%% %% Notes for the lectures of Peter Littelmann %% \documentstyle[11pt]{amsart} %% Page size and style \textheight220truemm \textwidth155truemm \headheight11pt \headsep16pt \baselineskip13pt %% Large bold font \catcode`\@=11 \def\larger{\@setsize\larger{18\p@}\xviipt\@xviipt} %% My Declarations - section is redefined to make section names bf \def\mysection{\@startsection{\@string\section}% 1\z@{9\p@\@plus12\p@}{6\p@}% {\defaultfont\bf\large}} \renewcommand{\section}{\mysection} \def\myabstract{% \vskip-4\p@\@plus-14\p@ \moveright 3pc\vtop \bgroup \advance \hsize -6pc\bgroup % extra \bgroup to localize \small \defaultfont\small \noindent {\bf\abstractname}.\enspace\ignorespaces \global\let\myabstract\relax} \renewcommand{\abstract}{\myabstract} \def\mybibliography#1{\section*\refname \defaultfont\normalsize\labelsep .5em\relax \list{\@arabic\c@enumi.}{\settowidth\labelwidth{#1.}% \leftmargin\labelwidth \advance\leftmargin\labelsep \usecounter{enumi}}% \sloppy \clubpenalty4000\relax \widowpenalty\clubpenalty \sfcode`\.\@m} %% End of my declarations %% Other declarations \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{thm}{Theorem}[subsection] \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{cor}[thm]{Corollary} \newtheorem{claim}{Claim} \renewcommand{\theclaim}{} \newtheorem{conj}{Conjecture} \renewcommand{\theconj}{} \newtheorem{demazure}{Demazure Type Character Formula} \renewcommand{\thedemazure}{} \newtheorem{weyl}{Weyl Character Formula} \renewcommand{\theweyl}{} \newtheorem{concat}{Concatenation} \renewcommand{\theconcat}{} \newtheorem{GLR}{Generalized Littlewood-Richardson rule} \renewcommand{\theGLR}{} \theoremstyle{remark} \newtheorem{demo}{Proof} \renewcommand{\thedemo}{} \newtheorem{demorem}{Some remarks on the proof} \renewcommand{\thedemorem}{} \newtheorem{demothm}{Proof of the theorem} \renewcommand{\thedemothm}{} \newtheorem{democor}{Proof of the corollary} \renewcommand{\thedemocor}{} \newtheorem{demoBT}{Proof of the Basis Theorem for path vectors} \renewcommand{\thedemoBT}{} \theoremstyle{definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{rem}[thm]{Remark} \newtheorem{defn}[thm]{Definition} %% Other definitions and abbreviations \def\char{\mathop{\text{Char}}} \def\End{\mathop{\hbox{\rm End}}} \def\Lie{\mathop{\hbox{\rm Lie}}} \def\Im{\mathop{\hbox{\rm Im}}} \def\sgn{\mathop{\hbox{\rm sgn}}} \def\rk{\mathop{\hbox{\rm rank}}} \def\Hom{\mathop{\hbox{\rm Hom}}} \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{\Gr}{\mbox{\rm Gr}} \newcommand{\Fl}{\mbox{\rm Fl}} \def\integ{\mathop{\hbox{\rm int}}} \def\Cal{\cal} \def\G{{\Cal G}} \def\HH{{\Cal H}} \def\L{{\Cal L}} \def \O{{\Cal O}} \def\A{\cal{A}} \newcommand{\II}{{\cal I}} \newcommand{\JJ}{{\cal J}} \def\T{{\bold T}} \def\p{{\bf p}} \def\q{{\bf q}} \def\v{{\bold v}} \def\kk{\bold k} \def\nn{\bold n} \def\ut{\bold{t}} \def\us{\bold{s}} \def\rr{\bold{r}} \def\mm{\bold{m}} \newcommand{\OO}{{\bf O}} %\def\bb{\Bbb} \def\NN{{\Bbb N}} \def\B{{\Bbb B}} \def\P{{\Bbb P}} \def\Z{{\Bbb Z}} \def\ZZ{{\Bbb Z}} \def\Q{{\Bbb Q}} \def\C{{\Bbb C}} \def\R{{\Bbb R}} \def\V{{\Bbb V}} \def\K{{\Bbb K}} \def\Pr{{\Bbb P}} \def\RR{{\Bbb R}} \def\TT{{\Bbb T}} \def\Fr{\frak} \def\km{{\Fr g}} \def\b{{\Fr b}} \def\l{{\Fr l}} \def\H{{\Fr H}} \def\h{{\Fr h}} \def\c{{\Fr c}} \def\labs{\vert\!\langle} \def\rabs{\rangle\!\vert} \def\<{\langle} \def\>{\rangle} \def\lam{\lambda} \def\D{\Delta} \def\d{\delta} \def\e{\epsilon} \def\om{{\omega}} \def\o{\omega} \def\Lam{\Lambda} \def\Om{\Omega} \def\avee{\alpha^\vee} \def\bvee{\beta^\vee} \def\gvee{\gamma^\vee} \def\gve{\gamma^{\vee,l}} \def\a{{\alpha}} \def\bb{{\beta}} \def\g{{\gamma}} \def\m{\mu} \def\n{\nu} \def\f{\phi} \def\s{\sigma} \def\0{{\theta}} \def\te{\theta} \def\t{\tau} \def\taur{\tau^{\rr}} \newcommand{\zi}{(\zeta^*)^{-1}} \newcommand{\ziJ}{\zi \sqrt{\JJ(\taur)}} \def\Ta{\underline{\tau}} \def\ulam{{\underline{\lam}}} \def\umu{\underline{\mu}} \def\ua{\underline{a}} \def\ud{\underline{d}} \def\uv{\underline{\mu}} \def\ub{\underline{b}} \def\unu{\underline{\nu}} \def\ui{\underline{i}} \def\uj{\underline{j}} \def\mn{\underline{m}} \def\ns{\underline{n}} \def\st{\underline{s}} \def\od{{\overline d}} \def\oa{{\overline a}} \def\oV{{\overline V}} \def\lbar{{\overline \ell}} \newcommand{\bi}{{\overline{i}}} \newcommand{\bj}{{\overline{j}}} \def\tlam{{\tilde \lam}} \def\tmu{{\tilde \mu}} \def\tR{{\tilde R}} \def\pt{{\tilde \pi}} \newcommand{\tJ}{\widetilde{\JJ}} \newcommand{\tA}{\widetilde{A}} \newcommand{\tw}{\widetilde{w}} \def\w{\wedge} \def\lt{\triangleleft} \def\lte{\underline{\lt}} \def\res{|} \def\ni{\noindent} \def\vs{\vskip} \def\taum{ \tau^{ \mbox{\tiny max} } } \def\sur{\twoheadrightarrow} \def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}} \def\mapri#1{\smash{\mathop{\longrightarrow}\limits^{#1}}} \def\mapro#1{\nearrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} \def\mapru#1{\searrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} \def\bij{\smash{\mathop{\sim}\limits^{1:1}}} \newcommand{\bdot}{ {\hspace{-.1em} \mbox{\bf .} } } \newcommand{\eqdef}{\stackrel{\rm def}{=}} \newcommand{\eqto}{\stackrel{\rm \sim}{\to}} \newcommand{\mo}{ { \!-\!1} } \newcommand{\po}{ { \!+\!1} } \pagestyle{myheadings} \def\@evenhead{\strut\rlap{\rm{\thepage}}\hfill {\em Lakshmibai, Littelmann and Magyar}} \def\@oddhead{\strut{\em Standard Monomial Theory and Applications} \hfill\llap{\rm{\thepage}}} %% Begin document here \begin{document} %% Topmatter - Titles, Authors etc { \thispagestyle{empty} \begin{center} \large {\bf\larger Standard Monomial Theory and Applications} \vskip3em Venkatramani LAKSHMIBAI\footnote{E-mail : lakshmibai@neu.edu} \vskip0.2em {\em Department of Mathematics, \\ Northeastern University, \\ Boston, MA 02115, U.S.A. %E-mail : lakshmibai@neu.edu } \vskip0.8em Peter LITTELMANN\footnote{E-mail : littelma@math.u-strasbg.fr} \vskip0.2em {\em Universit\'e Louis Pasteur et Institut Universitaire de France, \\ Institut de Recher\-che Math\'ematique Avanc\'ee, \\ 7 rue Ren\'e Descartes, \\ 67084 Strasbourg Cedex, France } %E-mail : littelma@math.u-strasbg.fr \vskip0.3em and \vskip0.3em Peter MAGYAR\footnote{Supported by a Postdoctoral Fellowship from the US National Science Foundation. E-mail : pmagyar@lynx.neu.edu} \vskip0.2em {\em Department of Mathematics, \\ Northeastern University, \\ Boston, MA 02115, U.S.A.\\ %E-mail : pmagyar@lynx.neu.edu }\vskip2em Notes by \vskip0.3em Rupert W.T. YU \end{center} \vskip2em } %% End Topmatter %% Abstract \begin{abstract} In these notes, we explain how one can construct Standard Monomial Theory for reductive algebraic groups by using the path models of their representations and quantum groups at a root of unity. As applications, we obtain a combinatorial proof of the Demazure character formula and representation theoretic proofs of geometrical properties of Schubert varieties, such as normality, vanishing theorems, ideal theory and so on. Further applications of Standard Monomial theory are made to prove geometrical properties of certain ladder determinantal varieties and certain quiver varieties. We sketch at the end an extension of the theory to Bott-Samelson varieties and configuration varieties. \end{abstract} %% Introduction \section{Introduction} In the theory of finite dimensional representations of complex reductive algebraic groups, the group $GL_n(\C)$ is singled out by the fact that besides the usual language of weight lattices, roots and characters, there exists an additional important combinatorial tool: the Young tableaux. To construct objects like the tableaux in a more general setting, consider the weight lattice $X$ of a complex semisimple Lie algebra (or, more general, symmetrizable Kac-Moody algebra) $\km$, and denote by $\Pi$ the set of all piecewise linear paths $\pi:[0,1]_\Q\rightarrow X_\Q$ starting in $0$ and ending in an integral weight. We associate to a simple root $\a$ operators $e_\a$ and $f_\a$ on $\Pi$, and, using these operators, we construct for a dominant weight $\lambda$ a set of paths $B(\lam)$ that can be viewed as a generalization of the Young tableaux: For example, the sum over the endpoints of all paths in $B(\lam)$ is the character of $V(\lam)$, and the Littlewood-Richardson rule can be generalized in a straightforward way. Though the theory of the paths is completely independent of the theory of quantum groups, they can be viewed as a geometric realization of the theory of crystals of representations. The next step is then to associate a basis of the representation to the paths. The starting point for the theory was a series of articles in which Lakshmibai, Musili and Seshadri initiated a program to construct a basis for the space $H^0(G/B,\L_\lam)$ with some particularly nice geometric properties. Here we suppose that $G$ is a reductive algebraic group defined over an algebraically closed field $k$, $B$ is a fixed Borel subgroup, and $\L_\lam$ is the line bundle on the flag variety $G/B$ associated to a dominant weight. The purpose of the program is to extend the Hodge-Young standard monomial theory for the group $GL(n)$ to the case of any semisimple linear algebraic group and, more generally, to Kac-Moody algebras. Using quantum groups at a root of unity, we define a basis of the representation such that each element of the basis can be viewed in some sense as a $\ell$-th root of a product of extremal weight vectors. As applications we get a straightforward construction of Standard Monomial Theory, a representation theoretic proof of the normality of Schubert varieties, a combinatorial proof of the Demazure character formula, the ``Good filtration'' property for tensor products in positive characteristic \cite{32a}, a reduced Groebner basis for the defining ideal of Schubert varieties in terms of generalized Pl\"ucker relations, $\ldots$ In the first two sections we recall the main facts concerning the path model. In the third and fourth section we give an introduction into the construction of the path vectors, the main tool here is the quantum Frobenius map for quantum groups at roots of unity. For simplicity we restrict ourself to the finite dimensional case, but the proofs hold, with the appropriate adaptions, also for arbitrary symmetrizable Kac-Moody algebras. For details see \cite{33}. In the next three sections we discuss the application to the geometry of Schubert varieties, we generalize the results in \cite{33} to unions of Schubert varieties. In addition to the geometric consequences for Schubert varieties such as normality, vanishing theorems, ideal theory etc, the Standard Monomial Theory has also led to the determination of the singular loci of Schubert varieties (cf. \cite{l1}, \cite{l2}, \cite{l-sa}, \cite{l-se2}, \cite{l-w}), and the results are recalled in \S \ref{sing}. As a further application of Standard Monomial Theory, one obtains (\cite{g-l}, \cite{l-ma}) the normality and Cohen-Macaulayness for two classes of affine varieties - certain ladder determinantal varieties (cf. \S \ref{ldv}) and certain quiver varieties (cf. \S \ref{quiv}). These results are proved by identifying them with the ``opposite cells'' in suitable Schubert varieties $X_Q(w)$ in suitable $SL(n)/Q$. Standard Monomial Theory grew out of, and applies to, the algebraic geometry of flag varieties and their subvarieties. We sketch an extension of the theory to a larger class of spaces, the Bott-Samelson varieties and configuration varieties. These varieties (like Schubert varieties) have a natural $B$-action, and the spaces of global sections of ample line bundles provide a class of $B$-representations whose description is the main goal of the theory. We have a generalized Bruhat order and a set of L-S paths fitting into a path model, which provide an indexing system for bases. Moreover, although we do not yet have a direct generalization of the basis $\{p_{\pi}\}$ corresponding to L-S paths, we do describe an analog of the ``standard tableau'' bases of Section \ref{Standard Monomials II}. %% Part 1 - P. Littelmann \section{An indexing system for a basis: The L-S paths} For a complex semisimple Lie algebra $\km$ fix a Cartan subalgebra $\h$, a Borel subalgebra $\b$, and denote by $X$ the weight lattice of $\km$. Corresponding to the choice of $\b$ let $X^+$ be the set of dominant weights. On $X_\R:=X\otimes_\Z\R$ denote by $(\cdot,\cdot)$ the Killing form, and for a root $\beta$ let $\bvee=2\beta/(\beta,\beta)$ be the co-root. Let $V(\lam)$ be the simple $\km$-module of highest weight $\lam$. The aim of this section is to describe an indexing system for a basis of $V(\lam)$ of $\h$-eigenvectors. Denote by $\pi_\lam:t\rightarrow t\lam$ the path that connects the origin with $\lam$ by a straight line. We are going to describe a set of paths obtained by bending $\pi_\lam$: the Lakshmibai-Seshadri paths. The definition given here is a ``translation'' of the definition in \cite{27} into the language of paths. Let $W$ be the Weyl group of $\km$, and for a dominant weight $\lam$ denote by $W_\lam$ the stabilizer of $\lam$ in $W$. Let ``$\le$'' be the Bruhat order on $W/W_\lam$. We identify a pair $\pi=(\Ta,\ua)$ of sequences: \begin{itemize} \item[$\bullet$] $\Ta:\tau_1>\tau_2>\ldots>\tau_r$ is a sequence of linearly ordered cosets in $W/W_\lam$ and \item[$\bullet$] $\ua:a_0:=0 \sigma$ be two elements of $W/W_\lam$ and let $0\kappa_1:=s_{\beta_1}\tau> \kappa_2:=s_{\beta_2}s_{\beta_1}\tau>\ldots >\kappa_s:=s_{\beta_s}\cdot\ldots\cdot s_{\beta_1}\tau=\sigma,\] where $\beta_1,\ldots,\beta_s$ are positive real roots and $l(\kappa_i)=l(\kappa_{i-1})-1$, $a(\kappa_i(\lam),\beta^\vee_i)\in\ZZ$ for all $i=1,\ldots,s$. \begin{definition} A pair $(\Ta,\ua)$ is called a {\it Lakshmibai-Seshadri} path of shape $\lam$ if for all $i=1,\ldots,r-1$ there exists an $a_i$-chain for the pair $(\tau_{i},\tau_{i+1})$. \end{definition} \begin{example} For $\sigma\in W/W_\lam$ let $\pi_{\sigma(\lam)}$ be the path $t\mapsto t\sigma(\lam)$ that connects $0$ with $\sigma(\lam)$ by a straight line. Then $\pi_{\sigma(\lam)}$ is the Lakshmibai-Seshadri path $(\sigma;0,1)$. \end{example} \begin{example} Let $\a$ be a simple root and suppose $\sigma\in W/W_\lam$ is such that $n=(\sigma(\lam),\avee)>0$. Then $(s_\a\sigma,\sigma;0,i/n,1)$ is an L-S path for $1\le i0$. Then $ii)$ and $iv)$ implies that that the map $\pi\mapsto f_\a^n\pi$ induces a bijection $B_\mu\rightarrow B_{s_\a(\mu)}$. Similarly, if $(\mu,\avee)=n<0$, then $\pi\mapsto e_\a^{\vert n\vert}\pi$ induces a bijection $B_\mu\rightarrow B_{s_\a(\mu)}$. \qed \end{democor} Denote by $C:=\{\nu\in X_\R\mid (\nu,\bvee)\ge 0 \hbox{\it\ for all positive roots\ }\beta\}$ the dominant Weyl chamber, and let $C^0$ be the interior of $C$. Let $\Pi^+\subset \Pi$ the set of paths $\eta$ such that the image $\Im\eta$ is contained in the dominant Weyl chamber $C$. Denote by $\rho\in X^+$ half the sum of the positive roots. If $B\subset \Pi$ is a finite subset such that $B\cup\{0\}$ is stable under the root operators $e_\a,f_\a$, then we have already seen that its character $\char B:=\sum_{\eta\in B} e^{\eta(1)}$ is stable under the action of the Weyl group $W$. For $\pi\in\Pi^+$ let $B(\pi)\subset\Pi$ be the subset of paths which can be obtained from $\pi$ by applying the operators, and let $G(\pi)$ be the colored, directed graph having as vertices the elements of $B(\pi)$, and we put an arrow $\eta\mapright{\a}\eta'$ with color a simple root $\a$ between $\eta,\eta'\in B_\pi$ if and only if $f_\a(\eta)=\eta'$ (or, equivalently, $e_\a(\eta')=\eta$). The structure of the set of paths $B(\pi)$ generated by a path $\pi\in\Pi^+$ is described by the following theorem (see \cite{30} for proofs): \begin{theorem} Suppose $\pi,\pi_1,\pi_2\in\Pi^+$. \begin{itemize} \item[a)] Integrality: $B(\pi)$ is integral, i.e., the minimum of the function $t\mapsto (\eta(t),\avee)$ is an integer for all $\eta\in B(\pi)$ and all simple roots $\a$. \item[b)] Highest weight path: $\pi$ is the only path in $B(\pi)$ such that $e_\a\pi=0$ for all simple roots. \item[c)] Isomorphism: $G(\pi_1)\simeq G(\pi_2)$ if and only if $\pi_1(1)=\pi_2(1)$. \item[d)] Weyl group: The action of the simple reflections $s_\a$ on $\Pi$ defined by: $$ s_\a(\eta):=\cases f_\a^p(\eta), & \hbox{if\ } p:=(\eta(1),\a^\vee)\ge 0,\cr e_\a^p(\eta), & \hbox{if\ }-p:=(\eta(1),\a^\vee) < 0,\cr \endcases $$ extends to an action of the Weyl group $W$ on $\Pi$ such that the $w(\eta)(1)=w(\eta(1))$. \end{itemize} \end{theorem} The independence of the graph structure of the choice of the starting path has as consequence that the graph is isomorphic to the crystal graph of the representation, see \cite{14} or \cite{15} for details. \begin{weyl} Let $\rho\in X$ be half the sum of the positive roots, and suppose $\pi\in\Pi^+$. Then $$ \sum_{\sigma\in W} \sgn (\sigma)e^{\sigma(\rho)}\char B(\pi) =\sum_{\sigma\in W} \sgn(\sigma) e^{\sigma(\rho +\pi (1))}. $$ In particular, $\char B(\pi)$ is equal to the character of the irreducible $\km$-module $V(\lam)$ of highest weight $\lam:=\pi(1)$. \end{weyl} The integrality property stated above seems at the first instance to be a ``technical'' fact without further consequences. But using the lemma above, it follows easily from the definition of the operators that: If $\pi,\eta$ are paths having the integrality property, then $e_\a(\pi*\eta)=\pi*(e_\a\eta)$ if there exists an $n>0$ such that $e^n_\a\eta\not=0$ but $f^n_\a\pi=0$, and it is equal to $(e_\a\pi)*\eta$ otherwise. Similarly, $f_\a(\pi*\eta)=(f_\a\pi)*\eta$ if there exists an $n>0$ such that $f^n_\a\pi\not=0$ but $e^n_\a\eta=0$, and it is equal to $\pi*(f_\a\eta)$ otherwise. So if $\pi_1,\pi_2\in \Pi^+$, then let $B(\pi_1)*B(\pi_2)$ be the set of all concatenations $\eta_1*\eta_2$, where $\eta_1\in B(\pi_1)$ and $\eta_2\in B(\pi_2)$. The rules above show that $B(\pi_1)*B(\pi_2)$ is stable under the root operators. It follows by the theorem above: \begin{concat} Suppose $\pi_1,\pi_2\in\Pi^+$. Then $$B(\pi_1)*B(\pi_2)=\bigcup B(\pi_1*\eta),$$ where the union runs over all paths $\eta\in B(\pi_2)$ such that $\pi_1*\eta\in\Pi^+$. \end{concat} Since $\char B(\pi_1) *B(\pi_2)=\char B(\pi_1) \char B(\pi_2)=\char V({\pi_1(1)})\otimes V({\pi_2(1)})$, we get as an immediate consequence of the theorem above and the character formula: \begin{GLR} For dominant weights $\lam,\mu$, let $\pi_1,\pi_2$ in $\Pi^+$ be such that $\pi_1(1)=\lam$ and $\pi_2(1)=\mu$. Then the tensor product of the irreducible representations $V(\lam)$, $V(\mu)$ of $\km$ of highest weight $\lam,\mu$ is isomorphic to the direct sum $$ V(\lam)\otimes V(\mu)\simeq\bigoplus V({\lam+\eta(1)}), $$ where the sum runs over all paths $\eta\in B(\pi_2)$ such that $\pi_1*\eta\in\Pi^+$. \end{GLR} The L-S paths discussed in the previous section are an example of such a set of paths, stable under the root operators. The character formula stated in the preceding section is an immediate consequence of the proposition below and the character formula above. For a proof of the following proposition see \cite{29}. \begin{proposition} Let $\pi_\lam:t\mapsto t\lam$ be the path that joins the origin with the dominant weight $\lam$ by a straight line. Then the set of paths $B(\pi_\lam)$, obtained from $\pi_\lam$ by applying all possible combinations of the root operators, is equal to $B(\lam)$, the set of L-S paths of shape $\lam$. \end{proposition} \section{A basis associated to the L-S paths} The character formula shows that we can use the set of L-S paths as an indexing system of a basis of $\h$-eigenvectors of $V(\lam)$. The next aim is to attach in a canonical way such a basis to $B(\lam)$. The idea of the construction is the following: Suppose for simplicity that $\km$ is of simply laced type. For a dominant weight $\lam$, let $V(\lam)$ be the irreducible module of $\km$ of highest weight $\lam$. Denote by $U_{\v}(\km)$ the quantum group at an $\ell$-th root of unity $\v$, let $N(\lam)$ be the Weyl module and by $L(\lam)$ the simple module for $U_\v(\km)$ of highest weight $\lam$. Lusztig \cite{34} has constructed a Frobenius map $Fr:U_{\v}(\km)\rightarrow U(\km)$ between the quantum group and the enveloping algebra $U(\km)$ of $\km$. Further, he has shown that if we consider $V(\lam)$ via the Frobenius $Fr$ as an $U_\v(\km)$-module, then this is the simple module $L(\ell\lam)$. This identification provides a $U_\v(\km)$-equivariant map $p:N(\ell\lam) \rightarrow V(\lam)$, the quotient of $N(\ell\lam)$ by its maximal proper $U_\v(\km)$-submodule. We are going to define a subspace $N(\ell\lam)^\ell$ which is naturally equipped with a $U(\km)$-action, and we will use this to define a section $s:V(\lam)\rightarrow N(\ell\lam)^\ell \subset N(\ell\lam)$ of the projection defined above. The dual map $s^*: N(\ell\lam)^{\ell,*}\rightarrow V(\lam)^*$ induces a map: $$ (N(\lam)^*)^{\otimes\ell}\rightarrow N(\ell\lam)^*\rightarrow N(\ell\lam)^{\ell,*}\rightarrow V(\lam)^* $$ Now once a highest weight vector $m_\lam\in N(\lam)$ is fixed, there is a canonical choice of extremal weight vectors $m_\tau\in N(\lam)$ of weight $\tau(\lam)$, $\tau\in W/W_\lam$, and corresponding dual vectors $b_\tau\in N(\lam)^*$ of weight $-\tau(\lam)$. Let now $\pi=(\tau_1,\ldots,\tau_r;0,a_1,a_2,\ldots,1)$ is an L-S path of shape $\lam$, and suppose $\ell$ is such that $\ell a_i\in\Z$ for all $i=1,\ldots,r$. Then the vector $b_\pi$ is well defined: $$ b_{\pi}:= \underbrace{b_{\tau_1}\otimes\ldots\otimes b_{\tau_1}}_{\ell a_1} \otimes \underbrace{b_{\tau_2}\otimes\ldots\otimes b_{\tau_2}}_{\ell(a_2-a_1)} \otimes \ldots\otimes \underbrace{b_{\tau_r}\otimes\ldots\otimes b_{\tau_r}}_{\ell(1-a_{r-1})}\in (N(\lam)^*)^{\otimes{\ell}} $$ Denote by $p_\pi\in V(\lam)^*$ its image, the {\it path vector} associated to $\pi$. To make the construction canonical we assume that the $\ell$ above is minimal with the property that $\ell a_i\in\Z$ for all $i$ and for all $\pi$ of shape $\lambda$. The construction presented here is actually characteristic free and works over the ring $\tR$ obtained from $\Z$ by adjoining all roots of unity. To make the construction more precise, we need first to fix some notation: Let $A=(a_{i,j})$ be the Cartan matrix of $\km$, and let $\km^t$ be the semisimple Lie algebra associated to the transposed matrix $A^t$. We fix $\ud=(d_1,\ldots,d_n)$ minimal such that $(d_i a_{i,j})$ is a symmetric matrix, and let $d$ be the smallest common multiple of the $d_j$. In the following we will often attach a $(\ )^t$ to some object associated to $\km^t$ to distinguish it from the corresponding $\km$ object. Let $\a_1,\ldots,\a_n$ be the simple roots of $\km$, and for $\km^t$ let the corresponding roots be $\gamma_1=\a_1/d_1,\ldots,\gamma_n=\a_n/d_n$. Let $U_q(\km^t)$ be the quantum group associated to $\km^t$ over the field $\Q(q)$, with generators $E_{\g_i}, F_{\g_i}, K_{\g_i}$ and $K_{\g_i}^{-1}$. We use the usual abbreviations ($\od_i:= d/d_i$) $$ [n]_i:={q^{\od_i n}-q^{-\od_i n}\over q^{\od_i}-q^{-\od_i}}, \ [n]_i!:=[1]_i\cdots[n]_i, \ {n\atopwithdelims[]m}_i:={[n]_i!\over [m]_i![n-m]_i!}, $$ where we define the latter to be zero for $n0$, then we consider $k$ as an $\tR$-module by extending the canonical map $\Z\rightarrow k$ to a map ${\tilde R}\rightarrow k$ (where the first map is given by the projection $\Z\rightarrow\Z/p\Z$ and the inclusion $\Z/p\Z\subset k$). Denote by $V_\tR(\lam)=V_\Z(\lam)\otimes_\Z \tR$ the corresponding Weyl module over the ring $\tR$, then the collection of vectors $$ \B(\lam):=\{p_\pi\mid\pi\in B(\lam)\}\subset V_\tR(\lam)^* $$ is well defined. By abuse of notation we write also $p_\pi$ for the image of the vector in $V_k(\lam)^*:=V_\tR(\lam)^*\otimes_\tR k$ for any algebraically closed field. \begin{theorem} The path vectors $\B(\lam)$ form a basis for the $U_\tR(\km)$-module $V_\tR(\lam)^*$. \end{theorem} Note that $\B(\lam)$ is a basis for $V_k(\lam)^*$ for any algebraically closed field $k$. The proof of the theorem will be given in the next section. The idea is to construct a basis $v_\pi$ of $V_\Z(\lam)$, indexed by L-S paths, such that $p_\pi(v_\pi)=1$, and for $\pi\not=\pi'$ we have $p_\pi(v_{\pi'})=1$ only if $\pi'\ge \pi$ in some partial order on the set of L-S paths. Note that this implies that $\B(\lam)$ will be, up to an upper triangular transformation, the dual basis of the basis given by the $v_\pi$, in particular, $\B(\lam)$ is a basis of $V_\tR(\lam)^*$. The disadvantage of the basis given by the $v_\pi$ is that it depends heavily on a choice of a reduced decomposition of $i(\pi)$. \section{A basis for $V_\Z(\lam)_\tau$} To construct the basis $D(\lam)$ of $V_\Z(\lam)$ we have first to introduce a partial order on weight vectors. For extremal weight vectors we write $m_\tau\ge m_\kappa$ if $\tau\ge \kappa$ in the Bruhat order on $W/W_\lam$. Similarly, we shall write $\pi\ge \eta$ for two L-S paths $\pi=(\tau_1,\ldots,\tau_r;0,a_1,\ldots,1)$ and $\eta = (\kappa_1,\ldots,\kappa_s;0,b_1,\ldots,1)$ of shape $\lam$ if $\tau_1>\kappa_1$ or $\tau_1=\kappa_1$ and $a_1>b_1$, or $\tau_1=\kappa_1$ and $a_1=b_1$ and $\tau_2>\kappa_2$, $\ldots$. Recall that $V_\Z(\lam)_\tau$ is the $U^+_R(\km)$ submodule of $V_\Z(\lam)$ generated by $m_\tau$, i.e., $V_\Z(\lam)_\tau=U^+_R(\km)m_\tau$. For an extremal weight vector $m_\tau$ and an arbitrary weight vector $m$ we write $m_\tau\ge m$ if $m\in V_\Z(\lam)_\tau$. We denote by ``$\ge$'' as well the induced lexicographic partial order on tensor products of weight vectors in $N_R(\lam)^{\otimes \lbar}$. We define a weaker order on weight vectors by saying that $m_\mu\succ m_\nu$ for two eigenvectors of weight $\nu,\mu$ if $\nu\succ\mu$ in the ususal weight ordering (i.e., $\nu-\mu$ is a sum of positive roots). We denote by ``$\succ$'' as well the induced lexicographic partial order on tensor products. Suppose $\lbar a_i\in\Z$ for all $i$, we denote by $m^\pi$ the tensor product $$ m^\pi=\underbrace{m_{\tau_1}\otimes\ldots\otimes m_{\tau_1}}_{\lbar a_1} \otimes \underbrace{m_{\tau_2}\otimes\ldots\otimes m_{\tau_2}}_{\lbar(a_2-a_1)} \otimes \ldots\otimes \underbrace{m_{\tau_r}\otimes\ldots\otimes m_{\tau_r}}_{\lbar(1-a_{r-1})}\in (N_R(\lam)^*)^{\otimes{\lbar}} $$ Fix a reduced decomposition $\tau_1=s_{i_1}\cdots s_{i_t}$. Let $s(\pi)=(n_1,\ldots,n_t)$ be the sequence of integers defined by the following procedure, which has been inspired by the article of K.~N.~Raghavan and P.~Sankaran \cite{41}. Fix $j$ minimal such that $s_{i_1}\tau_j>\tau_j$, and set $j=r+1$ if $s_{i_1}\tau_j\le \tau_j$ for all $j$. It is easy to see that $\pi'=(s_{i_1}\tau_1,\ldots,s_{i_1}\tau_{j-1}, \tau_j,\ldots,\tau_r;0,a_1,\ldots,1)$ is an L-S path of shape $\lam$ (it is understood that we omit $a_{j-1}$ if $s_{i_1}\tau_{j-1}=\tau_j$). It follows that $\pi'(1)-\pi(1)$ is an integral multiple of the simple root $\a_{i_1}$. Let $n_1\in\NN$ be such that $\pi'(1)-\pi(1)= n_1\a_{i_1}$. Note that $s_{i_1}\tau_1= s_{i_2}\ldots s_{i_r}$ is a reduced decomposition, and $s_{i_1}\tau_1<\tau_1$. Suppose we have already defined $s(\pi')=(n_2,\ldots,n_r)$ (where $s(id;0,1)$ is the empty sequence). We define the sequence for $\pi$ to be the one obtained by adding $n_1$ to the sequence for $\pi'$. \begin{definition} We denote by $v_\pi$ the vector $v_\pi:= Y_{i_1}^{(n_1)} Y_{i_2}^{(n_2)}\ldots Y_{i_t}^{(n_t)} v_\lam \in V_\Z(\lam)$. \end{definition} Recall that $V_\Z(\lam)_\tau$ can also be described as the subspace obtained from $V_\Z(\lam)_\kappa$ (where $\kappa=s_{i_1}\tau$) by applying $Y_{i_1}$, i.e., $V_\Z(\lam)_\tau=\sum_{n\ge 0} Y_{i_1}^{(n)}V_\Z(\lam)_\kappa$. It follows hence that $v_\pi\in V_\Z(\lam)_{i(\pi)}$. \begin{theorem} $D(\lam)_\tau:=\{ v_\pi\mid i(\pi)\le \tau\}$ is a basis for $V_\Z(\lam)_\tau$. \end{theorem} Recall that we can consider $V_R(\lam)$ as a submodule of $N(\lam)^{\otimes \lbar}$. A first step towards the proof of the theorem is the following: \begin{proposition} Suppose $\lbar a_i\in\Z$ for all $i$. Then $$ v_\pi=Y_{i_1}^{(n_1)} Y_{i_2}^{(n_2)}\ldots Y_{i_t}^{(n_t)}v_{\lambda} = m^\pi + \hbox{\rm\ tensor products\ } m_\mu$ in the strong partial order, so $m_\mu\in N_R(\lam)_{s_\a\tau_i}$. Then, for $n>0$, we have $F_\gamma^{(n)} m_\mu\in N_R(\lam)_{\tau_i}$. In particular, $m_{\tau_i}\ge F_\gamma^{(n)} m_\mu$. But note that, by weight consideration, we can have equality only if $m_{s_\a\tau_i}=Cm_\mu$ for some $C\in R$, so $m_{s_\a\tau_i}>m_\mu$ implies $m_{\tau_i}>F_\gamma^{(n)} m_\mu$. Combining this with the arguments above, one sees that applying $Y_\a^{(n_1)}$ to an arbitrary summand $\not= m^{\pi'}$ in the expression of $v_{\pi'}$ gives only tensors which are smaller in the partial order then $m^\pi$. It follows that $$ Y_\a^{(n_1)}v_{\pi'}= C m^{\pi} + \hbox{\rm\ tensor products\ } 0$ by assumption) we know that $(\mu,\avee)\in\Z$. Now $K_\gamma^{-\ell_{i_1}}$ applied to a weight vector of weight $\lbar\mu$ gives \begin{align*} K_\gamma^{-\ell_{i_1}} m_{\lbar\mu}= \v^{-\ell_{i_1}(d/d_{i_1})(\lbar\mu,\gvee)}m_{\lbar\mu}= \v^{-\ell \ell_{i_1}(\mu,\avee)}m_{\lbar\mu} &=\v^{(-2\ell)(\ell_{i_1}/2)(\mu,\avee)}m_{\lbar\mu}\\ &=m_{\lbar\mu}, \end{align*} because $\ell_{i_1}/2\in\Z$ and $(\mu,\avee)\in \Z$. So we see that the leading term is $$ \big(F_\gamma^{(\ell_{i_1} n_1)}(m_{s_\a\tau_1}\otimes\ldots\otimes m_{s_\a\tau_{j-1}})\big) \otimes m_{\tau_{j}}\otimes\ldots\otimes m_{\tau_r}. $$ Now it is easy to see that if $n=(\tau(\lam),\gvee),m=(\kappa(\lam),\gvee)\ge 0$, then $F_\gamma^{(n+m)}(m_\tau\otimes m_\kappa)=(F_\gamma^{(n)}m_\tau) \otimes (F_\gamma^{(m)}m_\kappa)$. By induction one can show that the leading term is hence equal to $m^\pi$, so the constant $C=1$. \qed \end{demo} \begin{demothm} Fix $\ell$ such that for all L-S paths $\pi=(\tau_1,\ldots,\tau_r;0,a_1,\ldots,1)$ and all $i$ we have $\lbar a_i\in\Z$, and consider the embedding $V_\R(\lam)\hookrightarrow N_R(\lam)^{\otimes\lbar}$. The leading term of $v_\pi$ is the tensor $m^\pi$. Since the $m^\pi$ are obviously linearly independent, the proposition above implies that the $v_\pi$ are also linearly independent. By the Weyl character formula (for representations and the path model, see section 3), we know hence that the $v_\pi$ span an $R$-lattice in $V_R(\lam)$ of maximal rank. The $m^\pi$ can be viewed as a subset of an $R$-basis for $N_R(\lam)^{\otimes\lbar}$. Since the coefficient of $m^\pi$ is $1$ the expression for $v_\pi$, it follows that the $v_\pi$ form an $R$-basis of $V_R(\lam)$. Since the $v_\pi\in V_\Z(\lam)$ by construction, it follows that the $v_\pi$ form in fact a $\Z$-basis of $V_\Z(\lam)$. It remains to prove $D(\lam)_\tau$ is a basis of $V_\Z(\lam)_\tau$. Denote by $V'_\tau$ the $\Z$-submodule spanned by $D(\lam)_\tau$. We have already pointed out that $V'_\tau\subset V_\Z(\lam)_\tau$. Since the extremal weight vector $v_{(\tau;0,1)}=v_\tau$ is an element of $V'_\tau$, to prove the theorem it suffices to prove that $V'_\tau$ is $U_\Z^+(\km)$-stable. This is a consequence of the following lemma, which finishes the proof of the theorem. \qed \end{demothm} \begin{lemma} $X_\a^{(n)} v_\pi=\sum a_{\pi,\eta} v_\eta$, where $a_{\pi,\eta}\not=0$ only if $\pi>\eta$. \end{lemma} \begin{demo} We consider $V_\Z(\lam)$ again as a subspace of $N_R(\lam)^{\otimes \lbar}$. We know that $v_\pi=m^\pi$ + terms strictly smaller in the partial order. It is now easy to see that $X_\a^{(n)} v_\pi$ is a sum of tensor products of weight vectors which are smaller then $m^\pi$ in the partial order. In particular, for any maximal $\eta$ such that $a_{\pi,\eta}\not=0$, we know that the coefficient of $m^\eta$ in the expression of $X_\a^{(n)} v_\pi$ is not zero, so we have necessarily $\pi>\eta$. \qed \end{demo} As an immediate consequence we get by the Demazure type character formula for the L-S paths (section 2): \begin{corollary} (Demazure character formula) $V_\Z(\lam)_\tau$ is a direct summand of $V_\Z(\lam)$, and for any reduced decomposition $\tau=s_{i_1}\ldots s_{i_r}$, the character $\char V_\Z(\lam)_\tau$ is given by the Demazure character formula $\char V_\Z(\lam)_\tau= \Lam_{i_1}\ldots \Lam_{i_r} e^\lam$. \end{corollary} \begin{demoBT} We have obviously $p_\pi(v_\eta)\not=0$ (where $\ell$ etc. has been chosen appropriately) only if $m^\eta$ occurs with non-zero coefficient in the expression of $v_\eta$ as element of $N(\lam)^{\otimes\lbar}$. But this is only possible if $\eta\ge\pi$. We have also seen above that the coefficient of $m^\pi$ is one in the expression of $v_\pi$, so $p_\pi(v_\pi)=1$. It follows that for any algebraically closed field, the path vectors $p_\pi$, $\pi$ an L-S path of shape $\lam$, form a basis of $V_k(\lam)^*$. \qed \end{demoBT} The fact that the basis given by the $v_\pi$ is compatible with the Demazure submodules $V_\Z(\lam)_\tau$ implies: \begin{corollary} The kernel of the restriction map $V_k(\lam)^*\rightarrow V_k(\lam)^*_\tau$ has as basis the $p_\pi$ such that $i(\pi)\not\le \tau$, and the images of the $p_\pi$ such that $i(\pi)\le \tau$, form a basis of $V_k(\lam)^*_\tau$. \end{corollary} \section{Schubert varieties} We apply now the results above to the geometry of Schubert varieties. We show how to obtain from the path basis the normality of Schubert varieties, the vanishing theorems, the reducedness of intersections of unions of Schubert varieties etc. These facts have been proved before, mostly using the machinery of Frobenius splitting (Andersen, Kumar, Mathieu, Mehta, Ramanan, Ramanathan), in some special cases proofs had been given before using standard monomial theory, (Lakshmibai, Musili, Rajeswari, Seshadri), see for example \cite{26}, \cite{27}, \cite{35}, \cite{39}, \cite{40} for a description of the development. Let $k$ be an algebraically closed field, we will omit the subscript $k$ whenever there is no confusion possible. Let $G$ be the simply connected semisimple group corresponding to $\km$, and, according to the choice of the triangular decomposition of $\km$, let $B\subset G$ be a Borel subgroup. Fix a dominant weight $\lam$ and let $P\supset B$ be the parabolic subgroup of $G$ associated to $\lam$. It is well-known that the space of global sections $\Gamma(G/P,\L_\lam)$ of the line bundle $\L_\lam:=G\times_P k_{-\lam}$ is, as a $G$-representation, isomorphic to $V(\lam)^*$. Let $\phi: G/P\hookrightarrow {\Bbb P}(V({\lam}))$ be the corresponding embedding. For $\tau\in W/W_\lam$ denote by $X(\tau)\subset G/P$ the Schubert variety. Let $Y=\bigcup_{i=1}^r X(\tau_i)$ be a union of Schubert varieties. By abuse of notation, we denote by $\L_\lam$ and $p_\pi$ also the restrictions $\L_\lam\vert_{Y}$ and $p_\pi\vert_{Y}$. Recall that the linear span of the affine cone over $X(\tau)$ in $V(\lam)$ is the submodule $V(\lam)_\tau$. The restriction map $\Gamma(G/P,\L_\lam)\rightarrow\Gamma(X(\tau),\L_\lam)$ induces hence an injection $V(\lam)^*_\tau\hookrightarrow\Gamma(X(\tau),\L_\lam)$. We call a path vector $p_\pi$ {\it standard on} $Y$ if $i(\pi)\le\tau_i$ for at least one $1\le i\le r$. Denote $\B(\lam)_Y$ the set of standard path vectors on $Y$. \begin{theorem} \begin{itemize} \item[a)] $\B(\lam)_Y$ is a basis of $\Gamma(Y,\L_\lam)$. \item[b)] $p_\pi\vert_Y\equiv 0$ if and only if $i(\pi)\not\le\tau_i$ for all $i=1,\ldots,r$. \end{itemize} \end{theorem} \begin{corollary} The restriction map $\Gamma(G/P,\L_\lam) \rightarrow \Gamma(Y,\L_\lam)$ is surjective. \end{corollary} Further, by the character formula presented in section 2 we get: \begin{corollary} For any reduced decomposition $\tau=s_{i_1}\ldots s_{i_r}$, $\char \Gamma(X(\tau),\L_\lam)^*$ is given by the Demazure character formula $\char \Gamma(X(\tau),\L_\lam)^*= \Lam_{i_1}\ldots \Lam_{i_r} e^\lam$. \end{corollary} The proof of the theorem is by induction on the dimension and the number of irreducible components of maximal dimension. Let $Y,Y_1,Y_2$ be unions of Schubert varieties. During the induction procedure we prove in addition: \begin{theorem} \begin{itemize} \item[i)] $H^i(Y,\L_\lam)=0$ for $i\ge 1$. \item[ii)] $X(\tau)$ is a normal variety. \item[iii)] The scheme theoretic intersection $Y_1\cap Y_2$ is reduced. \end{itemize} \end{theorem} \begin{demo} In the case of Schubert varieties, a proof is given in \cite{33}, we will give here only a rough sketch of the proof in this case and concentrate on the generalisation to the case of unions of Schubert varieties. The proof uses the ideas presented in \cite{26}, but since the construction of the basis is not anymore part of the induction procedure, these arguments can be applied in a straight forward manner. The theorems hold obviously if $Y$ is a point. Suppose first that $Y=X(\tau)$ is a Schubert variety of positive dimension, and let $\a$ be a simple root such that $\kappa:=s_\a\tau<\tau$. Denote by $SL_2(\a)$ the corresponding subgroup of $G$ with Borel subgroup $B_\a=B\cap SL_2(\a)$. The canonical map $\Psi: Z_\a:=SL_2(\a)\times_{B_\a} X(\kappa)\rightarrow X(\tau)$ is birational and has connected fibres. The map induces an injection $\Gamma(X(\tau),\L_\lam) \hookrightarrow\Gamma(Z_\a,\Psi^*\L_\lam)$. By induction hypothesis, we know that $H^i(X(\kappa),\L_\lam)=0$ for $i\ge 1$. Since the restriction of $\Psi^*\L_\lam$ to $X(\kappa)$ is again $\L_\lam$, the bundle map $Z_\a\rightarrow \Pr^1=SL_2(\a)/B_\a$ induces isomorphisms $H^i(Z_\a,\Psi^*\L_\lam) \rightarrow H^i(\Pr^1, \tilde\Gamma(X(\kappa),\L_\lam))$. (Here $\tilde\Gamma(X(\kappa),\L_\lam)$ denotes the vector bundle associated to the $B_\a$-module $\Gamma(X(\kappa),\L_\lam)$). The short exact sequence $0\rightarrow K \rightarrow V(\lam)^*_\tau\rightarrow V(\lam)^*_\kappa=\Gamma(X(\kappa),\L_\lam)\rightarrow 0$ of $B_\a$-modules induces a long exact sequence in cohomology: $$ \ldots \rightarrow H^i(\Pr^1,{\tilde K}) \rightarrow H^i(\Pr^1,{\tilde V}(\lam)^*_\tau)\rightarrow H^i(\Pr^1,{\tilde \Gamma}(X(\kappa),\L_\lam))\rightarrow \ldots $$ Since $V(\lam)^*_\tau$ is a $SL_2(\a)$-module, the higher cohomology groups vanish for ${\tilde V}(\lam)^*_\tau$ and hence also for ${\tilde \Gamma}(X(\kappa),\L_\lam)$. It follows that $H^i(Z_\a,\Psi^*\L_\lam)=0$ for $i>0$. Recall that if $M$ is a $B_\a$-module and ${\tilde M}$ the associated vector bundle on $\Pr^1$, then $$ \Lambda_\a\char M= \char \Gamma(\Pr_1,{\tilde M})- \char H^1(\Pr^1,{\tilde M}). $$ Since $H^1(\Pr^1,\tilde\Gamma(X(\kappa),\L_\lam))=0$, it follows that $$ \char \Gamma(Z_\a,\Psi^*\L_\lam) =\Lambda_\a \char \Gamma(X(\kappa),\L_\lam). $$ By induction, the character of $\Gamma(Z_\a,\Psi^*,\L_\lam)$ is hence given by the Demazure character formula. Since the same is true for $V(\lam)^*_\tau$ by the corollary in section 5, the inclusions $V(\lam)^*_\tau\hookrightarrow \Gamma(X(\tau),\L_\lam)\hookrightarrow \Gamma(Z_\a,\Psi^*\L_\lam)$ have to be isomorphisms. This proves the theorem for Schubert varieties. Since $\L_\lam$ is an arbitrary ample line bundle and $Z_\a$ is normal, one concludes easily from the isomorphism $V(\lam)^*_\tau\simeq \Gamma(X(\tau),\L_\lam)\simeq \Gamma(Z_\a,\Psi^*\L_\lam)$ that $X(\tau)$ has to be normal. A simple Leray spectral sequence argument shows then that we have in fact $H^i( X(\tau),\L_{\lam})\simeq H^i(Z_\a,\Psi^*\L_{\lam})$, which finishes the proof for Schubert varieties because we know already that $H^i(Z_\a,\Psi^*\L_{\lam})=0$ for all $i>0$. We show now by induction on the number of irreducible components and on the dimension the corresponding statements for unions of Schubert varieties. Let $b(\lam)_Y$ be the number of path vectors standard on $Y$, and denote by $h^0(Y,\L_\lam)$ the dimension of $H^0(Y,\L_\lam)$. Note that the path vectors $p_\pi$ which are standard on $Y$ remain linearly independent: The restriction of a linear dependance relation $\sum a_\pi\pi$ to any maximal irreducible component has to vanish by the results above, which means that all coefficients $a_\pi$ vanish. As a consequence we get: $h^0(Y,\L_\lam)\ge b(\lam)_Y$. Let $Y_1$ and $Y_2$ be unions of Schubert varieties such that $h^0(Y_i,\L_\lam)=b(\lam)_{Y_i}$ for all ample line bundles $\L_\lam$ on $G/P$. We have the following exact sequences of $\O_{G/P}$-modules: $$ 0\rightarrow \O_{Y_1\cup Y_2}\rightarrow \O_{Y_1}\oplus\O_{Y_2}\rightarrow \O_{Y_1\cap Y_2} \rightarrow 0, $$ where $Y_1\cap Y_2$ denotes the scheme theoretic intersection. Let $\L_\lam$ be an ample line bundle on $G/P$. If we tensor the sequence above with $\L_{m\lam}$, then we get for $m\gg 0$ by Serre's vanishing theorem and the long exact sequence in cohomology: $$ h^0(Y_1\cap Y_2,\L_{m\lam})+h^0(Y_1\cup Y_2,\L_{m\lam}) =h^0(Y_1,\L_{m\lam})+h^0(Y_2,\L_{m\lam}). $$ It is easy to see that $b(m\lam)_{(Y_1\cap Y_2)_{red}}+ b(m\lam)_{Y_1\cup Y_2}=b(m\lam)_{Y_1}+b(m\lam_{Y_2})$. Here $(Y_1\cap Y_2)_{red}$ is the intersection with the induced reduced structure, i.e., it is the union of all Schubert varieties contained in $Y_1$ and $Y_2$. Since $h^0(Y_1\cup Y_2,\L_{m\lam})\ge b(m\lam)_{Y_1\cup Y_2}$ and $$ h^0(Y_1\cap Y_2,\L_{m\lam})\ge h^0((Y_1\cap Y_2)_{red},\L_{m\lam}) \ge b(m\lam)_{Y_1\cap Y_2}, $$ it follows by the assumption $h^0(Y_i,\L_\lam)=b(\lam)_{Y_i}$ for $m\gg 0$: $$ b(m\lam)_{(Y_1\cap Y_2)_{red}}= h^0((Y_1\cap Y_2)_{red},\L_{m\lam})= h^0(Y_1\cap Y_2,\L_{m\lam}),$$ and $b(m\lam)_{Y_1\cup Y_2}=h^0(Y_1\cup Y_2,\L_{m\lam})$. The first equality implies that $Y_1\cap Y_2$ is reduced. We will now use the reducedness of $Y_1\cap Y_2$ to prove by induction the basis theorem and the vanishing theorem. Let $Y$ be a union of Schubert varieties, let $Y_1$ be an irreducible component of maximal dimension and let $Y_2$ be the union of the other maximal irreducible components. We assume by induction (on the dimension respectively the number of components) that $h^0(Y_i,\L_\lam)= b(\lam)_{Y_i}$, $b(\lam)_{Y_1\cap Y_2}=h^0(Y_1\cap Y_2,\L_{\lam})$ and $H^j(Y_i,\L_\lam)=H^j(Y_1\cap Y_2,\L_\lam)=0$ for $j>0$ and all ample line bundles $\L_\lam$ on $G/P$. The long exact sequence: $$ 0\rightarrow H^0(Y,\L_\lam)\rightarrow H^0(Y_1,\L_\lam) \oplus H^0(Y_2,\L_\lam) \rightarrow H^0(Y_1\cap Y_2,\L_\lam) \rightarrow H^1(Y,\L_\lam)\rightarrow 0 $$ implies that $H^j(Y,\L_\lam)=0$ for $j\ge 2$. The basis theorem shows that all global sections on $Y_1\cap Y_2$ can be lifted to global sections on $G/P$. But this means that the restriction map $H^0(Y_1,\L_\lam)\oplus H^0(Y_2,\L_\lam) \rightarrow H^0(Y_1\cap Y_2,\L_\lam)$ is surjective and hence $H^1(Y,\L_\lam)=0$. It follows that $h^0(Y,\L_\lam)=h^0(Y_1,\L_\lam)+h^0(Y_2,\L_\lam) -h^0(Y_1\cap Y_2,\L_\lam)$. So the additivity of $b(\lam)_{(\cdot)}$ implies again that $h^0(Y,\L_\lam)=b(\lam)_Y$, which finishes the proof of the theorems. \qed \end{demo} \section{Defining ideals, standard monomials and Groebner bases} For $\lam\in X^+$ let $\pi_1=(\tau^1_1,\ldots,\tau^1_{r_1};\ldots,1)$, $\ldots$, $\pi_s=(\tau^s_1,\ldots,\tau^s_{r_s};\ldots,1)$ be a collection of L-S paths of shape $\lam$, and let $p_{\pi_1},\ldots,p_{\pi_s}\in H^0(G/P,\L_\lam)$ be the corresponding sections. \begin{definition} The monomial $p_{\pi_1}\cdot\ldots\cdot p_{\pi_s}\in H^0(G/P,\L_{s\lam})$ and the concatenation ${\pi_1}*\ldots* \pi_s$ of paths are called {\it standard monomials} of degree $s$ if $$ \tau^1_1>\ldots>\tau^1_{r_1}\ge \tau^2_1>\ldots\ge \tau^s_1>\ldots>\tau^s_{r_s}. $$ The monomial is called standard on $X(\tau)\subset G/P$ if it is standard and $\tau\ge \tau_1^1$. \end{definition} \begin{theorem} The standard monomials of degree $s$ form a basis of $H^0(G/P,\L_{s\lam})$. The monomials standard on $X(\tau)$ form a basis of $H^0(X(\tau),\L_{s\lam})$, and the standard monomials which are not standard on $X(\tau)$ form a basis of $\ker\big(H^0(G/P,\L_{s\lam})\rightarrow H^0(X(\tau),\L_{s\lam})\big)$. \end{theorem} \begin{demorem} The idea of the proof is very similar to the proof of the basis theorem for the path vectors. The first step is to prove that the standard monomials $\pi_1*\ldots*\pi_s$ of degree $s$ are (up to reparametrization) exactly the L-S paths of shape $s\lam$. The bijection is given by $$ p_{\pi_1}\cdot\ldots\cdot p_{\pi_s} \rightarrow (\tau^1_1,\ldots,\tau^1_{r},\ldots,\tau^s_1,\ldots,\tau^s_{r_s}; 0,{a^1_1\over s},\ldots,{1\over s},{1+a^2_1\over s},\ldots,1). $$ It is understood that we omit $\tau^i_{r_i}$ if $\tau^i_{r_i}=\tau^{i+1}_1$. For details see \cite{33}. For simplicity we assume in the following that $s=2$. For $\pi=(\tau_1,\ldots;0,a_1,\ldots,1)$ and $\eta=(\kappa_1,\ldots;0,b_1,\ldots,1)$ let $\ell_1$ and $\ell_2$ be minimal such that they are divisible by $2d$ and $\lbar_1 a_i\in\Z$ for all $i$ and $\lbar_2 b_j\in\Z$ for all $j$. Consider the sequence of embeddings of $U_\tR(\km)$-modules: $$ V_\tR(2\lam)\hookrightarrow V_\tR(\lam)\otimes V_\tR(\lam) \hookrightarrow \big(N_\tR(\lam)^{\otimes\lbar_1}\big)^{\lbar_1} \otimes \big(N_\tR(\lam)^{\otimes\lbar_2}\big)^{\lbar_2} $$ The same procedure as in the preceding sections can be used to associate to an L-S paths $\pi*\eta$ of shape $2\lam$ (a standard monimial of degree 2) a vector $v_{\pi*\eta}$, and to prove that, considered as an element of the tensor product above, it can be expressed as $m^\pi\otimes m^\eta$ plus a sum of tensor products of weight vectors which are smaller in the (induced lexicographic) ordering. It follows for two standard monomials $\pi*\pi'$ and $\eta*\eta'$ that $p_\pi p_{\pi'}(v_{\eta*\eta'})\not=0$ only if $\pi*\pi'<\eta*\eta'$ in the ordering, and $p_\pi p_{\pi'}(v_{\pi*\pi'})=1$. So we can use the same arguments as before to deduce that the standard monomials of degree $s$ and standard on $X(\tau)$ form a basis of $H^0(X(\tau),\L_{s\lam})$, and the standard monomials, not standard on $X(\tau)$, form a basis of the kernel of the restriction map $H^0(G/B,\L_{s\lam})\rightarrow H^0(X(\tau),\L_{s\lam})$. \qed \end{demorem} It remains to consider products of path vectors that are not standard. We associate to a pair of L-S paths $(\pi,\pi')$, $\pi=(\tau_1,\ldots,1)$, $\pi'=(\kappa_1,\ldots,1)$, of shape $\lam$ a pair of sequences as follows: Fix a total order ``$\ge_t$'' on $W/W_\lam$ refining the Bruhat order. Then let $\pi\wedge\pi'=(\sigma_1,\ldots,\sigma_p;0,c_1,c_1+c_2\ldots,\sum_{i=1}^p c_i)$ be defined by: $\{\sigma_1,\ldots,\sigma_t\}= \{\tau_1,\ldots,\kappa_1,\ldots \}$, rewritten such that $\sigma_1\ge_t\ldots\ge_t\sigma_p$, and $c_i$ is equal to $(a_j-a_{j-1})/2$ if $\sigma_i=\tau_j$, $c_i=(b_j-b_{j-1})/2$ if $\sigma_i=\kappa_j$, respectively $c_i(a_j-a_{j-1}+b_{j'}-b_{j'-1})/2$ if $\sigma_i=\tau_j=\kappa_{j'}$. Note if $\pi*\pi'$ is standard, then obviously $\pi*\pi'=\pi\wedge\pi'$. More generally, we call a rational $\lam$-path a pair of sequences $(\sigma_1,\ldots,\sigma_r;0,c_1,\ldots,1)$ where $\sigma_i\in W/W_\lam$, the sequence is linearly ordered with respect to $\ge_t$, and $0_t\kappa_1$, or $\sigma_1=\kappa_1$ and $c_1>d_1$, etc. Similarly, we write ``$\ge_t^r$'' if we extend the total order reverse lexicographically, i.e., if $\sigma_r>_t\kappa_s$ or $\sigma_r=\kappa_s$ and $1-c_{r-1}>1-d_{s-1}$, or $\sigma_r=\kappa_s$ and $1-c_{r-1}=1-d_{s-1}$ and $\sigma_{r-1}>_t\kappa_{s-1}$, etc. We define two orderings on pairs of L-S paths of shape $\lam$ as follows: $(\pi,\pi')\ge_t (\eta,\eta')$ if $\pi\wedge\pi'\ge_t \eta\wedge\eta'$, and if $\pi\wedge\pi'= \eta\wedge\eta'$, then we define $(\pi,\pi')\ge_t (\eta,\eta')$ if $\pi\ge_t\eta$ respectively $\pi=\eta$ and $\pi'\ge_t\eta'$. We define a reverse version of the ordering by $(\pi,\pi')\ge_t^r (\eta,\eta')$ if $\pi\wedge\pi'\ge_t^r \eta\wedge\eta'$ in the reverse lexocigraphic ordering, and if $\pi\wedge\pi'= \eta\wedge\eta'$, then we define $(\pi,\pi')\ge_t^r (\eta,\eta')$ if $\pi'\ge_t^r\eta'$ respectively $\pi'=\eta'$ and $\pi\ge_t\eta$. \begin{proposition} If $\pi,\pi'$ are two L-S paths of shape $\lam$, then $p_\pi p_{\pi'}=\sum a_{\eta,\eta'}p_\eta p_{\eta'}$, where $p_\eta p_{\eta'}$ is standard and $a_{\eta,\eta'}\not=0$ only if $(\eta,\eta')\ge_t(\pi,\pi')\ge_t^r (\eta,\eta')$. \end{proposition} \begin{demo} The proposition is obviously correct if either $p_\pi p_{\pi'}$ or $p_{\pi'} p_\pi$ is standard. It remains to consider the case where none of the products are standard. We can repeat the procedure to construct a basis with a different algorithm. For $\pi=(\tau_1,\ldots,\tau_r; 0,a_1,\ldots,1)$ let $s_{\a_1}$ be such that $s_{\a_1}\tau_r>\tau_r$. Let $j$ be minimal such that $s_{\a_1}\tau_i\ge\tau_i$ for $i=j,\ldots,r$. It is easy to see that $$ \pi'=(\tau_1,\ldots,\tau_{j-1}, s_{\a_1}\tau_j,\ldots,s_{\a_1}\tau_r;0,a_1,\ldots,1) $$ is again an L-S path. Fix $n_1$ such that $\pi(1)-\pi'(1)=n_1\a_{i_1}$, and let $s(n_1,\ldots,n_t)$ be the sequence obtained from $\pi$ with respect to a reduced decomposition $w_0=s_{\a_t}\cdots s_{\a_1}\tau$ of the longest word in the Weyl group. As in section~5, one shows that $$ u_\pi:=X^{(n_1)}_{\a_{i_1}}\ldots X^{(n_t)}_{\a_{i_t}} v_{w_0} =m^\pi + \sum_{m>^r m^\pi} m \in N_R(\lam)^{\otimes \lbar} $$ for an appropriate $\ell$. The ordering $>^r$ is defined as follows: $m_\tau\ge m_\kappa$ if $\tau\ge\kappa$ in the Bruhat ordering, and for a weight vector $m_\nu\in N_R(\lam)$ we write $m_\nu\ge^r m_\kappa$ if $m_\nu\in U^-_\v(\km^t)m_\kappa$. On tensor products we take the induced reverse lexicographic partial order ``$\ge^r$''. A first observation to make is that we have defined the path vectors $p_\pi$ according to a minimal choice of an appropriate $\ell$, but, in fact, the definition makes sense for an arbitrary $\ell$ divisible by $2d$ and with the property that $\lbar a_i\in\Z$ for all $i$. Using the proposition describing the embedding of $v_\pi$ into $N(\lam)^{\otimes \lbar}$ in section~5 and the description of $u_\pi$ above, it is easy to check that such a vector $p_{\pi,\ell}$ has the property $p_{\pi,\ell}(v_\eta)\not=0$ only if $\eta\ge\pi$ and $p_{\pi,\ell}(u_\eta)\not=0$ only if $\pi\ge^r\eta$. Since $p_\pi$ has the same properties it follows that $p_{\pi,\ell}$ can be written as $p_\pi +$ a linear combination of $p_\eta$'s such that $\eta>\pi>^r\eta$. The second observation is that if $\eta\ge\pi\ge^r\eta$ and $\eta'\ge\pi'\ge^r\eta'$, then $\eta\wedge\eta'\ge_t \pi\wedge\pi'\ge_t^r \eta\wedge\eta'$, and hence $(\eta,\eta')\ge_t (\pi,\pi')\ge_t^r (\eta,\eta')$. It follows that it is sufficient to prove the proposition for the $p_{\pi,\ell}$ for some appropriate $\ell$: If the relation above is correct for the $p_{\pi,\ell}$, then we can replace them by the corresponding linear combination $p_\pi+\sum a_\eta p_\eta$. Of course, there may occur now again non-standard products $p_{\eta_i} p_{\eta'_j}$ after replacing the $p_{\eta,\ell}$ by their expression as linear combination of the $p_{\eta_i}$. But since $(\eta,\eta')\ge_t(\pi,\pi')\ge_t^r (\eta,\eta')$ and $\eta_i\ge\eta\ge^r\eta_i$, we know that all terms that occur have the property $(\eta_i,\eta_j')\ge_t(\pi,\pi')\ge_t^r (\eta_i,\eta_j')$ One may assume by induction that the relation holds for pairs that are $>_t (\pi,\pi')$ and $<_t^r (\pi,\pi')$ in the ordering. So after replacing these non-standard products by the linear combination of standard products provided by induction, we see that we get the desired relation. It remains to prove that the relation holds for some appropriate $\ell$. Let $\ell$ be such that $2d$ divides $\ell$ and $\lbar c_i\in\Z$ for all L-S paths of shape $\lam$. Then $p_{\pi,\ell} p_{\pi',\ell}(v_{\eta*\eta'})\not=0$ for a standard monomial $\eta*\eta'$ only if, in the expression of $v_{\eta*\eta'}$ as element of $N(2\lam)^{\otimes\lbar}\subset N(\lam)^{\otimes 2\lbar}$, the tensor $m^\pi\otimes m^{\pi'}$ occurs with a coefficient different from zero. The tensor product is not symmetric for quantum groups, but for the $F$'s it is symmetric up to multiplication with a root of unity. So to demand that $m^\pi\otimes m^{\pi'}$ occurs with a coefficient different from zero is equivalent to demand that $m^{\pi\wedge\pi'}$ occurs with a non-zero coefficient. Here the definition of $m^{\pi\wedge\pi'}$ is the same as for L-S paths of shape $2\lam$. The same arguments as before show that such a tensor can occur only if $m^{\eta*\eta'}\ge m^{\pi\wedge\pi'}$. By the definition of the ordering on the tensors this implies $p_{\pi,\ell} p_{\pi',\ell}(v_{\eta*\eta'})\not=0$ only if $\eta*\eta'\ge \pi\wedge\pi'$. In terms of the ordering on pairs this implies $(\eta,\eta')\ge_t (\pi,\pi')$ (because we assume that $\pi*\pi'$ is not standard). The same arguments apply to $u_{\eta*\eta'}$ and show that $p_{\pi,\ell} p_{\pi',\ell}(u_{\eta*\eta'})\not=0$ only if $\pi\wedge\pi'\ge^r\eta*\eta'$. Since $\pi*\pi'$ is not standard, this implies $(\pi,\pi')\ge^r_t(\eta,\eta')$. \qed \end{demo} There is a case where we can be a little more precise about one coefficient. We say that two L-S paths have the same support if the $\tau_i$ and $\kappa_j$ can be chosen out of one maximal chain in $W/W_\lam$. It is easy to see that in this case the element $\pi\wedge\pi'$ is an L-S path of shape $2\lam$. So in this case the set of standard monomials $\eta*\eta'$ such that $(\eta,\eta')\ge_t (\pi,\pi')$ admits a unique minimal element: $\pi\wedge\pi'$. The same arguments as above show that the coefficient of $m^{\pi\wedge\pi'}$ in the expression of $v_{\pi\wedge\pi'}$ as element of $N(2\lam)^{\otimes\lbar}\subset N(\lam)^{\otimes 2\lbar}$ is $1$. Let $\pi_1,\pi_1'$ be the two L-S paths of shape $\lam$ such that $\pi_1*\pi_1'=\pi\wedge\pi'$, then we get: \begin{corollary} $p_\pi p_{\pi'}=p_{\pi_1} p_{\pi_1'} + \sum a_{\eta,\eta'}p_\eta p_{\eta'}$, where $p_\eta p_{\eta'}$ is standard and $a_{\eta,\eta'}\not=0$ only if $(\eta,\eta')>_t(\pi,\pi')>_t^r (\eta,\eta')$. \end{corollary} Now $\pi$ and $\pi'$ have obviously the same support if $\pi=\pi'$. If $da_i\in\Z$ for all $a_i$ and $\pi=(\tau_1,\ldots\tau_r;0,a_1,\ldots,1)$, then the corollary shows that as smallest term in the expression of $p_\pi^d$ as a linear combination of standard monomials, with respect to $\ge_t$, we get the product of extremal weight vectors $p_{\tau_1}^{t a_1}\cdots p_{\tau_r}^{t (1-a_{r-1})}$. In that sense we can consider $p_\pi$ as an approximation of an $\lbar$-th root of the section $p_{\tau_1}^{\lbar a_1}\cdots p_{\tau_r}^{\lbar (1-a_{r-1})}$ in $H^0(G/P,\L_{\lbar\lam})$. The next theorem states that these relations define already the Schubert variety $X(\tau)$ scheme theoretically as a subvariety of $\P(V(\lam)_\tau)$. \begin{theorem} Denote by $S_\tau$ the free associative algebra $k\{x_\pi\mid i(\pi)\le\tau\}$, and let $I$ be the ideal obtainend as kernel of the canonical surjective map $S_\tau\rightarrow\bigoplus_{n\ge 0} H^0(X(\tau),\L_{n\lam})$, $x_\pi\mapsto p_\pi$. The relations: $$ \cases p_\pi p_{\pi'}-p_{\pi'} p_{\pi} &\hbox{\ if }p_{\pi'} p_{\pi} \hbox{\ is a standard monomial},\cr p_\pi p_{\pi'}=\sum_{(\eta,\eta')\ge_t(\pi,\pi')\ge_t^r(\eta,\eta')} a_{(\eta,\eta')} p_\eta p_{\eta'} &\hbox{\ if\ }p_{\pi'} p_{\pi},p_{\pi} p_{\pi'} \hbox{\ are not standard},\cr \endcases $$ form a non-commutative reduced Groebner basis for $I$. \end{theorem} \begin{remark} The fact that the relations provide a non-commutative Groebner basis for $I$ provides a new proof of the fact that the ring $\bigoplus_{n\ge 0} H^0(X(\tau),\L_{n\lam})$ is a Koszul ring. This has been proved before for example by S.~P.~Inamdar and V.~Mehta \cite{13}. Using standard arguments from Groebner basis theory one can use the results above to deform the affine cone over $X(\tau)$ into a union of affine spaces. If one takes into account the refined version given by the corollary, then one sees that one can deform the affine cone into a union of toric varieties, where the irreducible components are indexed by maximal chains in $\{\kappa\in W/W_\lam\mid\kappa\le\tau\}$. \end{remark} \begin{demo} To prove that the set is a generating system for the ideal we have to show that we can express any monomial as a linear combination of standard monomials just by using the relations above. Denote by $I'\subset S_\tau$ the ideal generated by the relations above. As a first step we define an ordering on the $n$-tuples of L-S paths satisfying $i(\pi)\le\tau$. We identify in the following $n$-tuples with monomials of degree $n$ in $S_\tau$. The notion $\pi_1\wedge\ldots\wedge\pi_r$ can be generalized in the obvious way, and we say $(\pi_1,\ldots,\pi_n)\ge^r_t (\eta_1,\ldots,\eta_n)$ if $\pi_1\wedge\ldots\wedge\pi_n >^r \eta_1\wedge\ldots\wedge\eta_n$, and if $\pi_1\wedge\ldots\wedge\pi_n =\eta_1\wedge\ldots\wedge\eta_n$, then we say $(\pi_1,\ldots,\pi_n)\ge_t^r (\eta_1,\ldots,\eta_n)$ if this is true in the induced reverse lexicographic ordering on the tuples. We extend this order to a total order by saying that a monomial of degree $n$ is strictly greater then a monomial of degree $m$ if $n>m$. It is easy to check that this total order is a left and right monomial order. Note that if we replace a couple $(\pi_i,\pi_{i+1})$ by a couple $(\pi_i,\pi_{i+1})>^r_t (\pi'_i,\pi'_{i+1})$, then $(\pi_1,\ldots,\pi_i,\pi_{i+1},\ldots)>^r_t (\pi_1,\ldots,\pi'_i,\pi'_{i+1},\ldots)$. Recall that, by the definition of a standard monomial, a monomial $\pi_1*\ldots*\pi_n$ is standard if and only if $\pi_i*\pi_{i+1}$ is standard for all $i=1,\ldots,n-1$. We call a monomial $(\eta_1,\ldots,\eta_n)\in S_\tau$ standard if $\eta_1*\ldots*\eta_n$ is standard. Start with an arbitrary monomial $(\pi_1,\ldots,\pi_n)$ in $S_\tau$ and suppose that $\pi_i*\pi_{i+1}$ is not standard, then, using the relations above, we may replace the monomial by a linear combination of monomials in $S_\tau$ that are strictly smaller with respect to $\ge^r_t$. Since there are only a finite number of monomials of a given degree, we obtain after a finite number of steps an expression $(\pi_1,\ldots,\pi_n)\equiv$ a sum of standard monomials $\mod I'$, where the standard monomials are all strictly smaller then $(\pi_1,\ldots,\pi_n)$ with respect to $\ge_t^r$. It follows that the map $S_\tau/I'\rightarrow \bigoplus_{n\ge 0} H^0(X(\tau),\L_{n\lam})$ is an isomorphism. It remains to prove that the generators form a Groebner basis. The leading terms of the generators are the non-standard monomials of the form $(\pi,\pi')$, so the ideal generated by the leading terms are the linear combinations of all non-standard monomials. Suppose $f$ is an element of $I'$, we have to show that its leading term with respect to $\ge_t^r$ is not a standard monomial. Suppose the contrary is true, so $f=s + $ smaller terms. Let $f'$ be the element obtained from $f$ by replacing all non-standard monomials by their corresponding expression as a sum of standard monomials, this gives a nonzero element of $S_\tau$ with leading term $s$. Modulo $I'$, these two elements are equal, so the image of $f'$ is zero in $\bigoplus_{n\ge 0} H^0(X(\tau),\L_{n\lam})$. On the other hand, $f'$ is a non-zero sum of standard monomials, so the image cannot be equal to zero. It follows that the leading term cannot be a standard monomial. From the description of the generating set it follows imediately that the basis is reduced. \qed \end{demo} \section{Standard monomials II}\label{Standard Monomials II} Let $\lam_1,\ldots,\lam_r$ be some dominant weights, set $\lam=\sum \lam_i$, and fix $\tau\in W/W_\lam$. For each $i$ let $\tau_i$ be the image of $\tau$ in $W/W_{\lam_i}$. A module $V_\lam$ (without specifying the underlying ring) is always meant to be the Weyl module of highest weight $\lam$ over an algebraically closed field. The inclusion $V_\lam\hookrightarrow V_{\lam_1}\otimes\ldots \otimes V_{\lam_r}$ induces a map $V_\lam(\tau)\hookrightarrow V_{\lam_1}(\tau_1)\otimes \ldots\otimes V_{\lam_r}(\tau_r)$, and hence in turn a map $V_{\lam_1}^*(\tau_1)\otimes\ldots\otimes V_{\lam_r}^*(\tau_r)\rightarrow V_\lam^*(\tau)$. We write $\pi_i$ and $\pi_\lam$ for the paths $t\mapsto t\lam_i$ respectively $t\mapsto t\lam$. Denote by $B_i$ the set of L-S paths of shape $\lam_i$, and by $B_\lam$ the set of paths of shape $\lam$. Recall that the associated graph $G(\pi_\lam)$ has as vertices the set $B_\lam$, and we put an arrow $\eta\mapright{\a}\eta'$ with colour a simple root $\a$ if $f_\a(\eta)=\eta'$. Denote by $B_1*\ldots*B_r$ the set of concatenations of all paths in $B_1,\ldots,B_r$. Remember that the set of paths is stable under the root operators, and the associated graph decomposes into the disjoint union of irreducible components. Denote by $G(\pi_1*\ldots*\pi_r)$ the irreducible component containing $\pi_1*\ldots*\pi_r$. Recall that the map $\pi_1*\ldots*\pi_r\mapsto\pi_\lam$ extends to an isomorphism of graphs $\phi:G(\pi_1*\ldots*\pi_r)\rightarrow G(\pi_\lam)$. A monomial $\eta_1*\ldots*\eta_r \in B_1*\ldots*B_r$ is called standard if it is in the irreducible component $G(\pi_1*\ldots*\pi_r)$, and in this case we define: $i(\eta_1*\ldots*\eta_r):= i(\phi(\eta_1*\ldots*\eta_r))$. \begin{definition} Let $\eta_1,\ldots,\eta_r$ be L-S paths of shape $\lam_1,\ldots,\lam_r$. A monomial of path vectors $p_{\eta_1}\cdots p_{\eta_r}$ is called {\it standard} if the concatenation $\eta_1*\ldots*\eta_r$ is standard. The standard monomial is called {\it standard with respect to} $\tau$ if $i(\eta_1*\ldots*\eta_r)\le \tau$. \end{definition} The proof of the following theorem is very similar to the proof of the corresponding theorem in the previous section. For details see \cite{33}. \begin{theorem} The set of standard monomials form a basis of $H^0(G/B,\L_\lam)$, and the set of monomials, standard with respect to $\tau$, form a basis of $H^0(X(\tau),\L_\lam)$. \end{theorem} %% Part 2 - V. Lakshmibai \section{Determination of the singular locus of $X(w)$}\label{sing} Let Sing $X(w)$ denote the singular locus of $X(w)$. In this section, we recall from \cite{l1}, \cite{l2}, \cite{l-sa}, \cite{l-se2}, \cite{l-w} the description of Sing $X(w)$. We first recall some generalities on $G/Q$. Let $G$ be a semisimple and simply connected algebraic group defined over an algebraically closed field $k$ of arbitrary characteristic. As above, let $T\subset G$ be a maximal torus, and $B\supset T$ be a Borel subgroup. Let $W$ be the Weyl group of $G$. Let $R$ be the root system of $G$ relative to $T$. Let $R^+$ (resp. $S$) be the system of positive (resp. simple) roots of $R$ with respect to $B$. Let $R^-$ be the corresponding system of negative roots. \subsection{The set $W_Q^{min}$ of minimal representatives of $W/W_Q$}. Let $Q$ be a parabolic subgroup of $G$ containing $B$, and $W_Q$ be the Weyl group of $Q$. In each coset $wW_Q$, there exists a unique element of minimal length (cf. \cite{bou}). Let $W_Q^{\text{min}}$ be this set of representatives of $W/W_Q$. The set $W_Q^{\text{min}}$ is called the {\em set of minimal representatives of} $W/W_Q$. We have $$ W_Q^{\text{min}}=\{w\in W\mid l(ww')=l(w)+l(w'), \text{ for all }w'\in W_{Q} \}. $$ The set $W_Q^{\text{min}}$ may also be characterized as $$W_Q^{\text{min}}=\{w\in W\mid w(\a)>0, \text{ for all } \a\in S_Q\}$$ (here by a root $\beta$ being $>0$ we mean $\bb\in R^+$). In the sequel, given $w\in W$, the minimal representative of $wW_Q$ in $W$ will be denoted by $w_Q^{\text{min}}$. \subsection{The set $W_Q^{max}$ of maximal representatives of $W/W_Q$} In each coset $wW_Q$ there exists a unique element of maximal length. Let $W_Q^{\text{max}}$ be the set of these representatives of $W/W_Q$. We have $$W_Q^{\text{max}}=\{w\in W\mid w(\a)<0\text{ for all } \a\in S_Q\}.$$ Further, if we denote by $w_Q$ the element of maximal length in $W_Q$, then we have $$W_Q^{\text{max}}=\{ww_Q\mid w\in W_Q^{\text{min}}\}.$$ In the sequel, given $w\in W$, the maximal representative of $wW_Q$ in $W$ will be denoted by $w_Q^{\text{max}}$. \subsection{The big cell and the opposite big cell}\label{1.7} The $B$-orbit $Be_{w_0}$ in $G/Q$ ($w_0$ being the unique element of maximal length in $W$) is called the {\em big cell} in $G/Q$. It is a dense open subset of $G/Q$, and it gets identified with $R_u(Q)$, the unipotent radical of $Q$, namely the subgroup of $B$ generated by $\{U_\a\mid\a\in R^+ \setminus R_Q^+\}$ (cf. \cite{bo}). Let $B^-$ be the Borel subgroup of $G$ opposite to $B$, i.e. the subgroup of $G$ generated by $T$ and $\{U_{\a}\mid\a\in R^-\}$. The $B^-$-orbit $B^-e_{\text{id},Q}$ is called the {\em opposite big cell} in $G/Q$. This is again a dense open subset of $G/Q$, and it gets identified with the unipotent subgroup of $B^-$ generated by $\{U_{\a}\mid \a\in R^- \setminus R_Q^-\}$. Observe that both the big cell and the opposite big cell can be identified with $\Bbb{A}^{N_Q}$, where $N_Q=\#\{R^+ \setminus R_Q^+\}$. For a Schubert variety $X_Q(w)\subset G/Q$, $Y_Q(w):=B^-e_{\text{id}}\cap X_Q(w)$ is called the {\em opposite cell} in $X_Q(w)$ (by abuse of language). In general, it is not a cell (except for $w=w_0$). It is a nonempty affine open subvariety of $X_Q(w)$, and a closed subvariety of the affine space $B^-e_{\text{id}}$. \subsection{Equations defining a Schubert variety}\label{1.8} Let $L$ be an ample line bundle on $G/Q$. Consider the projective embedding $G/Q\hookrightarrow {\text{Proj}}(H^0(G/Q,L))$. As a consequence of Standard Monomial Theory - abbreviated as SMT in the sequel - we have seen from the previous section that the homogeneous ideal of $G/Q$ for this embedding is generated in degree $2$, and any Schubert variety $X$ in $G/Q$ is scheme theoretically (even at the cone level) the intersection of $G/Q$ with all the hyperplanes in $\text{Proj}(H^0(G/Q,L))$ containing $X$. For a maximal parabolic subgroup $P_i$, let us denote the ample generator of $\text{Pic\,}(G/P_i)$ ($\simeq\Bbb{Z}$) by $L_i$. Given a parabolic subgroup $Q$, let us denote $S\setminus S_Q$ by $\{\a_1,\dots,\a_t\}$, for some $t$. Let \begin{align} R&=\bigoplus_{\underline{a}}H^0(G/Q, \bigotimes_iL_i^{a_i})\notag\\ R_w&=\bigoplus_{\underline{a}}H^0(X_Q(w), \bigotimes_iL_i^{a_i}),\notag \end{align} where $\underline{a}=(a_1,\dots,a_t)\in\Bbb{Z}_+^t$. We have that the natural map $$\bigoplus \cal{S}^{a_1}(H^0(G/Q, L_1))\otimes\dots\otimes \cal{S}^{a_1}(H^0(G/Q, L_t)) \to R$$ is surjective, and its kernel is generated as an ideal by elements of total degree $2$. Further, the restriction map $R\to R_w$ is surjective, and its kernel is generated as an ideal by elements of total degree $1$. \subsection{Sing $X(w)$} If $X(w)$ is not smooth, then Sing $ X(w)$ is a non-empty $B$-stable closed subvariety of $X(w)$. Given a point $x\in X(w)$, let $T(w,x)$ denote the Zariski tangent space to $X(w)$ at $x$. To decide if $x$ is a smooth point or not, it suffices (in view of Bruhat decomposition) to determine if the $T$-fixed point $e_\t$ of the $B$-orbit through $x$ is a smooth point or not. We shall denote $T(w,e_{\tau})$ by just $T(w,\tau)$. Recall that ${\text dim}\ T(w,\tau) \geq {\text dim}\ X(w)$ ($=l(w)$) with equality if and only if $e_{\tau}$ is a smooth point. \subsection{A canonical affine neighbourhood of a $T$-fixed point in $G/B$}\label{sub:neigh} Let $\t \in W$. Let $U^- _\t$ be the unipotent part of the Borel subgroup $B^- _\t\ $, opposite to $B_\t\ (=\ \t B\t^{-1})$ (it is the subgroup of $G$ generated $\{U_{\a}\ |\ \a\in \t(R^- )\}$). Then $U^- _\t e_\t$ is an affine neighbourhood of $e_\t$ in $G/B$, and can be identified with ${{\Bbb A}}^{N}$, where $N=\#\{R^+\}$. Let us denote it by ${\cal O}_\t^-$. For $w \in W,\ w\ge \t$, let us denote $Y(w,\t):={\cal O}_\t^-\cap X(w)$. It is a nonempty affine open subvariety of $X(w)$, and a closed subvariety of the affine space ${\cal O}_\t^-$. Let $I(w,\t)$ be the ideal defining $Y(w,\t)$ as a closed subvariety of ${\cal O}_\t^-$. As a consequence of SMT, we have \begin{prop}\label{ideal} Let ${\cal B}^d$ be the basis for $H^0(G/B, L_{\o_d}),\ 1\le d\le l$ as given by SMT (here, $l$ is the rank of $G$, and $\o_d$ is the $d^{{\text th}}$ fundamental weight). Then $I(w,\t)$ is generated by $\{u|_{Y(w,\t)},\ u \in {\cal B}^d,\ 1\le d \le l\ |\ u|_{X(w)}=0\}$. \end{prop} The problem of the determination of the singular locus of a Schubert variety was first solved by the first author (in collaboration with Seshadri (cf. \cite{l-se2}), for $G$ classical. The main idea in \cite{l-se2} is to write down the equations defining $Y(w,\t)$ as a closed subvariety of the affine space ${\cal O}_\t^-$ (as given by Proposition \ref{ideal}), and then use the Jacobian criterion for smoothness. Below, we recall the result of \cite{l-se2} for type {\bf A} and we refer the reader to \cite{l-se2}, \cite{l1}, \cite{l2} for results for other classical groups. \subsection{Description of Sing $X(w)$ for type A}\label{sub:sing.a} \begin{thm}(cf. \cite{l-se2}) Let $G=SL(n)$. Let $w,\t \in W, \t \le w$. Then $${\text dim}\ T(w, \t)= \# \{\a \in R^+\ |\ w\ge \t s_\a\}.$$ \end{thm} \subsection{A criterion for smoothness of Schubert varieties for type A in terms of permutations}\label{sub:criterion.a} Recall that for $G=SL(n),\ W={\cal S}_n$. First consider $G=SL(4)$. In this case $X(3412), X(4231)$ are the only singular Schubert varieties. The situation for a general $n$ turns out to be ``nothing more than this" as given by the following theorem. \begin{thm}(cf. \cite{l-sa}). Let $w \in {\cal S}_n$, say $w=(a_1,...,a_n)$. Then $X(w)$ is singular if and only if the following property holds \begin{equation*}\label{e:L-Sa1} \left\{ \begin{matrix} {{\text{there}}\,\,\, {\text{exist}}}\,\,\,i,j,k,l,\ 1\le i < j s_\bb \t $, then $\bb \in N_{w, \t}$ (necessarily). \item Let $\t $ either $\t s_{2\e_i}$, or $\t s_{2\e_j}$, then $\bb \in N_{w,\t}\ \Longleftrightarrow w \geq s_\bb \t $. \end{enumerate} \item Let $\t \t s_{\e_i - \e_j}$, then $\bb \in N_{w,\t}\ \Longleftrightarrow w \geq s_\bb\t s_{2\e_j } $. \end{enumerate} \end{enumerate} \end{thm} \begin{rem} One has similar descriptions of $ N_{w,\t}$ for types {\bf B} and {\bf D} (see \cite{l4} for details). \end{rem} \subsection{Irreducible components of Sing $X(w)$} The problem of the determination of the irreducible components of Sing $X(w)$ is open even for type {\bf A}. \vs.2cm\ni{\bf Known results for $G/P$.} The irreducible components of Sing $X(w)$ have been determined in \cite{l-w} for $X(w)$ in $G/P$, for $G$ classical, and $P$ certain parabolic subgroup. We recall this result below. \vskip1em \vs.1cm\ni{\bf TYPE A.} Let $G=SL(n)$, and $P=P_d$, the maximal parabolic subgroup ( with associated set of simple roots being $S\setminus \{\a_d\}$). Then it is well known that $G/P$ gets identified with the Grassmannian variety $G_{d,n}=$ the set of $d$- dimensional subspaces of $k^n$. It is well known that $W^{P_d}$, the set of minimal representatives may be identified as $$W^{P_d}=\{(a_1,\cdots ,a_d)\ |\ 1\le a_1n$ . The conditions on the $a_i$'s imply that the partition ${\bold a}$ is a {\it self-dual} partition contained in an $n\times n$ square. For a partition ${\bold a}=({\bold a}_1,\cdots ,{\bold a}_n)$, we shall denote by $X_{{\bold a}}$ the Schubert variety corresponding to $(a_1,\cdots ,a_n)$. Thus here again Schubert varieties in $G/P$ are indexed by self-dual partitions contained in $n^n$. \begin{thm}\label{th 17} (cf. \cite{l-w}). Let ${\bold a}$ be a self-dual partition. Then Sing $X_{{\bold a}}=\cup X_{{\bold b}}$, where ${\bold b}\subset {\bold a}$, and either ${\bold a}/{\bold b}$ is a disjoint sum of two hooks that are dual to each other, or, ${\bold a}/{\bold b}=(r+i, r^{r-1},1^i)\ /\ ((r-1)^{r-1})$ for some $r,i$ with $i>0$ (the sum of two hooks dual to each other connected at one box), or ${\bold a}/{\bold b}=(r^2,2^{r-2})\ /(0^r)$ for some $r>2$ (self-dual double hook). \end{thm} \vs.2cm\ni {\bf TYPE D.} Let $G=$ SO($2n$), and $P=P_n$, the maximal parabolic with associated set of simple roots being $S\setminus \{\a_n\}$ (notations being as in \cite{bou}). Then $G/P$ can be identified with the isotropic Grassmannian of $n$ spaces in a $2n$-dimensional space with a non-degenerate symmetric bilinear form (,). Then it can be seen easily that $W^{P_n}$ can be identified as $$ \left\{(a_1 \cdots a_n) \left | \begin{matrix} (1) & 1\leq a_1 < a_2 < \cdots < a_n \leq 2n \hfill \\ (2) & \#\{i,1\le i\le n\ |\ a_i>n\} \text{ is even}\\ (3) & {\text{for}}\,\,\, 1 \leq i \leq 2n,\,\,\,{\text{if}}\,\,\, i \in \{a_1,..., a_n \} \hfill \\ & {\text{then}}\,\,\, 2n+1-i \notin \{ a_1,..., a_n \} \hfill \end{matrix} \right. \right\}. $$ Let $P=P_n, Q=P_{n-1}$. Consider the map $\delta:W^P\rightarrow W^Q, \delta (a_1,\cdots , a_n)=(b_1,\cdots , b_{n-1}) $, where $(b_1,\cdots , b_{n-1})$ is obtained from $(a_1,\cdots , a_n)$ by replacing $n$ by $n'(=n+1)$ (resp. $n'$ by $n$) if $n$ (resp. $n'$) is present in $\{a_1,\cdots , a_n\}$. Note that if $a_n>n$, then precisely one of $\{n, n'\}$ is present in $(a_1,\cdots , a_{n-1})$; if $a_n=n$, then $(a_1,\cdots , a_n)=(1,\cdots ,n)$, and $\delta (a_1,\cdots , a_n)=(1,\cdots ,n-1)$. It is easily seen that $\delta$ is a bijection preserving the Bruhat order. In fact $\delta$ is induced by the isomorphism of the varieties $G/P\rightarrow G/Q$. Let us denote $W^{'}=W(SO(2n-1))$, and define $\te : W^{'P_{n-1}}\rightarrow W^P$ as $\te (a_1,\cdots ,a_{n-1}) = (a_1,\cdots , a_n) $, where $a_n=n$ or $n'$ and the choice is made so that $\#\{i,1\le i\le n\ |\ a_i>n\}$ is even (the $i'$ in $(a_1,\cdots , a_{n-1})$ (resp. $\te (a_1,\cdots , a_{n-1})$) should be understood as $2n-i$ (resp. $2n+1-i$)). Then it is easily seen that $\te$ is a bijection preserving the Bruhat order. In fact $\te$ is induced by the isomorphism of the varieties $SO(2n-1)/P_{n-1}\rightarrow SO(2n)/P$. In view of the isomorphisms $\te$ and $\delta$, we have results for Schubert varieties in $G/P, G/Q, G$ being $SO(2n)$ similar to Theorem \ref{th 17}. \begin{rem} For other related results on Sing $X(w)$, we refer the readers to \cite{ca}, \cite{ku} and \cite{po} \end{rem} \section{Applications to other varieties} In this section, we introduce two classes of affine varieties - certain ladder determinantal varieties (cf. \S\ref{ldv}) and certain quiver varieties (cf. \S\ref{quiv} ) - and we conclude (cf. \cite{g-l}, \cite{l-ma} ) that these varieties are normal, Cohen-Macaulay and have rational singularities by identifying them with $Y_Q(w)$ (cf. \S \ref{1.7}) for suitable Schubert varieties $X_Q(w)$ in suitable $SL(n)/Q$ (note that $Y_Q(w)$ is normal, Cohen-Macaulay and has rational singularities, since $X_Q(w)$ has all these properties). We first recall some facts on ``Opposite cells" in Schubert varieties in $SL(n)/Q$. %%%%%%%%%%%%%%%%%%%%%%%%% %%% SL(n)/B \subsection{Opposite cells in Schubert varieties in $SL(n)/B$}\label{s13} Let $G=SL(n)$, the special linear group of rank $n-1$. Let $T$ be the maximal torus consisting of all the diagonal matrices in $G$, and $B$ the Borel subgroup consisting of all the upper triangular matrices in $G$. It is well-known that $W$ can be identified with $\cal{S}_n$, the symmetric group on $n$ letters. Following \cite{bou}, we denote the simple roots by $\e_i-\e_{i+1}$, $1\le i\le n-1$ (note that $\e_i-\e_{i+1}$ is the character sending diag$(t_1,\dots,t_n)$ to $t_it_{i+1}^{-1}$). Then $R=\{\e_i-\e_j\mid 1\le i,j\le n\}$, and the reflection $s_{\e_i-\e_{i+1}}$ may be identified with the transposition $(i,j)$ in $\cal{S}_n$. For $\a=\a_i(=\e_i-\e_{i+1})$, we also denote $P_{\hat{\a}}$ (resp. $W_{P_{\hat{\a}}}^{\text{min}}$) by just $P_i$ (resp. $W^i$). \subsection{The partially ordered set $I_{d,n}$}\label{idn} Let $Q=P_d$. Then \begin{align} Q&=\left\{ A\in G\biggm| A= \begin{pmatrix} *&*\\ 0_{(n-d)\times d}&* \end{pmatrix}\right\},\notag\\ W_Q&=\cal{S}_d\times \cal{S}_{n-d}.\notag \end{align} Hence $$W_Q^{\text{min}}=\{(a_1\dots a_n)\in W\mid a_1<\dots a_t$, $l\le a_t$, $1\le t\le k$. Denote by $O^-$ the subgroup of $G$ generated by $\{U_\a\mid \a\in R^-\setminus R_Q^-\}$. Then $O^-$ consists of the elements of $G$ of the form $$ \begin{pmatrix} I_1&0 &0 &\cdots&0 &0\\ \ast &I_2&0&\cdots& 0&0\\ \vdots&\vdots&\vdots&&\vdots&\vdots\\ \ast &\ast &\ast &\cdots&I_k&0\\ \ast &\ast &\ast &\cdots&\ast& I_a \end{pmatrix}, $$ where $I_t$ is the $c_t\times c_t$ identity matrix, $1\le t\le k$, $I_a$ is the $a\times a$ identity matrix, and if $x_{ml}\ne 0$, with $m\ne l$, then $m>a_t$, $l\le a_t$ for some $t$, $1\le t\le k$. Further, the restriction of the canonical morphism $f:G\to G/Q$ to $O^-$ is an open immersion, and $f(O^-)\simeq B^-e_{\text{id},Q}$. Thus $B^-e_{\text{id},Q}={\cal O}^-$, the opposite big cell in $G/Q$ gets identified with $O^-$. \subsection{Pl\" ucker coordinates on the Grassmannian}\label{13.5} Let $G_{d,n}$ be the Grassmannian variety, consisting of $d$-dimensional subspaces of an $n$-dimensional vector space $V$. Let us identify $V$ with $k^n$, and denote the standard basis of $k^n$ by $\{e_i\mid 1\le i\le n\}$. Consider the Pl\" ucker embedding $f_d:G_{d,n}\hookrightarrow \Bbb{P}(\w^dV)$, where $\w^dV$ is the $d$-th exterior power of $V$. For $\ui=(i_1,\dots,i_d)\in I_{d,n}$, let $e_{\ui}=e_{i_1}\w\dots\w e_{i_d}$. Then the set $\{e_{\ui}\mid \ui\in I_{d,n}\}$ is a basis for $\w^dV$. Let us denote the basis of $(\w^dV)^\ast$ (the linear dual of $\w^dV$) dual to $\{e_{\ui}\mid \ui\in I_{d,n}\}$ by $\{p_{\uj}\mid \uj\in I_{d,n}\}$. Then $\{p_{\uj}\mid \uj\in I_{d,n}\}$ gives a system of coordinates for $\Bbb{P}(\w^dV)$. These are the so-called {\em Pl\" ucker coordinates}. \subsection{Schubert varieties in the Grassmannian}\label{sch} Let $Q=P_{d}$. We have $$G_{d,n}\simeq G/P_d.$$ Let $\ui=(i_1,\dots,i_d)\in I_{d,n}$. Then the $T$-fixed point $e_{\ui,P_d}$ is simply the $d$-dimensional span of $\{e_{i_1},\dots,e_{i_d}\}$. Thus $X_{P_d}(\ui)$ is simply the Zariski closure of $B[e_{i_1}\w\dots\w e_{i_d}]$ in $\Bbb{P}(\w^dV)$. In view of the Bruhat decomposition for $X_{P_d}(\ui)$, %(cf. \S\ref{1.6}), we have $$p_{\uj}\bigr|_{X_{P_d}(\ui)}\ne 0\iff\ui\ge\uj.$$ \subsection{Evaluation of Pl\" ucker coordinates on the opposite big cell in $G/P_d$} Consider the morphim $\f_d:G\to \Bbb{P}(\w^dV)$, where $\f_d=f_d\circ \te_d$, $\te_d$ being the natural projection $G\to G/P_d$. Then $p_{\uj} (\f_d(g))$ is simply the minor of $g$ consisting of the first $d$ columns and the rows with indices $j_1,\dots, j_d$. Now, denote by $Z_d$ the unipotent subgroup of $G$ generated by $\{U_\a\mid\a\in R^-\setminus R_{P_d}^-\}$. We have, as in \S \ref{3.6} $$Z_d=\left\{ \begin{pmatrix} I_{d}&0_{d\times (n-d)}\\ A_{(n-d)\times d}&I_{n-d} \end{pmatrix}\in G\right\}$$ As in \S \ref{3.6}, we identify $Z_d$ with the opposite big cell in $G/P_d$. Then, given $z\in Z_d$, the Pl\" ucker coordinate $p_{\uj}$ evaluated at $z$ is simply a certain minor of $A$, which may be explicitly described as follows. Let $\uj=(j_1,\dots,j_d)$, and let $j_r$ be the largest entry $\le d$. Let $\{k_1,\dots,k_{d-r}\}$ be the complement of $\{j_1,\dots,j_r\}$ in $\{1,\dots,d\}$. Then this minor of $A$ is given by column indices $k_1,\dots k_{d-r}$, and row indices $j_{r+1},\dots,j_d$ (here the rows of $A$ are indexed as $d+1,\dots,n$). Conversely, given a minor of $A$, say, with column indices $b_1,\dots,b_s$, and row indices $i_{d-s+1},\dots,i_d$, it is the evaluation of the Pl\" ucker coordinate $p_{\ui}$ at $z$, where $\ui=(i_1,\dots,i_d)$ may be described as follows: $\{i_1,\dots,i_{d-s}\}$ is the complement of $\{b_1,\dots,b_s\}$ in $\{1,\dots,d\}$, and $i_{d-s+1},\dots,i_d$ are simply the row indices (again, the rows of $A$ are indexed as $d+1,\dots,n$). \subsection{Evaluation of the Pl\" ucker coordinates on the opposite big cell in $G/Q$}\label{13.7} Consider $$f:G\to G/Q\hookrightarrow G/P_{a_1}\times\dots\times G/P_{a_k}\hookrightarrow\bold{P}_1\times\dots\times\bold{P}_k,$$ where $\bold{P}_t=\Bbb{P}(\w^{a_t}V)$. Denoting the restriction of $f$ to $O^-$ also by just $f$, we obtain an embedding $f:O^-\hookrightarrow \bold{P}_1\times\dots\times\bold{P}_k$, $O^-$ having been identified with the opposite big cell ${\cal O}^-$ in $G/Q$. For $z\in O^-$, the multi-Pl\" ucker coordinates of $f(z)$ are simply all the $a_t\times a_t$ minors of $z$ with column indices $\{1,\dots,a_t\}$, $1\le t\le k$. \subsection{Equations defining the cones over Schubert varieties in $G_{d,n}$}\label{4.10} Let $Q=P_d$. Given a $d$-tuple $\ui=(i_1,\dots,i_d)\in I_{d,n}$, let us denote the associated element of $W_{P_d}^{\text{min}}$ by $\te_{\ui}$. For simplicity of notation, let us denote $P_d$ by just $P$, and $\te_{\ui}$ by just $\te$. Then, by \S \ref{sch}, $X_P(\te)$ is simply the Zariski closure of $B[e_{i_1}\w\dots\w e_{i_d}]$ in $\Bbb{P}(\w^dV)$. Now using \S \ref{1.8}, we obtain that the restriction map $R\to R_{\te}$ is surjective, and the kernel is generated as an ideal by $\{p_{\uj}\mid\ui\not\ge\uj\}$. \subsection{Equations defining multicones over Schubert varieties in $G/Q$}\label{4.11} Let $X_Q(w)\subset G/Q$. Denoting $R$, $R_w$ as in \S \ref{1.8}, the kernel of the restriction map $R\to R_w$ is generated by the kernel of $R_1\to (R_{w})_1$; but now, in view of \S \ref{4.10}, this kernel is the span of $$\{p_{\ui}\mid\ui\in I_{d,n},d\in\{a_1,\dots,a_k\},\ w^{(d)}\not\ge\ui\},$$ where $w^{(d)}$ is the $d$-tuple corresponding to the Schubert variety which is the image of $X_Q(w)$ under the projection $G/Q\to G/P_{a_t}$, $1\le t\le k$. \subsection{Ideal of the opposite cell in $X_Q(w)$}\label{3.13} Let $Y_Q(w)=B^-e_{\text{id},Q}\cap X_Q(w)$. Then as in \S \ref{3.6}, we identify $B^-e_{\text{id},Q}$ with the unipotent subgroup $O^-$ generated by $\{U_\a\mid\a\in R^-\setminus R^-_Q\}$, and consider $Y_Q(w)$ as a closed subvariety of $O^-$. In view of \S \ref{4.11}, we obtain that the ideal defining $Y_Q(w)$ in $O^-$ is generated by $$\{p_{\ui}\mid\ui\in I_{d,n},d\in\{a_1,\dots,a_k\},\ w^{(d)}\not\ge\ui\}.$$ \subsection{The classical determinantal variety} Let $A=(x_{ij}),\ 1\le i\le m,\ 1\le j \le n$ be a $m\times n$ matrix of variables. Let $k$ be a positive integer such that $k\le {\text{min}}(m,n)$, and $D_k$ be the determinantal variety defined by the vanishing of all $k+1$ - minors of $A$. Then one knows (see \cite{l-se1} for example) that $D_k $ can be identified with $Y_Q(w)$ (cf. \S \ref{3.13}) for a suitable Schubert variety $X(w)$ in the Grassmannian $G_{n,m+n}$; in particular, one may conclude that $D_k$ is normal, Cohen-Macaulay and has rational singularities. \subsection{Ladder determinantal varieties}\label{ldv} Let $X=(x_{ba})$, $1\le b\le m$, $1\le a\le n$ be a $m\times n$ matrix of indeterminates. Given $1\le b_1<\dotst_{i-1}-t_i\text{ for }1t_i\text{ if }s_i\not\in\{b_1,\dots,b_h\}, 11$. $(2)$ if $i>1$, then $w^{(a_{i-1})}\subset w^{(a_i)}$. Note that $w^{(a_i)}$, $1\le i\le h$, is well defined in $W^i$, and $w$ is well defined as an element in $W_Q^{\text{min}}$. \begin{thm}\label{th1}(cf. \cite{g-l}) The variety $Z$ $(=D_{\us,\ut}(L)\times\Bbb{A}^r)$ identifies with the opposite cell in $X_Q(w)$, i.e. $Z=X_Q(w)\cap {\cal O}^-$ (scheme theoretically). \end{thm} The above theorem is proved using \S \ref{3.13}. As a consequence of the above Theorem, we obtain (cf. \cite{g-l}) \begin{thm} The variety $D_{\us,\ut}(L)$ is irreducible, normal, Cohen-Macaulay, and has rational singularities. \end{thm} \subsection{The varieties $V_i,\ 1\le i\le l$} Let $V_i$, $1\le i\le l$ be the subvariety of $D_{\us,\ut}(L)$ defined by the vanishing of all $(t_i-1)$-minors in $L(i)$, where $L(i)$ is as in \S\ref{sub}. In \cite{g-l} the singular locus of $D_{\us,\ut}(L)$ has also been determined, as described below. \begin{thm} $\text{Sing\,}D_{\us,\ut}(L)=\cup_{i=1}^l V_i$. \end{thm} \subsection{The varieties $Z_j,\ X_Q(\te_j),\ 1\le j\le l$}\label{3.21} Let us fix $j\in\{1,\dots, l\}$, and let $Z_j=V_j\times \Bbb{A}(H\setminus L)$. We shall now define $\te_j\in W_Q^{\text{min}}$ such that the variety $Z_j$ identifies with the opposite cell in the Schubert variety $X_Q(\te_j)$ in $G/Q$. Note that $w^{(a_r)}(a_r-t_j+1)=s_j-1$, and $s_j-1$ is the end of a block of consecutive integers in $w^{(a_r)}$, where $r=j^*$ is the largest integer such that $b_r\le s_j$. Also, the beginning of this block is $\ge 2$ (if the block started with $1$, we would have $a_r-t_j+1=s_j-1\ge b_r-1\ge a_r$, which is not possible, since $t_j\ge 2$). Let $u_j+1$ be the beginning of this block, where $u_j\ge 1$. Then it is easily seen that if $s_j-1$ is the end of a block in $w^{(a_i)}$, $1\le i\le h$, then the beginning of the block is $u_j+1$. For each $i$, $1\le i\le h$, such that $u_j\not\in w^{(a_i)}$, let $v_i$ be the smallest entry in $w^{(a_i)}$ which is bigger than $s_j-1$. Note that $v_i=w^{(a_i)}(a_k-t_j+2)$, where $k\in\{1,\dots,i\}$ is the largest such that $b_k\le s_j$. Define $\te_j{}^{(a_i)}$, $1\le i\le h$, as follows. If $s_j-1\not\in w^{(a_i)}$ (which is equivalent to $j>1$, $t_{j-1}=t_j$ and $ir$). Note that $\te_j$ is well defined as an element in $W_Q^{\text{min}}$, and $\te_j\le w$. \begin{rem} An equivalent description of $\te_j$ is the following. Let $t_{i_k}1$ and $t_{j-1}=t_j$), then for $ir$ and $u_j\in w^{(a_i)}$, $\te_j^{(a_i)}=w_j^{(a_i)}$; for $i>r$ and $u_j\not\in w^{(a_i)}$, $\te_j^{(a_i)}=w_j^{(a_i)}\setminus\{v_i\}\cup\{u_j\}$, where $v_i$ is the smallest entry in $w^{(a_i)}\setminus\te_j^{(a_{i-1})}$. (II) If $j\in\{i_1,\dots,i_m\}$, (i.e. $t_{j-1}>t_j$ if $j>1$), then for $i\le r$, $\te_j^{(a_i)}=w_j^{(a_i)}\setminus\{e_{i_k}\}\cup\{u_j\}$, where $u_j$ is the largest entry in $\{1,\dots,s_j-1\}\setminus w^{(a_r)}$; for $i>r$ and $u_j\in w^{(a_i)}$, $\te_j^{(a_i)}=w_j^{(a_i)}$; for $i>r$ and $u_j\not\in w^{(a_i)}$, $\te_j^{(a_i)}=w_j^{(a_i)}\setminus\{v_i\}\cup\{u_j\}$, where $v_i$ is the smallest entry in $w^{(a_i)}\setminus\te_j^{(a_{i-1})}$. \end{rem} \begin{thm}\label{th2}(cf. \cite{g-l}) The subvariety $Z_j\subset Z$ identifies with the opposite cell in $X_Q(\te_j)$, i.e. $Z_j=X_Q(\te_j)\cap {\cal O}^-$ (scheme theoretically). \end{thm} As a consequence of the above theorem, we obtain (cf. \cite{g-l}) \begin{thm}\label{th3} The irreducible components of $\text{Sing\,}D_{\us,\ut}(L)$ are precisely the $V_j$'s, $1\le j\le l$. \end{thm} Let $X(w^{\text{max}})$ (resp. $X(\te_j^{\text{max}})$, $1\le j\le l$) be the pull-back in $SL(n)/B$ of $X_Q(w)$ (resp. $X_Q(\te_j)$, $1\le j\le l$) under the canonical projection $\pi :SL(n)/B\to SL(n)/Q$ (here $B$ is a Borel subgroup of $SL(n)$ such that $B\subset Q$). Then using Theorems \ref{th1}, \ref{th2} and \ref{th3} above, we obtain (cf. \cite{g-l}) \begin{thm} The irreducible components of $\text{Sing\,}X(w^{\text{max}})$ are precisely $X(\te_j^{\text{max}})$, $1\le j\le l$. \end{thm} In \cite{g-l}, it is also shown that the conjecture of \cite{l-sa} on the irreducible components of Sing $X(\theta), \ \theta \in W$ holds for $X(w^{\text{max}})$. \begin{rem} Ladder determinantal varieties were first introduced by Abyankar (cf. \cite{ab}). \end{rem} \begin{rem} A similar identification as in Theorem \ref{th1} for the case $t_1=\cdots =t_l$ has also been obtained by Mulay (cf. \cite{mu}). \end{rem} \begin{rem} In \cite{g-l}, the theory of Schubert varieties and the theory of ladder determinantal varieties are complementing each other. To be more precise, geometric properties such as normality, Cohen-Macaulayness, etc., for ladder determinantal varieties are concluded by relating these varieties to Schubert varieties. The components of singular loci of Schubert varieties are determined by first determining them for ladder determinantal varieties, and then using the above mentioned relationship between ladder determinantal varieties and Schubert varieties. \end{rem} \subsection{Quiver varieties}\label{quiv} Fulton \cite{fu} and Buch-Fulton \cite{b-f} have recently given a theory of ``universal degeneracy loci'', characteristic classes associated to maps among vector bundles, in which the role of Schubert varieties is taken by certain degeneracy schemes. The underlying varieties of these schemes arise in the theory of quivers: they are the closures of orbits in the space of representations of the equioriented quiver $A_h$. Many other classical varieties also appear as quiver varieties, such as determinantal varieties and the variety of complexes (see \cite{d-s}, \cite{h-e}, \cite{m-s}.) In \cite{l-ma}, the quiver varieties (corresponding to the equioriented type A quiver) are shown to be normal and Cohen-Macaulay (in arbitrary characteristic) by identifying them with $Y_Q(w)$ (cf. \S \ref{1.7}) for suitable Schubert varieties $X_Q(w)$ in suitable $SL(n)/Q$. Fix an $h$-tuple of non-negative integers $\nn = (n_1,\ldots,n_h)$ and a list of vector spaces $V_1,\ldots, V_h$ over an arbitrary field $\kk$ with respective dimensions $n_1,\ldots,n_h$. Define $Z$, the {\it variety of quiver representations} (of dimension $\nn$, of the equioriented quiver of type $A_h$) to be the affine space of all $(h\!-\!1)$-tuples of linear maps $(f_1,\ldots,f_{h\!-\!1}):$ $$ V_1 \stackrel{f_1}{\to} V_2 \stackrel{f_2}{\to} \cdots \stackrel{f_{h\!-\!2}}{\to} V_{h\!-\!1} \stackrel{f_{h\!-\!1}}{\to} V_h \ . $$ If we endow each $V_i$ with a basis, we get $V_i \cong \kk^{n_i}$ and $$ Z \cong M(n_2 \!\times\! n_1) \!\times\! \cdots \times M(n_{h} \!\times\! n_{h\!-\!1}) , $$ where $M(l\!\times\! m)$ denotes the affine space of matrices over $\kk$ with $l$ rows and $m$ columns. The group $$ G_{\nn} = GL(n_1) \times \cdots \times GL(n_h) $$ acts on $Z$ by $$ (g_1,g_2,\cdots,g_h) \cdot (f_1,f_2,\cdots,f_{h\!-\!1}) = (g_2 f_1 g_1^{-1}, g_3 f_2 g_2^{-1},\cdots, g_{h}f_{h\!-\!1} g_{h\!-\!1}^{-1}), $$ corresponding to change of basis in the $V_i$. Now, let $\rr = (r_{ij})_{1 \leq i \leq j \leq h}$ be an array of non-negative integers with $r_{ii} = n_i$, and define $r_{ij} = 0$ for any indices other than $1\leq i\leq j \leq h$. Define the set $$ Z^{\circ}(\rr) = \{(f_1,\cdots,f_{h\!-\!1}) \in Z \ \mid\ \forall\, i\!<\!j,\ \rank (f_{j\!-\!1} \cdots f_i : V_i \to V_j) = r_{ij} \}. $$ (This set might be empty for a bad choice of $\rr$.) \begin{prop}(cf. \cite{ga}) {\it The $G_{\nn}$-orbits of $Z$ are exactly the sets $Z^{\circ}(\rr)$ for $\rr=(r_{ij})$ with $$ r_{ij}-r_{i,j\!+\!1}-r_{i\!-\!1,j}+r_{i\!-\!1,j\!+\!1} \geq 0,\quad \forall\ 1\! \leq\! i\! <\! j\! \leq\! h. $$ } \end{prop} \begin{defn} We define the {\it quiver variety} as the algebraic set $$ Z(\rr) =\{(f_1,\cdots,f_{h\!-\!1}) \in Z \mid \forall i,j,\ \rank (f_{j\!-\!1} \cdots f_i : V_i \to V_j) \leq r_{ij}\}. $$ \end{defn} \begin{rem} The variety $Z(\rr)$ is simply the Zariski closure of $Z^{\circ}(\rr)$ (cf. \cite{a-d-k}, \cite{l-ma}). \end{rem} \subsection{The Schubert varieties $X_Q(\taum),\ X_Q(\taur)$} Given $\nn=(n_1,\cdots,n_h)$, for $1 \leq i \leq h$ let $$ a_i = n_1 + n_2 + \cdots +n_i, \qquad a_0 = 0, \qquad \mbox{and} \qquad n = n_1 + \cdots + n_h \ . $$ For positive integers $i \leq j$, we shall frequently use the notations $$ [i,j] = \{ i, i+1, \ldots, j\}, \qquad [i] = [1,i], \qquad [0] = \{\} \ . $$ Let $\kk^n \cong V_1 \oplus \cdots \oplus V_h$ have basis $e_1,\ldots,e_n$ compatible with the $V_i$. Consider its general linear group $GL(n)$, the subgroup $B$ of upper-triangular matrices, and the parabolic subgroup $Q$ of block upper-triangular matrices $$ Q = \{ (a_{ij}) \in GL(n) \mid a_{ij}=0 \ \mbox{whenever}\ j\leq a_k j \\ \end{array} \right. $$ Namely, $$ \taur_i = \{\, \underbrace{1\ldots a_{i\!-\!1}}_ {\mbox{\small $a_{i\!-\!1}$}} \ \underbrace{. \ldots\ldots a_{i}}_ {\mbox{\small $r_{ii}\!-\!r_{i,i+1}$}} \ \underbrace{.\ldots\ldots a_{i+1}}_ {\mbox{\small $r_{i,i+1}\!-\!r_{i,i+2}$}} \ \underbrace{.\ldots\ldots a_{i+2}}_ {\mbox{\small $r_{i,i+2}\!-\!r_{i,i+3}$}}\ \ldots\ \ \underbrace{.\ldots\ldots n_{\mbox{}}}_ {\mbox{\small $r_{i,h}$}} \} $$ where we use the visual notation $$ \underbrace{\cdots\cdots a}_{\mbox{\small $b$}} = [a\!-\!b\!+\!1,a]. $$ Recall that $a_j = a_{j-1}+n_j$ and $0\leq r_{ij} -r_{i,j+1} \leq n_j$, so that each $\taur_i$ is an increasing list of integers. Also $r_{ij}-r_{i,j+1}\leq r_{i+1,j}-r_{i+1,j+1}$, so that $\taur_i \subset \taur_{i+1}$. Thus, $\ \taur$ are indeed subset-flags. \subsection{Examples} We give below four examples. \vs.2cm \ni {\bf Example 1} {\it A small generic case.} \ni Let $h=4$,\ $\nn = (2,3,2,2)$,\ $$ \rr= \begin{array}{|cccc|}\hline 2&2&0&0\\ &3&1&1\\ &&2&2\\ &&&2\\ \hline \end{array} $$ where $r_{ij}$ are written in the usual matrix positions. Then we get $(a_1,a_2,a_3,a_4) = (2,5,7,9)$,\ $n=9$, and $$ \taum = (89\subset 12589 \subset 1234589 \subset [9]), \qquad \taur = (45\subset 12459 \subset 1234589 \subset [9]), $$ which correspond to the cosets in $W/W_{\nn}$ $$ w^{\mbox{\tiny max}} = 89|125|34|67, \qquad w^{\rr} = 45|129|38|67. $$ (The minimal-length representatives of these cosets are the permutations as written; the other elements are obtained by permuting numbers within each block.) The partial flag variety is $\Fl = \{ U_1 \subset U_2 \subset U_3 \subset \kk^9 \mid \dim U_i = a_i \}$, and the Schubert varieties are: $$ X_Q(\taum)=\left\{ U\bdot \left| \begin{array}{c} \kk^2 \subset U_2\\ \kk^5 \subset U_3 \end{array} \right. \right\}, \quad X_Q(\taur)=\left\{ U\bdot \left| \begin{array}{c} U_1\!\subset\! \kk^5\!\subset\! U_3,\ \kk^2\! \subset\! U_2\\ \dim U_2 \cap \kk^5 \geq 4 \end{array} \right. \right\}. $$ The opposite cells $Y_Q(\tau)$ are defined by the extra conditions $U_i \cap E'_i = 0$. \vs.2cm \ni {\bf Example 2.} {\it Fulton's universal degeneracy schemes} (cf. \cite{fu}). \ni Given $m>0$, let $Z$ be the affine space associated to the quiver data $h=2m$,\ $\nn = (1,2,\ldots,m,m,\ldots,2,1)$. For each $w \in {\cal S}_{m\po}$, Fulton defines a ``degeneracy scheme'' $\Omega_{w} = Z(\rr)$ as follows. Denote $\bi = 2m+1-i$, and define $\rr = \rr(w) = (r_{ij})$ by: $$ \begin{array}{c} r_{ij} = r_{\bj\bi} = i \\[.2em] r_{i\bj} = \#\,[i]\cap w[j] \end{array} $$ for $1\leq i,j\leq m$. The associated Schubert varieties $X_Q(\taur)$ are given by $\taur = (\taur_1\subset \cdots \subset \taur_{\overline{1}})$ or by cosets $\tw = \tw_1 |\cdots |\tw_{\overline 1} \in W/W_{\nn}$ $$ \mbox{\hspace{-1em}} \begin{array}{c} \taur_i = [a_{i\mo}] \cup \{ a_{\overline{w^{\mo}(1)}}\,, a_{\overline{w^{\mo}(2)}}\,, \ldots, a_{\overline{w^{\mo}(i)}} \},\\[.5em] \taur_{\bi} = [a_{\bi}-\!1] \cup \{ a_{\overline{1}}, a_{\overline{2}}, \ldots, a_{\overline{m}} \} \end{array} \quad \begin{array}{c} \tw_i = [a_{i\!-\!2}+\!1,a_{i\mo}] \cup \{ a_{\overline{ w^{\mo}(i)}} \} \\[.2em] \tw_{\overline{m}} = [a_{m\!-\!1}\po,a_{m}\mo] \cup \{ a_{\overline{ w^{\mo}(m\po) }}\} \\[.2em] \tw_{\bj} = [a_{\bj\!-\!2}\po,a_{\bj\mo}] \end{array} $$ for $1\leq i\leq m$,\ \ $1\leq j\leq m-1$. Furthermore $\taum = \tau^{\rr(w)}$ and $\tw^{\mbox{\tiny max}} = \tw^{\rr(w)}$ for $w = e \in {\cal S}_{m\po}$, the identity permutation. % \vs.2cm \ni {\bf Example 3.} {\it The variety of complexes.} \ni For a given $h$ and $\nn$, the {\it variety of complexes} is defined as the union ${\cal C} = \cup_{\rr} Z(\rr)$ over all $\rr = (r_{ij})$ with $r_{i,i\!+\!2}=0$ for all $i$. The subvarieties $Z(\rr)$ correspond to the multiplicity matrices $\mm = (m_{ij})$ with $m_{ij}=0$ for all $i+2 \leq j$, and $m_{ii}+m_{i\mo,i}+ m_{i,i\po} = n_i$ for all $i$. In \cite{m-s}, Musili-Seshadri have shown that each component of ${\cal C}$, is isomorphic to the opposite cell in a Schubert variety. \vs.2cm \ni {\bf Example 4.} {\it The classical determinantal variety.} \ni The classical {\it determinantal variety} of $k \times l$ matrices of rank $\leq t$ is ${\cal D} = Z(r)$ for $\rr=$ {\footnotesize $\left( \begin{array}{@{\!}cc@{\!}}l&m\\0&k\end{array}\right)$} and $\mm=$ {\footnotesize $\left( \begin{array}{@{\!}cc@{\!}}l\!-\!m&m\\ 0&k\!-\!m\end{array}\right)$ } where $m = \min(t+1,k,l)$. Also $n=k+l$,\ $$\taum = ([k+1,k+l]\subset [n]),\quad \taur = ([m+1,l]\cup [k+l-m+1,k+l]\subset [n]) $$ $$ X(\taum) = \Fl = \Gr(l,\kk^n), \quad\ X(\taur) \cong \{ U\in \Gr(l,\kk^n)\mid U\cap \kk^l = l-m \}, $$ $$ {\cal D} = Z(\rr) \cong Y(\taur) = \{U\in \Gr(l,\kk^n)\mid U\cap \kk^l = l-m, \ U\cap E'=0\}, $$ where $E'=\langle e_{l\po},e_{l\!+\!2},\ldots,e_n\rangle$. \vs.2cm Denote a generic element of the quiver space $ Z = M(n_2\times n_1) \times \cdots \times M(n_{h}\times n_{h\!-\!1})$ by $(A_1,\ldots,A_{h-1})$, so that the coordinate ring of $Z$ is the polynomial ring in the entries of all the matrices $A_i$. Let $\rr = (r_{ij})$ index the quiver variety $Z(\rr) = \{(A_1,\ldots,A_{h-1}) \mid \rank\, A_{j-1}\cdots A_i \leq r_{ij}\}$. Let $\JJ(\rr) \subset \kk[Z]$ be the ideal generated by the determinantal conditions implied by the definition of $Z(\rr)$: $$ \JJ(\rr) = \left\langle \det(A_{j-1} A_{j-2} \cdots A_i)_ {\lambda\times\mu} \ \left| \ \begin{array}{c} j>i,\ \lambda \subset [n_j],\ \mu \subset [n_i] \\[.2em] \#\lambda = \#\mu = r_{ij}+1 \end{array} \right. \right\rangle\ . $$ Clearly $\JJ(\rr)$ defines $Z(\rr)$ set-theoretically. \begin{thm}(cf. \cite{l-ma}) $\JJ(\rr)$ is a prime ideal and is the vanishing ideal of $Z(\rr)\subset Z$. There are isomorphisms of reduced schemes $$ Z(\rr) = \mbox{Spec}(\kk[Z]\,/\,\JJ(\rr)) \cong \mbox{Spec}(\kk[{\cal O}^-]\,/\,\II(\taur)) = Y_Q(\taur). $$ That is, the quiver scheme $Z(\rr)$ defined by $\JJ(\rr)$ is isomorphic to the (reduced) variety $Y_Q(\taur)$, the opposite cell of a Schubert variety. \end{thm} In proving the above theorem again, one uses the standard monomial theory for Schubert varieties. \begin{rem} Over a field of characteristic $0$, the normality and Cohen-Macaulayness of $Z(\rr)$ also follow from \cite{a-d-k}. \end{rem} %% Part 3 - P. Magyar \newcommand{\proj}{\mathop{\text{proj}}} \newcommand{\PP}{{\Bbb P}} \newcommand{\ii}{ {\bold i} } \newcommand{\Ph}{ \widehat{P} } \newcommand{\al}{ \alpha } \renewcommand{\om}{ \varpi } \newcommand{\Zii}{ Z_{\ii} } \newcommand{\LL}{ {\cal L} } \renewcommand{\OO}{ {\cal O} } \newcommand{\LS}{ {\cal L\! S} } \newcommand{\tf}{ \tilde{f} } \renewcommand{\te}{ \tilde{e} } \newcommand{\tal}{ \tilde{\alpha} } \newcommand{\tpi}{ \tilde{\pi} } \newcommand{\tdel}{ \tilde{\delta} } \newcommand{\tLam}{ \tilde{\Lambda} } \newcommand{\tX}{ \tilde{X} } \section{Bott-Samelson varieties} Throughout this section, we once again take $G$ to be a simply connected semisimple algebraic group over an algebraically closed field $k$. \subsection{Geometry} The Bott-Samelson varieties are an important tool in the representation theory of the group $G$ and the geometry of the flag variety $G/B$. First defined in \cite{BS} as a desingularization of the Schubert varieties in $G/B$, they were exploited by Demazure \cite{Dem1} to analyze the singular cohomology or Chow ring $H^{\cdot}(G/B)$ (the Schubert calculus), and the projective coordinate ring $k[G/B]$. Since the irreducible representations of $G$ are embedded in the coordinate ring, Demazure was able to obtain a new iterative character formula for these representations. Bott-Samelson varieties are so useful because they ``factor'' the flag variety into a ``product'' of projective lines. More precisely, they are iterated $\PP^1$-fibrations and they each have a natural, birational map to $G/B$. The Schubert subvarieties themselves lift birationally to iterated $\PP^1$-fibrations under this map (hence the desingularization). The combinatorics of Weyl groups enters because a given $G/B$ can be ``factored'' in many ways, indexed by sequences $\ii = (i_1, i_2, \ldots, i_N)$ such that $w_0 = s_{i_1} s_{i_2} \cdots s_{i_N}$ is a reduced decomposition of the longest Weyl group element $w_0$ into simple reflections. More generally, we may define a Bott-Samelson variety $\Zii$ for an arbitrary reduced or non-reduced sequence of indices $\ii = (i_1, i_2, \ldots, i_N)$. Let $P_k \supset B$ be the minimal parabolic associated to the simple reflection $s_k$ so that $P_i/B \cong \PP^1$, the projective line. Then $$ \Zii = P_{i_1} \times \cdots \times P_{i_N} /B^N, $$ where $B^N$ acts on the right of the product via: $$ (p_1,p_2,\ldots,p_N)\cdot (b_1,b_2,\ldots,b_N) = (p_1 b_1, b_1^{-1} p_2 b_2,\ldots,b_{N-1}^{-1} p_N b_N). $$ Furthermore, $B$ acts on the left of $\Zii$ by multiplication of the first factor. Although we will not use it here, a key structure in analyzing the geometry of $\Zii$ (and hence $G/B$) is the {\it opposite big cell} $$ \begin{array}{ccc} k^N & \rightarrow & \Zii \\ (t_1,\ldots,t_N) & \mapsto & (\exp(t_1 F_{i_1}),\ldots,\exp(t_N F_{i_N})), \end{array} $$ where $t\mapsto\exp(t F_{i})$ is the exponential map onto the one-parameter unipotent subgroup corresponding to the negative simple root $\alpha_i$. The image of $k^N$ is a dense open cell in $\Zii$. We may embed $\Zii$ in a product of flag varieties by the iterated multiplication map: $$ \begin{array}{cccc} \mu: &\Zii & \to & (G/B)^{N+1} \\ &(p_1,\ldots,p_N) & \mapsto & (eB,p_1 B,p_1 p_2 B,\cdots, p_1\!\!\cdots\!p_N B). \end{array} $$ The embedding is compatible with the $B$-action on $\Zii$ and the diagonal $B$-action on $(G/B)^{N+1}$. The image of this embedding is a dual version of $\Zii$, a fiber product: $$ \mu(\Zii) = eB \times_{G/P_{i_1}} G/B \times_{G/P_{i_2}} \cdots \times_{G/P_{i_N}} G/B \subset (G/B)^{N+1}. $$ By composing $\mu$ with various projections of $(G/B)^{N+1}$, we obtain maps from $\Zii$. For example, the canonical map to the flag variety is $$ \begin{array}{ccc} \Zii & \rightarrow & G/B \\ (p_1,\ldots,p_N) & \mapsto & p_1 p_2 \cdots p_N B. \end{array} $$ which is a birational morphism exactly when $\ii$ is a reduced decomposition of the longest element of $W$. For general $\ii$ the image is the Schubert variety $X(s_{i_1}\cdots s_{i_N})$. Let $\Gr(\ii) = G/\Ph_{i_1} \times \cdots \times G/\Ph_{i_N}$, where $\Ph_{i}$ is the {\it maximal} parabolic subgroup associated to all the simple reflections {\it except} $s_i$. If we compose $\mu$ with the projection of $(G/B)^{N+1}$ to $\Gr(\ii)$, the result is still an embedding of $\Zii$: $$ \begin{array}{cccc} \bar{\mu}: &\Zii & \to & \Gr(\ii) \\ &(p_1,\ldots,p_N) & \mapsto & (p_1 \Ph_{i_1},p_1 p_2 \Ph_{i_2},\cdots, p_1\!\!\cdots\!p_N \Ph_{i_N}). \end{array} $$ That is, $\Zii \cong \mu(\Zii) \cong \bar{\mu}(\Zii)$. This gives an embedding of $\Zii$ in a conveniently small variety. Finally, if we project $\Gr(\ii)$ to any product of $G/\Ph_i$ with some of the $G/\Ph_{i_j}$ factors missing, the image of $\bar{\mu}(\Zii)$ is no longer isomorphic to $\Zii$: we call this image a {\it configuration variety}. \\[1em] Line bundles on $\Zii$ are indexed by sequences of integers $\mm = (m_1,\ldots,m_N)$. Define the line bundle $$ \LL_{\mm} = (P_{i_1} \times \cdots \times P_{i_N}) \times_{B^N} (k_{-m_1 \om_{i_1}}\otimes \cdots \otimes k_{-m_N \om_{i_N}}) $$ associated to the character $e^{-m_1 \om_1}\otimes\cdots\otimes e^{-m_N\om_{i_N}}:B^N\to k^{\times}$, where $\om_i$ denotes the $i$-th fundamental weight of $G$. We can also define $\LL_{\mm}$ in terms of the embedding $\bar{\mu}$. Let $\OO(1) = G \times^{\Ph_i} k_{-\om_i}$ denote the unique minimal ample line bundle on $G/\Ph_i$. Then $\LL_{\mm}$ is the pullback via $\bar{\mu}$ of the bundle $\OO(\mm) = \OO(1)^{\otimes m_1} \otimes \cdots \otimes \OO(1)^{\otimes m_N}$ over $\Gr(\ii)$. Our substitutes for Weyl modules and Demazure modules will be the spaces of global sections $$ V(\ii,\mm)^* = \Gamma(\Zii,\LL_{\mm}). $$ For appropriately chosen $\ii$ and $\mm$, the $B$-representations $V(\ii,\mm)^*$ are isomorphic to the dual Weyl modules $V(\lambda)^*$ and the Demazure modules $V(\lambda)^*_{\tau}$ considered previously. The vanishing theorem of Mathieu \cite{Mathieu} and Kumar \cite{Kumar} implies: \begin{thm} (i) The restriction map $\Gamma(\Gr(\ii),\OO(\mm))\to \Gamma(\Zii,\LL_{\mm})$ is surjective. (ii) The character of $\Gamma(\Zii,\LL_{\mm})^*$ is given by the Demazure formula: $$ \char\Gamma(\Zii,\LL_{\mm})^* = \Lam_{i_1} (e^{m_1 \om_{i_1}} \Lam_{i_2}(e^{m_2 \om_{i_2}}\ldots (\Lam_{i_N} e^{m_N \om_{i_N}})\ldots)). $$ \end{thm} It should be possible to prove this theorem by the same methods used above in the case of Schubert varieties. From the theorem, we see that $V(\ii,\mm)^*$ is a quotient of the tensor product $$ V(m_1 \om_{i_1})^* \otimes \cdots \otimes V(m_N \om_{i_N})^* = \Gamma(\Gr(\ii),\OO(\mm)). $$ {\bf Example.} Let $G = SL(n)$. Then $G/\Ph_i \cong {\text{Gr}}(i,k^n)$, the Grassmannian of $i$-planes in linear $n$-space. Let $E_i\in {\text{Gr}}(i,k^n)$ be the span of the first $i$ standard coordinate vectors in $k^n$. Then we may identify $\bar{\mu}(\Zii) \subset \Gr(\ii) = \Gr(i_1,k^n) \times\cdots\times \Gr(i_1,k^n)$ as the variety of $N$-tuples of subspaces $(V_1,\ldots,V_N)\in {\text{Gr}}(\ii)$ with $\dim V_j = i_j$, and subject to the following inclusions: if $i_{p}=i_{q}+1$, and $i_{r}\neq i_p, i_{q}$ for every $r$ between $p$ and $q$, then $V_{p} \subset V_{q}$; and if $i_{q} \neq i_p-1$ for $q< p$, then $E_{i_p-1} \subset V_p$; and if $i_{q}\neq i_p+1$ for $q 0} G/\Ph_{i_j}$, where we drop all factors with $m_j=0$. Then consider the set of all images $\{ \eta \bar{\mu}(Y) \mid \mbox{ $Y\subset \Zii$ a Bott-Samelson}\mbox{ subvariety }\}$, and order these varieties by inclusion. The resulting poset bears a relationship to $LS(\ii,\mm)$ similar to that of the usual Bruhat order to usual L-S paths. Now we consider a set of paths which will allow us to construct a basis in the framework of the previous section. Let $\tilde{\mu} = \tilde{\mu}(\ii,\mm)\in \tilde{\Pi}^+$ be the piecewise-linear path defined as a concatenation of $N$ straight line paths $$ \tilde{\nu} = \tpi_{m_1\tdel_1}* \cdots * \tpi_{m_N\tdel_N}. $$ so that $\tilde{\nu}(1) = \tdel$. Define the set of {\it pseudo standard tableaux} as the paths $\tilde{ST}(\ii,\mm)=B(\tilde{\nu})_{\ii}$. Now, for each path $\tpi$ in $\tilde{X}_{\RR}$, consider its projection $\proj \tpi(t)$ to $X_{\RR}$, and define the set of {\it standard tableaux} as $ST(\ii,\mm) = \proj \tilde{ST}(\ii,\mm)$, the projection of the pseudo standard tableaux. There is an obvious inclusion $$ \tilde{ST}(\ii,\mm)= B(\tilde{\nu})_{\ii} \subset B(m_1\tdel_1)_{\ii}* \cdots * B(m_N \tdel_N)_{\ii} $$ which projects to the inclusion $$ ST(\ii,\mm) \subset B(\om_{m_1 i_1})* \cdots *B(m_N \om_{i_N}). $$ We may also construct the standard tableaux $ST(\ii,\mm)$ using only the usual lowering operators $f_i$ in $X_{\RR}$ and another path version of Demazure's character formula: $$ ST(\ii,\mm) = \{\, f_{i_1}^{n_1} (\pi_{m_1 \om_{i_1}}* f_{i_2}^{n_2}(\pi_{m_1 \om_{i_2}}* \ldots (f_{i_N}^{n_N}(\pi_{m_N \om_{i_N}})\ldots))) \mid n_1,\ldots,n_N \geq 0\, \}. $$ Finally, we can characterize the paths in $ST(\ii,\mm)$ by certain standardness conditions (the factors must decrease in an appropriate analog of the Bruhat order). See \cite{LakMag1}, \cite{LakMag2}. \\[1em] Now we construct our basis for $V(\ii,\mm)^*$. Recall the path basis ${\Bbb B}(\lambda) = \{ p_{\pi} \mid \pi \in B(\lambda)\}$ for each $G$-representation $V(\lambda)^*$ with lowest weight $-\lambda \in X$. Now, for each standard tableau $$ \nu = \pi_1*\cdots*\pi_N \in ST(\ii,\mm) \subset B(m_1 \om_{i_1})* \cdots *B(m_N \om_{i_N}) $$ we may define $$ p_{\nu}= p_{\pi_1}\!\cdots p_{\pi_N}\in V(m_1\om_{i_1})^* \otimes \cdots \otimes V(m_N \om_{i_N})^*. $$ Let ${\Bbb B}(\ii,\mm) = \{p_{\nu} \in ST(\ii,\mm)\}$. \vskip1em \begin{thm} The set ${\Bbb B}(\ii,\mm)$ restricts to a basis of $V(\ii,\mm)^*$. \end{thm} \noindent{\bf Example.} Again taking $G = GL(4)$, $\ii=13212$, $\mm =(0,0,1,0,2)$, let us denote an extremal weight $w(\om_i)$ by a subset of $i$ elements in $\{1,2,3,4\}$, and use the same symbol to denote the straight-line path $\pi_{w(\om_i)}$ in $X_{\RR}$. Thus a path in $ST(\ii,\mm)$ is of the form $ab*cd*ef$, where $1\leq a,b,c,d,e,f\leq 4$ and $a