% % A triple intersection proof of the Pieri-type formula for % maximal isotropic Grassmannians. % % Frank Sottile % LaTeX2e document % Begun: May 10, 1997. % Completed: June 6 1997. % Submitted to Transactions of the AMS, 8 June 1997 % MSRI preprint # 1997-062 % % % % \documentclass[12pt]{amsart} \usepackage{amssymb} \usepackage{epsf} % Makes a nice wide page, good for sending out compact copies % %\textwidth 6.7in\oddsidemargin -.1in\evensidemargin -.1in %\textheight 9.2in\topmargin -0.6in \headheight=8pt \textheight=624pt \textwidth=432pt \oddsidemargin=18pt \evensidemargin=18pt \topmargin=0pt %\textwidth 5.3in\oddsidemargin .6in\evensidemargin .6in %\textheight 8in\topmargin 0in %\def\baselinestretch{1.15} \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{\Fdot}{{F\!_{\DOT}}} \newcommand{\Fpdot}{{{F\!_{\DOT}}'}} \newcommand{\Span}[1]{\langle #1 \rangle} \newcommand{\QED}{ \setlength{\unitlength}{1.0pt}% \begin{picture}(10,7.5) \put(0,-2.5){\rule{2.5pt}{2.5pt}} \put(0,0){\rule{5pt}{2.5pt}} \put(0,2.5){\rule{10pt}{2.5pt}} \end{picture}\vspace{10pt}} \newtheorem{thm}{Theorem}[section] \newtheorem{lemma}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} \newtheorem{prop}[thm]{Proposition} \newtheorem{ex}[thm]{Example} \newtheorem{defn}[thm]{Definition} \newtheorem{rem}[thm]{Remark} \newtheorem{athm}{Theorem}[subsection] \newtheorem{aex}[athm]{Example} \newtheorem{alem}[athm]{Lemma} \newtheorem{acor}[athm]{Corollary} \begin{document} \title[Pieri-type formula for maximal isotropic Grassmannians] {Pieri-type formulas for maximal isotropic Grassmannians via triple intersections} \author{Frank Sottile} \address{Department of Mathematics\\ University of Toronto\\ 100 St.~George Street\\ Toronto, Ontario M5S 3G3\\ Canada} \email{sottile@math.toronto.edu} \date{\today} \thanks{Research supported in part by NSF grant DMS-90-22140 and NSERC grant OPG0170279} \subjclass{14M15} \keywords{isotropic Grassmannian, Pieri's formula, Schubert varieties} \begin{abstract} We give an elementary proof of the Pieri-type formula in the cohomology of a Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic vector space. This proof proceeds by explicitly computing a triple intersection of Schubert varieties. The decisive step is an exact description of the intersection of two Schubert varieties, from which the multiplicities (which are powers of 2) in the Pieri-type formula are immediately obvious. \end{abstract} \maketitle \section*{Introduction} The goal of this paper is to give an elementary geometric proof of Pieri-type formulas in the cohomology of Grassmannians of maximal isotropic subspaces of odd orthogonal or symplectic vector spaces. For this, we explicitly compute a triple intersection of Schubert varieties, where one is a special Schubert variety. Previously, Sert\"oz~\cite{Sertoz} had studied such triple intersections in orthogonal Grassmannians, but was unable to determine the intersection multiplicities and obtain a formula. These multiplicities are either 0 or powers of 2. Our proof explains them as the intersection multiplicity of a linear subspace (defining the special Schubert variety) with a collection of quadrics and linear subspaces (determined by the other two Schubert varieties). This is similar to triple intersection proofs of the classical Pieri formula ({\em cf.}~\cite{Hodge_intersection}\cite[p. 203]{Griffiths_Harris}% \cite[\S 9.4]{Fulton_tableaux}) where the multiplicities (0 or 1) count the number of points in the intersection of linear subspaces. A proof of the Pieri-type formula for classical flag varieties~\cite{Sottile_pieri_schubert} was based upon those ideas. Similarly, the ideas here provide a basis for a proof of Pieri-type formulas in the cohomology of symplectic flag varieties~\cite{Bergeron_Sottile_Lagrangian_Pieri}. These Pieri-type formulas are due to Hiller and Boe~\cite{Hiller_Boe}, whose proof used the Chevalley formula~\cite{Chevalley91}. Another proof, using the Leibnitz formula for symplectic and orthogonal divided differences, was given by Pragacz and Ratajski~\cite{Pragacz_Ratajski_Operator}. These formulas also arise in the theory of projective representations of symmetric groups~\cite{Schur,Hoffman_Humphreys} as product formulas for Schur $P$- and $Q$-functions, and were first proven in this context by Morris~\cite{Morris}. The connection of Schur $P$- and $Q$-functions to geometry was established by Pragacz~\cite{Pragacz_88} (see also~\cite{Pragacz_S-Q}). \smallskip In Section 1, we give the basic definitions and state the Pieri-type formulas in both the orthogonal and symplectic cases, and conclude with an outline of the proof in the orthogonal case. Since there is little difference between the proofs in each case, we only do the orthogonal case in full. In Section 2, we describe the intersection of two Schubert varieties, which we use in Section 3 to complete the proof. \section{The Grassmannian of maximal isotropic subspaces} Let $V$ be a ($2n+1$)-dimensional complex vector space equipped with a non-degenerate symmetric bilinear form $\beta$ and $W$ a $2n$-dimensional complex vector space equipped with a non-degenerate alternating bilinear form, also denoted $\beta$. A subspace $H$ of $V$ or of $W$ is {\em isotropic} if the restriction of $\beta$ to $H$ is identically zero. Isotropic subspaces have dimension at most $n$. The {\em Grassmannian of maximal isotropic subspaces} $B_n$ or $B(V)$ (respectively $C_n$ or $C(W)$) is the collection of all isotropic $n$-dimensional subspaces of $V$ (respectively of $W$). The group $\mbox{\em So}_{2n+1}{\mathbb C} = {\rm Aut}(V,\beta)$ acts transitively on $B_n$ with the stabilizer $P_0$ of a point a maximal parabolic subgroup associated to the short root, hence $B_n = \mbox{\em So}_{2n+1}{\mathbb C}/P_0$. Similarly, $C_n= \mbox{\em Sp}_{2n}{\mathbb C}/P_0$, where $P_0$ is a maximal parabolic associated to the long root. Both $B_n$ and $C_n$ are smooth complex manifolds of dimension $\binom{n+1}{2}$. While they are not isomorphic if $n>1$, they have identical Schubert decompositions and Bruhat orders. Another, deeper, connection is discussed in Remark~\ref{rem:similarities}. We describe the Schubert decomposition. For an integer $j$, let $\overline{\jmath}$ denote $-j$. Choose bases $\{e_{\overline{n}},\ldots,e_n\}$ of $V$ and $\{f_{\overline{n}},\ldots,f_n\}$ of $W$ for which: $$ \beta(e_i,e_j)\ =\ \left\{ \begin{array}{ll} 1&\mbox{ \ if } i=\overline{\jmath}\\ 0&\mbox{ \ otherwise}\end{array}\right. \quad\mbox{and}\quad \beta(f_i,f_j)\ =\ \left\{ \begin{array}{ll} j/|j|&\mbox{ \ if } i=\overline{\jmath}\\ 0 &\mbox{ \ otherwise}\end{array}\right. . $$ For example, $\beta(e_1,e_0)=\beta(f_{\overline{2}},f_1)=0$ and $\beta(e_0,e_0)= \beta(f_{\overline{1}},f_1)=-\beta(f_1,f_{\overline{1}})= 1$. Schubert varieties are determined by sequences $$ \lambda:\ n\geq \lambda_1>\lambda_2>\cdots>\lambda_n\geq \overline{n} $$ whose set of absolute values $\{|\lambda_1|,\ldots,|\lambda_n|\}$ equals $\{1,2,\ldots,n\}$. Let ${\mathbb{SY}}_n$ denote this set of sequences. The Schubert variety $X_\lambda$ of $B_n$ is $$ \{H\in B_n\mid\dim H\cap\Span{e_{\lambda_j},\ldots,e_n}\geq j \mbox{\ for } 1\leq j\leq n\} $$ and the Schubert variety $Y_\lambda$ of $C_n$ $$ \{H\in C_n\mid\dim H\cap\Span{f_{\lambda_j},\ldots,f_n}\geq j \mbox{\ for } 1\leq j\leq n\}. $$ Both $X_\lambda$ and $Y_\lambda$ have codimension $|\lambda|:=\lambda_1+\cdots+\lambda_k$, where $\lambda_k>0>\lambda_{k+1}$. Given $\lambda,\mu\in{\mathbb{SY}}_n$, we see that $$ X_\mu\supset X_\lambda\ \Longleftrightarrow\ Y_\mu\supset Y_\lambda\ \Longleftrightarrow\ \mu_j\leq \lambda_j \mbox{ for } 1\leq j\leq n. $$ Define the {\em Bruhat order} by $\mu\leq \lambda$ if $\mu_j\leq \lambda_j$ for $1\leq j\leq n$. Note that $\mu\leq \lambda$ if and only if $\mu_j\leq \lambda_j$ for those $j$ with $0<\mu_j$. \begin{ex}{\rm Suppose $n=4$. Then $X_{3\,2\,\overline{1}\,\overline{4}}$ consists of those $H\in B_4$ such that $$ \dim H\cap\Span{e_3,e_4}\geq 1,\ \dim H\cap\Span{e_2,e_3,e_4}\geq 2,\ \mbox{and} \dim H\cap\Span{e_{\overline{1}},\ldots,e_4}\geq 3. $$ } \end{ex} Define $P_\lambda:=[X_\lambda]$, the cohomology class Poincar{\'e} dual to the fundamental cycle of $X_\lambda$ in the homology of $B_n$.\ Likewise set $Q_\lambda:=[Y_\lambda]$. Since Schubert varieties are closures of cells from a decomposition into (real) even-dimensional cells, these {\em Schubert classes} $\{P_\lambda\}$, $\{Q_\lambda\}$ form bases for integral cohomology: $$ H^*B_n\ =\ \bigoplus_{\lambda} P_\lambda \cdot {\mathbb Z} \qquad \mbox{and}\qquad H^*C_n\ =\ \bigoplus_{\lambda} Q_\lambda \cdot {\mathbb Z}. $$ Each $\lambda\in{\mathbb{SY}}_n$ determines and is determined by its diagram, also denoted $\lambda$. The diagram of $\lambda$ is a left-justified array of $|\lambda|$ boxes with $\lambda_j$ boxes in the $j$th row, for $\lambda_j>0$. Thus $$ 3\,2\,\overline{1}\,\overline{4} \ \longleftrightarrow\ \setlength{\unitlength}{.9pt}\begin{picture}(30,20)(0,7) \put( 0, 0){\line(0,1){20}}\put(0, 0){\line(1,0){20}} \put(10, 0){\line(0,1){20}}\put(0,10){\line(1,0){30}} \put(20, 0){\line(0,1){20}}\put(0,20){\line(1,0){30}} \put(30,10){\line(0,1){10}}\end{picture}\, \qquad\mbox{ and }\qquad 4\,2\,1\,\overline{3} \ \longleftrightarrow\ \setlength{\unitlength}{.9pt}\begin{picture}(40,20)(0,12) \put( 0, 0){\line(0,1){30}}\put(0,30){\line(1,0){40}} \put(10, 0){\line(0,1){30}}\put(0,20){\line(1,0){40}} \put(20,10){\line(0,1){20}}\put(0,10){\line(1,0){20}} \put(30,20){\line(0,1){10}}\put(0, 0){\line(1,0){10}} \put(40,20){\line(0,1){10}}\end{picture}\, . \raisebox{-15pt}{\rule{0pt}{5pt}} $$ The Bruhat order corresponds to inclusion of diagrams. Given $\mu\leq \lambda$, let $\lambda/\mu$ be their set-theoretic difference. For instance, $$ 4\,2\,1\,\overline{3}/3\,2\,\overline{1}\,\overline{4}\ \longleftrightarrow\ \setlength{\unitlength}{.9pt}\begin{picture}(40,20)(0,12) \put(30,20){\line(0,1){10}}\put(30,30){\line(1,0){10}} \put(40,20){\line(0,1){10}}\put(30,20){\line(1,0){10}} \put( 0, 0){\line(0,1){10}}\put( 0,10){\line(1,0){10}} \put(10, 0){\line(0,1){10}}\put( 0, 0){\line(1,0){10}} \put( 0,20){\dashbox{2}(10,10)[t]{}}\put( 0,10){\dashbox{2}(10,10)[t]{}} \put(10,20){\dashbox{2}(10,10)[t]{}}\put(10,10){\dashbox{2}(10,10)[t]{}} \put(20,20){\dashbox{2}(10,10)[t]{}} \end{picture} \qquad\mbox{ and }\qquad 3\,2\,\overline{1}\,\overline{4}/ 1\,\overline{2}\,\overline{3}\,\overline{4}\ \longleftrightarrow\ \setlength{\unitlength}{.9pt}\begin{picture}(30,20)(0,7) \put( 0, 0){\line(0,1){10}}\put( 0, 0){\line(1,0){20}} \put(10, 0){\line(0,1){20}}\put( 0,10){\line(1,0){30}} \put(20, 0){\line(0,1){20}}\put(10,20){\line(1,0){20}} \put(30,10){\line(0,1){10}}\put( 0,10){\dashbox{2}(10,10)[t]{}} \end{picture}\, . \raisebox{-15pt}{\rule{0pt}{5pt}} $$ Two boxes are connected if they share a vertex or an edge; this defines {\em components} of $\lambda/\mu$. We say $\lambda/\mu$ is a {\em skew row} if $\lambda_1\geq\mu_1\geq\lambda_2\geq\cdots\geq\mu_n$ equivalently, if $\lambda/\mu$ has at most one box in each column. Thus $4\,2\,1\,\overline{3}/3\,2\,\overline{1}\,\overline{4}$ is a skew row, but $3\,2\,\overline{1}\,\overline{4}/ 1\,\overline{2}\,\overline{3}\,\overline{4}$ is not. The {\em special Schubert class} $p_m\in H^*B_n$ ($q_m\in H^*C_n$) is the class whose diagram consists of a single row of length $m$. Hence, $p_2 = P_{2\,\overline{1}\,\overline{3}\,\overline{4}}$. A {\em special Schubert variety} $X_K$ ($Y_K$) is the collection of all maximal isotropic subspaces which meet a fixed isotropic subspace $K$ nontrivially. If $\dim K=n+1-m$, then $[X_K]=p_m$ and $[Y_K]=q_m$. \begin{thm}[Pieri-type Formula]\label{thm:pieri} For any $\mu\in{\mathbb{SY}}_n$ and $1\leq m\leq n$, \begin{enumerate} \item ${\displaystyle P_\mu\cdot p_m\ =\ \sum_{\lambda/\mu\ \mbox{\scriptsize skew row}} 2^{\delta(\lambda/\mu)-1}\, P_\lambda}$ \qquad and \item ${\displaystyle Q_\mu\cdot q_m\ =\ \sum_{\lambda/\mu\ \mbox{\scriptsize skew row}} 2^{\varepsilon(\lambda/\mu)}\, Q_\lambda}$, \end{enumerate} where $\delta(\lambda/\mu)$ counts the components of the diagram $\lambda/\mu$ and $\varepsilon(\lambda/\mu)$ counts the components of $\lambda/\mu$ which do not contain a box in the first column. \end{thm} \begin{ex}\label{ex:prod} {\em For example, \begin{eqnarray*} P_{3\,2\,\overline{1}\,\overline{4}}\cdot p_2 &=& 2\cdot P_{4\,2\,1\,\overline{3}}\ \,+ \ \ \, P_{4\,3\,\overline{2}\,\overline{1}}\qquad\mbox{ and}\\ Q_{3\,2\,\overline{1}\,\overline{4}}\cdot q_2 &=& 2\cdot Q_{4\,2\,1\,\overline{3}}\ +\ 2\cdot Q_{4\,3\,\overline{2}\,\overline{1}}. \end{eqnarray*} } \end{ex} For $\lambda,\mu,\nu\in{\mathbb{SY}}_n$, there exist integral constants $g^\lambda_{\mu,\nu}$ and $h^\lambda_{\mu,\nu}$ defined by the identities $$ P_\mu\cdot P_\nu\ =\ \sum_\lambda g^\lambda_{\mu,\nu}\,P_\lambda \qquad \mbox{and}\qquad Q_\mu\cdot Q_\nu\ =\ \sum_\lambda h^\lambda_{\mu,\nu}\,Q_\lambda. $$ These constants were first given a combinatorial formula by Stembridge~\cite{Stembridge_shifted}.\smallskip Define $\lambda^c$ by $\lambda^c_j:=\overline{\lambda_{n+1-j}}$. Let $[\mbox{pt}]$ be the class dual to a point. The Schubert basis is self-dual with respect to the intersection pairing: If $|\lambda|=|\mu|$, then \begin{equation}\label{eq:poincare} P_\mu\cdot P_{\lambda^c} \ =\ Q_\mu\cdot Q_{\lambda^c} \ =\ \left\{\begin{array}{ll} [\mbox{pt}]&\ \ \mbox{if}\ \lambda=\mu\\ 0 &\ \ \mbox{otherwise}\end{array}\right.. \end{equation} Define the Schubert variety $X'_{\lambda^c}$ to be $$ \{H\in B_n\mid \dim H\cap\Span{e_{\overline{n}},\ldots,e_{\lambda_j}}\geq n+1-j \ \mbox{for}\ 1\leq j \leq n\}. $$ This is a translate of $X_{\lambda^c}$ by an element of $\mbox{\em So}_{2n+1}{\mathbb C}$. In a similar fashion, define $Y'_{\lambda^c}$, a translate of $Y_{\lambda^c}$ by an element of $\mbox{\em Sp}_{2n}{\mathbb C}$. For any $\lambda,\mu$, $X_\mu\bigcap X'_{\lambda^c}$ is generically transverse~\cite{Kleiman}. This is because if $X_\mu$ and $X'_{\lambda^c}$ are {\em any} Schubert varieties in general position, then there is a basis for $V$ such that these varieties and the quadratic form $\beta$ are as given. We see that to establish the Pieri-type formula, it suffices to compute the degree of $$ X_\mu\bigcap X'_{\lambda^c}\bigcap X_K \quad\mbox{and}\quad Y_\mu\bigcap Y'_{\lambda^c}\bigcap Y_K $$ where $K$ is a general isotropic ($n+1-m$)-plane and $|\lambda|=|\mu|+m$.\smallskip We only do the (more difficult) orthogonal case of Theorem~\ref{thm:pieri} in full, and indicate the differences for the symplectic case. We first determine when $X_\mu\bigcap X'_{\lambda^c}\neq\emptyset$. Let $\mu,\lambda\in{\mathbb{SY}}_n$. Then, by the definition of Schubert varieties, $H\in X_\mu\bigcap X'_{\lambda^c}$ implies $\dim H\bigcap \Span{e_{\mu_j},\ldots,e_{\lambda_j}}\geq 1$, for every $1\leq j\leq n$. Hence $\mu\leq\lambda$ is necessary for $X_\mu\bigcap X'_{\lambda^c}$ to be nonempty. In fact, $$ X_\mu \bigcap X'_{\lambda^c}\ =\ \left\{ \begin{array}{ll} \Span{e_{\lambda_1},\ldots,e_{\lambda_n}}&\mbox{ if } \lambda=\mu\\ \emptyset &\mbox{ otherwise}, \end{array}\right. $$ which establishes~(\ref{eq:poincare}). Suppose $\mu\leq \lambda$ in ${\mathbb{SY}}_n$. For each component $d$ of $\lambda/\mu$, let col$(d)$ index the columns of $d$ together with the column just to the left of $d$, which is $0$ if $d$ meets the first column, in that it has a box in the first column. For each component $d$ of $\lambda/\mu$, define a quadratic form $\beta_d$: $$ \beta_d\ :=\ \sum_{j\in \mbox{\scriptsize col}(d)}x_j x_{\overline{\jmath}}, $$ where $x_{\overline{n}},\ldots,x_n$ are coordinates for $V$ dual to the basis $e_{\overline{n}},\ldots,e_n$. For each {\em fixed point} of $\lambda/\mu$ ($j$ such that $\lambda_j=\mu_j$), define the linear form $\alpha_j:=x_{\overline{\lambda_j}}$. If there is no component meeting the first column, then we say that $0$ is a fixed point of $\lambda/\mu$ and define $\alpha_0:=x_0$. Let $Z_{\lambda/\mu}$ be the common zero locus of these forms $\alpha_j$ and $\beta_d$. \begin{lemma}\label{lemma:vanish} Let $\mu\leq\lambda$ and $H\in X_\mu\bigcap X'_{\lambda^c}$. Then $H\subset Z_{\lambda/\mu}$. \end{lemma} Let ${\mathcal Q}$ be the isotropic points in $V$, the zero locus of $\beta$. For each $0\leq i\leq n$, there is a unique form among the $\alpha_j$, $\beta_d$ in which one (or both) of the coordinates $x_{i},x_{\overline{\imath}}$ appears. Thus $\beta$ is in the ideal generated by these forms $\alpha_j$, $\beta_d$ and we see that they are dependent on ${\mathcal Q}$. However, if $\delta=\delta(\lambda/\mu)$ counts the components of $\lambda/\mu$ and $\varphi$ the number of fixed points, then the collection of $\varphi$ forms $\alpha_j$ and $\delta-1$ of the forms $\beta_d$ {\em are} independent on ${\mathcal Q}$. Moreover, Lemma~\ref{lem:columns} shows that $$ n +1 \ =\ \varphi+\delta+\#\mbox{\rm columns of $\lambda/\mu$}. $$ Thus, if $m=|\lambda|-|\mu|$, then $\varphi+\delta-1\leq n-m$, with equality only when $\lambda/\mu$ is a skew row. Since ${\mathcal Q}$ has dimension $2n$, it follows that a general isotropic ($n+1-m$)-plane $K$ meets $Z_{\lambda/\mu}$ only if $\lambda/\mu$ is a skew row. We deduce \begin{thm}\label{thm:geom} Let $\mu,\lambda\in{\mathbb{SY}}_n$ and suppose $K$ is a general isotropic $(n+1-m)$-plane with $|\mu|+m = |\lambda|$. Then $$ X_\mu \bigcap X'_{\lambda^c}\bigcap X_K\ \neq \ \emptyset $$ only if $\mu\leq \lambda$ and $\lambda/\mu$ is a skew row. \end{thm} Under the hypotheses of Theorem~\ref{thm:geom}, the forms $\alpha_j$ and $\beta_d$ define $2^{\delta(\lambda/\mu)-1}$ isotropic lines in $K$. Theorem~\ref{thm:unique} asserts that a general isotropic line in $Z_{\lambda/\mu}$ determines a unique $H\in X_\mu \bigcap X'_{\lambda^c}$, which completes the proof of Theorem~\ref{thm:pieri}. \begin{ex} {\em We show that if $K\subset {\mathcal Q}$ is a general 3-plane, then $$ \#\, X_{3\,2\,\overline{1}\,\overline{4}}\bigcap X'_{(4\,2\,1\,\overline{3})^c} \bigcap X_K\ =\ 2. $$ Note that 2 is the coefficient of $P_{4\,2\,1\,\overline{3}}$ in the product $P_{3\,2\,\overline{1}\,\overline{4}}\cdot p_2$ of Example~\ref{ex:prod}. First, the local coordinates for $X_{3\,2\,\overline{1}\,\overline{4}}\bigcap X'_{(4\,2\,1\,\overline{3})^c}$ described in Lemma~\ref{lem:loc_coords} show that, for any $x,z\in{\mathbb{C}}$, the row span $H$ of the matrix with rows $g_i$ and columns $e_j$ $$ \begin{array}{l|cccc|c|cccc} {} &e_{\overline{4}}&e_{\overline{3}} &e_{\overline{2}}&e_{\overline{1}}&e_0&e_1&e_2&e_3&e_4\\ \hline g_1&0&0&0&0&0&0&0&-x&1\\ g_2&0&0&0&0&0&0&1&0&0\\ g_3&0&0&0&1&2z&-2z^2&0&0&0\\ g_4&x&1&0&0&0&0&0&0&0 \end{array} $$ is in $X_{3\,2\,\overline{1}\,\overline{4}}\bigcap X'_{(4\,2\,1\,\overline{3})^c}$. Suppose $K$ is the row span of the matrix with rows $v_i$ $$ \begin{array}{l|cccc|c|cccc} {} &e_{\overline{4}}&e_{\overline{3}} &e_{\overline{2}}&e_{\overline{1}}&e_0&e_1&e_2&e_3&e_4\\ \hline v_1&0&1&0&1&0& 0&1&0&1\\ v_2&1&1&0&1&2&-2&1&1&-1\\ v_3&0&0&1&0&0&-1&0&0&0 \end{array} $$ Then $K$ is an isotropic 3-plane, and the forms \begin{eqnarray*} \beta_0&=& 2x_{\overline{1}}x_1 + x_0^2\\ \beta_d&=&x_{\overline{4}}x_4 + x_{\overline{3}}x_3\\ \alpha_2&=& x_{\overline{2}} \end{eqnarray*} define the 2 isotropic lines $\Span{v_1}$ and $\Span{v_2}$ in $K$. Lastly, for $i=1,2$, there is a unique $H_i\in X_{3\,2\,\overline{1}\,\overline{4}}\bigcap X'_{(4\,2\,1\,\overline{3})^c}$ with $v_i\in H_i$. In these coordinates, $$ H_1\ :\ x=z=0\quad\mbox{and}\quad H_2\ :\ x=z=1. $$ } \end{ex} In the symplectic case, isotropic $K$ are not contained in a quadric ${\mathcal Q}$, the form $\alpha_0=x_0$ does not arise, only components which do not meet the first column give quadratic forms $\beta_d$, and the analysis of Lemma~\ref{lem:components}~(2) in Section~\ref{sec:triple} is (slightly) different. \section{The intersection of two Schubert varieties} We study the intersection $X_\mu\bigcap X'_{\lambda^c}$ of two Schubert varieties. Our main result, Theorem~\ref{thm:main}, expresses $X_\mu\bigcap X'_{\lambda^c}$ as a product whose factors correspond to components of $\lambda/\mu$, and each factor is itself an intersection of two Schubert varieties. These factors are described in Lemmas~\ref{lemma:0comp} and~\ref{lem:offdiag}, and in Corollary~\ref{cor:classical}. These are crucial to the proof of the Pieri-type formula that we complete in Section 3. Also needed is Lemma~\ref{lem:subspace}, which identifies a particular subspace of $H\cap \Span{e_1,\ldots,e_n}$ for $H\in X_\mu\bigcap X'_{\lambda^c}$. For Lemma~\ref{lem:subspace}, we work in the (classical) Grassmannian $G_k(V^+)$ of $k$-planes in $V^+:=\Span{e_1,\ldots,e_n}$. For basic definitions and results see any of ~\cite{Hodge_Pedoe,Griffiths_Harris,Fulton_tableaux}. Schubert subvarieties $\Omega_\sigma,\Omega'_{\sigma^c}$ of $G_k(V^+)$ are indexed by partitions $\sigma\in{\mathbb{Y}}_k$, where $n - k\geq \sigma_1\geq\cdots\geq\sigma_k\geq0$. For $\sigma\in{\mathbb{Y}}_k$ define $\sigma^c\in{\mathbb{Y}}_k$ by $\sigma^c_j=n-k-\sigma_{k+1-j}$. For $\sigma,\tau\in{\mathbb{Y}}_k$, define \begin{eqnarray*} \Omega_\tau&:=& \{H\in G_k(V^+)\mid \dim H\cap\Span{e_{k+1-j+\tau_j},\ldots,e_n}\geq j,\ 1\leq j\leq k\}\\ \Omega'_{\sigma^c}&:=& \{H\in G_k(V^+)\mid \dim H\cap\Span{e_1,\ldots,e_{j+\sigma_{k+1-j}}}\geq j,\ 1\leq j\leq k\}. \end{eqnarray*} Let $\lambda,\mu\in\mathbb{SY}_n$ with $\mu\leq \lambda$, and suppose $\mu_k>0>\mu_{k+1}$. Define partitions $\sigma$ and $\tau$ in ${\mathbb{Y}}_k$ (which depend upon $\lambda$ and $\mu$) by \begin{eqnarray*} \tau&:=& \mu_1-k\geq \cdots\geq \mu_k-1\geq 0\\ \sigma&:=& \lambda_1-k\geq \cdots\geq \lambda_k-1>0. \end{eqnarray*} \begin{lemma}\label{lem:subspace} Let $\mu\leq \lambda\in{\mathbb{SY}}_n$, and define $\sigma,\tau\in{\mathbb{Y}}_k$, and $k$ as above. If $H\in X_\mu\bigcap X'_{\lambda^c}$, then $H\cap V^+$ contains a $k$-plane $L\in\Omega_\tau\bigcap\Omega'_{\sigma^c}$. \end{lemma} \noindent{\bf Proof. } Suppose first that $H\in X_\mu$ with $\dim H\cap \Span{e_{\mu_j},\ldots,e_n}=j$ for $j=k$ and $k+1$. Since $\mu_k>0>\mu_{k+1}$, we see that $L:= H\cap V^+$ has dimension $k$. If $H\in X'_{\lambda^c}$ in addition, it is an exercise in the definitions to verify that $L\in\Omega_\tau\bigcap\Omega'_{\sigma^c}$. The lemma follows as such $H$ are dense in $X_\mu$. \QED The first step towards Theorem~\ref{thm:main} is the following combinatorial lemma. \begin{lemma}\label{lem:columns} Let $\varphi$ count the fixed points and $\delta$ the components of $\lambda/\mu$. Then $$ n +1\ =\ \varphi+\delta+\#\mbox{\rm columns of $\lambda/\mu$}, $$ and $\mu_j>\lambda_{j+1}$ precisely when $|\mu_j|$ is an empty column of $\lambda/\mu$. \end{lemma} \noindent{\bf Proof. } Suppose $k$ is a column not meeting $\lambda/\mu$. Thus, there is no $i$ for which $\mu_i\lambda_{j+1}$ so that $|\mu_j|$ is an empty column of $\lambda/\mu$. Then the definitions of Schubert varieties imply $$ H\ =\ H\cap\Span{e_{\overline{n}},\ldots,e_{\lambda_{j+1}}}\oplus H\cap\Span{e_{\mu_j},\ldots,e_n}. $$ Suppose $d$ is a component not meeting the first column. If the rows of $d$ are $j,\ldots,k$, then \begin{eqnarray*} H\cap V^+_d&=& H\cap\Span{e_{\mu_k},\ldots,e_{\lambda_j}}\\ &=& H \cap\Span{e_{\overline{n}},\ldots,e_{\lambda_j}} \cap\Span{e_{\mu_k},\ldots,e_n}, \end{eqnarray*} and so has dimension at least $k-j+1$. Similarly, if $l,\ldots,m$ are the indices $i$ with $\overline{\lambda_{j}}\leq \mu_i,\lambda_i\leq \overline{\mu_k}$, then $H\bigcap V^-_d$ has dimension at least $l-m+1$. Hence $\dim V_d/2 = \#\mbox{col}(d)=k+m-l-j+2$, as $\lambda_j,\ldots,\lambda_k,\overline{\lambda_l},\ldots,\overline{\lambda_m}$ are the columns of $d$. Since $H$ is isotropic, $\dim H^+_d + \dim H^-_d \leq \#\mbox{col}(d)$, which proves the first part of (3). Moreover, $H\bigcap V^+_d=\left(H\bigcap V^-_d\right)^\perp$: Since $H$ is isotropic, we have $\subset$, and equality follows by dimension counting. Similar arguments prove the other statements. \QED For $H\in X_\mu\bigcap X'_{\lambda^c}$, define $H_\varphi:= H\bigcap V_\varphi$, $H_0:= H\bigcap V_0$, $H^+_d := H\bigcap V^+_d$, and $H^-_d := H\bigcap V^-_d$. Then $H_\varphi\subset V_\varphi$ is the zero locus of the linear forms $\alpha_j$, $H_0$ is isotropic in $V_0$, and, for each component $d$ of $\lambda/\mu$ not meeting the first column, $H_d:= H^+_d\oplus H^-_d$ is isotropic in $V_d$, which proves Lemma~\ref{lemma:vanish}. Moreover, $H$ is the orthogonal direct sum of $H_\varphi$, $H_0$, and the $H_d$. \begin{thm}\label{thm:main} The map $$ \{H_0\mid H\in X_\mu\bigcap X'_{\lambda^c}\}\ \times \prod_{0\not\in\mbox{\scriptsize col}(d)} \{H_d\mid H\in X_\mu\bigcap X'_{\lambda^c}\} \longrightarrow\ X_\mu\bigcap X'_{\lambda^c} $$ defined by $$ (H_0,\ldots,H_d,\ldots) \longmapsto \Span{H_\varphi,H_0,\ldots,H_d,\ldots} $$ is an isomorphism. \end{thm} \noindent{\bf Proof. } By the previous discussion, it is an injection. For surjectivity, note that both sides have the same dimension. Indeed, $\dim X_\mu\bigcap X'_{\lambda^c}=|\lambda|-|\mu|$, the number of boxes in $\lambda/\mu$. Lemmas~\ref{lemma:0comp} and \ref{lem:offdiag} show that the factors of the domain each have dimension equal to the number of boxes in the corresponding components. \QED Suppose there is a component, $d_0$, meeting the first column. Let $l$ be the largest column in $d_0$, and define $\lambda(0),\mu(0)\in{\mathbb{SY}}_l$ as follows: Let $j$ be the first row of $d_0$ so that $l=\lambda_j$. Then, since $d_0$ is a component, for each $j\leq i\cdots >\mu_{j+l-1}\\ \lambda(0)&:=& \lambda_j>\cdots >\lambda_{j+l-1} \end{eqnarray*} Define $\lambda(0)^c$ by $\lambda(0)^c_p:= \overline{\lambda(0)_{l+1-p}} = \overline{\lambda_{j+l-p}}$. The following lemma is a straightforward consequence of these definitions. \begin{lemma}\label{lemma:0comp} With the above definitions, $$ \{H_0\mid H\in X_\mu\bigcap X'_{\lambda^c}\}\ =\ X_{\mu(0)}\bigcap X'_{\lambda(0)^c}, $$ as subvarieties of $B_k\simeq B(V_0)$, and $\lambda(0)/\mu(0)$ has a unique component meeting the first column and no fixed points. \end{lemma} We similarly identify $\{H_d\mid H\in X_\mu\bigcap X'_{\lambda^c}\}$ as an intersection $X_{\mu(d)}\cap X'_{\lambda(d)^c}$ of Schubert varieties in $B_{\#\mbox{\scriptsize col}(d)}\simeq B(\Span{e_0,V_d})$. Let $j,\ldots,k$ be the rows of $d$ and $l,\ldots,m$ be the indices $i$ with $\overline{\lambda_j}\leq \mu_i,\lambda_i\leq \overline{\mu_k}$, as in the proof of Lemma~\ref{lemma:intersection}. Let $p=\#\mbox{col}(d)$ and define $\lambda(d),\mu(d)\in {\mathbb{SY}}_p$ as follows. Set $a = \mu_k$, and define \begin{eqnarray*} \mu(d)&:=& \mu_j-a+1>\cdots> \hspace{24pt}1\hspace{24pt} >\mu_l+a-1>\cdots>\mu_m+a-1\\ \lambda(d)&:=& \lambda_j-a+1>\cdots>\lambda_k-a+1> \lambda_l+a-1>\cdots>\lambda_m+a-1 \end{eqnarray*} As with Lemma~\ref{lemma:0comp}, the following lemma is straightforward. \begin{lemma}\label{lem:offdiag} With these definitions, $$ \{H_d\mid H\in X_\mu\bigcap X'_{\lambda^c}\}\ \simeq\ X_{\mu(d)}\bigcap X'_{\lambda(d)^c} $$ as subvarieties of $B_p\simeq B(\Span{e_0,V_d})$ and $\lambda(d)/\mu(d)$ has a unique component not meeting the first column and no fixed points. \end{lemma} Suppose now that $\mu,\lambda\in{\mathbb{SY}}_n$ where $\lambda/\mu$ has a unique component $d$ not meeting the first column and no fixed points. Suppose $\lambda$ has $k$ rows. A consequence of Lemma~\ref{lemma:intersection} is that the map $H^+_d \mapsto \Span{H^+_d,\left(H^+_d\right)^\perp}$ gives an isomorphism \begin{equation}\label{isomorphism} \{ H^+_d\mid H\in X_\mu\bigcap X'_{\lambda^c}\}\ \stackrel{\sim}{\longrightarrow}\ X_\mu\bigcap X'_{\lambda^c}. \end{equation} The following corollary of Lemma~\ref{lem:subspace} identifies the domain. \begin{cor}\label{cor:classical} With $\mu,\lambda$ as above and $\sigma,\tau$, and $k$ as defined in the paragraph preceding Lemma~\ref{lem:subspace}, we have: $$ \{H^+_d\mid H\in X_\mu\bigcap X'_{\lambda^c}\}\ =\ \Omega_{\tau}\bigcap\Omega_{\sigma^c}, $$ as subvarieties of $G_k(V^+)$. \end{cor} \begin{rem}\label{rem:similarities} {\em The symplectic analogs of Lemma~\ref{lem:offdiag} and Corollary~\ref{cor:classical}, which are identical save for the necessary replacement of $Y$ for $X$ and $C_p$ for $B_p$, show an unexpected connection between the geometry of the symplectic and orthogonal Grassmannians. Namely, suppose $\lambda/\mu$ has no component meeting the diagonal. Then the projection map $V \twoheadrightarrow W$ defined by $$ e_i\ \longmapsto\ \left\{\begin{array}{ll}0&\ \mbox{if}\ i=0\\ f_i&\ \mbox{otherwise} \end{array}\right. $$ and its right inverse $W\hookrightarrow V$ defined by $f_j\mapsto e_j$ induce isomorphisms $$ X_\mu\bigcap X'_{\lambda^c}\ \stackrel{\sim}{\longleftrightarrow}\ Y_\mu\bigcap Y'_{\lambda^c}. $$ } \end{rem} \section{Pieri-type intersections of Schubert varieties}\label{sec:triple} Fix $\lambda/\mu$ to be a skew row with $|\lambda|-|\mu|=m$. Let $Z_{\lambda/\mu}\subset{\mathcal Q}$ be the zero locus of the forms $\alpha_j$ and $\beta_d$ of \S 2. If $\lambda/\mu$ has $\delta$ components, then as a subvariety of ${\mathcal Q}$, $Z_{\lambda/\mu}$ is the generically transverse intersection of the zero loci of the forms $\alpha_j$ and any $\delta-1$ of the forms $\beta_d$. It follows that a general $(n+1-m)$-plane $K\subset{\mathcal Q}$ meets $Z_{\lambda/\mu}$ in $2^{\delta-1}$ lines. Thus if $\Span{v}\subset Z_{\lambda/\mu}$ is a general line, then $$ \# X_\mu\bigcap X'_{\lambda^c}\bigcap X_K \ =\ 2^\delta\cdot \# X_\mu\bigcap X'_{\lambda^c}\bigcap X_{\Span{v}}. $$ Theorem~\ref{thm:pieri} is a consequence of this observation and the following: \begin{thm}\label{thm:unique} Let $\lambda/\mu$ be a skew row, $Z_{\lambda/\mu}$ be as above, and $\Span{v}$ a general line in $Z_{\lambda/\mu}$. Then $X_\mu\bigcap X'_{\lambda^c}\bigcap X_{\Span{v}}$ is a singleton. \end{thm} \noindent{\bf Proof. } Let ${\mathcal Q}_0$ be the cone of isotropic points in $V_0$ and ${\mathcal Q}_d$ the cone of isotropic points in $V_d$ for $d\neq d_0$. Since $$ Z_{\lambda/\mu}\ =\ H_\varphi \oplus {\mathcal Q}_0\oplus \bigoplus_{0\not\in\mbox{\scriptsize col}(d)} {\mathcal Q}_d, $$ we see that a general non-zero vector $v$ in $Z_{\lambda/\mu}$ has the form $$ v\ =\ \sum_{\mu_j=\lambda_j} a_je_{\mu_j}\ +\ v_0\ + \sum_{0\not\in\mbox{\scriptsize col}(d)} v_d, $$ where $a_j\in {\mathbb C}^\times$ and $v_0\in {\mathcal Q}_0$, $v_d\in {\mathcal Q}_d$ are general vectors. Thus, if $H\in X_\mu\bigcap X'_{\lambda^c}\bigcap X_{\Span{v}}$, we see that $v_0\in H_0$ and $v_d\in H_d$. By Theorem~\ref{thm:main}, $H$ is determined by $H_0$ and the $H_d$, thus it suffices to prove that $H_0$ and the $H_d$ are uniquely determined. The identifications of Lemmas~\ref{lemma:0comp} and~\ref{lem:offdiag} show that this is just the case of the theorem when $\lambda/\mu$ has a single component, which is Lemma~\ref{lem:components} below. \QED \begin{lemma}\label{lem:components} Suppose $\lambda,\mu\in{\mathbb{SY}}_n$ where $\lambda/\mu$ is a skew row with a unique component and no fixed points. \begin{enumerate} \item If $\lambda/\mu$ does not meet the first column and $v\in {\mathcal Q}_d$ is a general vector, then $X_\mu\bigcap X'_{\lambda^c}\bigcap X_{\Span{v}}$ is a singleton. \item If $\lambda/\mu$ meets the first column and $v\in {\mathcal Q}$ is general, then $X_\mu\bigcap X'_{\lambda^c}\bigcap X_{\Span{v}}$ is a singleton. \end{enumerate} \end{lemma} \noindent{\bf Proof of (1). } Let $v\in{\mathcal Q}_d$ be a general vector. Since ${\mathcal Q}_d\subset V^+\oplus V^-$, $v=v^+\oplus v^-$ with $v^+ \in V^+$ and $v^-\in V^-$. Suppose $\mu_k>0>\mu_{k+1}$. Consider the set $$ \{H^+\in G_k(V^+)\mid v\in H^+\oplus\left(H^+\right)^\perp\} \ =\ \{H^+\mid v^+\in H^+\subset (v^-)^\perp\}. $$ This is a Schubert variety $\Omega''_{h(n-k,k)}$ of $G_kV^+$, where $h(n-k,k)$ is the partition of hook shape with a single row of length $n-k$ and a single column of length $k$. Under the isomorphisms of (\ref{isomorphism}) and Lemma~\ref{lem:offdiag}, and with the identification of Corollary~\ref{cor:classical}, we see that $$ X_\mu\bigcap X'_{\lambda^c}\bigcap X_{\Span{v}}\ \simeq\ \Omega_\tau\bigcap\Omega'_{\sigma^c}\bigcap\Omega''_{h(n-k,k)}, $$ where $\sigma,\tau$ are as defined in the paragraph preceding Corollary~\ref{cor:classical}. For $\rho\in{\mathbb{Y}}_k$, let $S_\rho:=[\Omega_\rho]$ be the cohomology class Poincar\'e dual to the fundamental cycle of $\Omega_\rho$ in $H^*G_kV^+$. The multiplicity we wish to compute is \begin{equation}\label{Schur:calc} \deg (S_\tau\cdot S_{\sigma^c}\cdot S_{h(n-k,k)}). \end{equation} By a double application of the classical Pieri's formula (as $S_{h(n-k,k)}=S_{n-k}\cdot S_{1^{k-1}}$), we see that (\ref{Schur:calc}) is either 1 or 0, depending upon whether or not $\sigma/\tau$ has exactly one box in each diagonal. But this is the case, as the transformation $\mu,\lambda \longmapsto \tau,\sigma$ takes columns to diagonals. \QED Our proof of Lemma~\ref{lem:components} (2) uses a system of local coordinates for $X_\mu\bigcap X'_{\lambda^c}$. Let $\lambda/\mu$ be as in Lemma~\ref{lem:components} (2), and suppose $\lambda_{k+1}=1$. For $y_2,\ldots,y_n,x_0,\ldots,x_{n-1}\in{\mathbb C}$, define vectors $g_j\in V$ as follows: \begin{equation}\label{loc_coords} g_j\ := \ \left\{\begin{array}{ll} {\displaystyle e_{\lambda_j} + \sum_{i=\mu_j}^{\lambda_j-1}x_i\,e_i}&\ j\leq k\\ {\displaystyle -2x_0^2e_1 + 2x_0e_0+ e_{\overline{1}} + \sum_{i=\mu_{k+1}}^{\overline{2}}y_i\,e_i}&\ j= k+1\\ {\displaystyle e_{\lambda_j} + \sum_{i=\mu_j}^{\lambda_j-1}y_i\,e_i}&\ j>k+1\\ \end{array}\right.. \end{equation} \begin{lemma}\label{lem:loc_coords} Let $\lambda,\mu\in{\mathbb{SY}}_n$ where $\lambda/\mu$ is a skew row meeting the first column with no fixed points, and define $\tau,\sigma\in{\mathbb{Y}}_k$, and $k$ as for Lemma~\ref{lem:subspace}. Then \begin{enumerate} \item For any $x_1,\ldots,x_{n-1}\in{\mathbb C}$, we have $\Span{g_1,\ldots,g_k}\ \in\ \Omega_\tau\bigcap\Omega'_{\sigma^c}$. \item For and $x_0,\ldots,x_{n-1}\in{\mathbb C}$ with $x_{\overline{\mu_{k+1}}},\ldots,x_{\overline{\mu_{n-1}}}\neq 0$, the condition that $H:= \Span{g_1,\ldots,g_n}$ is isotropic determines a unique $H\in X_\mu\bigcap X'_{\lambda^c}$. \end{enumerate} Moreover, these coordinates parameterize dense subsets of the intersections. \end{lemma} \noindent{\bf Proof. } The first statement is immediate from the definitions. For the second, note that each $g_j\in{\mathcal Q}$. The conditions that $\Span{g_1,\ldots,g_n}$ is isotropic are $$ \beta(g_i,g_j)\ =\ 0\quad \mbox{for}\quad i\leq k< j. $$ Only $n-1$ of these are not identically zero. Indeed, for $i\leq k