%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% LaTeX file %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt]{amsart} \input epsf \oddsidemargin=16pt \evensidemargin=16pt \topmargin=16pt \headheight=8pt \textheight=571pt \textwidth=436pt \theoremstyle{plain} \newtheorem{thm}[subsubsection]{Theorem} \newtheorem{lem}[subsubsection]{Lemma} \newtheorem{prop}[subsubsection]{Proposition} \newtheorem{cor}[subsubsection]{Corollary} \newtheorem{thrm}{Theorem} \renewcommand{\thethrm}{} \theoremstyle{definition} \newtheorem{rem}[subsubsection]{Remark} \newtheorem{defn}[subsubsection]{Definition} \newtheorem{ex}[subsubsection]{Example} \renewcommand{\theex}{} \def\v{\vee} \def\w{\wedge} \def\mn{\underline{m}} \def\ns{\underline{n}} \def\st{\underline{s}} \def\ui{\underline{i}} \def\lt{\triangleleft} \def\lte{\underline{\lt}} \def\res{|} \def\L\mathcal{{L}} \def\A\mathcal{{A}} \def\a{\alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\f{\phi} \def\h{\chi} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\o{\omega} \def\O{\Omega} \def\s{\sigma} \def\t{\tau} \def\te{\theta} \def\Te{\Theta} \def\sur{\twoheadrightarrow} \def\ui{\underline{i}} \def\uj{\underline{j}} \def\ul{\underline{l}} \def\uk{\underline{k}} \def\ue{\underline{e}} \def\uf{\underline{f}} \def\ut{\mathbf{t}} \def\us{\mathbf{s}} \def\ub{\mathbf{b}} \def\ni{\noindent} \def\vs{\vskip} \newcommand{\bdot}{ {\hspace{-.1em} \mbox{\bf .} } } \newcommand{\kk}{{\bf k}} \newcommand{\nn}{{\bf n}} \newcommand{\rr}{{\bf r}} \newcommand{\mm}{{\bf m}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\NN}{{\mathbb N}} \newcommand{\RR}{\mathcal{R}} \newcommand{\AAA}{\mathbb{A}} \newcommand{\PPP}{\mathbb{P}} \newcommand{\OO}{{\bf O}} \newcommand{\II}{\mathcal{I}} \newcommand{\JJ}{\mathcal{J}} \newcommand{\tJ}{\widetilde{\JJ}} \newcommand{\tA}{\widetilde{A}} \newcommand{\tw}{\widetilde{w}} \newcommand{\ep}{\epsilon} \newcommand{\al}{\alpha} \newcommand{\lam}{\lambda} \newcommand{\sig}{\sigma} \newcommand{\Om}{\Omega} \newcommand{\taum}{ \tau^{ \mbox{\tiny max} } } \newcommand{\taur}{\tau^{\rr}} \newcommand{\eqdef}{\stackrel{\rm def}{=}} \newcommand{\eqto}{\stackrel{\rm \sim}{\to}} \newcommand{\zi}{(\zeta^*)^{-1}} \newcommand{\ziJ}{\zi \sqrt{\JJ(\taur)}} \newcommand{\bi}{{\overline{i}}} \newcommand{\bj}{{\overline{j}}} \newcommand{\Gr}{\mbox{\rm Gr}} \newcommand{\Fl}{\mbox{\rm Fl}} \newcommand{\Span}{\mbox{\rm Span}} \newcommand{\Stab}{\mbox{\rm Stab}} \newcommand{\rank}{\mbox{\rm rank}} \newcommand{\codim}{\mbox{\rm codim}} \newcommand{\Block}{\mbox{\rm Block}} \newcommand{\mo}{ { \!-\!1} } \newcommand{\po}{ { \!+\!1} } \DeclareMathOperator{\MMod}{\mathrm{mod }} \DeclareMathOperator{\Mult}{\mathrm{mult}} \DeclareMathOperator{\Spec}{\mathrm{Spec}} \DeclareMathOperator{\Sing }{\mathrm{Sing}} \begin{document} \title{Richardson Varieties in the Grassmannian} \author {Victor Kreiman} \address{Department of Mathematics\\ Northeastern University\\ Boston, MA 02115} \email{vkreiman@lynx.neu.edu} \author[V. Lakshmibai]{V. Lakshmibai${}^{\dag}$} \address{Department of Mathematics\\ Northeastern University\\ Boston, MA 02115} \email{lakshmibai@neu.edu} \dedicatory{Dedicated to Professor J. Shalika on his sixtieth birthday} \thanks{${}^{\dag}$ Partially supported by NSF grant DMS 9971295.} \begin{abstract} The Richardson variety $X_w^v$ is defined to be the intersection of the Schubert variety $X_w$ and the opposite Schubert variety $X^v$. For $X_w^v$ in the Grassmannian, we obtain a standard monomial basis for the homogeneous coordinate ring of $X_w^v$. We use this basis first to prove the vanishing of $H^i(X_w^v,L^m)$, $i > 0 $, $m \geq 0$, where $L$ is the restriction to $X_w^v$ of the ample generator of the Picard group of the Grassmannian; then to determine a basis for the tangent space and a criterion for smoothness for $X_w^v$ at any $T$-fixed point $e_\t$; and finally to derive a recursive formula for the multiplicity of $X_w^v$ at any $T$-fixed point $e_\t$. Using the recursive formula, we show that the multiplicity of $X_w^v$ at $e_\t$ is the product of the multiplicity of $X_w$ at $e_\t$ and the multiplicity of $X^v$ at $e_\t$. This result allows us to generalize the Rosenthal-Zelevinsky determinantal formula for multiplicities at $T$-fixed points of Schubert varieties to the case of Richardson varieties. \end{abstract} \maketitle %%%%%%%%%%%%%%%%%%%%%%%% \setcounter{tocdepth}{1} \tableofcontents \section*{Introduction} Let $G$ denote a semisimple, simply connected, algebraic group defined over an algebraically closed field $K$ of arbitrary characteristic. Let us fix a maximal torus $T$ and a Borel subgroup $B$ containing $T$. Let $W$ be the Weyl group ($ N(T)/T$, $N(T)$ being the normalizer of $T$). Let $Q$ be a parabolic subgroup of $G$ containing $B$, and $W_Q$, the Weyl group of $Q$. For the action of $G$ on $G/Q$ given by left multiplication, the $T$-fixed points are precisely the cosets $e_w:=wQ$ in $G/Q$. For $w \in W/W_Q$, let $X_w$ denote the {\it Schubert variety} (the Zariski closure of the $B$-orbit $Be_w$ in $G/Q$ through the $T$-fixed point $e_w$), endowed with the canonical structure of a closed, reduced subscheme of $G/Q$. Let $B^-$ denote the Borel subgroup of $G$ opposite to $B$ (it is the unique Borel subgroup of $G$ with the property $B\cap B^-=T$). For $v \in W/W_Q$, let $X^v$ denote the {\it opposite Schubert variety}, the Zariski closure of the $B^-$-orbit $B^-e_v$ in $G/Q$. Schubert and opposite Schubert varieties play an important role in the study of the generalized flag variety $G/Q$, especially, the algebraic-geometric and representation-theoretic aspects of $G/Q$. A more general class of subvarieties in $G/Q$ is the class of Richardson varieties; these are varieties of the form $X^v_w:= X_w\cap X^v $, the intersection of the Schubert variety $X_w$ with opposite Schubert variety $X^v$. Such varieties were first considered by Richardson in (cf. \cite{Ri}), who shows that such intersections are reduced and irreducible. Recently, Richardson varieties have shown up in several contexts: such double coset intersections $BwB\cap B^-xB$ first appear in \cite{K-Lu$_1$}, \cite{K-Lu$_2$}, \cite{Ri}, \cite{Ri-Ro-S}. Very recently, Richardson varieties have also appeared in the context of K-theory of flag varieties (\cite{Br}, \cite{L-L}). They also show up in the construction of certain degenerations of Schubert varieties (cf. \cite{Br}). In this paper, we present results for Richardson varieties in the Grassmannian variety. Let $G_{d,n}$ be the Grassmannian variety of $d$-dimensional subspaces of $K^n$, and $p:G_{d,n}\hookrightarrow \mathbb{P}^N \,(=\mathbb{P}(\wedge^dK^n))$, the Pl\"ucker embedding (note that $G_{d,n}$ may be identified with $G/P, G=SL_n(K), P$ a suitable maximal parabolic subgroup of $G$). Let $X:=X_w\cap X^v$ be a Richardson variety in $G_{d,n}$. We first present a Standard monomial theory for $X$ (cf. Theorem \ref{mainth}). Standard monomial theory (SMT) consists in constructing an explicit basis for the homogeneous coordinate ring of $X$. SMT for Schubert varieties was first developed by the second author together with Musili and Seshadri in a series of papers, culminating in \cite{g/p5}, where it is established for all classical groups. Further results concerning certain exceptional and Kac--Moody groups led to conjectural formulations of a general SMT, see \cite{ls2}. These conjectures were then proved by Littelmann, who introduced new combinatorial and algebraic tools: the path model of representations of any Kac-Moody group, and Lusztig's Frobenius map for quantum groups at roots of unity (see \cite{Li$_1$,Li$_2$}); recently, in collaboration with Littelmann (cf. \cite{L-L}), the second author has extended the results of \cite{Li$_2$} to Richardson varieties in $G/B$, for any semisimple $G$. Further, in collaboration with Brion (cf. \cite{b-l}), the second author has also given a purely geometric construction of standard monomial basis for Richardson varieties in $G/B$, for any semisimple $G$; this construction in loc. cit. is done using certain flat family with generic fiber $\cong$ diag$(X^v_w)\subset X^v_w\times X^v_w$, and the special fiber $\cong \cup_{v\le x\le w}X^v_x\times X_w^x$. If one is concerned with just Richardson varieties in the Grassmannian, one could develop a SMT in the same spirit as in \cite{Mu} using just the Pl\"ucker coordinates, and one doesn't need to use any quantum group theory nor does one need the technicalities of \cite{b-l}. Thus we give a self-contained presentation of SMT for unions of Richardson varieties in the Grassmannian. We should remark that Richardson vatieties in the Grassmannian are also studied in \cite{st}, where these varieties are called {\it skew Schubert varieties}, and standard monomial bases for these varieties also appear in loc. cit. (Some discussion of these varieties also appears in \cite{hp2}.) As a consequence of our results for unions of Richardson varieties, we deduce the vanishing of $H^i(X,L^m), i\ge 1, m\ge 0, L$, being the restriction to $X$ of ${\mathcal O}_{\mathbb{P}^N}(1)$ (cf. Theorem \ref{main}); again, this result may be deduced using the theory of Frobenius-splitting (cf. \cite{m-r}), while our approach uses just the classical Pieri formula. Using the standard monomial basis, we then determine the tangent space and also the multiplicity at any $T$-fixed point $e_\tau$ on $X$. We first give a recursive formula for the multiplicity of $X$ at $e_\tau$ (cf. Theorem \ref{8.8.5.3}). Using the recursive formula, we derive a formula for the multiplicity of $X$ at $e_\tau$ as being the product of the multiplicities at $e_\tau$ of $X_w$ and $X^v$ (as above, $X=X_w\cap X^v$) (cf. Theorem \ref{prodmult}). Using the product formula, we get a generalization of Rosenthal-Zelevinsky determinantal formula (cf. \cite{rose-zel}) for the multiplicities at singular points of Schubert varieties to the case of Richardson varieties (cf. Theorem \ref{rzr}). It should be mentioned that the multiplicities of Schubert varieties at $T$-fixed points determine their multiplicities at all other points, because of the $B$-action; but this does not extend to Richardson varieties, since Richardson varieties have only a $T$-action. Thus even though, certain smoothness criteria at $T$-fixed points on a Richardson variety are given in Corollaries \ref{smcrit1} and \ref{smcrit2}, the problem of the determination of singular loci of Richardson varieties still remains open. In \S \ref{gdn}, we present basic generalities on the Grassmannian variety and the Pl\"ucker embedding. In \S \ref{sor}, we define Schubert varieties, opposite Schubert varieties, and the more general Richardson varieties in the Grassmannian and give some of their basic properties. We then develop a standard monomial theory for a Richardson variety $X_w^v$ in the Grassmannian in \S \ref{smt1} and extend this to a standard monomial theory for unions and nonempty intersections of Richardson varieties in the Grassmannian in \S \ref{unn}. Using the standard monomial theory, we obtain our main results in the three subsequent sections. In \S \ref{van}, we prove the vanishing of $H^i(X_w^v,L^m)$, $i > 0 $, $m \geq 0$, where $L$ is the restriction to $X_w^v$ of the ample generator of the Picard group of the Grassmannian. In \S \ref{tss}, we determine a basis for the tangent space and a criterion for smoothness for $X_w^v$ at any $T$-fixed point $e_\t$. Finally, in \S \ref{mas}, we derive several formulas for the multiplicity of $X_w^v$ at any $T$-fixed point $e_\t$. We are thankful to the referee for many valuable comments and suggestions, especially for the alternate proof of Theorem \ref{prodmult}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The Grassmannian Variety $G_{d, n}$}\label{gdn} Let $K$ be the base field, which we assume to be algebraically closed of arbitrary characteristic. Let $d$ be such that $1 \le d < n$. The {\it Grassmannian} $G_{d,n}$ is the set of all $d$-dimensional subspaces of $K^n$. Let $U$ be an element of $G_{d,n}$ and $\{ a_1, \ldots a_d \}$ a basis of $U$, where each $a_j$ is a vector of the form \vspace{-.05in} \begin{displaymath} a_j = \left(\begin{array}{c} a_{1j} \\ a_{2j} \\ \vdots \\ a_{nj} \end{array}\right),\, \textrm{with } a_{ij} \in K. \end{displaymath} Thus, the basis $\{a_1, \cdots, a_d\}$ gives rise to an $n \times d$ matrix $A = (a_{ij})$ of rank $d$, whose columns are the vectors $a_1, \cdots, a_d$. We have a canonical embedding \begin{equation*} p:G_{d,n}\hookrightarrow \mathbb{P}(\wedge^dK^n) \ , \ U \mapsto [a_1 \wedge \cdots \wedge a_d] \end{equation*} called the {\it Pl\" ucker embedding}. It is well known that $p$ is a closed immersion; thus $G_{d, n}$ acquires the structure of a projective variety. Let \begin{equation*} I_{d,n}=\{\underline{i}=(i_1,\dots,i_d) \in \mathbb{N}^d \, : \, 1\le i_1<\dots$ both $\ui$ and $\uj$, and $\b <$ both $\ui$ and $\uj$. \end{thm} \subsection{Identification of $G/P_d$ with $G_{d,n}$}\label{fd} Let $G=SL_n(K)$. Let $P_d$ be the maximal parabolic subgroup \begin{equation*} P_d=\left\{ A\in G\biggm| A= \begin{pmatrix} *&*\\ 0_{(n-d)\times d}&* \end{pmatrix}\right\}. \end{equation*} For the natural action of $G$ on $\mathbb{P}(\wedge^d K^n)$, we have, the isotropy at $[e_1\wedge \cdots \wedge e_d]$ is $P_d$ while the orbit through $[e_1\wedge \cdots\wedge e_d]$ is $G_{d,n}$. Thus we obtain a surjective morphism $\pi:G\rightarrow G_{d,n},\ g\mapsto g\cdot a$, where $a=[e_1\wedge \cdots \wedge e_d]$. Further, the differential $d\pi_e :$ Lie$G \rightarrow T(G_{d,n})_a$ (= the tangent space to $G_{d,n}$ at $a$) is easily seen to be surjective. Hence we obtain an identification $f_d:G/P_d\cong G_{d,n}$ (cf. \cite{borel1}, Proposition 6.7). \subsection{Weyl Group and Root System}\label{weylroot} Let $G$ and $P_d$ be as above. Let $T$ be the subgroup of diagonal matrices in $G$, $B$ the subgroup of upper triangular matrices in $G$, and $B^-$ the subgroup of lower triangular matrices in $G$. Let $W$ be the Weyl group of $G$ relative to $T$, and $W_{P_d}$ the Weyl group of $P_d$. Note that $W = S_n$, the group of permutations of a set of $n$ elements, and that $W_{P_d} = S_d \times S_{n-d}$. For a permutation $w$ in $S_n$, $l(w)$ will denote the usual length function. Note also that $I_{d, n}$ can be identified with $W / W_{P_d}$. In the sequel, we shall identify $I_{d,n}$ with the set of ``minimal representatives" of $W / W_{P_d}$ in $S_n$; to be very precise, a $d$-tuple $\ui\in I_{d,n}$ will be identified with the element $(i_1,\dots,i_d,j_1,\dots,j_{n-d})\in\mathcal{S}_n$, where $\{j_1,\dots,j_{n-d}\}$ is the complement of $\{i_1,\dots,i_d\}$ in $\{1,\dots,n\}$ arranged in increasing order. We denote the set of such minimal representatives of $S_n$ by $W^{P_d}$. Let $R$ denote the root system of $G$ relative to $T$, and $R^+$ the set of positive roots relative to $B$. Let $R_{P_d}$ denote the root system of $P_d$, and $R_{P_d}^+$ the set of positive roots. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Schubert, Opposite Schubert, and Richardson Varieties in $G_{d,n}$}\label{sor} For $1\le t\le n$, let $V_t$ be the subspace of $K^n$ spanned by $\{ e_1,\dots,e_t\}$, and let $V^t$ be the subspace spanned by $\{ e_n, \ldots, e_{n-t+1}\}$. For each $\ui\in I_{d,n}$, the {\em Schubert variety $X_{\ui}$} and {\em Opposite Schubert variety $X^{\ui}$} associated to $\ui$ are defined to be \[ X_{\ui}=\{U\in G_{d,n}\mid \dim (U\cap V_{i_t})\ge t\ ,\ 1\le t\le d\}, \] \[ X^{\ui}=\{U\in G_{d,n}\mid \dim (U\cap V^{n-i_{(d-t+1)}+1})\ge t\ ,\ 1\le t\le d\}. \] For ${\ui}, {\uj} \in I_{d, n}$, the {\em Richardson Variety $X_{\ui}^{\uj}$}is defined to be $X_{\ui} \cap X^{\uj}$. For ${\ui}, {\uj}, {\ue}, {\uf} \in I_{d,n}$, where ${\ue}=(1, \ldots, d)$ and ${\uf}=(n+1-d, \ldots, n)$, note that $G_{d,n}=X_{\uf}^{\ue}$, $X_{\ui}=X_{\ui}^{\ue}$, and $X^{\ui}=X_{\uf}^{\ui}$. For the action of $G$ on $\mathbb{P}(\wedge^dK^n)$, the $T$-fixed points are precisely the points corresponding to the $T$-eigenvectors in $\wedge^dK^n$. Now \[ \wedge^dK^n=\bigoplus_{\ui\in I_{d,n}} Ke_{\ui} \text{\rm ,\quad as $T$-modules,} \] where for $\ui =(i_1,\cdots ,i_d),\ e_{\ui}=e_{i_1}\wedge \cdots \wedge e_{i_d}$. Thus the $T$-fixed points in $\mathbb{P}(\wedge^dK^n)$ are precisely $\left[ e_{\ui}\right]$, $\ui\in I_{d,n}$, and these points, obviously, belong to $G_{d,n}$. Further, the Schubert variety $X_{\ui}$ associated to $\ui$ is simply the Zariski closure of the $B$-orbit $B\left[ e_{\ui}\right]$ through the $T$-fixed point $\left[ e_{\ui}\right]$ (with the canonical reduced structure), $B$ being as in \S \ref{weylroot}. The opposite Schubert variety $X^{\ui}$ is the Zariski closure of the $B^-$-orbit $B^-\left[ e_{\ui}\right]$ through the $T$-fixed point $\left[ e_{\ui}\right]$ (with the canonical reduced structure), $B^-$ being as in \S \ref{weylroot}. \subsection{Bruhat Decomposition}\label{bruh} Let $V=K^n$. Let $\ui\in I_{d,n}$. Let $C_{\ui}=B[e_{\ui}]$ be the {\em Schubert cell} and $C^{\ui}=B^-[e_{\ui}]$ the {\em opposite Schubert cell} associated to $\ui$. The $C_{\ui}$'s provide a cell decomposition of $G_{d,n}$, as do the $C^{\ui}$'s. Let $X=V\oplus\dots\oplus V$ (d times). Let $$\pi:X\to\w^dV, (u_1,\dots,u_d)\mapsto u_1\w\dots\w u_d,$$ and $$p:\w^dV\setminus\{0\}\to\mathbb{P}(\w^dV), u_1\w\dots\w u_d\mapsto[u_1\w\dots\w u_d].$$ Let $v_{\ui}$ denote the point $(e_{i_1},\dots,e_{i_d})\in X$. Identifying $X$ with $M_{n\times d}$, $v_{\ui}$ gets identified with the $n\times d$ matrix whose entries are all zero except the ones in the $i_j$-th row and $j$-th column, $1\le j\le d$, which are equal to $1$. We have $$B\cdot v_{\ui}=\{ A\in M_{n\times d}\mid x_{ij}=0,i>i_j,\textrm{ and } \prod_t x_{i_t t}\ne 0\},$$ $$B^-\cdot v_{\ui}=\{ A\in M_{n\times d}\mid x_{ij}=0,ii_j\}$. Further, $\pi(B\cdot v_{\ui})=p^{-1}(C_{\ui})$, $\pi(D_{\ui})=\widehat{X_{\ui}}$, the cone over $X_{\ui}$. Denoting $\overline{B^-\cdot v_{\ui}}$ by $D^{\ui}$, we have $D^{\ui}=\{A\in M_{n\times d}\mid x_{ij}=0,i0$. Let \begin{equation*} 0 = \sum_{i=1}^rc_i F_i,\ c_i\in K^*\, ,\tag{$*$} \end{equation*} be a linear relation of standard monomials $F_i$ of degree $m$. Let $F_i=p_{w_{i1}}\dots p_{w_{im}}$. Suppose that $w_{i1}d(n-d)$ ($=\dim G_{d,n}$) and $l(w)=\text{dim}X_w$. If $F$ is standard, there is nothing to prove. Let $t$ be the first violation of standardness, i.e. $p_{w_1}\dots p_{w_{t-1}}$ is standard, but $p_{w_1}\dots p_{w_t}$ is not. Hence $w_{t-1}\not\ge w_t$, and using the quadratic relations (cf. Theorem \ref{T:greq}) \begin{equation*} p_{w_{t-1}}p_{w_t}=\sum_{\a,\b}\pm p_\a p_\b,\tag{*} \end{equation*} $F$ can be expressed as $F=\sum F_i$, with $N_{F_i}>N_F$ (since $\a>w_{t-1}$ for all $\a$ on the right hand side of $(*)$). Now the required result is obtained by decreasing induction on $N_F$ (the starting point of induction, i.e. the case when $N_F$ is the largest, corresponds to standard monomial $F=p_\te^m$, where $\te=(n+1-d,n+2-d,\cdots ,n)$, in which case $F$ is clearly standard). \end{proof} Combining Theorems \ref{indep} and \ref{generation}, we obtain \begin{thm}\label{mainth} Standard monomials on $X_w^v$ of degree $m$ give a basis for $R_w^v$ of degree $m$. \end{thm} As a consequence of Theorem \ref{mainth} (or also Theorem \ref{T:greq}), we have a qualitative description of a typical quadratic relation on a Richardson variety $X_w^v$ as given by the following \begin{prop}\label{qualitative} Let $w,\t,\varphi, v \in I_{d,n},\ w>\t,\varphi$ and $\t, \varphi >v$. Further let $\t,\varphi$ be non-comparable (so that $p_\t p_\varphi$ is a non-standard degree $2$ monomial on $X_w^v$). Let \begin{equation*} p_{\t}p_{\varphi}=\sum_{\a,\b}\ c_{\a,\b} p_\a p_\b,\ c_{\a,\b}\in k^*\, \tag{*} \end{equation*} be the expression for $p_{\t}p_{\varphi}$ as a sum of standard monomials on $X_w^v$. Then for every $\a,\b$ on the right hand side we have, $\a>$ both $\t\text{ and }\varphi$, and $\b<$ both $\t\text{ and }\varphi$. \end{prop} Such a relation as in (*) is called a {\em straightening relation}. \subsection{Equations Defining Richardson Varieties in the Grassmannian}\label{defneqns} Let \ni $w, v \in I_{d,n}$, with $w \ge v$. Let $\pi_w^v$ be the map $R_0\to R_w^v$ (the restriction map). Let $\ker \pi_w^v=J_w^v$. Let $Z_w^v=\{\text{all standard monomials }F\ |\ F \text{ contains some } p_\varphi$ for some $w \not\ge \varphi \text{ or } \varphi \not\ge v \}$. We shall now give a set of generators for $J_w^v$ in terms of Pl\"ucker coordinates. \begin{lem}\label{ideal} Let $I_w^v=(p_\varphi,w \not\ge \varphi \text{ or } \varphi \not\ge v)$ (ideal in $R_0$). Then $Z_w^v$ is a basis for $I_w^v$. \end{lem} \begin{proof} Let $F\in I_w^v$. Then writing $F$ as a linear combination of standard monomials $$F=\sum a_i F_i +\sum b_j G_j,$$ where in the first sum each $F_i$ contains some $p_{\t}$, with $w \not\ge \t \text{ or } \t \not\ge v $, and in the second sum each $G_j$ contains only coordinates of the form $p_{\t}$, with $w \ge \t \ge v$. This implies that $\sum a_i F_i\in I_w^v$, and hence we obtain $$\sum b_jG_j\in I_w^v.$$ This now implies that considered as an element of $R_w^v$, $\sum b_jG_j$ is equal to $0$ (note that $I_w^v \subset J_w^v$). Now the linear independence of standard monomials on $X_w^v$ implies that $b_j=0$ for all $j$. The required result now follows. \end{proof} \begin{prop}\label{eqns} Let $w, v \in I_{d,n}$ with $w \ge v$. Then $R_w^v = R_0/I_w^v$. \end{prop} \begin{proof} We have, $R_w^v = R_0/J_w^v$ (where $J_w^v$ is as above). We shall now show that the inclusion $I_w^v \subset J_w^v$ is in fact an equality. Let $F\in R_0$. Writing $F$ as a linear combination of standard monomials $$F=\sum a_i F_i+\sum b_j G_j,$$ where in the first sum each $F_i$ contains some term $p_{\t}$, with $w \not\ge \t \text{ or } \t \not\ge v $, and in the second sum each $G_j$ contains only coordinates $p_{\t}$, with $w \ge \t \ge v$, we have, $\sum a_i F_i\in I_w^v$, and hence we obtain \ni $F\in J_w^v$ \ni $\Longleftrightarrow \sum b_j G_j \in J_w^v$ (since $\sum a_i F_i \in I_w^v$, and $I_w^v \subset J_w^v$) \ni $ \Longleftrightarrow \pi_w^v(F)$ $(=\sum b_jG_j)$ is zero \ni $\Longleftrightarrow \sum b_jG_j$ (= a sum of standard monomials on $X_w^v$) is zero on $X_w^v$ \ni $\Longleftrightarrow b_j=0$ for all $j$ (in view of the linear independence of standard monomials on $X_w^v$) \ni $\Longleftrightarrow F=\sum a_i F_i$ \ni $\Longleftrightarrow F\in I_w^v$ . \ni Hence we obtain $J_w^v=I_w^v$. \end{proof} \ni {\bf Equations defining Richardson varieties:} Let $w, v \in I_{d,n}$, with $w \ge v$. By Lemma \ref{ideal} and Proposition \ref{eqns}, we have that the kernel of $(R_0)_1\to (R_w^v)_1$ has a basis given by $\{p_{\t} \mid w \not\ge \t \text{ or } \t \not\ge v \}$, and that the ideal $J_w^v$ (= the kernel of the restriction map $R_0\to R_w^v$) is generated by $\{p_{\t} \mid w \not\ge \t \text{ or } \t \not\ge v \}$. Hence $J_w^v$ is generated by the kernel of $(R_0)_1\to (R_w^v)_1$. Thus we obtain that $X_w^v$ is scheme-theoretically (even at the cone level) the intersection of $G_{d,n}$ with all hyperplanes in $\mathbb{P}(\wedge^dk^n)$ containing $X_w^v$. Further, as a closed subvariety of $G_{d,n}$, $X_w^v$ is defined (scheme-theoretically) by the vanishing of $\{ p_{\t} \mid w \not\ge \t \text{ or } \t \not\ge v\}$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section[Standard Monomial Theory for a Union of Richardson Varieties]{Standard Monomial Theory for a Union of Richardson Varieties}\label{unn} In this section, we prove results similar to Theorems \ref{indep} and \ref{mainth} for a union of Richardson varieties. Let $X_i$ be Richardson varieties in $G_{d,n}$. Let $X=\cup X_i$. \begin{defn} A monomial $F$ in the Pl\" ucker coordinates is {\em standard} on the union $X=\cup X_i$ if it is standard on some $X_i$. \end{defn} \subsection{Linear Independence of Standard Monomials on $X=\cup X_i$} \begin{thm}\label{indep'} Monomials standard on $X=\cup X_i$ are linearly independent. \end{thm} \begin{proof} If possible, let \begin{equation*} 0 = \sum_{i=1}^r a_i F_i, \ a_i \in K^*\tag{*} \end{equation*} be a nontrivial relation among standard monomials on $X$. Suppose $F_1$ is standard on $X_j$. Then restricting $(*)$ to $X_j$, we obtain a nontrivial relation among standard monomials on $X_j$, which is a contradiction (note that for any $i,\ F_i|_{X_j}$ is either $0$ or remains standard on $X_j$; further, $F_1|_{X_j}$ is non-zero). \end{proof} \subsection{Standard Monomial Basis} \begin{thm}\label{union} Let $X=\cup_{i=1}^r\,X_{w_i}^{v_i}$, and $S$ the homogeneous coordinate ring of $X$. Then the standard monomials on $X$ give a basis for $S$. \end{thm} \begin{proof} For $w, v \in I_{d,n}$ with $w \ge v$, let $I_w^v$ be as in Lemma \ref{ideal}. Let us denote $I_t=I_{w_t}^{v_t}, X_t=X_{w_t}^{v_t}, \, 1\le t\le r$. We have $R_{w_t}^{v_t}=R_0/I_t$ (cf. Proposition \ref{eqns}). Let $S=R_0/I$. Then $I=\cap I_t$ (note that being the intersection of radical ideals, $I$ is also a radical ideal, and hence the set theoretic equality $X=\cup X_i$ is also scheme theoretic). A typical element in $R_0/I$ may be written as $\pi (f)$, for some $f\in R_0$, where $\pi$ is the canonical projection $R_0\to R_0/I$. Let us write $f$ as a sum of standard monomials $$f=\sum a_j G_j+\sum b_lH_l,$$ where each $G_j$ contains some $p_{\t_j}$ such that $w_i \not\ge \t_j \text { or } \t_j \not\ge v_i$, for $1\le i\le r$; and for each $H_l$, there is some $i_l$, with $1\le i_l\le r$, such that $H_l$ is made up entirely of $p_{\t}$'s with $w_{i_l} \ge \t \ge v_{i_l}$. We have $\pi(f)=\sum b_l H_l$ (since $\sum a_j G_j\in I$). Thus we obtain that $S$ (as a vector space) is generated by monomials standard on $X$. This together with the linear independence of standard monomials on $X$ implies the required result. \end{proof} \subsection{Consequences}\label{consq} \begin{thm}\label{reduced} Let $X_1$, $X_2$ be two Richardson varieties in $G_{d,n}$. Then $(1)$ $X_1\cup X_2$ is reduced. $(2)$ If $X_1\cap X_2\ne\emptyset$, then $X_1\cap X_2$ is reduced. \end{thm} \begin{proof} $(1)$ Assertion is obvious. $(2)$ Let $X_1=X_{w_1}^{v_1}$, $X_2=X_{w_2}^{v_2}$, $I_1=I_{w_1}^{v_1}$, and $I_2=I_{w_2}^{v_2}$. Let $A$ be the homogeneous coordinate ring of $X_1\cap X_2$. Let $A=R_0/I$. Then $I=I_1+I_2$. Let $F\in I$. Then by Lemma \ref{ideal} and Proposition \ref{eqns}, in the expression for $F$ as a linear combination of standard monomials $$F=\sum a_j F_j,$$ each $F_j$ contains some $p_\t$, where either ($(w_1 \text{ or } w_2) \not \ge \t$) or ($\t \not\ge (v_1 \text{ or } v_2)$). Let $X_1\cap X_2= X_{\m}^{\n}$ set theoretically, where $\m = w_1 \wedge w_2$ and $\n = v_1 \vee v_2$ (cf. Remark \ref{intersect}). If $B=R_0/\sqrt{I}$, then by Lemma \ref{ideal} and Proposition \ref{eqns}, under $\pi:R_0\to B$, $\ker\pi$ consists of all $f$ such that $f=\sum c_k f_k$, $f_k$ being standard monomials such that each $f_k$ contains some $p_{\varphi}$, where $\m \not\ge\varphi\text{ or } \varphi\not\ge \n$. Hence either ($(w_1 \text{ or } w_2) \not \ge \varphi$) or ($\varphi \not\ge (v_1 \text{ or } v_2)$). Hence $\sqrt{I}=I$, and the required result follows from this. \end{proof} \begin{defn} Let $w > v$. Define $\partial^+X_w^v := \bigcup\limits_{w > w' \geq v}X_{w'}^v$, and $\partial^-X_w^v := \bigcup\limits_{w \geq v' > v}X_w^{v'}$. \end{defn} \begin{thm}\label{pieri}(Pieri's formulas) Let $w > v$. \begin{enumerate} \item $X_w^v\cap\{p_w=0\}= \partial^+ X_w^v,\quad\text{scheme theoretically}.$ \item $X_w^v\cap\{p_v=0\}= \partial^- X_w^v,\quad\text{scheme theoretically}.$ \end{enumerate} \end{thm} \begin{proof} Let $X=\partial^+X_w^v$, and let $A$ be the homogeneous coordinate ring of $X$. Let $A=R_w^v/I$. Clearly, $(p_w) \subseteq I,\ (p_w)$ being the principal ideal in $R_w^v$ generated by $p_w$. Let $f\in I$. Writing $f$ as $$f=\sum b_i G_i+\sum c_j H_j,$$ where each $G_i$ is a standard monomial in $R_w^v$ starting with $p_w$ and each $H_j$ is a standard monomial in $R_w^v$ starting with $p_{\te_{j1}}$, where $\te_{j1} v$. Then $$ \begin{aligned} s(w,v,m)&=s(w,v,m-1)+s\left(\partial^+X_w^v,m\right) \\ &= s(w,v,m-1)+s\left(\partial^-X_w^v,m\right) \end{aligned} $$. \end{lem} \ni (1) and (2) are easy consequences of the results of the previous section. Let $X$ be a closed subvariety of $G_{d,n}$. Let $L=p^*(\mathcal{O}_{\mathbb{P}}(1))$, where $\mathbb{P}=\mathbb{P}(\wedge^dK^n)$, and $p:X\hookrightarrow\mathbb{P}$ is the Pl\"ucker embedding restricted to $X$. \begin{prop}\label{inductive} Let $r$ be an integer $\le d(n-d)$. Suppose that all Richardson varieties $X$ in $G_{d,n}$ of dimension at most $r$ satisfy the following two conditions: $(1)$ $H^i(X,L^m)=0$, for $i\ge 1$, $m\ge 0$. $(2)$ The set $S(X,m)$ is a basis for $H^0(X,L^m),\ m\ge 0$. \ni Then any union of Richardson varieties of dimension at most $r$ which have nonempty intersection, and any nonempty intersection of Richardson varieties, satisfy $(1)$ and $(2)$. \end{prop} \begin{proof} The proof for intersections of Richardson varieties is clear, since any nonempty intersection of Richardson varieties is itself a Richardson variety (cf. Lemma \ref{inter} and Theorem \ref{reduced}). We will prove the result for unions by induction on $r$. Let $S_r$ denote the set of Richardson varieties $X$ in $G_{d,n}$ of dimension at most $r$. Let $Y=\cup_{j=1}^t X_j$, $X_j\in S_r$. Let $Y_1=\cup_{j=1}^{t-1} X_j$, and $Y_2=X_t$. Consider the exact sequence $$0\to\mathcal{O}_Y\to\mathcal{O}_{Y_1}\oplus \mathcal{O}_{Y_2}\to\mathcal{O}_{Y_1\cap Y_2}\to 0,$$ where $\mathcal{O}_Y\to\mathcal{O}_{Y_1}\oplus \mathcal{O}_{Y_2}$ is the map $f\mapsto (f\res_{Y_1},f\res_{Y_2})$ and $\mathcal{O}_{Y_1}\oplus \mathcal{O}_{Y_2}\to\mathcal{O}_{Y_1\cap Y_2}$ is the map $(f,g)\mapsto (f-g)\res_{Y_1\cap Y_2}$. Tensoring with $L^m$, we obtain the long exact sequence $$\to H^{i-1}(Y_1\cap Y_2,L^m)\to H^i(Y,L^m)\to H^i(Y_1,L^m)\oplus H^i(Y_2,L^m)\to H^i(Y_1\cap Y_2,L^m)\to$$ Now $Y_1\cap Y_2$ is reduced (cf. Theorem \ref{reduced}) and $Y_1\cap Y_2\in S_{r-1}$. Hence, by the induction hypothesis $(1)$ and $(2)$ hold for $Y_1\cap Y_2$. In particular, if $m\ge 0$, then $(2)$ implies that the map $H^0(Y_1,L^m)\oplus H^0(Y_2,L^m)\to H^0(Y_1\cap Y_2,L^m)$ is surjective. Hence we obtain that the sequence $$0\to H^0(Y,L^m)\to H^0(Y_1,L^m)\oplus H^0(Y_2,L^m)\to H^0(Y_1\cap Y_2, L^m)\to 0$$ is exact. This implies $H^0(Y_1\cap Y_2, L^m)\to H^1(Y,L^m)$ is the zero map; we have, $H^1(Y,L^m)\to H^1(Y_1,L^m)\oplus H^1(Y_2,L^m)$ is also the zero map (since by induction $H^1(Y_1,L^m)=0=H^1(Y_2,L^m)$). Hence we obtain $H^1(Y,L^m)=0, \ m\ge 0$, and for $i\ge 2$, the assertion that $H^i(Y,L^m)=0,\ m\ge 0$ follows from the long exact cohomology sequence above (and induction hypothesis). This proves the assertion (1) for $Y$. To prove assertion (2) for $Y$, we observe $$ \begin{aligned} h^0(Y,L^m)&=h^0(Y_1,L^m)+h^0(Y_2,L^m)-h^0(Y_1\cap Y_2,L^m) \\ &=s(Y_1,m)+s(Y_2,m)-s(Y_1\cap Y_2,m). \end{aligned} $$ Hence Lemma \ref{count} implies that $$h^0(Y,L^m)=s(Y,L^m).$$ This together with linear independence of standard monomials on $Y$ proves assertion (2) for $Y$. \end{proof} \begin{thm}\label{main} Let $X$ be a Richardson variety in $G_{d,n}$. Then (a) $H^i(X,L^m)=0$ for $i\ge 1$, $m\ge 0$. (b) $S(X,m)$ is a basis for $H^0(X,L^m)$, $m\ge 0$. \end{thm} \begin{proof} We prove the result by induction on $m, \text{ and }\dim X$. If $\dim X=0$, $X$ is just a point, and the result is obvious. Assume now that $\dim X\ge 1$. Let $X=X_w^v, \, w > v$. Let $Y=\partial^+{X_w^v}$. Then by Pieri's formula (cf. \S \ref{pieri}), we have, $$Y=X(\t)\cap\{p_\t=0\}\quad\text{\rm (scheme theoretically)}.$$ Hence the sequence $$0\to\mathcal{O}_X(-1)\to\mathcal{O}_X\to\mathcal{O}_Y\to 0$$ is exact. Tensoring it with $L^m$, and writing the cohomology exact sequence, we obtain the long exact cohomology sequence $$\dots\to H^{i-1}(Y,L^m)\to H^i(X,L^{m-1})\to H^i(X,L^m)\to H^i(Y,L^m)\to\cdots.$$ Let $m\ge 0$, $i\ge 2$. Then the induction hypothesis on $\dim X$ implies (in view of Proposition \ref{inductive}) that $H^i(Y,L^{m})=0$, $i\ge 1$. Hence we obtain that the sequence $0\to H^i(X,L^{m-1})\to H^i(X,L^{m})$, $i\ge 2$, is exact. If $i=1$, again the induction hypothesis implies the surjectivity of $H^0(X, L^m)\to H^0(Y, L^m)$. This in turn implies that the map $H^0(Y,L^m)\to H^1(X,L^{m-1})$ is the zero map, and hence we obtain that the sequence $0\to H^1(X, L^{m-1})\to H^1(X,L^m)$ is exact. Thus we obtain that $0\to H^i(X,L^{m-1})\to H^i(X,L^{m})$, $m\ge 0$, $i\ge 1$ is exact. But $H^i(X,L^m)=0$, $m\gg 0$, $i\ge 1$ (cf. \cite{fac}). Hence we obtain \begin{equation*} H^i(X,L^m)=0 \text{ for } i\ge 1,\ m\ge 0,\tag{1} \end{equation*} and \begin{equation*} h^0(X,L^m)=h^0(X,L^{m-1})+h^0(Y,L^m).\tag{2} \end{equation*} In particular, assertion (a) follows from (1). The induction hypothesis on $m$ implies that $h^0(X,L^{m-1})=s(X,m-1)$. On the other hand, the induction hypothesis on $\dim X$ implies (in view of Proposition \ref{inductive}) that $h^0(Y,L^m)=s(Y,m)$. Hence we obtain \begin{equation*} h^0(X,L^m)=s(X,m-1)+s(Y,m).\tag{3} \end{equation*} Now $(3)$ together with Lemma \ref{count},(2) implies $h^0(X,L^m)=s(X,m)$. Hence (b) follows in view of the linear independence of standard monomials on $X_w^v$ (cf. Theorem \ref{indep}). \end{proof} \begin{cor}\label{6.6.0.7} We have \ni 1. $R_w^v=\bigoplus\limits_{m\in \mathbb{Z}^+}\ H^0(X_w^v,L^{m}),\ w \geq v.$ \ni 2. $\dim H^0 (\partial^+ X_w^v, L^m) = \dim H^0 (\partial^- X_w^v, L^m), \,w>v, \, m \ge 0.$ \end{cor} \begin{proof} Assertion 1 follows immediately from Theorems \ref{mainth} and \ref{main}(b). Assertion 2 follows from Lemma \ref{count}, Theorem \ref{inductive}(2), and Theorem \ref{main}(b). \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Tangent Space and Smoothness}\label{tss} \subsection{The Zariski Tangent Space} Let $x$ be a point on a variety $X$. Let $\mathfrak{m}_x$ be the maximal ideal of the local ring $\mathcal{O}_{X,\,x}$ with residue field $K(x)$($=\mathcal{O}_{X,\,x}/\mathfrak{m}_x$). Note that $K(x)=K$ (since $K$ is algebraically closed). Recall that the Zariski tangent space to $X$ at $x$ is defined as $$ \begin{aligned} T_x(X)&=\text{\rm Der}_K(\mathcal{O}_{X,\,x},K(x))\\ &=\{D:\mathcal{O}_{X,\,x}\to K(x), \ K\text{\rm -linear such that }D(ab)=D(a)b+aD(b)\} \end{aligned} $$ (here $K(x)$ is regarded as an $\mathcal{O}_{X,\,x}$-module). It can be seen easily that $T_x(X)$ is canonically isomorphic to $\text{\rm Hom}_{K\text{\rm -mod}}(\mathfrak{m}_x/\mathfrak{m}_x^2,K)$. \subsection{Smooth and Non-smooth Points} A point $x$ on a variety $X$ is said to be a {\em simple} or {\em smooth} or {\em nonsingular point of $X$} if $\mathcal{O}_{X,\,x}$ is a regular local ring. A point $x$ which is not simple is called a {\em multiple} or {\em non-smooth} or {\em singular point} of $X$. The set $\Sing X=\{x\in X\mid x\ \text{\rm is a singular point}\}$ is called the {\em singular locus of $X$}. A variety $X$ is said to be {\em smooth} if $\text{\rm Sing}X=\emptyset$. We recall the well known \begin{thm}\label{1.2.37.1} \label{greater} Let $x\in X$. Then $\dim_K T_x(X)\ge\dim\mathcal{O}_{X,\,x}$ with equality if and only if $x$ is a simple point of $X$. \end{thm} \subsection{The Space $T_{w,\t}^v$} Let $G,T,B,P_d,W,R,W_{P_d}, R_{P_d}$ etc., be as in \S \ref{weylroot}. We shall henceforth denote $P_d$ by just $P$. For $\a \in R$, let $X_\a$ be the element of the Chevalley basis for $\mathfrak{g} \ (={\rm Lie}G$), corresponding to $\a$. We follow \cite{bou} for denoting elements of $R,R^{+}$ etc. For $w \geq \t \geq v$, let $T_{w, \t}^v $ be the Zariski tangent space to $X_w^v$ at $e_{\t} $. Let $w_0$ be the element of largest length in $W$. Now the tangent space to $G$ at $e_{id}$ is $\mathfrak g$, and hence the tangent space to $G/P$ at $e_{id}$ is $\bigoplus\limits_{\b\in R^+\setminus R_P^+}\ {{{\mathfrak g}}}_{-\b}$. For $\t\in W$, identifying $G/P$ with $G/^{\t}P$ (where $^{\t}P=\t P\t ^{-1}$) via the map $gP\mapsto (n_\t gn_\t^{-1}) {}^{\t}P,\ n_\t$ being a fixed lift of $\t$ in $N_G(T)$, we have, the tangent space to $G/P$ at $e_\t$ is $\bigoplus\limits_{\b \in \t(R^+)\setminus \t(R_P^+)}\ {{{\mathfrak g}}}_{-\b}$, i.e., $$T_{w_0,\t}^{\text{id}} =\bigoplus\limits_{\b \in \t(R^+)\setminus \t(R_P^+)}\ {{{\mathfrak g}}}_{-\b}. $$ Set $$N_{w,\t}^v=\{ \b \in \t (R^+)\setminus \t(R_P^+)\ |\ X_{-\b }\in T_{w,\t}^v \} .$$ Since $T_{w, \t}^v$ is a $T$-stable subspace of $T_{w_0,\t}^v$, we have $$T_{w,\t}^v =\text{ the span of }\{X_{-\b},\ \b \in N_{w,\t}^v\}.$$ \subsection{Certain Canonical Vectors in $T_{w,\t}^v$} \label{curves} For a root $\a \in R^+\setminus R_P^+$, let $Z_\a$ denote the $SL(2)$-copy in $G$ corresponding to $\a$; note that $Z_\a$ is simply the subgroup of $G$ generated by $U_\a$ and $U_{-\a}$. Given $x\in W^P$, precisely one of $\{U_\a, U_{-\a}\}$ fixes the point $e_x$. Thus $Z_\a\cdot e_x$ is a $T$-stable curve in $G/P$ (note that $Z_\a\cdot e_x\cong {\mathbb{P}}^1$), and conversely any $T$-stable curve in $G/P$ is of this form (cf. \cite{C2}). Now a $T$-stable curve $Z_\a\cdot e_x$ is contained in a Richardson variety $X_w^v$ if and only if $e_x, e_{s_\a x}$ are both in $X_w^v$. \begin{lem}\label{curve} Let $w,\t,v \in W^P,\ w\ge \t \ge v$. Let $\b \in \t(R^+\setminus R_P^+)$. If $w\ge s_\b\t \ge v \, (\MMod W_P)$, then $X_{-\b} \in T_{w,\t}^v$. \end{lem} \ni (Note that $s_{\b}\t$ need not be in $W^P$.) \begin{proof} The hypothesis that $w\ge s_\b\t \ge v (\MMod W_P)$ implies that the curve $Z_\b\cdot e_\t$ is contained in $X_w^v$. Now the tangent space to $Z_\b\cdot e_\t$ at $e_\t$ is the one-dimensional span of $X_{-\b}$. The required result now follows. \end{proof} We shall show in Theorem \ref{tgt} that $w,\t,v$ being as above, $T_{w,\t}^v$ is precisely the span of $\{X_{-\b},\ \b \in \t(R^+\setminus R_P^+ )\ |\ w\ge s_\b\t \ge v \, (\MMod W_P) \}$. \subsection{A Canonical Affine Neighborhood of a $T$-fixed Point }\label{neigh} Let $\t \in W$. Let $U^{-}_\t$ be the unipotent subgroup of $G$ generated by the root subgroups $U_{-\b},\ \b \in \t(R^+)$ (note that $U^-_{\t}$ is the unipotent part of the Borel sub group $^{\t}B^-$, opposite to $^{\t}B\ (= \t B\t^{-1})$). We have $$U_{-\b} \cong \mathbb{G}_a,\ U^{-}_\t \cong \prod\limits_{\b \in \t(R^+)}\ U_{-\b}. $$ Now, $U^-_{\t}$ acts on $G/P$ by left multiplication. The isotropy subgroup in $U^-_{\t}$ at $e_\t$ is $\Pi_{\b \in \t(R_P^+)}\ U_{-\b}$. Thus $U_{\t}^- e_\t \cong \Pi_{\b \in \t(R^+\setminus R_P^+)}\ U_{-\b}$. In this way, $U^-_{\t} e_\t$ gets identified with $\mathbb{A}^{N}$, where $N=\# (R^+\setminus R_P^+)$. We shall denote the induced coordinate system on $U^{-}_\t e_\t$ by $\{x_{-\b},\ \b \in \t(R^+\setminus R_P^+)\}$. In the sequel, we shall denote $U^- _{\t} e_{\t}$ by ${\mathcal O}^{-}_{\t}$ also. Thus we obtain that ${\mathcal O}^{-}_{\t}$ is an affine neighborhood of $e_{\t}$ in $G/P$. \subsection{The Affine Variety $Y_{w,\tau}^v$}\label{ywt} For $w, \t, v \in W,\ w\ge \t \ge v$, let us denote $Y_{w,\t}^v:={\mathcal O}^{-}_{\t}\cap X_{w, \t}^v$. It is a nonempty affine open subvariety of $X_w^v$, and a closed subvariety of the affine space ${\mathcal O}^{-}_{\t}$. Note that $L$, the ample generator of $\textrm{Pic}(G/P)$, is the line bundle corresponding to the Pl\"ucker embedding, and $H^0(G/P,L)=(\wedge^dK^n)^*$, which has a basis given by the Pl\"ucker coordinates $ \{ p_{\te}, \, \te \in I_{d, n} \}$. Note also that the affine ring ${\mathcal O}^{-}_{\t}$ may be identified as the homogeneous localization $(R_0)_{(p_{\t})}$, $R_0$ being as in \S \ref{sec:sm3.1}. We shall denote $p_{\te}/p_{\t}$ by $f_{\te,\t}$. Let $I_{w, \t}^v$ be the ideal defining $Y_{w, \t}^v$ as a closed subvariety of ${\mathcal O}^{-}_{\t}$. Then $I_{w, \t}^v$ is generated by $\{f_{\te, \t} \mid w \not\geq \te \text{ or } \te \not\le v \}$. \subsection{Basis for Tangent Space \& Criterion for Smoothness of $X_w^v$ at $e_\t$} Let $Y$ be an affine variety in $\mathbb{A}^n$, and let $I(Y)$ be the ideal defining $Y$ in $\mathbb{A}^{n}$. Let $I(Y)$ be generated by $\{f_1, f_2, \ldots, f_r\}$. Let $J$ be the Jacobian matrix $(\frac{\partial f_i }{\partial x_j})$. We have (cf. Theorem \ref{greater}) the dimension of the tangent space to $Y$ at a point $P$ is greater than or equal to the dimension of $Y$, with equality if and only if $P$ is a smooth point; equivalently, rank $J_P \leq \text{codim}_{\mathbb{A}^n}Y$ with equality if and only if $P$ is a smooth point of $Y$ (here $J_P$ denotes $J$ evaluated at $P$). Let $w,\t, v \in W,\ w\ge\t\ge v$. The problem of determining whether or not $e_{\tau}$ is a smooth point of $X_w^v$ is equivalent to determining whether or not $e_{\tau}$ is a smooth point of $Y_{w,\tau}^v$ (since $Y_{w,\t}^v$ is an open neighborhood of $e_{\tau}$ in $X_w^v$). In view of Jacobian criterion, the problem is reduced to computing $(\partial f_{\te, \t} / \partial x_{-\b})_{e_\t}, \, w \not\ge \te \text{ or } \te \not\ge v$ (the Jacobian matrix evaluated at $e_\tau$). To carry out this computation, we first observe the following: Let $V$ be the $G$-module $H^0(G/P,L) \ ( = (\wedge^dK^n)^*)$. Now $V$ is also a ${\mathfrak g}$-module. Given $X$ in ${\mathfrak g}$, we identify $X$ with the corresponding right invariant vector field $D_X$ on $G$. Thus we have $D_X p_{\te} = X p_{\te}$, and we note that \begin{equation*} (\partial f_{\te, \t} / \partial x_{-\b})({e_\t}) = X_{-\b}\,p_{\te} (e_\t), \ \b \in \t(R^+\setminus R_P^+), \end{equation*} where the left hand side denotes the partial derivative evaluated at $e_\t$. We make the folowing three observations: (1) For $\te,\mu\in W^P$, $p_\te(e_\mu)\ne 0\iff\te=\mu$, where, recall that for $\te =(i_1\cdots i_d) \in W^P,\,\ e_\te$ denotes the vector $e_{i_1}\wedge \cdots \wedge e_{i_d}$ in $\wedge^d K^n$, and $p_\te$ denotes the Pl\"ucker coordinate associated to $\te$. (2) Let $X_\a$ be the element of the Chevalley basis of $\mathfrak{g}$, corresponding to $\a \in R$. If $X_\a p_\mu\ne 0$, $\mu\in W^P$, then $X_\a p_\mu=\pm p_{s_\a \mu}$, where $s_\a$ is the reflection corresponding to the root $\a$. (3) For $\a\ne \b$, if $X_\a p_\mu,\ X_\b p_\mu$ are non-zero, then $X_\a p_\mu \ne X_\b p_\mu$. The first remark is obvious, since $\{p_\te\mid\te\in W^P\}$ is the basis of $(\wedge^P K^n)^* \ (=H^0(G/P,L))$, dual to the basis $\{e_\varphi,\ \varphi \in I_{d,n}\}$ of $\wedge^d K^n$. The second remark is a consequence of $SL_2$ theory, using the following facts: (a) $|\langle\chi,\a^*\rangle|=\left|\frac{2(\chi,\a)}{(\a,\a)}\right|=0\ \text{\rm or } 1$, $\chi$ being the weight of $p_\mu$. (b) $p_\mu$ is the lowest weight vector for the Borel subgroup ${}^\mu B=\mu B\mu^{-1}$. The third remark follows from weight considerations (note that if $X_\a p_\mu \ne 0$, then $X_\a p_\mu$ is a weight vector (for the $T$-action) of weight $\chi +\a$, $\chi$ being the weight of $p_\mu$). \begin{thm}\label{jacobian} Let $w, \t, v \in W^P,\ w \ge \t \ge v$. Then $$\dim T_{w,\t}^v=\#\{\g\in \t(R^+\setminus R_P^+)\mid w\ge s_\g \t \ge v \, (\MMod W_P)\}.$$ \end{thm} \begin{proof} By Lemma \ref{ideal} and Proposition \ref{eqns}, we have, $I_{w,\t}^v$ is generated by $\{f_{\te, \t} \mid w \not\ge\te \text{ or } \te \not \ge v \}$. Denoting the affine coordinates on $\mathcal{O}^{-}_\t $ by $x_{-\b},\ \b \in \t(R^+\setminus R_P^+)$, we have the evaluations of $\frac{\partial f_{\te, \t}}{\partial x_\b}$ and $X_\b p_\te$ at $e_\t$ coincide. Let $J_w^v$ denote the Jacobian matrix of $Y_{w,\t}^v$ (considered as a subvariety of the affine space $\mathcal{O}^{-}_\t $). We shall index the rows of $J_w^v$ by $\{f_{\te, \t} \mid w \not\ge\te \text{ or } \te \not \ge v \}$ and the columns by $x_{-\b},\ \b \in \t(R^+\setminus R_P^+)$. Let $J_w^v(\t)$ denote $J_w^v$ evaluated at $e_\t$. Now in view of (1) \& (2) above, the $(f_{\te,\t},x_{-\b})$-th entry in $J_w^v(\t)$ is non-zero if and only if $X_\b p_\te=\pm p_\t$. Hence in view of (3) above, we obtain that in each row of $J_w^v(\t)$, there is at most one non-zero entry. Hence rank$J_w^v(\t)=$ the number of non-zero columns of $J_w^v(\t)$. Now, $X_\b p_\te=\pm p_\t$ if and only if $\te \equiv s_\b \t \, (\MMod W_P)$. Thus the column of $J_w^v(\t)$ indexed by $x_{-\b}$ is non-zero if and only if $w\not\ge s_\b \t \, (\MMod W_P)$ or $s_\b \t \not\ge v \, (\MMod W_P) $. Hence rank$J_w^v(\t)=\#\{\g \in \t(R^+\setminus R_P^+)\ |\ w\not\ge s_\g \t \, (\MMod W_P) \text{ or } s_\g \t \not\ge v \, (\MMod W_P)\}$ and thus we obtain $$\dim T_{w,\t}^v=\#\{\g\in \t(R^+\setminus R_P^+)\mid w\ge s_\g \t \ge v \, (\MMod W_P) \}.$$ \end{proof} \begin{thm}\label{tgt} Let $w, \t, v$ be as in Theorem \ref{jacobian}. Then $\{X_{-\b},\ \b \in \t(R^+\setminus R_P^+)\ |\ w\ge s_\b \t \ge v \, (\MMod W_P)\}$ is a basis for $T_{w,\t}^v$. \end{thm} \begin{proof} Let $\b \in \t(R^+\setminus R_P^+)$ be such that $w\ge s_\b \t \ge v \, (\MMod W_P)$. We have (by Lemma \ref{curve}), $X_{-\b} \in T_{w,\t}^v$. On the other hand, by Theorem \ref{jacobian}, \ni $\dim T_{w,\t}^v=\#\{\b\in \t(R^+\setminus R_P^+)\mid w\ge s_\b \t \ge v \, (\MMod W_P)\}.$ The result follows from this. \end{proof} \begin{cor}\label{smcrit1} $X_w^v$ is smooth at $e_\t$ if and only if $l(w)-l(v)=\#\{\a \in R^+\setminus R_P^+ \mid w\ge \t s_\a \ge v \, (\MMod W_P) \}$. \end{cor} \begin{proof} We have, $X_w^v$ is smooth at $e_\t$ if and only if $\dim T_{w,\t}^v=\dim X_w^v$, and the result follows in view of Corollary \ref{morerich} and Theorem \ref{jacobian} (note that if $\b=\t(\a)$, then $s_\b \t=\t s_\a \, (\MMod W_P)$). \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Multiplicity at a Singular Point}\label{mas} \subsection{Multiplicity of an Algebraic Variety at a Point} Let $B$ be a graded, affine $K$-algebra such that $B_1$ generates $B$ (as a $K$-algebra). Let $X = \textrm{Proj}(B)$. The function $h_B(m)$ (or $h_X(m)$) $= \textrm{dim}_KB_m$, $m \in \mathbb{Z}$ is called the {\it Hilbert function} of $B \,(\textrm{or } X)$. There exists a polynomial $P_B(x)$ (or $P_X(x)$) $\in \mathbb{Q}[x]$, called the {\it Hilbert polynomial} of $B\, (\textrm{or } X)$, such that $f_B(m) = P_B(m)$ for $m \gg 0$. Let $r$ denote the degree of $P_B(x)$. Then $r = \textrm{dim}(X)$, and the leading coefficient of $P_B(x)$ is of the form $c_B/r!$, where $c_B \in \mathbb{N}$. The integer $c_B$ is called the {\it degree of $X$}, and denoted $\textrm{deg}(X)$ (see \cite{eis} for details). In the sequel we shall also denote $\textrm{deg}(X)$ by $\textrm{deg}(B)$. Let $X$ be an algebraic variety, and let $P \in X$. Let $A = \mathcal{O}_{X, P}$ be the stalk at $P$ and $\mathfrak{m}$ the unique maximal ideal of the local ring $A$. Then the {\it tangent cone} to $X$ at $P$, denoted $\textrm{TC}_P(X)$, is $\textrm{Spec}(\textrm{gr}(A,\mathfrak{m}))$, where $\textrm{gr}(A,\mathfrak{m}) = \oplus_{j=0}^{\infty}\mathfrak{m}^j/\mathfrak{m}^{j+1}$. The {\it multiplicity} of $X$ at $P$, denoted $\Mult_P(X)$, is $\textrm{deg}(\textrm{Proj}(\textrm{gr}(A,\mathfrak{m})))$. (If $X \subset K^n$ is an affine closed subvariety, and $m_P \subset K[X]$ is the maximal ideal corresponding to $P \in X$, then $\textrm{gr}(K[X], m_P) = \textrm{gr}(A,\mathfrak{m})$.) \subsection{Evaluation of Pl\" ucker Coordinates on $U_\t^-e_\t$}\label{evpl} Let $X=X_w^v$. Consider a $\t\in W^P$ such that $w \ge \t \ge v$. \ni I. Let us first consider the case $\t=\text{\rm id}$. We identify $U^-e_{\text{\rm id}}$ with $$ \left\{ \begin{pmatrix} &\text{\rm Id}_{d\times d}\\ x_{d+1\, 1} &\dots &x_{d+1\, d}\\ \vdots &&\vdots\\ x_{n\, 1}&\dots&x_{n\,d} \end{pmatrix},\quad x_{ij}\in k,\quad d+1\le i\le n, 1\le j\le d \right\}. $$ Let $A$ be the affine algebra of $U^-e_{\text{\rm id}}$. Let us identify $A$ with the polynomial algebra $k[x_{-\b},\b\in R^+\setminus R_P^+]$. To be very precise, we have $R^+\,\setminus\, R_P^+ =\{\e_j-\e_i$, $1\le j\le d,\ d+1\le i\le n\}$; given $\b\in R^+\setminus R_P^+$, say $\b=\e_j-\e_i$, we identify $x_{-\b}$ with $x_{ij}$. Hence we obtain that the expression for $f_{\te,\textrm{id}}$ in the local coordinates $x_{-\b}$'s is homogeneous. \begin{ex} Consider $G_{2,4}$. Then $$ U^-e_{\text{\rm id}}= \left\{ \begin{pmatrix} 1&0\\ 0&1\\ x_{31}&x_{32}\\ x_{41}&x_{42} \end{pmatrix},\quad x_{ij}\in k \right\}. $$ On $U^-e_{\text{\rm id}}$, we have $p_{12}=1$, $p_{13}=x_{32}$, $p_{14}=x_{42}$, $p_{23}=x_{31}$, $p_{24}=x_{41}$, $p_{34}=x_{31}x_{42}-x_{41}x_{32}$. \end{ex} Thus a Pl\"ucker coordinate is homogeneous in the local coordinates $x_{ij},\ d+1\le i\le n,\ 1\le j\le d$. \ni II. Let now $\t$ be any other element in $W^P$, say $\t=(a_1,\dots,a_n)$. Then $U_\t^-e_\t$ consists of $\{N_{d,n}\}$, where $N_{d,n}$ is obtained from $\begin{pmatrix}\text{\rm Id}\\X\end{pmatrix}_{n\times d}$ (with notations as above) by permuting the rows by $\t^{-1}$. (Note that $U_\t^-e_\t=\t U^-e_{\text{\rm id}}$.) \begin{ex} Consider $G_{2,4}$, and let $\t=(2314)$. Then $\t^{-1}=(3124)$, and $$U_\t^-e_\t= \left\{ \begin{pmatrix} x_{31}&x_{32}\\ 1&0\\ 0&1\\ x_{41}&x_{42} \end{pmatrix},\quad x_{ij}\in k \right\}. $$ We have on $U_\t^-e_\t$, $p_{12}=-x_{32}$, $p_{13}=x_{31}$, $p_{14}=x_{31}x_{42}-x_{41}x_{32}$, $p_{23}=1$, $p_{24}=x_{42}$, $p_{34}=-x_{41}$. \end{ex} As in the case $\t=\text{\rm id}$, we find that for $\te\in W^P$, $f_{\te,\t}:=$ $p_\te\res_{U_\t^-e_\t}$ is homogeneous in local coordinates. In fact we have \begin{prop} Let $\te\in W^P$. We have a natural isomorphism $$k[x_{-\b},\b\in R^+\setminus R_P^+]\cong k[x_{-\t(\b)}, \b\in R^+\setminus R_P^+],$$ given by $$f_{\te,\textrm{id}}\mapsto f_{\t\te,\t}.$$ \end{prop} \ni The proof is immediate from the above identifications of $U^-e_{\text{\rm id}}$ and $U_\t^-e_\t$. As a consequence, we have \begin{cor}\label{8.8.3.5} Let $\te\in W^P$. Then the polynomial expression for $f_{\te,\t}$ in the local coordinates $\{x_{-\t(\b)}, \b\in R^+\setminus R_P^+\}$ is homogeneous. \end{cor} \subsection{The algebra $A_{w,\t}^v$}\label{8.8.4} As above, we identify $A_\t$, the affine algebra of $U_\t^-e_\t$ with the polynomial algebra $K[x_{-\b},\b\in\t(R^+\setminus R_P^+)]$. Let $A_{w,\t}^v=A_\t/I_{w,\t}^v$, where $I_{w\t}^v$ is the ideal of elements of $A_\t$ that vanish on $X_w^v \cap U_\t^-e_\t$. Now $I(X_w^v)$, the ideal of $X_w^v$ in $G/P$, is generated by $\{p_\te,\te\in W^P\mid w\not\ge\te \text{ or } \te \not\ge v\}$. Hence we obtain (cf. Corollary \ref{8.8.3.5}) that $I_{w,\t}^v$ is homogeneous. Hence we get \begin{equation*} \text{gr\,}(A_{w,\t}^v,M_{w,\t}^v)=A_{w,\t}^v,\tag{*} \end{equation*} where $M_{w,\t}^v$ is the maximal ideal of $A_{w,\t}^v$ corresponding to $e_\t$. In particular, denoting the image of $x_{-\b}$ under the canonical map $A_\t\to A_{w,\t}^v$ by just $x_{-\b}$, the set $\{x_{-\b}\mid\b\in \t(R^+\setminus R_P^+)\}$ generates $A_{w,\t}^v$. Let $R_w^v$ be the homogeneous coordinate ring of $X_w^v$ (for the Pl\" ucker embedding), $Y_{w,\t}^v=X_w^v\cap U_\t^-e_\t.$ Then $K[Y_{w, \t}^v]=A_{w,\t}^v$ gets identified with the homogeneous localization ${(R_w^v)}_{(p_\t)}$, i.e. the subring of ${(R_w^v)}_{p_\t}$ (the localization of $R_w^v$ with respect to $p_\t$) generated by the elements $$\{\frac{p_\te}{p_\t},\te\in W^P,w \ge \te \ge v\}.$$ \subsection{The Integer $\deg_{\t}(\te)$} Let $\te\in W^P$. We define $\deg_{\t}(\te)$ by $$\deg_{\t}(\te):=\deg f_{\te,\t}$$ (note that $f_{\te,\t}$ is homogeneous, cf. Corollary \ref{8.8.3.5}). In fact, we have an explicit expression for $\deg_{\t}(\te)$, as follows (cf. \cite{L-G}): \begin{prop} Let $\te\in W^P$. Let $\t=(a_1,\dots,a_n)$, \ $\te=(b_1,\dots,b_n)$. Let $r=\#\{a_1,\dots,a_d\}\cap\{b_1,\dots,b_d\}$. Then $\deg_{\t}(\te)=d-r$. \end{prop} \subsection{A Basis for the Tangent Cone} Let $Z_\t=\{\te \in W^P \,|\,\text{either }\te\ge \t\,\text{or }\t\ge \te\}.$ \begin{thm} With notations as above, given $r\in\mathbb{Z}^+$, $$\{f_{\te_1,\t}\dots f_{\te_m,\t}\mid w\ge\te_1\ge...\ge\te_m\ge v ,\, \te_i\in Z_\t,\ \sum_{i=1}^m\deg_{\t}(\te_i)=r \}$$ is a basis for $(M_{w,\t}^v)^r/(M_{w,\t}^v)^{r+1}$. \end{thm} \begin{proof} For $F=p_{\te_1}\cdots p_{\te_m}$, let deg$F$ denote the degree of $f_{{\te_1},\t}\cdots f_{{\te_m},\t}$. Let $$A_r=\{F=p_{\te_1}\cdots p_{\te_m},\ w\ge \te_i\ge v \,|\,\text{deg}F=r\}.$$ Then in view of the relation (*) in \S \ref{8.8.4}, we have, $A_r$ generates $(M^v_{w,\t})^r/(M^{v}_{w,\t})^{r+1}$. Let $F\in A_r$, say $F=p_{\t_1}\cdots p_{\t_m}$. From the results in \S \ref{smt1}, we know that $p_{\t_1}\cdots p_{\t_m}$ is a linear combination of standard monomials $p_{\te_1}\cdots p_{\te_m},\ w\ge \te_i \ge v$. We {\em claim} that in each $p_{\te_1}\cdots p_{\te_m},\ \te_i \in Z_\t, \text{ for all }i$. Suppose that for some $i,\te_i \not\in Z_\t$. This means $\te_i$ and $\t$ are not comparable. Then using the fact that $f_{\t,\t}=1$, on $Y_{w,\t}^v$, we replace $p_{\te_i}$ by $p_{\te_i}p_\t$ in $p_{\te_1}\cdots p_{\te_m}$. We now use the straightening relation (cf. Proposition \ref{qualitative}) $p_{\te_i}p_\t=\sum c_{\a,\b}p_\a p_\b$ on $X_w^v$, where in each term $p_\a p_\b$ on the right hand side, we have $\a$ both $\te_i$ and $\t$, and $\b <$ both $\te_i$ and $\t$; in particular, we have, in each term $p_\a p_\b$ on the right hand side, $\a, \b$ belong to $Z_{\t}$. We now proceed as in the proof of Theorem \ref{generation} to conclude that on $Y_{w,\t}^v,\ F$ is a linear combination of standard monomials in $p_\te$'s, $\te \in Z_{\t}$ which proves the Claim. Clearly, $\{f_{\te_1,\t}\dots f_{\te_m,\t}\mid w\ge\te_1\ge...\ge\te_m\ge v , \, \te_i\in Z_\t \}$ is linearly independent in view of Theorem \ref{indep} (Since $p_{\t}^l f_{\te_1,\t} \cdots f_{\te_m,\t} = p_{\t}^{l-m}p_{\te_1} \cdots p_{\te_m}$ for $l\ge m$, and the monomial on the right hand side is standard since $\te_i \in Z_{\t}$). \end{proof} \subsection{Recursive Formulas for $\Mult_\t{X_w^v}$} \begin{defn} If $w > \t \geq v$, define $\partial_{w,\t}^{v, +} := \{ w' \in W^P \mid w > w' \geq \t \geq v, \, l(w')=l(w)-1\}$. If $w \geq \t >v$, define $\partial_{w,\t}^{v, -} := \{ v' \in W^P \mid w \geq \t \geq v' > v, l(v')=l(v)+1\}$. \end{defn} \begin{thm}\label{8.8.5.3} \begin{enumerate} \item Suppose $w > \t \ge v$. Then $$(\Mult_\t{X_w^v})\deg_\t w=\sum\limits_ {w' \in \partial_{w, \t}^{v, +}} \Mult_\t X_{w'}^v.$$ \item Suppose $w \ge \t > v$. Then $$(\Mult_\t{X_w^v})\deg_\t v=\sum\limits_ {v' \in \partial_{w, \t}^{v, -}} \Mult_\t X_{w}^{v'}.$$ \item $\Mult_\t{X_\t^\t} = 1$. \end{enumerate} \end{thm} \begin{proof} Since $X_{\t}^{\t}$ is a single point, (3) is trivial. We will prove (1); the proof of (2) is similar. Let $H_{\t} = \bigcup\limits_{w' \in \partial_{w, \t}^{v, + }}X_{w'}^v$. Let $\varphi_w^v(r)$ (resp. $\varphi_{H_{\t}}(r)$) be the Hilbert function for the tangent cone of $X_w^v$ (resp. $H_{\t}$) at $e(\t)$, i.e. $$\varphi_w^v(r)=\dim ((M_{w,\t}^v)^r/(M_{w,\t}^v)^{r+1}).$$ Let $$ \mathcal{B}_{w,\t}^v(r)=\left\{p_{\t_1}\dots p_{\t_m},\,\t_i\in Z_{\t}\mid (1)\, w\ge\t_1\ge\dots\ge\t_m\ge v, (2) \sum \deg_{\t}(\t_i)=r\right\}. $$ Let $$ \begin{aligned} \mathcal{B}_1&=\{p_{\t_1}\dots p_{\t_m}\in\mathcal{B}_{w,\t}^v(r)\mid\t_1=w\},\\ \mathcal{B}_2&=\{p_{\t_1}\dots p_{\t_m}\in\mathcal{B}_{w,\t}^v(r)\mid\t_1 \t > v$. Then $$(\Mult_{\t}{X_w^v}) (\deg_{\t}w + \deg_{\t}v) = \sum\limits_ {w' \in \partial_{w, \t}^{v, +}} \Mult_\t{X_{w'}^v} + \sum\limits_ {v' \in\partial_{w, \t}^{v,- } }\Mult_\t{X_w^{v'}}.$$ \end{cor} \begin{thm}\label{prodmult} Let $w \ge \t \ge v$. Then $\Mult_{\t}{X_w^v} = (\Mult_\t{X_w})\cdot(\Mult_\t{X^v})$ \end{thm} \begin{proof} We proceed by induction on $\dim X_w^v$. If $\dim X_w^v = 0$, then $w = \t = v$. In this case, by Theorem {\ref{8.8.5.3}} (3), we have that $\Mult_\t{X_\t^\t} = 1$. Since $e_{\t} \in Be_{\t } \subseteq X_w$, and $Be_{\t}$ is an affine space open in $X_w$, $e_{\t}$ is a smooth point of $X_w$, i.e. $\Mult_\t{X_w} = 1$. Similarly, $\Mult_\t{X^v} = 1$. Next suppose that $\dim X_w^v > 0$, and $w > \t \ge v$. By Theorem {\ref{8.8.5.3}} (1), $$\begin{aligned} \Mult_\t{X_w^v} &= \frac{1}{\deg_{\t}w}\sum\limits_ {w' \in {\partial_{w,\t}^{v,+}}}\Mult_\t{X_{w'}^v} \\ &= \frac{1}{\deg_{\t}w}\sum\limits_ {w' \in {\partial_{w,\t}^{v,+}}}\Mult_\t{X_{w'}}\cdot \Mult_\t{X^v} \\ &= \left(\frac{1}{\deg_{\t}w}\sum\limits_ {w' \in {\partial_{w,\t}^{v,+}}}\Mult_\t{X_{w'}}\right)\cdot \Mult_\t{X^v} \\ &= \left(\frac{1}{\deg_{\t}w}\sum\limits_ {w' \in {\partial_{w,\t}^{v,+}}}\Mult_{\t}{X_{w'}^ {\text{id}}}\right)\cdot \Mult_\t{X^v} \\ &= \left(\Mult_{\t}{X_w^{\text{id}}}\right)\cdot \Mult_\t{X^v} = \Mult_\t{X_w} \cdot \Mult_\t{X^v}. \end{aligned}$$ The case of $\dim X_w^v > 0$ and $w = \t > v$ is proven similarly. \end{proof} \begin{cor}\label{smcrit2} Let $w \ge \t \ge v$. Then $X_w^v$ is smooth at $e_\t$ if and only if both $X_w$ and $X^v$ are smooth at $e_\t$. \end{cor} \begin{rem} The following alternate proof of Theorem \ref{prodmult} is due to the referee, and we thank the referee for the same. \vskip.3cm \begin{quote}{\it Identify $\mathcal{O}^-_{\tau}$ with the affine space $\AAA ^N$ where $N=d(n-d)$, by the coordinate functions defined in \S \ref{neigh}. Then $X_w\cap \mathcal{O}^-_{\tau}$ and $X^v\cap \mathcal{O}^-_{\tau}$ are closed subvarieties of that affine space, both invariant under scalar multiplication (e.g. by Corollary \ref{8.8.3.5}). Moreover, $X_w$ and $X^v$ intersect properly along the irreducible subvariety $X_w^v$; in addition, the Schubert cells $C_w$ and $C^v$ intersect transversally (by \cite{Ri}). Now let $Y$ and $Z$ be subvarieties of $\AAA^N$, both invariant under scalar multiplication, and intersecting properly. Assume in addition that they intersect transversally along a dense open subset of $Y\cap Z$. Then $$ \Mult_{\bf{o}}(Y\cap Z)=\Mult_{\bf{o}}(Y) \cdot \Mult_{\bf{o}}(Z) $$ where ${\bf{o}}$ is the origin of ${\AAA ^N}$. To see this, let $\PPP(Y)$, $\PPP(Z)$ be the closed subvarieties of $\PPP(\AAA^N)=\PPP^{N-1}$ associated with $Y$, $Z$. Then $\Mult_{\bf{o}}(Y)$ equals the degree ${\rm deg}(\PPP(Y))$, and likewise for $Z$, $Y\cap Z$. Now $$ {\rm deg}(\PPP(Y\cap Z))= {\rm deg}(\PPP (Y)\cap\PPP(Z)) ={\rm deg}(\PPP(Y))\cdot{\rm deg}(\PPP(Z)) $$ by the assumptions and the Bezout theorem (see \cite{ful}, Proposition 8.4 and Example 8.1.11).} \end{quote} \vskip.3cm It has also been pointed out by the referee that the above alternate proof in fact holds for Richardson varieties in a minuscule $G/P$, since the intersections of Schubert and opposite Schubert varieties with the opposite cell are again invariant under scalar multiplication (the result analogous to Corollary \ref{8.8.3.5} for a minuscule $G/P$ follows from the results in \cite{l-w}). Recall that for $G$ a semisimple algebraic group and $P$ a maximal parabolic subgroup of $G$, $G/P$ is said to be {\em minuscule} if the associated fundamental weight $\o$ of $P$ satisfies $$(\o,\b^*)\, (\, =2(\o,\b)/(\o,\b))\, \leq 1$$ for all positive roots $\b$, where $(\ , \, )$ denotes a $W$-invariant inner product on $X(T)$. \end{rem} \subsection{Determinantal Formula for $\Mult X_w^v$} In this section, we extend the \ni Rosenthal-Zelevinsky determinantal formula (cf. \cite{rose-zel}) for the multplicity of a Schubert variety at a $T$-fixed point to the case of Richardson varieties. We use the convention that the binomial coefficient ${a \choose b} = 0$ if $b<0$. \begin{thm}\label{ros-zel} {(Rosenthal-Zelevinsky)} Let $w = (i_1,\ldots,i_d)$ and $\t=(\t_1,\ldots,\t_d)$ be such that $w\ge\t$. Then $$\Mult_{\tau}{X_w} = (-1)^{\kappa_1 + \cdots + \kappa_d} \left| \begin{matrix}{i_1 \choose -\kappa_1} & \cdots & {i_d \choose -\kappa_d} \cr {i_1 \choose 1-\kappa_1} & \cdots & {i_d \choose 1-\kappa_d} \cr \vdots & & \vdots \cr {i_1 \choose d-1-\kappa_1} & \cdots & {i_d \choose d-1-\kappa_d} \end{matrix} \right|,$$ where $\kappa_q := $ $\# \{ \t_p \mid \t_p > i_q \}$, for $q = 1, \ldots, d$. \end{thm} \begin{lem}\label{oppreg} $\Mult_{\t}{X^v} = \Mult_{w_o\t}{X_{w_0v}}$, where $w_0 = (n+1-d,\ldots,n)$. \end{lem} \begin{proof} Fix a lift $n_0$ in $N(T)$ of $w_0$. The map $f:X^v\to n_0X^v$ given by left multiplication is an isomorphism of algebraic varieties. We have $f(e_\t)=e_{w_0\t}$, and $n_0X^v=n_0\overline{B^-e_v}$ $=n_0\overline{n_0Bn_0e_v}=\overline{Bn_0e_v}$ $=\overline{Be_{w_0v}}=X_{w_0v}$. \end{proof} \begin{thm}\label{rzr} Let $w = (i_1,\ldots,i_d)$, $\t=(\t_1,\ldots,\t_d)$, and $v=(j_1,\ldots,j_d)$ be such that $w\ge\t\ge v$. Then $$\Mult_{\tau}{X_w^v} = (-1)^c \left|\left( \begin{matrix}{i_1 \choose -\kappa_1} & \cdots & {i_d \choose -\kappa_d} \cr {i_1 \choose 1-\kappa_1} & \cdots & {i_d \choose 1-\kappa_d} \cr \vdots & & \vdots \cr {i_1 \choose d-1-\kappa_1} & \cdots & {i_d \choose d-1-\kappa_d} \end{matrix} \right)\cdot\left( \begin{matrix}{n+1-j_d \choose -\gamma_d} & \cdots & {n+1-j_1 \choose -\gamma_1} \cr {n+1-j_d \choose 1-\gamma_d} & \cdots & {n+1-j_1 \choose 1-\gamma_1} \cr \vdots & & \vdots \cr {n+1-j_d \choose d-1-\gamma_d} & \cdots & {n+1-j_1 \choose d-1-\gamma_1} \end{matrix} \right)\right|,$$ where $\kappa_q := $ $\# \{ \t_p \mid \t_p > i_q \}$, for $q = 1, \ldots, d$, and $\gamma_q := $ $\# \{ \t_p \mid \t_p < j_q \}$, for $q = 1, \ldots, d$, and $c=\kappa_1 + \cdots + \kappa_d + \gamma_1 + \cdots + \gamma_d$. \end{thm} \begin{proof} Follows immediately from Theorems \ref{prodmult}, \ref{ros-zel}, and Lemma \ref{oppreg}, in view of the fact that $w_0\t=(n+1-\t_d,\ldots,n+1-\t_1)$ and $w_0v=(n+1-j_d,\ldots,n+1-j_1)$. \end{proof} \begin{thebibliography}{4} \bibitem{borel1} A. 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