}} \newcommand{\Edot}{{E_{\,\DOT}}} \newcommand{\Epdot}{{E_{\,\DOT}'}} \newcommand{\Etdot}{{\wt{E}_{\DOT}}} \newcommand{\Etpdot}{{\wt{E}_{\DOT}'}} \newcommand{\Fdot}{{F\!_{\DOT}}} \newcommand{\Fpdot}{{F\!_{\DOT}\,'}} \newcommand{\Gdot}{{G_{\DOT}}} \newcommand{\Gpdot}{{G_{\DOT}'}} \newcommand{\ol}[1]{{\overline{#1}}} \newcommand{\wh}[1]{{\widehat{#1}}} \newcommand{\wt}[1]{{\widetilde{#1}}} \newcommand{\precdot}{{\prec\!\!\!\cdot\,}} \newcommand{\Len}{{\mathcal L}} \newcommand{\Span}[1]{{\langle #1 \rangle}} \newcommand{\Perm}[1]{{\langle #1 ]}} \newcommand{\supp}{{\mbox{\rm supp}}} \newcommand{\Sln}{{\mbox{\it SL}_n}} \newcommand{\Spn}{{\mbox{\it Sp}_{2n}{\mathbb C}}} \newcommand{\Son}{{\mbox{\it So}_{2n+1}{\mathbb C}}} \newcommand{\tridot}% {{\,\vartriangleleft\hspace{-8.5pt}\cdot\hspace{3.5pt}}} \newcommand{\tridoto}% {{\,\vartriangleleft\hspace{-8.5pt}\cdot\hspace{2pt}_0\,\,}} \newcommand{\tridotk}% {{\,\vartriangleleft\hspace{-8.5pt}\cdot\hspace{2pt}_k\,\,}} \font\parenfont=cmr7 \font\subfont=cmr5 \newcommand{\pceq}{{\,\preceq\hspace{-5.2pt}\raisebox{2.4pt}{\parenfont )}\,}} \newcommand{\pcdot}% {{\,\prec\hspace{-6.5pt}\cdot\hspace{-2.pt}\raisebox{1.26pt}{\parenfont )}\,}} \newcommand{\prcc}{{\,\prec\hspace{-5.2pt}\raisebox{1.26pt}{\parenfont )}\,}} \newcommand{\prcs}{{\prec\hspace{-1.5pt}\raisebox{.6pt}{\subfont )}\,}} \newcommand{\QED}{ \setlength{\unitlength}{1.0pt}% \begin{picture}(10,7.5) \put(5,-2.5){\rule{2.5pt}{2.5pt}} \put(2.5,0){\rule{5pt}{2.5pt}} \put(0,2.5){\rule{10pt}{2.5pt}} \end{picture}\vspace{10pt}} \newtheorem{thm}{Theorem}[section] \newtheorem{prop}[thm]{Proposition} \newtheorem{lem}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} \newtheorem{example}[thm]{Example} \newtheorem{alg}[thm]{Algorithm} \newtheorem{defth}[thm]{Definition-Theorem} \newtheorem{defcor}[thm]{Definition-Corollary} \newtheorem{deflem}[thm]{Definition-Lemma} \newenvironment{ex}{\begin{example}\sl}{\end{example}} \newtheorem{definition}[thm]{Definition} \newenvironment{defn}{\begin{definition}\sl}{\end{definition}} \newtheorem{remark}[thm]{Remark} \newenvironment{rem}{\begin{remark}\rm}{\end{remark}} \begin{document} \title[Pieri-type formula]{A Pieri-type formula for isotropic flag manifolds} \author{Nantel Bergeron \and Frank Sottile} \address{Department of Mathematics and Statistics\\ York University\\ Toronto, Ontario M3J 1P3\\ CANADA} \email[Nantel Bergeron]{bergeron@mathstat.yorku.ca} \urladdr[Nantel Bergeron]{http://www.math.yorku.ca/bergeron} \address{Department of Mathematics\\ University of Wisconsin\\ Van Vleck Hall\\ 480 Lincoln Drive\\ Madison, Wisconsin 53706-1388\\ USA} \email[Frank Sottile]{sottile@math.wisc.edu} \urladdr[Frank Sottile]{http://www.math.wisc.edu/\~{}sottile} \date{2 July 1998} \thanks{First author supported in part by NSERC and CRM grants} \thanks{Second author supported in part by NSERC grant OGP0170279} \thanks{\ research at MSRI supported by NSF grant DMS-9701755} \subjclass{14M15, 05E15, 05E05, 06A07, 14N10} \keywords{Pieri formula, isotropic flag manifold, Bruhat order, Schubert variety, Lagrangian Grassmannian, Schubert polynomial, Schur $P$-function} \begin{abstract} We give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from a Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a type $B$ (respectively, type $C$) Schubert polynomial by the Schur $P$-polynomial $p_m$ (respectively, the Schur $Q$-polynomial $q_m$). Geometric constructions and intermediate results allow us to ultimately deduce this from formulas for the classical flag manifold. These intermediate results are concerned with the Bruhat order of the Coxeter group ${\mathcal B}_\infty$, identities of the structure constants for the Schubert basis of cohomology, and intersections of Schubert varieties. We show these identities follow from the Pieri-type formula, except some `hidden symmetries' of the structure constants. Our analysis leads to a new partial order on the Coxeter group ${\mathcal B}_\infty$ and formulas for many of these structure constants. \end{abstract} %\mbox{ \ }\vspace{-30pt} \maketitle \mbox{ \ }\vspace{-20pt} \tableofcontents \section*{Introduction} The cohomology of a flag manifold $G/B$ has an integral basis of Schubert classes ${\mathfrak S}_w$ indexed by elements $w$ of the Weyl group of $G$. The algebraic structure of these rings is known~\cite{Borel_cohomology} with respect to a monomial basis, and there are methods (Schubert polynomials) for expressing the ${\mathfrak S}_w$ in terms of this basis~\cite{BGG,BH,De74,FK_Bn,Fu96,LS82a,PrRa97}. Moreover, their multiplicative structure with respect to the Schubert basis is determined by Chevalley's formula~\cite{Chevalley91}. Despite this, it remains an open problem to give a closed or bijective formula for the integral structure constants $c^w_{u\,v}$ defined by the identity $$ {\mathfrak S}_u\cdot{\mathfrak S}_v \ =\ \sum_w c^w_{u\,v}\,{\mathfrak S}_w. $$ These $c^w_{u\,v}$ are non-negative as they count the flags in a suitable triple intersection of Schubert varieties. They are expected to be related to the enumeration of chains in the Bruhat order of the Weyl group (see~\cite{BS98a} and the references therein). Of particular interest are Pieri-type formulas which describe the constants $c^w_{u\,v}$ when ${\mathfrak S}_v$ is a special Schubert class pulled back from a Grassmannian projection ($G/P$, $P$ maximal parabolic), as these determine the ring structure for the cohomology of $G/P$ when $P$ is any parabolic subgroup. When $G$ is $\Sln{\mathbb C}$, a Pieri-type formula for multiplication by a special Schubert class was described~\cite{LS82a} in terms of the Weyl group element $wu^{-1}$. A formula in terms of chains in the Bruhat order was conjectured~\cite{BB} and given a geometric proof~\cite{Sottile96}. Our main results are the analogous formulas when $G$ is $\Spn$ or $\Son$ and ${\mathfrak S}_v$ is a special Schubert class pulled back from a Grassmannian of maximal isotropic subspaces. These are common generalizations of the Pieri-type formulas for $\Sln{\mathbb C}$, Chevalley's formula, and Pieri-type formulas for Grassmannians of maximal isotropic subspaces~\cite{Hiller_Boe}. \smallskip Our proof uses results on the Bruhat order, identities of these structure constants, a decomposition of intersections of Schubert varieties, and formulas in the cohomology of the $\Sln{\mathbb C}$-flag manifold to explicitly determine a triple intersection of Schubert varieties. This shows the coefficients in the Pieri-type formula are the intersection number of a linear space with a collection of quadrics. Some intermediate results, including a fundamental identity and some additional `hidden symmetries' of the structure constants, are deduced from constructions on $\Sln{\mathbb C}$-flag manifolds~\cite{BS98a}. This analysis leads to other results, including a new partial order on the infinite Coxeter group ${\mathcal B}_\infty$ and a monoid for chains in this order as in~\cite{BS_monoid}. We show how the Pieri-type formula implies our fundamental identity, use the identities to express many structure constants in terms of the Littlewood-Richardson coefficients for the multiplication of Schur $P$-- (or $Q$--) functions~\cite{Stembridge_shifted}, and apply the hidden symmetries to the enumeration of chains in the Bruhat order. \section{Statement of results} Schubert classes in the cohomology of the flag manifolds $\Son/B$ and $\Spn/B$ form integral bases indexed by elements of the Weyl group ${\mathcal B}_n$. We represent ${\mathcal B}_n$ as the group of permutations $w$ of $\{-n,\ldots,-2,-1,1,\ldots,n\}$ satisfying $w(-a)=-w(a)$ for $1\leq a\leq n$. Let ${\mathfrak B}_w$ denote the Schubert class indexed by $w\in{\mathcal B}_n$ in $H^*\Son/B$ and ${\mathfrak C}_w$ that in $H^*\Spn/B$. The degree of these classes is $2\cdot\ell(w)$, where the {\em length} $\ell(w)$ of $w$ is $$ \#\{0< iw(j)\}\quad + \quad \sum_{i>0>w(i)}|w(i)|. $$ For an integer $i$, let $\ol{\imath}$ denote $-i$. For each $1\leq m\leq n$, define $v_m\in {\mathcal B}_n$ by $$ \ol{m}=v_m(1)<0a_{i+1}$, where $a_1,a_2,\ldots, a_m$ is the sequence of edge labels in $\gamma$. A {\it descent} is an index $ia_{i+1}$ and an {\it ascent} is an index $i0$. Then \begin{enumerate} \item[I.] {\rm (Odd-orthogonal Pieri-type formula) } $$ {\mathfrak B}_u\cdot p_m\ =\ \sum {\mathfrak B}_{{\rm end}(\gamma)}, $$ the sum over all chains $\gamma$ in the labeled $0$-Bruhat order of ${\mathcal B}_n$ which begin at $u$, have length $m$, and no peaks. \item[II.] {\rm (Symplectic Pieri-type formula)} $$ {\mathfrak C}_u\cdot q_m\ =\ \sum {\mathfrak C}_{{\rm end}(\gamma)}, $$ \begin{enumerate} \item the sum over all chains $\gamma$ in the labeled $0$-Bruhat r\'eseau of ${\mathcal B}_n$ which begin at $u$, have length $m$, and no descents. \item the same sum, except with no ascents. \end{enumerate} \end{enumerate} }\medskip This generalizes Chevalley's formula and the Pieri-type formulas for $\Sln{\mathbb C}/B$, which are expressed in~\cite{Sottile96} as a sum of certain labeled chains in the Bruhat order on the symmetric group ${\mathcal S}_n$ with no ascents/no descents. The duality of these two formulas, one in terms of peaks for an order, and the other in terms of descents/ascents for an enriched structure on that order has connections with other dualities in combinatorics. These include Fomin's duality of graded graphs~\cite{Fomin_dualityI,Fomin_dualityII} and Stembridge's theory of enriched $P$-partitions~\cite{stembridge_enriched}, where peak and descent sets play a complementary role. These relations are explored in~\cite{BMSvW}, which extends the theory developed in~\cite{BS_hopf} to the ordered structures of this manuscript. \begin{ex}\label{ex:pieri} Represent permutations $w\in{\mathcal B}_3$ by their values $w(1)\,w(2)\,w(3)$. Consider the products ${\mathfrak B}_{3\ol{1}2}\cdot p_2$ and ${\mathfrak C}_{3\ol{1}2}\cdot q_2$. Figure~\ref{fig:height2} shows the part of the 0-Bruhat r\'eseau of \begin{figure}[htb] $$\epsfxsize=3.in \epsfbox{fig1.eps}$$ \caption{Chains above $3\ol{1}2$\label{fig:height2}} \end{figure} height 2 above $3\ol{1}2$ in ${\mathcal B}_3$. (Erase edges with negative labels to obtain its analog in the 0-Bruhat order.) Chains of length 2 are peakless, so by Theorem~A I, we have $$ {\mathfrak B}_{3\ol{1}2}\cdot p_2\ =\ 2\,{\mathfrak B}_{\ol{3}\,\ol{2}1}+{\mathfrak B}_{2\ol{3}1}+ {\mathfrak B}_{3\ol{2}\,\ol{1}}+{\mathfrak B}_{\ol{1}\,\ol{3}2}. $$ Every chain in Figure~\ref{fig:height2} with increasing labels may be paired with a chain with decreasing labels having the {\it same} underlying permutations, and this pairing exhausts all chains. Thus, by Theorem~A II, we have $$ {\mathfrak C}_{3\ol{1}2}\cdot q_2\ =\ 2\,{\mathfrak C}_{\ol{3}\,\ol{2}1}+2\,{\mathfrak C}_{2\ol{3}1}+ {\mathfrak C}_{3\ol{2}\,\ol{1}}+{\mathfrak C}_{\ol{1}\,\ol{3}2}. $$ \end{ex}\medskip If $\lambda$ is a {\it strict partition} (decreasing integral sequence $n\geq \lambda_1>\lambda_2>\cdots>\lambda_k>0$), then $\lambda$ determines a unique {\em Grassmannian permutation} $v(\lambda)\in{\mathcal B}_n$ where $v(i)=\ol{\lambda_i}$ for $i\leq k$ and $00>w(i)\}$) in $w$. Then the map ${\mathfrak C}_w\mapsto 2^{s(w)}{\mathfrak B}_w$ embeds $H^*\Spn/B$ into $H^*\Son/B$ and induces an isomorphism of their rational cohomology rings. Thus it suffices to work in $H^*\Spn/B$. This is fortunate, as a key geometric result, Theorem~\ref{thm:geom_facts}(2), holds only for $\Spn/B$. Let $f^w_u$ count the saturated chains in the interval $[u,w]_0$ and $g^w_u$ count the saturated chains in the r\'eseau $[u,w]_0$. Iterating Chevalley's formula~(\ref{eq:Chevalley}) with $u=e$, the identity permutation, we obtain the following expressions. $$ p_1^m\ =\ \sum_{|\lambda|=m} f^{v(\lambda)}_e\, P_\lambda \qquad\mbox{and}\qquad q_1^m\ =\ \sum_{|\lambda|=m} g^{v(\lambda)}_e\, Q_\lambda $$ Multiplying the first expression by ${\mathfrak B}_u$ and collecting the coefficients of ${\mathfrak B}_w$ in the resulting expansion (likewise for the second expression) gives the following proposition. \begin{prop}\label{prop:chains} Let $u,w\in{\mathcal B}_n$. Then $$ f^w_u\ =\ \sum_{|\lambda|=\ell(w)-\ell(u)} f^{v(\lambda)}_e\,b^w_{u\,\lambda} \qquad\mbox{and}\qquad g^w_u\ =\ \sum_{|\lambda|=\ell(w)-\ell(u)} g^{v(\lambda)}_e\,c^w_{u\,\lambda}. $$ \end{prop} \medskip Theorem~A and Proposition~\ref{prop:chains} show a close connection between chains in the 0-Bruhat order/r\'eseau and the structure constants $b^w_{u\,\lambda}$ and $c^w_{u\,\lambda}$. This justifies an elucidation of the basic properties of the $0$-Bruhat order and r\'eseau, which we do in Sections 2 and 6. These structures have a remarkable property and there are related fundamental identities among the structure constants. \medskip \noindent{\bf Theorem B. } {\it Suppose $u<_0w$ and $x<_0z$ in ${\mathcal B}_n$ with $wu^{-1}=zx^{-1}$. Then \begin{enumerate} \item[(1)] The map $v \mapsto vu^{-1}x$ induces an isomorphism of labeled intervals in the 0-Bruhat order and 0-Bruhat r\'eseau $[u,w]_0\stackrel{\sim}{\longrightarrow}[x,z]_0$. \item[(2)] For any strict partition $\lambda$, $$ b^w_{u\,\lambda}\ =\ b^z_{x\,\lambda} \qquad\mbox{and}\qquad c^w_{u\,\lambda}\ =\ c^z_{x\,\lambda}. $$ \end{enumerate} } \medskip We prove Theorem~B(1) in Section 2.1 using combinatorial methods. Theorem~B(2) is a consequence of a geometric result (Theorem~\ref{thm:geometric_shapes}) proven in Section 4. Both parts of Theorem~B are key to our proof of the Pieri-type formula. Interestingly, the Pieri-type formula and Theorem~B(1) together imply Theorem~B(2): For any composition $\alpha=(\alpha_1,\ldots,\alpha_s)$ with each $\alpha_i\geq 0$, let $p_\alpha:= p_{\alpha_1}\cdots p_{\alpha_s}$, $q_\alpha:= q_{\alpha_1}\cdots q_{\alpha_s}$, and $I(\alpha):= \{\alpha_1,\alpha_1+\alpha_2,\ldots,\alpha_1+\cdots+\alpha_{s-1}\}$. The {\it peak set} of a (maximal) chain in a labeled order is the set of indices of peaks in the chain. Given a chain in a labeled r\'eseau, its {\it descent set} (respectively {\it ascent set}) is the set of indices of descents (respectively ascents) in the chain. \begin{cor}\label{cor:peak-set} Let $u,w,x,z\in{\mathcal B}_n$. \begin{enumerate} \item[(1)] Let $\alpha$ be any composition. Then the coefficient of\/ ${\mathfrak B}_w$ in the product ${\mathfrak B}_u\cdot p_\alpha$ is the number of chains in the interval $[u,w]_0$ in the labeled $0$-Bruhat order with peak set contained in $I(\alpha)$. \item[(2)] Let $\alpha$ be any composition. Then the coefficient of\/ ${\mathfrak C}_w$ in the product ${\mathfrak C}_u\cdot q_\alpha$ is the number of chains in the interval $[u,w]_0$ in the labeled $0$-Bruhat r\'eseau with descent set contained in $I(\alpha)$. This is also the number with ascent set contained in $I(\alpha)$. \item[(3)] Suppose the Pieri-type formula (Theorem A) holds. Then the intervals $[u,w]_0$ and $[x,z]_0$ have the some number of chains with peak set $I(\alpha)$ for every composition $\alpha$ if and only if for every strict partition $\lambda$, $b^w_{u\,\lambda}=b^z_{x\,\lambda}$. The same statement holds for ascent/descent sets for chain in the r\'eseaux and the coefficients $c^w_{u\,\lambda},c^z_{x\,\lambda}$. In particular, Theorem~B(1) implies Theorem~B(2). \end{enumerate} Moreover, the numbers in 1 and 2 depend only upon the multiset $\{\alpha_1,\ldots,\alpha_s\}$. \end{cor} Parts 1 and 2 follow from Theorem A. For 3, note that the Schur $P$-polynomials (respectively $Q$-polynomials) are linear combinations of the $p_\alpha$ (respectively the $q_\alpha$)~\cite[III.8.6]{Macdonald95}. This linear combination gives a formula for $b^w_{u\,\lambda}$ (respectively $c^w_{u\,\lambda}$) in terms of chains with given peak sets (respectively, given ascent/descent sets). Let $\zeta\in{\mathcal B}_n$. By Theorem~B(1), we may define $\eta\preceq\zeta$ if there is a $u\in{\mathcal B}_n$ with $u\leq_0\eta u\leq_0\zeta u$ and $\Len(\zeta):= \ell(\zeta u)-\ell(u)$ whenever $u\leq_0 \zeta u$. Then $({\mathcal B}_n,\prec)$ is a graded partial order with rank function $\Len(\cdot)$. By the identity of Theorem~B(2), we may define $b^\zeta_\lambda:=b^{\zeta u}_{u\,\lambda}$ and $c^\zeta_\lambda:=c^{\zeta u}_{u\,\lambda}$ for any $u\in{\mathcal B}_n$ with $u\leq_0\zeta u$ and $|\lambda|=\Len(\zeta)$. These coefficients satisfy one obvious identity, $c^\zeta_\lambda=c^{\zeta^{-1}}_\lambda$, as $c^w_{u\, v}=c^{\omega_0w}_{\omega_0u\, v}$ where $\omega_0\in{\mathcal B}_n$ is the longest element. They also satisfy two others, which we call hidden symmetries. Let $\rho\in{\mathcal B}_n$ be the permutation defined by $\rho(i)=i-1-n$ for $1\leq i\leq n$. Then $\rho$ is the element with largest rank in $({\mathcal B}_n,\prec)$. Let $\gamma\in{\mathcal B}_n$ be defined by $\gamma(1)=2,\gamma(2)=3,\ldots,\gamma(n)=1$, so that $\gamma=(\ol{1},\,\ol{2},\,\ldots,\ol{n})(1,\,2,\,\ldots,\,n)$. \medskip \noindent{\bf Theorem C. } {\it For any $\zeta\in{\mathcal B}_n$, \begin{enumerate} \item[(1)] $\Len(\zeta)=\Len(\rho \zeta \rho)$ and for any strict partition $\lambda$, we have $b^\zeta_\lambda \ =\ b^{\rho\zeta\rho}_\lambda$ and $c^\zeta_\lambda \ =\ c^{\rho\zeta\rho}_\lambda$. \item[(2)] If $a\cdot \zeta(a)>0$ for all $a$, then $\Len(\zeta)=\Len(\gamma\zeta\gamma^{-1})$ and for any strict partition $\lambda$, we have $b^\zeta_\lambda \ =\ b^{\gamma\zeta\gamma^{-1}}_\lambda$ and $c^\zeta_\lambda \ =\ c^{\gamma\zeta\gamma^{-1}}_\lambda$. \end{enumerate} }\medskip We prove a strengthening of Theorem~B, relaxing the condition of equality of $wu^{-1}$ and $zx^{-1}$ to that of {\it shape equivalence}. Permutations $\eta,\zeta\in{\mathcal B}_n$ are shape equivalent if there exist sets $I: 00$ for all $a$, and one of $\zeta,\zeta^\gamma,\zeta^{\gamma^2},\ldots,\zeta^{\gamma^{n-1}}$ is shape equivalent to $v(\kappa)v(\mu)^{-1}$. \end{enumerate} \end{defn} \begin{cor} If $u\leq_0 w$ are permutations in ${\mathcal B}_n$ and $wu^{-1}$ has a skew shape $\kappa/\mu$, then for any strict partition $\lambda$ we have $$ b^w_{u\,\lambda}\ =\ b^\kappa_{\mu\,\lambda} \qquad\mbox{and}\qquad c^w_{u\,\lambda}\ =\ c^\kappa_{\mu\,\lambda}. $$ \end{cor} We call the partial order $\prec$ the {\em Lagrangian order} and transfer the labeling from the $0$-Bruhat order to obtain the labeled Lagrangian order. In the same fashion, we may transfer the labeling and multiple edges of the 0-Bruhat r\'eseau to $({\mathcal B}_n,\prec)$, obtaining the (labeled) {\em Lagrangian r\'eseau}. By Corollary~\ref{cor:peak-set}(3), Theorem~C has a purely enumerative corollary. \begin{cor}\label{cor:hidden-peaks} For any $\zeta\in{\mathcal B}_n$, \begin{enumerate} \item[(1)] For any subset $S$ of $\{2,\ldots, \Len(\zeta)-1\}$, the intervals $[e,\zeta]_\prec$ and $[e,\rho\zeta\rho]_\prec$ in the Lagrangian order have the same number of chains with peak set $S$. \item[(2)] For any subset $S$ of $\{1,\ldots, \Len(\zeta)-1\}$, the intervals $[e,\zeta]_\prec$ and $[e,\rho\zeta\rho]_\prec$ in the Lagrangian r\'eseau order have the same number of chains with descent set $S$ and the same number of chains with ascent set $S$, and these two numbers are equal. \item[(3)] If $a\cdot \zeta(a)>0$ for all $a$, then the same is true for $[e,\zeta]_\prec$ and $[e,\gamma\zeta\gamma^{-1}]_\prec$. \end{enumerate} \end{cor} In general, $[e,\zeta]_\prec\not\simeq [e,\rho\zeta\rho]_\prec$, (See Figure~\ref{fig:cyclic} in Example~\ref{ex:cyclic}) and if $a\cdot \zeta(a)>0$ for all $a$, then in general, $[e,\zeta]_\prec\not\simeq [e,\gamma\zeta\gamma^{-1}]_\prec$. (See Figure~\ref{fig:hidden} in Example~\ref{ex:no-iso}.) Let $\supp(\zeta):= \{a>0\mid \zeta(a)\neq a\}$, the {\it support} of $\zeta$. A permutation $\zeta\in {\mathcal B}_n$ is {\em reducible} if it has a non-trivial factorization $\zeta=\eta\cdot\xi$ with $\Len(\zeta)=\Len(\eta)+\Len(\xi)$ where $\eta$ and $\xi$ have disjoint supports ($\eta\cdot\xi=\xi\cdot\eta$). If $\eta\cdot\xi=\xi\cdot\eta$ with $\Len(\xi\cdot\eta)=\Len(\eta)+\Len(\xi)$, then $\eta\cdot\xi$ is a {\em disjoint product}. Permutations $w\in {\mathcal B}_n$ have unique factorizations into irreducibles. We represented ${\mathcal B}_n\subset{\mathcal S}_{\pm [n]}$, the group of permutations of $\{-n,\ldots,-1,1,\ldots,n\}$. We also have ${\mathcal S}_n\hookrightarrow {\mathcal B}_n$, and the image consists of those $\zeta$ with $a\cdot\zeta(a)>0$ for every $a$: For $\eta\in {\mathcal S}_n$, let $\overline{\eta}\in{\mathcal S}_{-[n]}$ be the permutation such that $\overline{\eta}(-i)=-\eta(i)$. Then $\eta \ol{\eta}\in {\mathcal B}_n$. For $\zeta\in{\mathcal B}_n$, define $\delta(\zeta)=1$ if $\zeta$ is in the image of ${\mathcal S}_n$, and $\delta(\zeta)=0$ otherwise. In Section~\ref{sec:minimal} we establish the following result. \begin{lem} Let $\zeta\in {\mathcal B}_n$ and suppose $\zeta$ is irreducible. Then $\Len(\zeta)\geq\# \supp(\zeta)-\delta(\zeta)$. If $\Len(\zeta)=\# \supp(\zeta)-\delta(\zeta)$, then either \begin{enumerate} \item[(1)] we have $\delta(\zeta)=1$ and there exists a cycle $\eta\in {\mathcal S}_n$ with $\zeta=\eta \ol{\eta}$, or \item[(2)] we have $\delta(\zeta)=0$ and $\zeta$ is a single cycle in ${\mathcal S}_{\pm[n]}$. \end{enumerate} \end{lem} \begin{defn} If every irreducible factor $\eta$ of $\zeta$ satisfies $\Len(\eta)=\# \supp(\eta)-\delta(\eta)$, then we say that $\zeta$ is {\it minimal}. If $\zeta\in{\mathcal B}_n$ is minimal, then set \begin{eqnarray*} \theta(\zeta)&:=& 2^{\#\{\mbox{\scriptsize irreducible factors of $\zeta$}\} -1},\\ \chi(\zeta)&:=& 2^{\#\{\mbox{\scriptsize irreducible factors $\eta$ of $\zeta$ with $\delta(\eta)=1$}\}}. \end{eqnarray*} If $\zeta$ is not minimal, then set $\theta(\zeta)=\chi(\zeta)=0$. \end{defn} We state the Pieri-type formula in terms of the permutation $wu^{-1}$. \medskip \noindent{\bf Theorem D. } {\em Let $u,w\in {\mathcal B}_n$ and $m\leq n$. Then $$ {\mathfrak B}_u\cdot p_m\ =\ \sum \theta(wu^{-1}){\mathfrak B}_w \qquad\mbox{and}\qquad {\mathfrak C}_u\cdot q_m\ =\ \sum \chi(wu^{-1}){\mathfrak C}_w, $$ the sum over all $w\in {\mathcal B}_n$ with $u\leq_0 w$ and $\ell(w)-\ell(u)=m$. }\medskip This is similar to the form of the Pieri-type formula for $\Sln{\mathbb C}/B$ in~\cite{LS82a}, which is in terms of the cycle structure of the permutation $wu^{-1}$. This also generalizes the form of the the Pieri formula for Grassmannians of maximal isotropic subspaces~\cite{Hiller_Boe}. Our proof for $Sp_{2n}{\mathbb C}/B$ shows these multiplicities to arise from the intersection of a linear subspace of ${\mathbb P}^{2n-1}$ with a collection of quadrics, one for each irreducible factor $\eta$ of $wu^{-1}$ with $\delta(\eta)=1$, similar to the proof of the Pieri-type formula for maximal isotropic Grassmannians in~\cite{sottile_maximal}. We relate the two formulations (Theorems A and D) of the Pieri-type formula in sections~\ref{sec:lagr-or} and~\ref{sec:lagr-re}. \begin{ex} We return to Example~\ref{ex:pieri}. For $u=3\ol{1}2$ and $w$ equal to each of\/ $\ol{3}\,\ol{2}1$, $2\ol{3}1$, $3\ol{2}\,\ol{1}$, and $\ol{1}\,\ol{3}2$ in turn, $wu^{-1}$ is the permutation in ${\mathcal S}_{\pm[3]}$: $$ (12)(\ol{1}\,\ol{2})\,(3\ol{3}),\quad (132)(\ol{1}\,\ol{3}\,\ol{2}),\quad (12\ol{1}\,\ol{2}),\quad \mbox{and}\quad (13\ol{1}\,\ol{3}). $$ As permutations in ${\mathcal B}_3$, the first has 2 irreducible factors, for these we have $\delta((12)(\ol{1}\,\ol{2}))=1$ and $\delta((3\ol{3}))=0$. The other permutations are irreducible with $\delta((132)(\ol{1}\,\ol{3}\,\ol{2}))=1$ and $\delta((12\ol{1}\,\ol{2}))=\delta((13\ol{1}\,\ol{3}))=0$. Thus the values of $\theta$ are $2,1,1,1$ and of $\chi$ are $2,2,1,1$, which shows the two forms of the Pieri-type formula agree on this example. We remark that the last two permutations, $(12\ol{1}\,\ol{2})$ and $(13\ol{1}\,\ol{3})$, are shape equivalent. \end{ex} While we use the cohomology rings of complex varieties, our results and methods are valid for the Chow rings~\cite{Fulton_intersection} and $l$-adic ({\'e}tale) cohomology~\cite{Deligne_SGA4.5} of these same varieties over any field not of characteristic 2. This paper is organized as follows: Section 2 contains basic combinatorial definitions and properties of the Bruhat order on ${\mathcal B}_\infty$ analogous to those of the symmetric group established in~\cite{BS98a,BS_monoid}. Section 3 contains the basic geometric definitions. In Section 4, we use geometry to establish the main identity, Theorem~B(2). In Section 5, we establish additional geometric results and prove Theorem~C. In Section 6, we establish further combinatorial properties of the Lagrangian order and r\'eseau needed for the proof of the Pieri-type formula, which is given in Section 7. \section{Orders on ${\mathcal B}_\infty$}\label{sec:orders} We derive the basic properties of the 0-Bruhat order on ${\mathcal B}_\infty$ analogous to properties of the $k$-Bruhat order on ${\mathcal S}_\infty$. Further properties are developed in Section 6. Let $\#S$ be the cardinality of a finite set $S$. For an integer $j$, its absolute value is $|j|$ and let $\ol{\jmath}:=-j$. Likewise, for a set $P$ of integers, define $\ol{P}:=\{\ol{\jmath}\mid j\in P\}$ and $\pm P:= P\cup \ol{P}$. Set $[n]:=\{1,\ldots,n\}$ and let ${\mathcal S}_{\pm[n]}$ be the group of permutations of $\pm[n]$. Let $e$ be the identity permutation in ${\mathcal S}_{\pm[n]}$ and $\omega_0$ the longest element in ${\mathcal S}_{\pm[n]}$: $\omega_0(i)=\ol{\imath}$. Then ${\mathcal B}_n$ is the subgroup of ${\mathcal S}_{\pm[n]}$ for which $\omega_0 w \omega_0 = w$ and $\omega_0\in {\mathcal B}_n$. We also have ${\mathcal B}_n\subset{\mathcal S}_{[\ol{n},n]}$, the symmetric group on $[\ol{n},n]:=\pm[n]\cup\{0\}$. We refer to elements of these groups as permutations. Permutations $w\in{\mathcal B}_n$ are often represented by their values $w(1)\,w(2)\,\ldots\,w(n)$. For example, $2\,4\,\ol{3}\,\ol{1}\in{\mathcal B}_4$. The {\em length} $\ell(w)$ of $w\in {\mathcal B}_n$ is $$ \ell(w)\quad =\quad \#\{0w(j)\}\ +\ \sum_{i>0> w(i)}|w(i)|. $$ Thus $\ell(2\,4\,\ol{3}\,\ol{1})=4+4=8$. Note that $\omega_0$ is the longest element in ${\mathcal B}_n$. An important class of permutations are the {\em Grassmannian permutations}, those $v\in {\mathcal B}_n$ for which $v(1)\ol{v(2)}>\cdots>\ol{v(k)}$. Note that $\ell(v)=\ol{v(1)}+\cdots+\ol{v(k)} =: |\lambda(v)|$. Likewise, given a decreasing sequence $\mu$ of positive integers (a {\it strict partition}) with $n\geq\mu_1$, let $v(\mu)$ be the Grassmannian permutation with $\lambda(v(\mu))=\mu$. We write $\mu\subset\lambda$ for strict partitions $\mu,\lambda$ if $\mu_i\leq\lambda_i$ for all $i$, equivalently, if $v(\mu)\leq_0v(\lambda)$. The inclusion $\pm[n]\hookrightarrow\pm[n{+}1]$ induces inclusions ${\mathcal B}_n\hookrightarrow{\mathcal B}_{n+1}$ and ${\mathcal S}_{\pm[n]}\hookrightarrow{\mathcal S}_{\pm[n+1]}$. Define ${\mathcal B}_\infty:= \bigcup_n{\mathcal B}_n$ and ${\mathcal S}_{\pm\infty}:= \bigcup_n{\mathcal S}_{\pm[n]}$. In this representation, ${\mathcal B}_\infty$ has three types of reflections, which are, as elements of ${\mathcal S}_{\pm\infty}$: $$ \begin{array}{lcl} t_{i\,j}&:=& (\ol{\jmath},\,\ol{\imath})(i,\,j)\\ t_{j} &:=& (\ol{\jmath},\,j)\\ t_{\ol{\imath}\,j} &:=& (\ol{\jmath},\,i)(\ol{\imath},\,j) \end{array}\qquad \mbox{for } 0w(j)\}. $$ The Bruhat order ($\vartriangleleft$) on ${\mathcal S}_{\pm\infty}$ is defined by its covers: $u\tridot w$ if and only if $wu^{-1}$ is a transposition and $l(w)=l(u)+1$. If $k\in{\mathbb Z}$, this is a cover (written $\tridotk$\!\!) in the k-Bruhat order ($\vartriangleleft_k$) on ${\mathcal S}_{\pm\infty}$ (or ${\mathcal S}_{\pm[n]}$) if $wu^{-1}=(a,\,b)$ with $aw(b)$, then $a0$. \begin{ex}\label{ex:not_nice} Consider the following two intervals in the 1-Bruhat order on ${\mathcal B}_4$: $$\epsfxsize=2.2in \epsfbox{fig4.eps}$$ Note that $(3\ol{2}14)^{-1}\cdot(3\ol{4}12) = (1\ol{2}43)^{-1}\cdot(1\ol{4}23) = (\ol{2},\,\ol{4})(2\,,4)$ and the two intervals are not isomorphic. \end{ex} \begin{rem}\label{rem:X} For any $\zeta\in{\mathcal B}_\infty$, there is a $u\in{\mathcal B}_\infty$ with $u\leq_0\zeta u$: Suppose $\{a\in \pm{\mathbb N}\mid a>\zeta(a)\}=\{a_1,\ldots,a_m\}$ with $\zeta(a_1)<\zeta(a_2)<\cdots<\zeta(a_m)$. Further let $\{a_{m+1}0\mid \zeta(a)\neq a\}$, then $\zeta u$ is a Grassmannian permutation. \end{rem} By Theorem~B(1), we may define a new partial order on ${\mathcal B}_\infty$, which we call the {\em Lagrangian order:} For $\eta,\zeta\in{\mathcal B}_\infty$, set $\eta\preceq \zeta$ if there is a $u\in{\mathcal B}_\infty$ with $u\leq_0\eta u\leq_0\zeta u$. By Remark~\ref{rem:X}, it has a unique minimal element $e$. This order is graded by the rank, $\Len(\zeta)$, where $\Len(\zeta):=\ell(\zeta u)-\ell(u)$ whenever $u\leq_0\zeta u$. These notions have definitions independent of $\leq_0$: \begin{defth}[{\it cf.}\/ Definition 3.2.2~\cite{BS98a}]\label{defth:2.6} Let $\eta,\zeta\in{\mathcal B}_\infty$. \begin{enumerate} \item[(1)] Then $\eta\preceq\zeta$ if and only if \begin{enumerate} \item[(i)] $a\in \pm {\mathbb N}$ with $a>\eta(a) \Longrightarrow \eta(a)\geq \zeta(a)$, and \item[(ii)] $a,b\in \pm {\mathbb N}$ with $a\zeta(a),b>\zeta(b)$, and $\zeta(a)<\zeta(b) \Longrightarrow \eta(a)<\eta(b)$.\vspace{-10pt} \end{enumerate} \vspace{15pt} \item[(2)] ${\displaystyle \Len(\zeta)\ =\ \sum_{a,0>\zeta(a)} |\zeta(a)| \;\ -\;\ \#\{(a,b)\mid 0\zeta(b)\}}$ \vspace{5pt} ${\displaystyle \qquad\ -\;\ \#\{(a,b)\mid a\zeta(a),b>\zeta(b),\zeta(a)>\zeta(b)\} \;\ -\ \sum_{0>a>\zeta(a)} |a|}$. \end{enumerate} \end{defth} \noindent{\bf Proof. } Let $u$ be the permutation with $u\leq_0 \zeta u$ constructed from $\zeta$ in Remark~\ref{rem:X}. If $u\leq_0 \eta u\leq_0\zeta u$, then $\eta$ satisfies the conditions in ({\em i}), and conversely. For ({\em ii}), consider the difference $\ell(\zeta u)-\ell(u)$. The length of $\zeta u$ is the first sum, plus the number of inversions of the form $0\zeta u(j)=u(j)$. In the construction of $u$, each of these is also an inversion in $u$ involving positions $00>u(i)} |u(i)|$. \QED The Lagrangian order is the ${\mathcal B}_\infty$-counterpart of the Grassmann-Bruhat order $\prcc$ on ${\mathcal S}_\infty$ of~\cite{BS98a,BS_monoid}. This is defined as follows: Let $\eta,\zeta\in{\mathcal S}_\infty$. Then $\eta\prcc\zeta$ if and only if there is a $u\in{\mathcal S}_\infty$ with $u\vartriangleleft_0 \eta u\vartriangleleft_0 \zeta u$. The Grassmann-Bruhat order is ranked with $||\eta||:=l(\eta u)-l(u)$ whenever $u\vartriangleleft_0\eta u$. Let $s(\zeta)$ count the sign changes $\{a>0\mid 0>\zeta(a)\}$ in $\zeta$. We have the following relation between these two orders. \begin{cor}\label{cor:compare_orders} \mbox{ } \begin{enumerate} \item[(1)] $({\mathcal B}_\infty,\prec)$ is an induced suborder of $({\mathcal S}_{\pm\infty},\prcc)$. \item[(2)] For $\zeta\in{\mathcal B}_\infty(\subset{\mathcal S}_{\pm\infty})$, we have $\Len(\zeta)=(||\zeta|| + s(\zeta))/2$. \end{enumerate} \end{cor} \noindent{\bf Proof. } The first statement is a consequence of Theorem~\ref{thm:induced_order}. For the second statement, consider any maximal chain in $[e,\zeta]_\prec$ (in ${\mathcal B}_\infty$). By Theorem~\ref{thm:induced_order}, this gives a maximal chain in $[e,\zeta]_\prcs$ (in ${\mathcal S}_{\pm\infty}$), where covers of the form $\eta\precdot t_{ab}\,\eta$ are replaced by $\eta\pcdot (a,b)\eta\pcdot (a,b)(\ol{a},\ol{b})\eta$. Thus $||\zeta||=\Len(\zeta)+\tau$, where $\tau$ counts the covers in that chain if the form $\eta\precdot t_{ab}\,\eta$. Since only covers of the form $\eta\precdot t_b\eta$ contribute to $s(\zeta)$, we have $||\zeta||=2\Len(\zeta)-s(\zeta)$. \QED Let $\eta,\zeta\in{\mathcal B}_\infty$. If $\zeta\cdot\eta=\eta\cdot\zeta$ with $\Len(\eta\cdot\zeta)=\Len(\eta)+\Len(\zeta)$, and neither of $\zeta$ or $\eta$ is the identity, then $\eta\cdot\zeta$ is the {\em disjoint product} of $\eta$ and $\zeta$. (In general $\Len(\eta\cdot\zeta)\leq\Len(\eta)+\Len(\zeta)$.) If a permutation cannot be factored in this way, it is {\em irreducible}. Permutations $\zeta\in{\mathcal B}_\infty$ factor uniquely into irreducibles. This is described in terms of non-crossing partitions~\cite{Kreweras}: (A non-crossing partition of $\pm{\mathbb N}$ is a set partition such that if $a\zeta(b)$. \item[2] Choose $a$ minimal subject to $a\leq \zeta(b)<\zeta(a)$. \item[3] $\zeta:=\zeta t_{a\,b}$, output $\zeta$. \end{enumerate} Before every execution of \/ {\rm 3}, $\zeta t_{a\,b}\precdot \zeta$. Moreover, this algorithm terminates in $\Len(\zeta)$ iterations and the reverse of the sequence produced is a chain in $[e,\zeta]_\prec$. \end{alg} The Lagrangian order has a theory of reduced decompositions, analogous to the usual theory for the Coxeter group $B_\infty$ with respect to the weak order. (Also analogous to that for the Grassmann-Bruhat order on ${\mathcal S}_\infty$~\cite{BS_monoid}.) We express this in the context of monoids. Define a monoid ${\mathcal M}$ with 0 and generators ${\bf t}_{a\, b}, {\bf t}_b$ for integers $0j$. Let $g_{\ol{\jmath}}(\Fdot)$ be the intersection of $E'_{\ol{w(j)}}$ with the affine space $$ W_j\ :=\ f_{\ol{\jmath}}(\Fdot)\ +\ \Span{g_{\ol{\imath}}(\Fdot) \mid i>j\mbox{ and } w(i)j$ and $w(i)\cdots>\lambda_k$ of positive integers. The projection map $\pi: G/B \twoheadrightarrow G/P_0$ maps Schubert varieties to Schubert varieties, with $\pi Y_u = \Upsilon_\lambda$, where $\lambda$ consists of the positive numbers among $\{\ol{u(1)},\ol{u(2)},\ldots,\ol{u(n)}\}$ arranged in decreasing order. For a strict partition $\lambda$, let $v(\lambda)$ be the Grassmannian permutation whose (initial) negative values are $\ol{\lambda_1}<\ol{\lambda_2}<\cdots<\ol{\lambda_k}$. Let $\lambda^c$ be the decreasing sequence obtained from the integers in $[n]$ which do not appear in $\lambda$. Then $Y_{v(\lambda)}=\pi^{-1}\Upsilon_\lambda$ and $$ \pi\ :\ Y_{\omega_0v(\lambda)} \ \longrightarrow\ \Upsilon_{\lambda^c} $$ is birational. One may see this by considering typical elements of their Schubert cells. Set $P_\lambda := [\Upsilon_\lambda]$ in $H^*\Son/P_0$ (equivalently $P_\lambda := [Y_{v(\lambda)}]$ in $H^*\Son/B$) and let $Q_\lambda$ be the corresponding class in the symplectic case. For $1\leq m\leq n$, the {\em special Schubert variety} $\Upsilon_{(m)}$ is the collection of all maximal isotropic subspaces which meet a fixed $(n+1-m)$-dimensional isotropic subspace. Let $p_m$ (respectively $q_m$) denote the class $P_{(m)}$ in either $H^*\Son/B$ or $H^*\Son/P_0$ (respectively the class $Q_{(m)}$ in either $H^*\Spn/B$ or $H^*\Spn/P_0$). We are particularly interested in the constants $b^w_{u\,\lambda}:=b^w_{u\,v(\lambda)}$ and $c^w_{u\,\lambda}:=c^w_{u\,v(\lambda)}$ which give the structure of the cohomology of $G/B$ as a module over the cohomology of $G/P_0$. Using the intersection pairing and the projection formula ({\it cf.}~\cite[8.1.7]{Fulton_intersection}), we have \begin{eqnarray*} c^w_{u\,\lambda} &=& \deg({\mathfrak C}_u\cdot{\mathfrak C}_{\omega_0w}\cdot \pi^*(Q_\lambda))\\ &=&\deg (\pi_*({\mathfrak C}_u\cdot{\mathfrak C}_{\omega_0w})\cdot Q_\lambda), \end{eqnarray*} and a similar formula for $b^w_{u\,\lambda}$. Our main technique will be to find formulas for $\pi_*({\mathfrak C}_u\cdot{\mathfrak C}_{\omega_0w})$ by studying the effect of the map $\pi$ on the cycle $Y_u\cap Y'_{\omega_0w}$. To that end, define ${\mathcal Y}^w_u:=\pi(Y_u\cap Y'_{\omega_0w})$. These cycles ${\mathcal Y}^w_u$ are, like Schubert varieties, defined up to translation by $G$. In the theorems below, write ${\mathcal Y}^w_u={\mathcal Y}^z_x$ to mean that the cycles may be carried onto each other by an element of $G$. (The proofs are more explicit.) The main result of Section~\ref{sec:identity} concerns these cycles. \begin{thm}\label{thm:geometric_shapes} Let $u,w\in{\mathcal B}_n$ with $u\leq_0w$. Then \begin{enumerate} \item[(1)] The map $\pi : Y_u\cap Y'_{\omega_0w} \rightarrow {\mathcal Y}^w_u$ has degree 1. \item[(2)] If we have $x,z\in{\mathcal B}_n$ with $z\leq_0z$ and $wu^{-1}$ shape equivalent to $zx^{-1}$, then ${\mathcal Y}^w_u={\mathcal Y}^z_x$. \end{enumerate} \end{thm} This implies the identity of Theorem~B(2). As a consequence of Theorem~\ref{thm:geometric_shapes}(2), define ${\mathcal Y}_\zeta:={\mathcal Y}^{\zeta u}_u$ for any $\zeta,u\in{\mathcal B}_n$ with $u\leq_0\zeta u$. These cycles satisfy more identities. Let $W$ be a $2m$-dimensional symplectic space. We study the manifolds of maximal isotropic subspaces of $V, W$, and $V\oplus W$, together with a map $$ \Xi\ :\ \Spn/P_0 \times \mbox{\it Sp}_{2m}{\mathbb C}/P_0 \ \longrightarrow\ {\it Sp}_{2n+2m}{\mathbb C}/P_0, $$ defined by $(H,K)\mapsto H\oplus K$. Also define $\rho\in{\mathcal B}_n$ by $\rho(i)=i{-}1{-}n$ for $1\leq i\leq n$ and $\gamma\in{\mathcal B}_n$ by $\gamma(i)=i{+}1$ for $1\leq i< n$ and $\gamma(n)=1$. \begin{thm}\label{thm:geom_facts} \mbox{ } \begin{enumerate} \item[(1)] For any $\zeta\in{\mathcal B}_n$, ${\mathcal Y}_{\zeta}\ =\ {\mathcal Y}_{\zeta^{-1}}$. \item[(2)] If $\eta\cdot\zeta$ is a disjoint product in ${\mathcal B}_{n+m}$ with $\eta'\in{\mathcal B}_n$ shape equivalent to $\eta$ and $\zeta'\in{\mathcal B}_m$ shape equivalent to $\zeta$, then $$ \Xi( {\mathcal Y}_{\eta'}\times{\mathcal Y}_{\zeta'})\ =\ {\mathcal Y}_{\eta\cdot\zeta}. $$ \item[(3)] For any $\zeta\in{\mathcal B}_n$, $$ {\mathcal Y}_{\zeta}\ =\ {\mathcal Y}_{\rho\zeta\rho}. $$ \item[(4)] For any $\zeta\in{\mathcal B}_n$ with $a\cdot\zeta(a)>0$ for every $a$, $$ {\mathcal Y}_{\zeta}\ =\ {\mathcal Y}_{\gamma\zeta\gamma^{-1}}. $$ \end{enumerate} \end{thm} The first statement follows from the observation that $Y_u\bigcap Y_{\omega_0\zeta u}= Y_{\omega_0\zeta u}\bigcap Y_{\omega_0\zeta^{-1}(\omega_0\zeta u)}$. Since $\Len(\zeta)=\dim {\mathcal Y}_\zeta$, Theorem~C(1) follows from part (3), by the projection formula. Similarly, Theorem~C(2) follows from part (4). We remark that part (2) is true only for the symplectic case, while (1), (3), and (4) hold also for the odd orthogonal flag manifold. Statements (2), (3), and (4) are proven in Section~\ref{sec:more_identities}. \section{Identities of structure constants} \label{sec:identity} We establish Theorem~\ref{thm:geometric_shapes} which implies Theorem~B(2). As in Section~\ref{sec:orders}, many results and methods are similar to those of~\cite{BS98a} for analogous results about $\Sln{\mathbb C}/B$. Our discussions are therefore brief. The results here hold for both $\Son/B$ and $\Spn/B$, with nearly identical proofs. We only provide justification for $\Spn/B$. Let $H_2=\Span{h,\ol{h}}\simeq {\mathbb C}^2$ be a symplectic space of dimension 2. Then the orthogonal direct sum $V\oplus H_2$ is a symplectic space of dimension $2n+2$. For each $1\leq p\leq n+1$, define embeddings $\psi_p,\psi_{\ol{p}}\ :\ \Spn/B \ \hookrightarrow\ \mbox{\it Sp}_{2n+2}{\mathbb C}/B$, the space of isotropic flags in $V\oplus H_2$, by $$ \left(\psi_p \Edot\right)_j\ = \ \left\{\begin{array}{ccl} E_{j+1}&& j\leq \ol{p}\: (<0)\\ \Span{E_j,h}&& \ol{p} p \mbox{ and } |w(j)| < |q|\\ w(j-1)-1 && j > p \mbox{ and } w(j) \leq -|q|\\ w(j-1)+1 && j > p \mbox{ and } w(j) \geq |q| \end{array}\right.. $$ Let $/_p:{\mathcal B}_{n+1}\rightarrow{\mathcal B}_n$ be the left inverse of $\varepsilon_{p,q}$, defined by $\varepsilon_{p,w(p)}(w/_p)=w$. If we represent permutations as permutation matrices, then the effect of $/_p$ on $w\in{\mathcal B}_{n+1}$ is to erase the $p$th and $\ol{p}$th columns and the $w(p)$th and $\ol{w(p)}$th rows. The effect of $\varepsilon_{p,q}$ on $w$ is to expand its permutation matrix with new $p$th, $\ol{p}$th columns and $q$th, $\ol{q}$th rows filled with zeroes, except for 1s at positions $(q,p)$ and $(\ol{q},\ol{p})$. For example: $$ \begin{array}{c} \varepsilon_{3,\ol{2}}(\ol{2}\,3\,4\,1)\ =\ \ol{3}\,4\,\ol{2}\,5\,1\\ \varepsilon_{3,2}(\ol{2}\,3\,4\,1)\ =\ \ol{3}\,4\,2\,5\,1 \end{array} \qquad\mbox{and}\qquad 4\,\ol{1}\,5\,\ol{2}\,3/_4\ =\ 3\,\ol{1}\,4\,2. $$ These definitions imply the following proposition ({\it cf.}~\cite{Sottile96}, Lemma 12). \begin{prop}\label{prop:subsets} Let $w\in{\mathcal B}_n$, $1\leq p, |q|\leq n+1$, and $\Edot$ any isotropic flag. Then $$ \psi_p\, Y_w \Edot \ \subset\ Y_{\varepsilon_{p,q}(w)}\psi_{\ol{q}}\Edot. $$ \end{prop} Recall that $e$ is the identity permutation and $Y_e$ is the flag manifold $G/B$. \begin{cor}\label{cor:intersection} Let $\Edot,\Epdot$ be opposite isotropic flags. Then for any $q\in \pm[n+1]$, $\psi_q\Edot, \psi_{\ol{q}}\Epdot$ are opposite flags, and for any $1\leq p\leq n+1$, $\psi_p\, Y_w\Edot$ equals either of the following cycles $$ Y_{\varepsilon_{p,\ol{n+1}}(w)}\psi_{n+1}\Edot \bigcap Y_{\varepsilon_{p,n+1}(e)}\psi_{\ol{n+1}}\Epdot \qquad\quad Y_{\varepsilon_{p,n+1}(w)}\psi_{\ol{n+1}}\Edot \bigcap Y_{\varepsilon_{p,\ol{n+1}}(e)}\psi_{n+1}\Epdot. $$ \end{cor} \noindent{\bf Proof. } It is straightforward to check that the flags are opposite. Moreover, by Proposition~\ref{prop:subsets}, $\psi_p\, Y_w\Edot$ is a subset of either intersection, as $Y_e=\Spn/B$. Since $$ \ell(\varepsilon_{p,\ol{n+1}}(w))\ =\ \ell(w)+n+p, \qquad\quad \ell(\varepsilon_{p,n+1}(w))\ =\ \ell(w) +n+1-p, $$ and $\dim \Spn/B=n^2$, Proposition~\ref{prop:transverse} implies that all three cycles are irreducible with the same dimension, proving their equality. \QED \begin{cor}\label{cor:push-forward} For any $w\in{\mathcal B}_n$ and $1\leq p\leq n$, we have $$ (\psi_p)_*{\mathfrak C}_w\ =\ {\mathfrak C}_{\varepsilon_{p,\ol{n+1}}(w)}\cdot {\mathfrak C}_{\varepsilon_{p,n+1}(e)}\ =\ {\mathfrak C}_{\varepsilon_{p,n+1}(w)}\cdot {\mathfrak C}_{\varepsilon_{p,\ol{n+1}}(e)}. $$ \end{cor} \begin{rem} As in Section~4.2 of~\cite{BS98a}, the Schubert classes ${\mathfrak C}_{\varepsilon_{p,n+1}(e)}$ and ${\mathfrak C}_{\varepsilon_{p,\ol{n+1}}(e)}$ are certain special Schubert classes from Grassmannian projections. Were the corresponding Pieri-type formulas known, we could deduce formulas for $(\psi_p)_*{\mathfrak C}_w$. Only one of these classes is a special Schubert class from $G/P_0$: $$ {\mathfrak B}_{\varepsilon_{1,\ol{n+1}}(e)}\ =\ p_{n+1} \qquad\mbox{and}\qquad {\mathfrak C}_{\varepsilon_{1,\ol{n+1}}(e)}\ =\ q_{n+1}. $$ We deduce formulas for $(\psi_1)_*{\mathfrak B}_w$ and $(\psi_1)_*{\mathfrak C}_w$ from this. Recall that $H^*\Spn/B$ is generated by $x_1,\ldots,x_n$. These classes are Chern classes of certain line bundles on $\Spn/B$: Let ${\mathcal E}_{\DOT}\rightarrow\Spn/B$ be the flag of bundles whose fibre at $\Edot$ is $\Edot$. Then $x_i=-c_1({\mathcal E}_{\ol{i+1}}/{\mathcal E}_{\ol{\imath}})$. Thus $$ \psi_p^*(x_i)\ =\ \left\{\begin{array}{lcl} x_i&\ &ip \end{array}\right.. $$ \end{rem} \begin{thm} Let $v\in{\mathcal B}_{n+1}$. Then \begin{enumerate} \item[(1)] In $H^*\Son/B$, $\psi_1^*{\mathfrak B}_v= \sum \theta(\varepsilon_{1,\ol{n+1}}(y)v^{-1})\ {\mathfrak B}_y$, the sum over all $y\in{\mathcal B}_n$ with $\ell(v)=\ell(y)$. \item[(2)] In $H^*\Spn/B$, $\psi_1^*{\mathfrak C}_v= \sum \chi(\varepsilon_{1,\ol{n+1}}(y)v^{-1})\ {\mathfrak C}_y$, the sum over all $y\in{\mathcal B}_n$ with $\ell(v)=\ell(y)$. \end{enumerate} \end{thm} \noindent{\bf Proof. } These are consequences of the projection formula and Corollary~\ref{cor:push-forward}. For the first, $$ \psi_1^*{\mathfrak B}_v\ =\ \sum_{y\in{\mathcal B}_n} \deg(\psi_1^*{\mathfrak B}_v\cdot{\mathfrak B}_{\omega_0 y}) {\mathfrak B}_y. $$ But $$ \deg(\psi_1^*{\mathfrak B}_v\cdot{\mathfrak B}_{\omega_0 y})\ =\ \deg({\mathfrak B}_v\cdot(\psi_1)_*({\mathfrak B}_{\omega_0 y}))\ =\ \deg({\mathfrak B}_v\cdot{\mathfrak B}_{\varepsilon_{1,n+1}(\omega_0y)} \cdot q_{n+1}) $$ Since $\varepsilon_{1,n+1}(\omega_0y)=\omega_0(\varepsilon_{1,\ol{n+1}}(y))$ and $\ell(\varepsilon_{1,\ol{n+1}}(y))=\ell(y)+n+1$, the result follows by the Pieri-type formula (Theorem~D). \QED \begin{lem}\label{lem:reduction} Suppose $u<_0 w$ in ${\mathcal B}_{n+1}$ and $u(p)=w(p)=q$ for some $1\leq p\leq n+1$. Then \begin{enumerate} \item[(1)] $u/_p<_0 w/_p$ and $\ell(w)-\ell(u) = \ell(w/_p)-\ell(u/_p)$. \item[(2)] For any opposite isotropic flags $\Edot,\Epdot$ in $V$, $$ \psi_p\left( Y_{u/_p}\Edot \bigcap Y_{\omega_0 w/_p}\Epdot\right) \ =\ Y_u\psi_{\ol{q}}\Edot \bigcap Y_{\omega_0 w}\psi_q\Epdot. $$ \end{enumerate} \end{lem} \noindent{\bf Proof. } Since $u<_0w$ and $u(p)=w(p)$, Proposition~\ref{prop:thmA} implies that $u/_p<_0 w/_p$. Moreover, $wu^{-1}$ is shape equivalent to $w/_p(u/_p)^{-1}$, so the first statement follows from Theorem~B(1). For (2), Proposition~\ref{prop:subsets} gives the inclusion $\subset$. By Corollary~\ref{cor:intersection}, $\psi_{\ol{q}}\Edot$ and $\psi_q\Epdot$ are opposite flags. Thus, by Proposition~\ref{prop:transverse} and (1), both sides are irreducible and have the same dimension, proving their equality. \QED \begin{thm}\label{thm:elem_equiv} Suppose $u<_0 w$ in ${\mathcal B}_{n+1}$ and $u(p)=w(p)=q$ for some $1\leq p\leq n+1$. Then for any strict partition $\lambda$, we have $$ b^w_{u\,\lambda}\ =\ b^{w/_p}_{u/_p\,\lambda} \qquad\mbox{and}\qquad c^w_{u\,\lambda}\ =\ c^{w/_p}_{u/_p\,\lambda}. $$ \end{thm} \noindent{\bf Proof. } We first study the map $\Psi : \Spn/P_0 \hookrightarrow \mbox{\it Sp}_{2n+2}{\mathbb C}/P_0$, defined by $K \mapsto \Span{K,h}$. Here, ${\it Sp}_{2n+2}{\mathbb C}/P_0$ is the Grassmannian of maximal isotropic subspaces of $V\oplus H_2$. If $\Edot,\Epdot$ are opposite isotropic flags in $V$, the analog of Corollary~\ref{cor:intersection} is $$ \Psi (\Upsilon_\lambda\Edot)\ =\ \Upsilon_\lambda \psi_{n+1}\Edot \bigcap \Upsilon_{(n+1)}\psi_{\ol{n+1}}\Epdot, $$ where $(n{+}1)$ is a decreasing sequence of length 1. We leave this to the reader. As this intersection is generically transverse, $\Psi_* Q_\lambda = Q_\lambda \cdot q_{n+1}$. Thus $$ \Psi^* Q_\lambda \ =\ \left\{\begin{array}{lcl} Q_\lambda &\quad&\lambda_1