% AMS-TeX,  amsppt style

\define\eqdef{\overset \text{def}\to =}
%\define\eqdef{\raise1.5mm \hbox{${\scriptstyle de\!f \atop =}$} }
\define\LL{\Cal L}
\define\ZZ{\Bbb Z}
\define\Ph{{\widehat{P}}}
\define\al{{\alpha}}
\define\lam{{\lambda}}
\define\Del{\Delta}
\define\om{{\varpi}}
\define\ep{{\epsilon}}
\define\ii{\bold i}
%\define\tildii{  \tilde{\boldsymbol{\imath}} }
\define\tildii{  \widetilde{\bold i} }
\define\mm{\bold m}
\define\Zii{{Z_{\ii}}}
\define\Xim{{X_{\, \ii}^{\!\mm}}}
\define\OO{{\Cal O}}
\define\Lm{{\LL_{\mm}}}
\define\LS{\Cal{L\!S}}
\define\Gr{\roman{Gr}}
\define\Grim{\Gr(\ii)^{\mm}}
\define\FF{\Bbb F}
\define\timesB{\mathop{\times}^B}
\define\jh{\hat{\jmath}}
\define\Ess{\text{Ess}}
\define\TT{\Cal T}
\define\V{V^*}
%\define\{\}


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\topmatter

\title % French
Th\'eorie des mon\^omes standard pour les vari\'et\'es de Bott-Samelson
\endtitle

\title % English
Standard Monomial Theory for Bott-Samelson Varieties
\endtitle

\author Venkatramani Lakshmibai et Peter Magyar \endauthor
\author V. Lakshmibai and Peter Magyar \endauthor

\address{ Department of Mathematics, 
Northeastern University, Boston, MA 02115, USA}
\endaddress

\email lakshmibai\@neu.edu \quad
pmagyar\@lynx.neu.edu \endemail

\thanks Both authors partially supported by the 
National Science Foundation. \endthanks

\abstract
We construct a standard monomial basis for the space of
sections $H^0(Z,\LL)$, where $Z$ is a Bott-Samelson variety 
and $\LL$ a positive line bundle over $Z$.  As a special case,
we recover and complete the classical Standard Monomial Theory for
an arbitrary semisimple algebraic group.

\endabstract

\keywords {Young tableau, LS paths, crystal graphs, Schur modules} 
\endkeywords

%\subjclass {17B20,17B70}
%\endsubjclass
\endtopmatter

\document

%\centerline{\bf Th\'eorie des mon\^omes standard pour 
%les vari\'et\'es de Bott-Samelson}
\centerline{\bf TH\'EORIE DES MON\^OMES STANDARD}
\centerline{\bf POUR LES VARI\'ET\'ES DE BOTT-SAMELSON}

\medskip \noindent 
{\bf R\'esum\'e}. Nous construisons une th\'eorie des mon\^omes standard pour
l'espace des sections $H^0(Z,\LL)$,
o\`u $Z$ est une vari\'et\'e de Bott-Samelson
et o\`u $\LL$  est un fibr\'e en droites positif sur $Z$.  En particulier,
nous retrouvons et compl\'etons la th\'eorie des mon\^omes standard
classique pour un groupe alg\'ebrique semisimple arbitraire.

\subhead Version fran\c caise abr\'eg\'ee \endsubhead
%
Soient $G$ un groupe alg\'ebrique semisimple d\'efini sur un corps
alg\'ebriquement clos de caract\'eristique arbitraire (ou sur
$\ZZ$), $W$ son groupe de Weyl engendr\'e par les r\'eflections
simples $\{s_1,\ldots,s_r\}$,
$B$ un sous-groupe de Borel, $P_i \supset B$ le sous-groupe parabolique
minimal associ\'e \`a $s_i$, et $\Ph_i \supset B$ le sous-groupe
parabolique maximal associ\'e \`a $\{s_1, \ldots, \widehat{s_i}, \ldots, s_r\}$.
Notons par $\al_i$ les racines simples, et par $\om_i$ les poids
fondamentaux (voir [Bo, Ja]).

Choisissons une suite {\it arbitraire} de r\'eflections simples
$(s_{i_1},s_{i_2},\ldots,s_{i_l})$, que nous identifierons au mot
$\ii = (i_1,i_2,\cdots,i_l)$.  La vari\'et\'e de Bott-Samelson associ\'ee
[De, Ma2] est le quotient
$$
\Zii = P_{i_1} \times P_{i_2} \times \cdots \times P_{i_l} \ / \ B^l \ ,
$$
o\`u $B^l$ op\`ere \`a droite sur $P_{i_1} \times \cdots \times P_{i_l}$
par
$$
(p_1, p_2, \ldots, p_l)\cdot (b_1,\ldots,b_l) =
(p_1 b_1, b_1^{-1} p_2 b_2,\ldots, b_{l-1}^{-1} p_l b_l).
$$
Soit $\mm = (m_1,\ldots,m_l)$ avec $m_j \in \ZZ^+$.
D\'efinissons le plongement
$$
{\iota : \atop }\!\! 
{ \Zii \atop (p_1,p_2, \dots ,p_l) }\!
{\ \rightarrow \ \atop \ \mapsto\ }\!
{(G/\Ph_{i_1})^{m_1} \atop 
(\underbrace{\overline{p_1},\ldots,\overline{p_1}}_{m_1 \ \text{fois}}}\!\!
{\times \atop ,}\!\!
{(G/\Ph_{i_2})^{m_2} \atop 
\underbrace{\overline{p_1 p_2},\ldots,\overline{p_1 p_2}}_{m_2 \ \text{fois}}}\!\!
{\times \atop ,}\! { \cdots \atop \ldots}\! {\times \atop ,}\!\!
{(G/\Ph_{i_l})^{m_l} \atop
\underbrace{\overline{p_1\!\! \cdots \!p_l},
\ldots,\overline{p_1\!\! \cdots \!p_l}}_
{m_l \ \text{fois}}) }\!\!\!
{\eqdef \ \Xim  \atop },
$$
o\`u $\overline{p}$ d\'esigne la classe de l'\'el\'ement $p$.

Soit $\OO(1)$ le fibr\'e en droites ample de degr\'e minimal sur
$\Xim$; sa restriction \`a $\Zii$ est not\'ee  $\Lm = \iota^* \OO(1)$.
On note encore $\OO(1)$ le fibr\'e ample minimal sur $G/\Ph_i$.

Notre r\'esultat principal construit une base de l'espace des sections
globales $H^0(\Zii, \Lm)$ form\'ee de ``mon\^omes standard'',
restrictions de certaines sections particuli\`eres dans l'espace
$$
H^0(\Xim, \OO(1)) \ \cong \ \V(\om_{i_1})^{\otimes m_1} \otimes
\ldots \otimes \V(\om_{i_l})^{\otimes m_l},
$$
o\`u $\V(\om_{i}) \cong H^0(G/\Ph_i, \OO(1))$
d\'esigne le dual de la
$i$-i\`eme repr\'esentation fondamentale de $G$  (un module de Weyl
dual, qui est irr\'eductible sur un corps de caract\'eristique nulle,
mais non en g\'en\'eral).

Le premier auteur a construit une base [La] pour $\V(\om_i)$, et en
fait pour chaque $\V(\lambda)$.  (Plus pr\'ecis\'ement, la base
donn\'ee dans [La] est pour l'espace dual $V(\om_i)$, et ici nous
consid\'erons sa base duale).
La base de  $\V(\om_i)$ est index\'ee par certaines suites de poids
extr\'emaux et de nombres rationnels, les suites de Lakshmibai-Seshadri
$\LS(\om_i)$ (voir [Li]). Par exemple, lorsque $G$ est un groupe
classique, les nombres rationnels sont superflus, et $\LS(\om_i)$
est form\'e des couples
$\pi = (\tau, \tau')$ de poids $\tau = w(\om_i)$,
$\tau' = w'(\om_i)$,  pour lesquels il existe une suite
$\tau = \tau_0, \tau_1, \ldots, \tau_q = \tau'$ telle que
pour tout $j$, $\tau_{j+1} = s_k(\tau_j)$ pour un certain $k$ et
$\tau_{j+1} - \tau_j = 2 \al_k$.
(Littelmann identifie un tel couple avec un chemin lin\'eaire par
morceaux dans l'espace des poids, allant de $0$ \`a ${1\over 2}\tau$
puis \`a ${1\over 2}\tau + {1\over 2}\tau'$.)

L'\'el\'ement de la base de $\V(\om_i)$ associ\'e \`a
$\pi$ est not\'e $p^{\om_i}_{\pi}$.
Ce base est compatible avec les vari\'et\'es de Schubert
dans $G/\Ph_i$ (ce point est essentiel).
Nous pouvons maintenant construire une base de
$H^0(\Xim, \OO(1))$ index\'ee par les suites
$\pi = (\pi_{11},\ldots,
\pi_{1m_1}, \pi_{21},
\ldots,\pi_{lm_l})$
avec $\pi_{km} \in \LS(\om_{i_k})$,
et qui est form\'ee des mon\^omes
$p_{\pi} = p^{\om_{i_1}}_{\pi_{11}} \otimes \cdots \otimes
p^{\om_{i_l}}_{\pi_{lm_l}}$.
(Nous pouvons encore identifier les suites $\pi$ avec des chemins
lin\'eaires par morceaux, par concat\'enation comme dans [Li].)

Un mon\^ome $p_{\pi}$ est appel\'e {\it standard}
si $\pi$ poss\`ede un ``rel\`evement $\ii$-compatible''.
Dans le cas des groupes classiques, cela signifie que pour
$\pi = (\tau_{11},\tau'_{11},\ldots,\tau_{lm_l},\tau'_{lm_l})$,
il existe une suite de sous-mots de $\ii = (i_1,\ldots,i_l)$
donn\'ee par des sous-ensembles
$\{1,\ldots,l\} \supset J_{11} \supset J'_{11} \supset \cdots
\supset J_{lm_l} \supset J'_{lm_l}$
telle que pour tous $k, m$,
$$
\tau_{km} =
(\prod_{j \in J_{km} \atop j \leq k} s_{i_j})(\om_{i_k}),
$$
et de m\^eme pour $\tau'_{km}$ et $J'_{km}$ (c\.f\. [LS]).
Le poids d'une suite $\pi$ (le caract\`ere du tore maximal agissant sur
l'\'el\'ement associ\'e $p_{\pi}$ de la base)
est l'oppos\'e de l'extr\'emit\'e du chemin associ\'e:
$-({1\over 2} \tau_{11} +
{1\over 2} \tau'_{11} + \ldots +
{1\over 2} \tau'_{lm_l} )$
(le signe moins est d\^u \`a la dualit\'e).
La d\'efinition pour un groupe $G$ arbitraire est analogue, et en
particulier elle ne fait jamais intervenir les nombres rationnels dans
la suite de LS.

\proclaim {Th\'eor\`eme}  Les mon\^omes standard $p_{\pi}$ sur $\Xim$
se restreignent en une base de $H^0(\Zii, \Lm)$.
\endproclaim
\medskip

\noindent {\bf Note.}  Parmi les fibr\'es en droites $\Lm$ figurent les
images inverses des fibr\'es en droites sur la vari\'et\'e de Schubert
$X_w$ o\`u $w = s_{i_1}\cdots s_{i_l}$. Ainsi, le r\'esultat pr\'ec\'edent
contient comme cas particulier la construction de bases pour les modules
de Demazure $\V_w(\lam) \cong H^0(X_w,\LL_{\lam})$,
ce qui est le sujet de la th\'eorie des mon\^omes standard classique.

\subhead French Summary \endsubhead
%
Let $G$ be a semi-simple algebraic group defined over 
an algebraically closed  
field of arbitrary characteristic (or over $\ZZ$), $W$ its Weyl group
generated by the simple reflections $\{s_1,\ldots,s_r\}$,
$B$ a Borel subgroup, $P_i \supset B$ the minimal parabolic
associated to $s_i$, and $\Ph_i \supset B$ the maximal 
parabolic associated to $\{s_1, \ldots, \widehat{s_i}, \ldots, s_r\}$.
Denote the simple roots by $\al_i$ and the fundamental weights by $\om_i$.
(See [Bo, Ja].)

Choose an {\it arbitrary} sequence of simple reflections 
$(s_{i_1},s_{i_2},\ldots,s_{i_l})$, which we identify with
the word $\ii = (i_1,i_2,\cdots,i_l)$.  The corresponding 
Bott-Samelson variety [De, Ma2] is the quotient space
$$
\Zii = P_{i_1} \times P_{i_2} \times \cdots \times P_{i_l} \ / \ B^l \ ,
$$
where $B^l$ acts on the right of $P_{i_1} \times \cdots \times P_{i_l}$
by 
$$
(p_1, p_2, \ldots, p_l)\cdot (b_1,\ldots,b_l) = 
(p_1 b_1, b_1^{-1} p_2 b_2,\ldots, b_{l-1}^{-1} p_l b_l).
$$
Let $\mm = (m_1,\ldots,m_l)$ with $m_j \in \ZZ^+$,
and define the embedding
%
%$$
%\matrix
%\iota : & 
%\Zii & \rightarrow & 
%((G/\Ph_{i_1})^{m_1} & \times & (G/\Ph_{i_2})^{m_2} & \times & \cdots &
%\times & (G/\Ph_{i_l})^{m_l} & \eqdef \Xim \\
%& (p_1,p_2, \dots ,p_l) & \mapsto & 
%(\underbrace{p_1,\ldots,p_1}_{m_1 \text{ times}} & , &
%\underbrace{p_1 p_2,\ldots,p_1 p_2}_{m_2 \text{ times}} & , &
%\ldots & , &
%\underbrace{p_1\!\! \cdots\! p_l,\ldots,p_1\!\! \cdots\! p_l}_{m_l \text{ times}}) & 
%\endmatrix
%$$
%
$$
{\iota : \atop }\!\! 
{ \Zii \atop (p_1,p_2, \dots ,p_l) }\!
{\ \rightarrow \ \atop \ \mapsto\ }\!
{(G/\Ph_{i_1})^{m_1} \atop 
(\underbrace{\overline{p_1},\ldots,\overline{p_1}}_{m_1 \ \text{times}}}\!\!
{\times \atop ,}\!\!
{(G/\Ph_{i_2})^{m_2} \atop 
\underbrace{\overline{p_1 p_2},\ldots,\overline{p_1 p_2}}_{m_2 \ \text{times}}}\!\!
{\times \atop ,}\! { \cdots \atop \ldots}\! {\times \atop ,}\!\!
{(G/\Ph_{i_l})^{m_l} \atop
\underbrace{\overline{p_1\!\! \cdots \!p_l},
\ldots,\overline{p_1\!\! \cdots \!p_l}}_
{m_l \ \text{times}}) }\!\!\!
{\eqdef \ \Xim  \atop },
$$
where $\overline{p}$ means the coset of the group element $p$ in the
appropriate $G/P$.

Let $\OO(1)$ be the minimal-degree ample line bundle on
$\Xim$, and denote its restriction to $\Zii$  by $\Lm = \iota^* \OO(1)$.
We also denote by $\OO(1)$ the minimal ample bundle on each $G/\Ph_i$.

Our main 
result is to construct a basis of the global sections
$H^0(\Zii, \Lm)$ consisting of ``standard monomials'',
the restrictions of certain special sections in the larger space
$$
H^0(\Xim, \OO(1)) \ \cong \ \V(\om_{i_1})^{\otimes m_1} \otimes
\ldots \otimes \V(\om_{i_l})^{\otimes m_l},
$$
where $\V(\om_{i}) \cong H^0(G/\Ph_i, \OO(1))$ 
denotes the dual of the 
$i^{\text{th}}$ fundamental 
representation of $G$  (a dual Weyl module,
which is irreducible over a field of 
characteristic zero but not in general).

The first author has given a basis [La] for $\V(\om_i)$, and in fact
for any $\V(\lambda)$.  (More precisely, the basis given there
is for the dual space $V(\lambda)$, so here we take the dual basis.)
The basis of $\V(\om_i)$ is indexed by certain sequences
of extremal weights and rational numbers, the Lakshmibai-Seshadri 
sequences $\LS(\om_i)$ (see [Li]).
For example, in the case of any classical group $G$ the rational
numbers are redundant, and $\LS(\om_i)$ consists of all pairs
$\pi = (\tau, \tau')$ of weights $\tau = w(\om_i)$, 
$\tau' = w'(\om_i)$,  for which there exists a sequence
$\tau = \tau_0, \tau_1, \ldots, \tau_q = \tau'$ such that
for each $j$, $\tau_{j+1} = s_k(\tau_j)$ for some $k$ and
$\tau_{j+1} - \tau_j = 2 \al_k$.
(Littelmann identifies such a pair with a piecewise-linear path
in the weight lattice going from $0$ to ${1\over 2}\tau$ to
${1\over 2}\tau + {1\over 2}\tau'$.)

The basis element in $\V(\om_i)$ corresponding 
to $\pi$ is denoted $p^{\om_i}_{\pi}$.
Crucially, this basis is compatible with 
the Schubert varieties in $G/\Ph_i$.
Now we can give a basis of $H^0(\Xim, \OO(1))$ 
indexed by sequences 
$\pi = (\pi_{11},\ldots,
\pi_{1m_1}, \pi_{21},
\ldots,\pi_{lm_l})$
with $\pi_{km} \in \LS(\om_{i_k})$,
consisting of all monomials
$p_{\pi} = p^{\om_{i_1}}_{\pi_{11}} \otimes \cdots \otimes p^{\om_{i_l}}_{\pi_{lm_l}}$.
(We may once again identify sequences $\pi$ with piecewise-linear
paths by head-to-tail concatenation of the entries as in [Li].)

We call a monomial $p_{\pi}$ {\it standard} 
if $\pi$ possesses an ``$\ii$-compatible lifting''.
Again for classical groups, this means that for
$\pi = (\tau_{11},\tau'_{11},\ldots,\tau_{lm_l},\tau'_{lm_l})$,
there exists a chain of subwords of $\ii = (i_1,\ldots,i_l)$
given by subsets 
$\{1,\ldots,l\} \supset J_{11} \supset J'_{11} \supset \cdots
\supset J_{lm_l} \supset J'_{lm_l}$ 
such that for all $k, m$,
$$
\tau_{km} = 
(\prod_{j \in J_{km} \atop j \leq k} s_{i_j})(\om_{i_k}),
$$
and similarly for $\tau'_{km}$ and $J'_{km}$ (c\.f\. [LS]).
The weight of a sequence $\pi$ (the character of the maximal torus
acting on the corresponding basis element $p_{\pi}$) 
is negative the endpoint of its path:
$-({1\over 2} \tau_{11} + 
{1\over 2} \tau'_{11} + \ldots +
{1\over 2} \tau'_{lm_l} )$.
(The minus sign is due to dualization.)
The definition for a general $G$ is similar, and in particular
never involves the rational-number data in the LS sequence.

\proclaim {Theorem}  The standard monomials $p_{\pi}$ on $\Xim$ 
restrict to a basis of $H^0(\Zii, \Lm)$.
\endproclaim
\medskip

\noindent {\bf Note.}  The line bundles $\Lm$ include those
pulled back from the Schubert variety $X_w$, where
$w = s_{i_1}\cdots s_{i_l}$, so the above theorem includes
as a special case the construction of bases for the (dual)
Demazure modules $\V_w(\lam) \cong H^0(X_w,\LL_{\lam})$,
which is the subject of classical Standard Monomial Theory [LS].

\medskip

\subhead English Version \endsubhead
%
For ease of exposition, we suppose $G$ to be of type $A_{n-1}$.
Let $G = SL(n,\FF)$  (where $\FF$ is an algebraically closed field
of arbitrary characteristic or $\FF = \ZZ$); consider the
subgroups $B = $ upper triangular matrices, 
$P_i =$ matrices which are upper triangular except for the position
$(i+1,i)$, $W = $ permutation matrices; 
let $s_i = $ the transposition $(i,i+1)$;
and note that $G/\Ph_i = \Gr(i,\FF^n) = \Gr(i)$ 
the Grassmannian of $i$-planes in $n$-space.  

For any word $\ii = (i_1,\ldots,i_l)$, $1\leq i_j \leq n-1$, 
reduced or non-reduced, 
and any sequence
$\mm = (m_1,\ldots,m_l)$, $m_j \in \ZZ^+$,
the Bott-Samelson variety 
$$\Zii = P_{i_1} \timesB \cdots \timesB P_{i_l} / B$$
embeds into the product of Grassmannians
$$
\Grim  = \Xim = \Gr(i_1)^{m_1} \times \cdots \times \Gr(i_l)^{m_l}
$$ 
by
$$
\matrix 
\iota :\,& \Zii & \rightarrow & \Grim \\
&(p_1,\ldots,p_l) & \mapsto & 
(\underbrace{p_1 \FF^{i_1},\ldots,p_1 \FF^{i_1}}_{m_1 \text{ times}},
\ldots,
\underbrace{p_1\!\! \cdots\! p_l \FF^{i_l},\ldots,
            p_1\!\! \cdots\! p_l \FF^{i_l}}_{m_l \text{ times}}) 
\endmatrix
$$
where $0 \subset \FF^1 \subset \cdots \subset \FF^n$ is the standard
flag.

Now, each Grassmannian has a minimal-degree ample line bundle $\OO(1)$,
and the tensor product of these for each $i_j$ is the minimal
ample line bundle on $\Grim$ which we again denote
$\OO(1) \eqdef \OO(1,\ldots,1)$.
Denote its restriction to $\Zii$ by
 $\Lm = \iota^* \OO(1)$.

\proclaim{Problem} Find an explicit basis for $H^0(\Zii,\Lm)$.
\endproclaim

\noindent
{\bf Note.}  The line bundles $\Lm$ include those
pulled back from the Schubert variety $X_w$, where
$w = s_{i_1}\cdots s_{i_l}$, so the above problem includes
as a special case the construction of bases for the 
(dual) Demazure modules $\V_w(\lam) \cong H^0(X_w,\LL_{\lam})$,
which is the subject of classical Standard Monomial Theory [LS].

\medskip \noindent 
%We solve the Problem by restricting certain elements from
%the obvious basis of $H^0(\Grim,\OO(1))$.
%
We first consider the fundamental Weyl module 
$H^0(\Gr(i),\OO(1)) \cong \V(\om_i)$.
This has a well-known basis consisting
of the Pl\"ucker coordinates,
$$
\V(\om_i) = \text{Span}_{\FF}
\left\{ \Del_I \mid I \, \subset [n] \text{ and } |I|=i \right\},
$$
where $[n] = \{1,2,\ldots,n\}$.
Here $\Del_I$ denotes the $i \times i$ minor
on the rows $I$ 
of the $n \times i$ matrix of
homogeneous coordinates on the Grassmannian $\Gr(i)$.
Thus, we get a basis of
$$
\align
H^0(\Grim,\OO(1)) & \cong \V(\om_{i_1})^{\otimes m_1} \otimes 
\cdots\otimes \V(\om_{i_l})^{\otimes m_l} \\
&= \text{Span} 
\left\{ \Del_{\pi} 
\ \left| \ \matrix 
\pi = (I_{11}, \ldots, I_{1m_1}, I_{21}, \ldots, I_{lm_l}) \\
\forall\, k,m, \ I_{km} \subset [n]  \text{ and } 
|I_{km}|=i_k
\endmatrix \right.
\right\},
\endalign
$$
where $\Del_{\pi} = p_{\pi} = 
\Del_{I_{11}} \otimes \cdots \otimes \Del_{I_{lm_l}}$.
We call $\Del_{\pi}$ a {\it monomial} and $\pi$ a {\it tableau}.
We may again picture tableaux as paths.

\medskip \noindent
{\bf Example.}  For $G=SL(3)$,
$\ii = (1,2,1)$, $m = (1,1,1)$,
the tableau $\pi = (2,13,3) = 2.13.3$ is identified 
with the piecewise-linear path in $\ZZ^3$ with vertices $0$,\ $e_2$,\
$e_2+e_1+e_3$,\ $e_2+e_1+e_3+e_3$, where $e_j$ denotes a 
coordinate vector.  The corresponding basis element is 
$$
\Del_{\pi} = \Del_2(x)\, \Del_{13}(y)\, \Del_3(z) =
x_2\, (y_{11} y_{32} - y_{31} y_{12})\, z_3
$$
in the coordinates
$$
\pmatrix x_1 \\ x_2 \\ x_3 \endpmatrix \times
\pmatrix y_{11} & y_{12} \\ y_{21} & y_{22} \\ y_{31} & y_{32}
\endpmatrix \times
\pmatrix z_1 \\ z_2 \\ z_3 \endpmatrix
\in \Gr(1) \times \Gr(2) \times \Gr(1) .
$$

\medskip \noindent {\bf Definition.}
Let $[k] = \{1,2,\ldots,k\}$, and allow $W$ to
act elementwise on subsets of $[n]$.
We say a tableau $\pi$ is {\it standard} if it possesses
a ``compatible lifting'' with respect to $\ii$.  
That is, for 
$\pi = (I_{11},\ldots,I_{lm_l})$, there exists
a decreasing sequence of subwords of $\ii = (i_1,\ldots,i_l)$,
indexed by subsets 
$\{1,\ldots,l\} \supset J_{11} \supset \cdots \supset J_{lm_l}$,
such that for all $k,m$,
$$
(\!\!\prod_{j \in J_{km} \atop j\leq k}\!\! s_{i_j}) [i_k] = I_{km}.
$$
A monomial is standard if its defining tableau is standard.

\medskip \noindent
{\bf Note.}  There is an efficient recursive algorithm 
for generating the standard tableaux, given by a refined Demazure
character formula based on Littelmann's path operators [Li]
(first defined for $SL(n)$ by Lascoux and Sch\"utzenberger [LSch]).

\proclaim{Theorem}  The standard monomials $\Del_{\pi}$ 
on $\Grim$ restrict to a basis of $H^0(\Zii,\Lm)$.
\endproclaim

\noindent
{\bf Example.} In the above case, $\pi = 2.13.3$ is standard
since the sequence $J_{11} = \{1,2,3\}$,\, $J_{21} = \{2,3\}$,\, 
$J_{31} =\{2, 3\}$
gives $s_{i_1}[i_1] = s_1[1]=2$,\ $s_{i_2}[i_2] = s_2[2] = 13$,
and $s_{i_2}s_{i_3}[i_3]=s_2 s_1[1] = 3$.
There are thirteen standard tableaux:
$1.12.1$,\
$1.12.2$,\
$1.13.2$,\
$1.13.3$,\
$2.13.2$,\
$2.13.3$,\
$2.23.3$,\
$1.13.1$,\
$2.12.1$,\
$2.12.2$,\
$2.13.1$,\
$2.23.1$,\
$2.23.2$.\

\medskip \noindent
{\bf Sketch of Proof.}  We prove the Theorem by induction on
the dimension of $\Zii$ (= the length of $\ii$) and on the degree
of $\Lm$.  
We show that both the space
of sections and the span of the standard monomials 
can be built up using the same recurrences, 
and hence they must be identical spaces.

Let $\TT(\ii,\mm)$ denote the set of standard tableaux
for the space $H^0(\Zii,\Lm)$.  
Suppose $\mm = (m_1,\ldots,m_l)$ and let $k$ be the smallest
value with $m_k > 0$.  Without loss of generality, we may assume
that the initial subword $(i_1,\ldots,i_k)$ is reduced.
That is, let $\tildii$ be a subword of $\ii$ with some of the
first $k$ letters removed so as to make $(i_1,\ldots,i_k)$ reduced.
Then we can show that $H^0(Z_{\tildii},\Lm) \cong H^0(\Zii,\Lm)$
and that $\TT(\tildii,\mm) = \TT(\ii,\mm)$.

Next, let $Z = \Zii$ and for $1 \leq j \leq l$, let $Z_{\jh}$
denote the Bott-Samelson variety of the word $\ii$
with the $j^{\text{th}}$ entry removed,
$\ii - i_j = (i_1,\ldots,\widehat{\imath_j},\ldots,i_l)$.
Then $Z_{\jh}$ embeds naturally as a divisor in $Z$.
Further, let $\LL_{\ep(k)}$ denote the line bundle $\Lm$ with
$\mm = \ep(k) = (0,\ldots,1,\ldots,0)$  \ \
(the 1 in the $k^{\text{th}}$ place).

Now define $J(k) = s_{i_1}\!\!\! \cdots\! s_{i_k} [i_k] \subset [n]$.
Then the work of Berenstein, Fomin, and Zelevinsky [BFZ, BZ] implies
that
$$
(\text{zero set of } \Del_{J(k)} )\ \cap Z  
\ = \!\!\!\bigcup_{j \in \Ess(k)}\!\!\!\! Z_{\jh} \ \ \ \ \ \text{(scheme theoretically)}
$$
where 
$$
\Ess(k) = \{ j \leq k \mid \,
s_{i_1}\!\! \cdots \!\widehat {s_{i_j}}\! \cdots\! s_{i_k} [i_k] 
\neq J(k) \}.
$$
Since $\Del_{J(k)}$ is a section of $\LL_{\ep(k)}$,
this (together with a Kempf-type vanishing theorem for $Z$
due to Mathieu [Mat1],[Mat2])
leads to the exact sequence
$$
0 \rightarrow H^0(Z, \Lm \otimes \LL_{\ep(k)}^{-1}) 
\overset{\Del_{J(k)}} \to {\longrightarrow} 
H^0(Z, \Lm) 
\overset \text{rest}\to {\longrightarrow} 
H^0(\bigcup_{j \in \Ess(k)}\!\!\!\! Z_{\jh}, \Lm)
\rightarrow 0 ,
$$
where the left-hand map is multiplication by the section
$\Del_{J(k)}$, and the right-hand map is restriction.

We can show the corresponding recurrence for the tableaux:
$$
\TT(\ii,\mm) = J(k)\,.\,\TT(\ii, \mm - \ep(k)) \ \ \ \sqcup \ 
\bigcup_{j \in \Ess(k)}\!\!\!\! \TT(\ii - i_j, \mm) ,
$$
where $J(k)\,.\,\TT$ denotes the concatenation of $J(k)$ 
at the beginning of all the tableaux in $\TT$,
and $\TT(\ii-i_k,\mm) \eqdef \TT(\ii-i_k,\mm')$ 
for a suitable $\mm' = (m'_1,\ldots,m'_{l-1})$.

Finally, the sections over the union $\cup_j Z_{\jh}$ are 
analyzed via the Mayer-Vietoris sequence, and the tableaux
again satisfy the same recurrence 
(provided we define a monomial to be standard on a union
if it is standard on one of the components).

From this the Theorem follows immediately by induction.
Namely, the standard monomials in 
$\Del_{J(k)}\! \cdot\! H^0(Z,\LL_{\mm-\ep(k)})$
form a basis of this space by induction on the degree of $\Lm$; 
and the standard monomials in $H^0(\cup_j Z_{\jh}, \Lm)$ form a
basis of this space by induction on dimension; 
and the two give complementary subspaces of
$H^0(Z,\Lm)$ by the exact sequence.   Hence the standard
monomials in $H^0(Z,\Lm)$ are a basis.

We thus show the Theorem not only for Bott-Samelson
varieties, but also for the unions 
$\cup_{j \in \Ess(k)} Z_{\jh}$.
Note, however, that it does
not hold for arbitrary unions even for the case of $SL(3)$.

The Theorem and proof are carried out similarly for a general
reductive $G$ in the spirit of [LS], and one may
also extend the analysis to the case of symmetrizable
Kac-Moody groups and quantum groups.

\Refs
\widestnumber\key{LSch}

\ref
\key BFZ
\by A. Berenstein, S. Fomin, and A. Zelevinsky
\pages
\paper Parametrizations of canonical bases 
and totally positive matrices
\yr 1996
\vol
\jour Adv. Math.
\endref

\ref
\key BZ
\by A. Berenstein and A. Zelevinsky
\paper Total positivity in Schubert varieties
\yr 1996
\jour preprint
\endref

\ref
\key Bo
\by A. Borel
\book  Linear Algebraic Groups, 
{\rm GTM 126} 
\publ Springer Verlag
\year 1991
\endref

\ref
\key De
\by M. Demazure
\pages 53-88
\paper D\'esingularisation des
 vari\'et\'es
 de Schubert g\'en\'eralis\'es
\yr 1974
\vol 7
\jour Ann. Sci. Ec. Norm. Sup.
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\ref
\key Fu
\by W. Fulton
\book Young Tableaux
with Applications to Representation Theory and
Geometry
\yr 1996
\publ Cambridge Univ. Press
\endref

\ref
\key Ja
\by J. C. Jantzen
\book Representations of Algebraic Groups,
{\rm Pure and App. Math, Vol 116}
\yr 1987
\publ Academic Press
\publaddr London
\endref

\ref
\key La
\by V. Lakshmibai
\paper Bases for quantum Demazure modules
\inbook in Representations of Groups, Banff 1994,
{\rm CMS Conf. Proc., Vol. 16}
\yr 1995
\eds B. N. Allison and G. H. Cliff
\publ Canadian Math. Soc.
\endref

\ref
\key LS
\by V. Lakshmibai and  C. S. Seshadri
\pages 462-557
\paper Geometry of $G/P$ -- V
\yr 1986
\vol 100
\jour J.  Alg.
\endref

\ref
\key LSch
\by A. Lascoux and M.-P. Sch\"utzenberger
\pages 125-144
\paper Keys and standard bases
\yr 1990
\inbook in Tableaux and Invariant Theory,
{\rm  IMA Vol. in Math. and App., Vol 19}
\publ IMA
\eds  D. Stanton
\endref

\ref
\key Li
\by P. Littelmann
\pages  329-346
\paper A Littlewood-Richardson
rule for symmetrizable Kac-Moody algebras
\yr 1994
\vol 116
\jour Inv. Math.
\endref

\ref
\key Ma1
\by P. Magyar
\paper Borel--Weil 
theorem for Schur modules  and configuration varieties
\yr 1994
\jour preprint alg-geom/9411014
\endref

\ref
\key Ma2
\bysame
\paper Bott-Samelson varieties and configuration spaces
\yr 1996
\jour preprint
\endref

\ref
\key Mat1
\by O. Mathieu
\paper Filtrations of $B$-modules
\yr 1989
\vol 59
\pages 421-442
\jour Duke Math. J.
\endref

\ref
\key Mat2
\by O. Mathieu
\paper Filtrations of $G$-modules
\yr 1990
\jour Ann. Ec. Norm. Sup.
\vol 23
\pages 625-644
\endref

\endRefs

\enddocument