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\begin{document}
\title[Singular loci of LDV's and Schubert varieties]{SINGULAR LOCI OF\\ LADDER DETERMINANTAL
VARIETIES\\ AND SCHUBERT VARIETIES}
\author{N. Gonciulea}
\address{Department of Mathematics\\ Northeastern University\\ Boston, MA 02115}
\email{ngonciul@@lynx.neu.edu}
\author[V. Lakshmibai]{V. Lakshmibai${}^{\dag}$}
\email{ngonciul@@lynx.neu.edu, lakshmibai@@neu.edu}
\thanks{${}^{\dag}$Partially supported by NSF Grant DMS 9502942\\
\indent ${}^{\dag}$Partially suported by Northeastern University RSDF 95-96.}
\maketitle
\begin{abstract}
We relate certain  ladder determinantal varieties (associated to one-sided ladders) to certain
Schubert varieties in
$SL(n)/Q$, for a suitable $n$ and a suitable parabolic subgroup $Q$, and we determine the
singular loci of these varieties. We state a conjecture on the irreducible
components of the singular locus of a Schubert variety in the flag variety, which is a
refinement of the conjecture of \cite{LS}. We prove the conjecture for a certain class of
Schubert varieties.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%

\section*{Introduction}
Let $k$ be the base field which we assume to be algebraically closed of arbitrary
charcateristic. Let $X=(x_{ba})$, $1\le b,a\le n$ be a matrix of variables, and $L\subset X$ an
{\em one-sided ladder}  with outside corners $(b_1,a_1),\dots,(b_h,a_h)$, i.e.
$$L=\{x_{ba}\mid\text{ there exists }1\le i\le h\text{ such that } b_i\le b\le m,1\le
a\le a_i\},$$
where $1\le b_1<\dots<b_h<n$, $1<a_1<\dots<a_h\le n$.
We suppose that $n$ is large enough so that
$b_i>a_i$,  for all $i$, $1\le i\le h$. Let $k[L]$ denote the polynomial ring $k[x_{ba},\
x_{ba}\in L]$, and let $\Bbb{A}(L)=\Bbb{A}^{|L|}$ be the associated affine space. For $1\le
i\le l$, let  $i^*$ denote the largest integer in $\{1,\dots,h\}$ such that
 $b_{i^*}\le s_i$. Let
$\us =(s_1,\dots,s_l)\in\Bbb{Z}_+^l$, $\ut=(t_1,\dots,t_l)\in\Bbb{Z}_+^l$ be such that $b_1=
s_1<\dots<s_l\le n$, $t_1\ge\dots\ge t_l$, $1\le t_i\le \min \{n-s_i+1,a_{i^*}\}$ for $1\le i\le
l$, and
$s_i-s_{i-1}>t_{i-1}-t_i$ for $1<i\le l$.  For each $1\le i\le l$, let
$L_i=\{x_{ba}\mid s_i\le b\le n\}$. Let $I_{\us,\ut}(L)$ be the ideal of
$k[L]$ generated by all the
$t_i$-minors in $L_i$, $1\le i\le l$. Let $D_{\us,\ut}(L)\subset \Bbb{A}(L)$ be the variety
defined by
$I_{\us,\ut}(L)$, and we call it a  {\em ladder determinantal variety} (the ladder being
one-sided). The variety
$D_{\us,\ut}(L)$ is isomorphic to $D_{\us',\ut'}(L')\times \Bbb{A}^d$, for suitable $l'$-tuples
$\us'$, $\ut'$, a suitable one-sided ladder $L'\subset L$ in $X$ defined by the outside corners
$(b_1', a_1'),\dots,(b_h', a_{h'}')$ such that
$\{b_1',\dots,b_{h'}'\}\subset\{s_1',\dots,s_{l'}'\}$ and
$d=|L|-|L'|$ (see Section \ref{1} for details). Thus it is enough to study the variety
$D_{\us,\ut}(L)$ under the assumption
$\{b_1,\dots,b_h\}\subset\{s_1,\dots,s_l\}$. Without loss of generality, we can also assume
that $t_l\ge 2$, and $t_{i-1}>t_i$ if $s_i\not\in\{b_1,\dots,b_h\}$ for $1<i\le l$.

 For each $1\le i\le l$, let
$L(i)=\{x_{ba}\mid s_i\le b\le n,1\le a\le a_{i^*}\}$. It is easy to see  that the ideal
$I_{\us,\ut}(L)$ is generated by the $t_i$-minors of $X$ contained in $L(i)$, $1\le i\le l$.
\begin{figure}
\centerline{\agraph}
\caption{The one-sided ladder $L$}
\end{figure}
First we relate the  ladder determinantal varieties (associated to one-sided ladders) to Schubert
varieties as given by the following (cf. Theorem \ref{w})
\begin{th}
The variety $D_{\us,\ut}(L)\times \Bbb{A}^r$ gets identified with the ``opposite cell" in a
certain Schubert variety $X(w)$ in $SL(n)/Q$, for a suitable parabolic subgroup $Q$ of $SL(n)$,
where $r=\text{dim\,} SL(n)/Q-|L|$.
\end{th}
As a consequence, we obtain (cf. Theorem \ref{normal})
\begin{thh}
The variety $D_{\us,\ut}(L)$ is irreducible, normal, Cohen-Macaulay, and has rational
singularities.
\end{thh}
We also determine the singular locus of $D_{\us,\ut}(L)$, as described below.  Let $V_j$, $1\le j\le l$ be
the subvariety of $D_{\us,\ut}(L)$ defined by the vanishing of all $(t_j-1)$-minors in $L(j)$.
We prove (cf. Theorem \ref{sing})
\begin{thhh}
We have $\text{Sing\,}D_{\us,\ut}(L)=\cup_{j=1}^l V_j$.
\end{thhh}
We further prove  the following (cf Theorem \ref{te})
\begin{thhhh}
For $1\le j\le l$, the subvariety $V_j\times \Bbb{A}^r$ of $D_{\us,\ut}(L)\times\Bbb{A}^r$ ($r$
being as above) gets identified with the ``opposite cell" in a certain Schubert subvariety
$X(\te_j)$ of $X(w)$.
\end{thhhh}
As a consequence, we obtain (cf. Theorem \ref{comp})
\begin{thhhhh}
The irreducible components of $\text{Sing\,}D_{\us,\ut}(L)$ are precisely the $V_j$'s, $1\le j\le l$.
\end{thhhhh}
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 $X(w)$ (resp. $X(\te_j)$, $1\le j\le l$) under the canonical projection
$\pi :SL(n)/B\to SL(n)/Q$ (here $B$ is a Borel suybgroup of $SL(n)$ such that $B\subset Q$).
Then using Theorems 1, 3 and 4, we obtain (cf. Theorem \ref{8.4})
\begin{thhhhhh}
The irreducible components of $\text{Sing\,}X(w^{\text{max}})$ are precisely
$X(\te_j^{\text{max}})$,
$1\le j\le l$.
\end{thhhhhh}
We state a conjecture on the irreducible components of the singular locus of a Schubert
variety in $SL(n)/B$, which is a refinement of the conjecture in \cite{LS} (see Section \ref{9}
for the statement of the conjecture). Using Theorem 6, we prove (cf. Theorem \ref{last})
\begin{thhhhhhh}
The conjecture  holds for $X(w^{\text{max}})$.
\end{thhhhhhh}

We now briefly describe how the above Theorems are proved. Let $Q=\cap_{i=1}^h P_{a_i}$, where
$P_{a_i}$ is the maximal parabolic subgroup of $SL(n)$ obtained by ``omitting" the simple root
$\a_{a_i}$, the simple roots being indexed as in \cite{Bou} (see Section \ref{s12} for details).
Let
$O^-$ be the ``opposite big cell" in $G/Q$ (see Section \ref{s12} for details).
We identify $O^-$ ($\simeq\Bbb{A}^N$, $N=\text{dim\,}G/Q$) as a subvariety of the variety of
lower triangular matrices in $SL(n)$. This in turn gives raise to an embedding
$\Bbb{A}(L)\subset O^-$.
 Let $Z_w=X(w)\cap O^-$ be the ``opposite cell" in
$X(w)$, and  $I_w$  the ideal defining  $Z_w$ in $O^-$ . Then one knows that the Pl\" ucker
coordinates vanishing on $Z_w$ generate $I_w$. Let $I_{\us,\ut}^*(L)$ be the ideal generated by
$I_{\us,\ut}(L)$ in
$k[\Bbb{A}^N]$. We prove Theorem 1 by showing that the Pl\" ucker coordinates vanishing on
$Z_w$ belong to $I_{\us,\ut}^*(L)$ and conversely, a typical $t_i$-minor in $L(i)$, $1\le i\le
l$, belongs to $I_w$.
Theorem 2 is a consequence of Theorem 1 and the fact that Schubert varieties are
irreducible, normal, Cohen-Macaulay, and have rational singularities (cf. \cite{KR},
\cite{RR}, \cite{R1}, \cite{R2}). Theorem 3 is proved using the Jacobian criterion for
smoothness. Towards this end, we first construct a Gr\" obner basis for
$I_{\us,\ut}(L)$, which then enables us to compute the codimension of $D_{\us,\ut}(L)$ in $\Bbb{A}(L)$. Theorem 4
is proved in the same spirit as Theorem 1. As one sees, Theorem 5 is an immediate consequence
of Theorems 3 and 4, and Theorem 6 is an immediate consequence of Theorems 1, 3 and 4.
Theorem 7 is proved through a relative study of $X(w^{\text{max}})$ and $X(\te_j^{\text{max}})$.
Thus we have used the theory of Schubert varieties to prove results on ladder determinantal
varities, and vice versa. To be more precise, geometric properties such as normality,
Cohen-Macaulayness, etc., for ladder determinantal varities 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.

A similar identification as in Theorem 1 for the case $t_1=\dots =t_l$ has also been obtained by
Mulay (see \cite{M}). Results similar to those of Theorem 2 for certain other ladder
determinantal varieties have been obtained by several authors (see  \cite{Conca}, \cite{CH},
\cite{GS},\cite{MS}, \cite{Na}). To the best of our knowledge, Theorem 5 is the only result in
the literature on the determination of the singular locus of a  ladder determinantal variety,
except for the case of the classical determinantal variety, i.e. $h=1$ and $l=1$ (see \cite{LW},
\cite{La}, \cite{Sv}).

The sections are organized as follows.
In section 1 we define ladder determinantal varieties and set up a few notations. In Section 2,
we recall some generalities on $G/Q$. In Section 3, we recall some generalities on Schubert
varieties in the flag variety. In Section 4, we prove two lemmas related to the evaluation of
Pl\" ucker coordinates on the ``opposite big cell". In Section 5, we bring out the relationship
between ladder determinantal varieties and Schubert varieties. In section 6, we compute the
dimension of ladder determinantal varities by constructing Gr\" obner bases for their
defining ideals. In Section 7, we determine the singular loci of ladder detrminantal varieties.
In section 8, we determine the irreducible components of the singular loci of ladder
determinantal varieties. In Section 9, we state a conjecture on the irreducible components of
the singular locus of a Schubert variety in $SL(n)/B$, and prove it for a certain class of
Schubert varieties, namely those Schubert varieties which are related to ladder determinanatal
varieties as in Section 5. This conjecture is a refinement of the conjecture in \cite{LS}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Ladder determinantal varieties}\label{1}
Let $X=(x_{ba})$, $1\le b\le m$, $1\le a\le n$ be a $m\times n$ matrix of indeterminates.

Given $1\le b_1<\dots<b_h< m$, $1< a_1<\dots
<a_h\le n$, we consider the subset of
$X$, defined by
$$L=\{x_{ba}\mid\text{ there exists }1\le i\le h\text{ such that } b_i\le b\le m,1\le
a\le a_i\}.$$
 We call $L$ an {\em one-sided ladder} in $X$, defined by the {\em outside corners}
$\o_i=x_{b_ia_i}$, $1\le i\le h$. For simplicity of notation, we identify the variable $x_{ba}$
with just $(b,a)$.

For $1\le i\le l$, let
$i^*$ be the largest integer such that $b_{i^*}\le s_i$.

Let  $\us=(s_1,s_2 \dots,s_l)\in \Bbb{Z}_+^l$, $\ut=(t_1,t_2 \dots,t_l)\in \Bbb{Z}_+^l$ such
that

\begin{gather}
b_1= s_1<s_2<\dots<s_l\le m,\notag\\
t_1\ge t_2\ge\dots\ge t_l,\ 1\le t_i\le \min \{ m-s_i+1,a_{i^*}\}\text{ for }1\le i\le l,
\text{ and}\tag{L1}\\
 s_i-s_{i-1}>t_{i-1}-t_i\text{ for }1<i\le l.\notag
\end{gather}

For  $1\le i\le l$, let
$$L_i=\{x_{ba}\in L\mid s_i\le b\le m\}.$$
Let $k[L]$ denote the polynomial ring $k[x_{ba}\mid x_{ba}\in L]$,
and let
$\Bbb{A}(L)=\Bbb{A}^{|L|}$ be the  associated affine space. Let $I_{\us,\ut}(L)$ be the ideal
in $k[L]$ generated by all the $t_i$-minors contained in $L_i$, $1\le i\le l$, and
$D_{\us,\ut}(L)\subset\Bbb{A}(L)$ the variety defined by the ideal $I_{\us,\ut}(L)$. We call
$D_{\us,\ut}(L)$ a {\em ladder determinantal variety} (associated to an one-sided ladder).

Let $\O=\{\o_1,\dots,\o_h\}$.
For each $1< j\le l$, let $$\O_j=\{\o_i\mid 1\le i\le h\text{ such that }s_{j-1}<
b_i<s_{j}\text{ and }s_{j}-b_i\le t_{j-1}-t_{j}\}.$$ Let
$$\O'=(\O\setminus\bigcup_{j=2}^l\O_j)\bigcup_{\O_j\ne\emptyset}\{(s_{j},a_{j^*})\}.$$
  Let
$L'$ be the one-sided ladder in $X$ defined by the set of outside corners $\O'$.
 Then it is easily  seen that
$D_{\us,\ut}(L)\simeq D_{\us,\ut}(L')\times \Bbb{A}^d$, where $d= |L|-|L'|$.

Let $\o_k'=(b_k',a_k')\in\O'$, for some $k$, $1\le k\le h'$, where  $h'=|\O'|$. If
$b_k'\not\in\{s_1,\dots,s_l\}$, then
$b_k'=b_i$ for some $i$, $1\le i\le h$, and we define $s_{j^-}=b_i$, $t_{j^-}=t_{j-1}$,
$s_{j^+}=s_j$,
$t_{j^+}=t_j$, where
$j$ is the unique integer such that $s_j< b_i<s_{j+1}$. Let
$\us'$ (resp. $\ut'$) be the sequence obtained from $\us$ (resp. $\ut$) by replacing $s_j$ (resp.
$t_j$) with $s_{j^-}$ and $s_{j^+}$ (resp. $t_{j^-}$ and $t_{j^+}$) for all $k$ such that
$b_k'\not\in\{s_1,\dots,s_l\}$, $j$ being the unique integer such that $s_{j-1}< b_i<
s_{j}$, and $i$ being given by $b_k'=b_i$. Let
$l'=|\us'|$. Then
$\us'$ and $\ut'$ satisfy (L1), and in addition we have
$\{b_1',\dots,b_{h'}'\}\subset
\{s_1',\dots,s_{l'}'\}$. It is easily seen that
$D_{\us,\ut}(L')=D_{\us',\ut'}(L')$, and hence $$D_{\us,\ut}(L)\simeq D_{\us',\ut'}(L')\times
\Bbb{A}^d.$$

Therefore it is enough to study $D_{\us,\ut}(L)$ with $\us,\ut\in\Bbb{Z}_+^l$ such that
\begin{equation*}
\{s_1,\dots,s_l\}\supset\{b_1,\dots,b_h\}.\tag{L2}
\end{equation*}
Without loss of generality, we can also assume that
\begin{equation}
t_l\ge 2,\text{ and }t_{i-1}>t_i\text{ if }s_i\not\in\{b_1,\dots,b_h\}, 1<i\le l.\tag{L3}
\end{equation}

 For  $1\le i\le l$, let
$$L(i)=\{x_{ba}\mid s_i\le b\le m,1\le a\le a_{i^*}\}.$$
Note that the ideal $I_{\us,\ut}(L)$ is
generated by the $t_i$-minors of $X$ contained in $L(i)$, $1\le i\le l$.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Generalities on $G/Q$}\label{s12}

Let $G$ be a semisimple and simply connected  algebraic group defined over an algebraically
closed field of arbitrary characteristic. Let
$T\subset G$ be a maximal torus, and  $B\supset T$ be a Borel subgroup. 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 Chevalley-Bruhat order}Let $w\in W$. A minimal expression for $w$ as a product of
simple reflections is called a reduced expression for $w$. We denote by $l(w)$ the length of a
reduced expression for $w$ (as a product of simple refelections). We have a partial order on
$W$, the well-known  Chevalley-Bruhat order, namely $w_1\ge w_2$ if a reduced expression for
$w_1$ contains a  subexpression which is a reduced expression for $w_2$.

\subsection{The Weyl subgroup $W_Q$} Let $Q$ be a parabolic subgroup of $G$ containing
$B$.\label{W_Q} Associated to
$Q$, there is a subset $S_Q$ of $S$ such that $Q$ is the subgroup of $G$ generated by $B$ and
$\{U_{-\a}\mid \a\in R^+_Q\}$, where $R^+_Q=\{\a\in R^+\mid \a=\sum_{\b\in S_Q} a_\b\b\}$ (here,
for $\b\in R$, $U_\b$ denotes the $1$ dimensional unipotent subgroup of $G$ associated to $\b$).
Let
$W_Q$ be the Weyl group of $Q$ (note that $W_Q$ is simply the subgroup of $W$ generated by
$\{s_\a\mid\a\in S_Q\}$; here, for $\a\in S$, $s_\a$ denotes the simple reflection (considered
as an element of $W$), associated to $\a$).

\subsection{The set $W_Q^{min}$ of minimal representatives of $W/W_Q$}
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 reperesentatives of} $W/W_Q$. We have
$$W_Q^{\text{min}}=\{w\in W\mid l(ww')=l(w)+l(w'),\text{ for all }w'\in Q\}.$$
The set $W_Q^{\text{min}}$ may be 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 being $>0$ we mean $\b\in R^+$).

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{Maximal parabolic subgroups}
The set of maximal parabolic subgroups is in one-to-one correpondence with $S$, namely given
$\a\in S$, the parabolic subgroup $Q$ where $S_Q=S\setminus\{\a\}$ is a maximal parabolic
subgroup, and conversely. We shall denote $Q$, where $S_Q=S\setminus\{\a\}$ by
$P_{\widehat{\a}}$, and refer to it as the {\em maximal parabolic subgroup obtained by
omitting}$\ \a$.

\subsection{Schubert varieties in $G/Q$}\label{12.4}
For $w\in W$, let us denote the point in $G/Q$ corresponding to the coset $wQ$ by $e_{w,Q}$. Then
the set of
$T$-fixed points in
$G/Q$ for the action given by left multiplication is presisely $\{e_{w,Q}\mid w\in W\}$.
Let $w\in W$, and let $X_Q(w)$ be the Zariski closure of $Be_{w,Q}$ in $G/Q$. Then
$X_Q(w)$  with the canonical reduced structure is called the Schubert variety in $G/Q$ associated
to $wW_Q$. In particular, we have bijections between $W_Q^{\text{min}}$ and the set of Schubert
varieties in $G/Q$, and between $W_Q^{\text{max}}$ and the set of Schubert varieties in $G/Q$.
We have the well-known Bruhat decomposition
$$G/Q=\dot{\cup}Be_{w,Q},\qquad X_Q(\te)=\dot{\cup}_{w\le\te}Be_{w,Q},\quad \te\in W.$$

As above, let $w_Q^{\text{min}}$ (resp. $w_Q^{\text{max}}$) denote the minimal (resp. maximal)
representative of $wW_Q$. Let $\pi :G/B\to G/Q$ be the canonical projection. Then it can be
easily seen that
$$\pi\bigr|_{X_B(w^{\text{max}})}:X_B(w_Q^{\text{max}})\to X_Q(w)$$ is a fibration with fiber
 $\simeq Q/B$, while $$\pi\bigr|_{X_B(w^{\text{min}})}:X_B(w_Q^{\text{min}})\to X_Q(w)$$ is
birational. In particular, we have $\dim X_Q(w)=\dim X_B(w_Q^{min})$.


\subsection{The big cell and the opposite big cell}\label{12.5}
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{B}). 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 $X(w)\subset G/Q$, $B^-e_{\text{id}}\cap X(w)$ is called the {\em opposite
 cell} in $X(w)$ (by abuse of language). In general, it is not a cell (except for $w=w_0$). It
is a nonempty affine open subvariety of $X(w)$, and a closed subvariety of the affine space
$B^-e_{\text{id}}$.

\subsection{Equations defining  a Schubert variety}\label{12.6}
Let $L$ be an ample line bundle on $G/Q$. Consider the projective embedding
$G/Q\hookrightarrow \text{Proj}(H^0(G/Q,L))$.
We recall (cf. \cite{R2}) 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 recall (cf.  \cite{KR}) 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 $1$.

%%%%%%%%%%%%%%%%%%%%%%%%%
%%% SL(n)/B

\section{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 $P_{\widehat{\a}}$ (resp.
$W_{P_{\widehat{\a}}}^{\text{min}}$) 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
$$W_Q^{\text{min}}=\{(a_1\dots a_n)\in W\mid a_1<\dots <a_d,\quad a_{d+1}<\dots <a_n\}.$$
Thus $W_Q^{\text{min}}$ may be identified with
$$I_{d,n}:=\{\underline{i}=(i_1,\dots,i_d)\mid 1\le i_1<\dots <i_d\le n\}.$$

Given $\underline{i},$, $\underline{j}\in I_{d,n}$, let $X_{\underline{i}}$, $X_{\underline{j}}$
be the associated Schubert varieties in $G/P_d$. We define
$\ui\ge\uj\iff X_{\ui}\supseteq X_{\uj}$ (in other words, the partial order $\ge$ on $I_{d,n}$
is induced by the Chevalley-Bruhat order on the set of Schubert varieties, via the bijection
in \S\ref{12.4}). In particular, we have
$$\ui\ge\uj\iff i_t\ge j_t\text{ for all } 1\le t\le d.$$


\subsection{The Chevalley-Bruhat order on $\cal{S}_n$}
For $w_1$, $w_2\in W$, we have
$$X(w_1)\subset X(w_2)\iff \pi_d(X(w_1))\subset \pi_d(X(w_2)),\text{ for all } 1\le d\le n-1,$$
where $\pi_d$ is the canonical projection $G/B\to G/P_d$. Hence we obtain that for
 $(a_1\dots a_n)$, $(b_1\dots b_n)\in\cal{S}_n$,
$$(a_1\dots a_n)\ge (b_1\dots b_n)\iff (a_1\dots a_d)\uparrow\ge (b_1\dots b_d)\uparrow,
\text{ for all } 1\le d\le n-1$$
(here, for  a $d$-tuple $(t_1\dots t_d)$ of distinct integers,
$(t_1\dots t_d)\uparrow$ denotes the ordered $d$-tuple obtained from $\{t_1,\dots, t_d\}$ by
arranging its elements in ascending order).

\subsection{The partially ordered set $I_{a_1,\dots,a_k}$}
Let $Q$ be a parabolic subgroup in $SL(n)$. Let  $1\le a_1<\dots <a_k\le n$, such that
$S_Q=S\setminus\{\a_{a_1},\dots,\a_{a_k}\}$ (we follow \cite{Bou} for indexing the simple roots).
Then $Q=P_{a_1}\cap\dots\cap P_{a_k}$, and
$W_Q=\cal{S}_{a_1}\times\cal{S}_{a_2-a_1}\times\dots\times\cal{S}_{n-a_k}$.
Let
$$I_{a_1,\dots,a_k}=\{(\ui_1,\dots,\ui_k)\in I_{a_1,n}\times\dots\times I_{a_k,n}\mid
\ui_t\subset \ui_{t+1}\text{ for all } 1\le t\le k-1\}.$$
Then it is easily seen that $W_Q^{\text{min}}$ may be identified with $I_{a_1,\dots, a_k}$.

The partial order on the set of Schubert varieties in $G/Q$ (given by inclusion)  induces a
partial  order $\ge$ on $I_{a_1,\dots,a_k}$, namely, for $\bold{i}=(\ui_1,\dots,\ui_k)$,
$\bold{j}=(\uj_1,\dots,\uj_k)\in I_{a_1,\dots,a_k}$, $\bold{i}\ge\bold{j}\iff\ui_t\ge\uj_t$
for all  $1\le t\le k$.

\subsection{The minimal and maximal representatives as  permutations}
Let $w\in W_Q$, and let $\bold{i}=(\ui_1,\dots,\ui_k)$ be the  element in $I_{a_1,\dots,a_k}$
which corresponds to
$w_Q^{\text{min}}$.
As a permutation, the element  $w_Q^{\text{min}}$ is given by $\ui_1$, followed by $\ui_2
\setminus\ui_1$ arranged in ascending order, and so on, ending with
$\{1,\dots,n\}\setminus \ui_k$ arranged in ascending order. Similarly, as a permutation, the element
$w_Q^{\text{max}}$ is given by $\ui_1$ arranged in descending order, followed by
$\ui_2\setminus\ui_1$ arranged in descending order, etc..

\subsection{The opposite big cell in $G/Q$}\label{13.4}
Let $Q=\cap_{t=1}^kP_{a_t}$. Let $a=n-a_k$, and $Q$ be the parabolic
subgroup consisting of all the elements of $G$ of the form
$$
\begin{pmatrix}
A_1&\ast &\ast &\cdots&\ast&\ast\\
0 &A_2&\ast &\cdots&\ast &\ast\\
\vdots&\vdots&\vdots&&\vdots&\vdots\\
0 &0 &0 &\cdots&A_k&\ast\\
0 &0 &0 &\cdots&0& A
\end{pmatrix},
$$
where $A_t$ is a matrix of size $c_t\times c_t$,   $c_t=a_t-a_{t-1}$, $1\le t\le k$ (here
$a_0=0$),
$A$ is a matrix of size
$a\times a$, and
$x_{ml}=0$, $m>a_t$, $l\le a_t$, $1\le t\le k$. 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  $B^-e_{\text{id},Q}$ 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{12.4}), 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{13.4}
$$Z_d=\left\{
\begin{pmatrix}
I_{d}&0_{d\times (n-d)}\\
A_{(n-d)\times d}&I_{n-d}
\end{pmatrix}\in G\right\}$$
As in \S \ref{13.4}, 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 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 $G_{d,n}$}
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{12.6}, 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{13.10}
Let $Q$ be as in \S \ref{13.4}. Let $X_Q(w)\subset G/Q$. Denoting $R$, $R_w$ as in \S \ref{12.6},
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{sch}, 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 $X_Q(w)$}
Let us denote $B^-e_{\text{id},Q}\cap X_Q(w)$ by just $A_w$. Then as in \S \ref{12.5}, 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 $A_w$ as a closed subvariety of $O^-$. In view of \S \ref{13.10},
we obtain that the ideal defining $A_w$ in $O^-$ is generated by
$$\{p_{\ui}\mid\ui\in I_{d,n},d\in\{a_1,\dots,a_k\},\ w^{(d)}\not\ge\ui\}.$$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Two lemmas related to the evaluation of Pl\" ucker coordinates on the opposite
cell of a Schubert variety in $G/Q$}\label{4}

 Let $G=SL(n)$, $1\le a_1<\dots <a_h\le n$, $Q=P_{a_1}\cap\dots\cap P_{a_h}$. Let
$O^-$ be the opposite big cell in $G/Q$. Let $X=(x_{ba})$, $1\le b,a\le n$ be a generic
$n\times n$ matrix and $H$  the one-sided ladder in $X$ defined by the outside corners
$(a_i+1,a_i)$, $1\le i\le h$.  Clearly,
$\Bbb{A}(H)\simeq O^-$. Let
$X^-=(x_{ba}^-)$,
$1\le b,a\le n$, where
$$x_{ba}^-=
\begin{cases}x_{ba},&\text{if }(b,a)\in H\\
1,&\text{if }b=a\\
0,&\text{otherwise}.
\end{cases}
$$
Note that, given $\t\in W^{a_i}$, for some $i$, $1\le i\le h$, the function $p_\t\res_{O^-}$
represents the determinant of the $a_i\times a_i$ submatrix $T$ of $X^-$ whose row indices are
$\{\t(1),\dots,\t(a_i)\}$, and column indices are $\{1,\dots,a_i\}$.

Let $H_i=\{ x_{ba}\mid a_i+1\le b\le n,1\le a\le a_i\}$, $1\le i\le h$.

\begin{lem}\label{one}
Let $M$ be a $t\times t$ matrix contained in $H_i$, for some $i$, $1\le i\le h$, with row
indices
$r_1<\dots <r_t$. Then $\det M$ belongs to the ideal of $k[H]$ generated by
$p_\f\res_{O^-}$, with $\f\in W^{a_i}$ such that
$\{\f(1),\dots,\f(a_i)\}\cap\{a_i+1,\dots,n\}=\{ r_1,\dots,r_t\}$.
\end{lem}
\begin{pf} Denote by $c_1<\dots <c_t$ the column indices of $M$.
Let $\t=(\{1,\dots,a_i\}\setminus\{c_1,\dots,c_t\})\cup\{r_1,\dots,r_t\}$. Then $\t\in W^{a_i}$,
and $p_\t\res_{O^-}=\det T$, where $T$ is the $a_i\times a_i$ submatrix of $X^-$ with row
indices $\{\t(1),\dots,\t(a_i)\}$ and column indices $\{1,\dots,a_i\}$. Using Laplace expansion
with respect to the last $t$ rows of $T$, we obtain
\begin{equation*}
\det T=\sum \pm \det N_{c_1',\dots,c_t'}\det M_{c_1',\dots,c_t'},\tag{$*$}
\end{equation*}
the sum being taken over all subsets with $t$
elements $\{c_1',\dots,c_t'\}$ of $\{1,\dots,a_i\}$ , where $ N_{c_1',\dots,c_t'}$ is the
$(a_i-t)\times (a_i-t)$ submatrix of $X^-$ with row indices
$\{1,\dots,a_i\}\setminus\{c_1,\dots,c_t\}$ and column indices
$\{1,\dots,a_i\}\setminus\{c_1',\dots,c_t'\}$, and $M_{c_1',\dots,c_t'}$ is the $t\times t$
submatrix of $X^-$ with row indices $\{r_1,\dots,r_t\}$ and column indices
$\{c_1',\dots,c_t'\}$. Note that $M_{c_1,\dots,c_t}=M$, and $N_{c_1,\dots,c_t}$ is a lower
triangular matrix, with all diagonal entries equal to $1$, and hence $\det M$ appears in $(*)$,
and its coefficient is $\pm 1$.
 Also note that $N_{c_1',\dots,c_t'}$ is obtained from $N_{c_1,\dots,c_t}$ by
replacing the columns with indices $c_1',\dots,c_t'$ by the columns with indices
$c_1,\dots,c_t$.

Let $\ge$ denote the partial order on $I_{t,a_i}$ as in \S \ref{idn}, namely $(d_1,\dots,d_t)\ge
(c_1,\dots,c_t)$ if $d_j\ge c_j$ for all $1\le j\le t$. We prove the  lemma by decreasing
induction with respect to the order $\ge$ on the
$t$-tuple $(c_1,\dots,c_t)$ consisting of the column indices of $M$.

If $c_j>a_{i-1}$ for all $1\le j\le t$, then for $\{c_1',\dots,c_t'\}\ne\{c_1,\dots,c_t\}$ we
have
$\det N_{c_1',\dots,c_t'}=0$, since at least one of $c_1,\dots,c_t$ is an index for a column in
$N_{c_1',\dots,c_t'}$, and all entries of this column are $0$. Thus, in this case $(*)$ reduces
to $\det T=\pm\det M$, i.e. $\det M=\pm p_\t\res_{O^-}$, with $\t\in W^{a_i}$ such that
$\{\t(1),\dots,\t(a_i)\}\cap \{a_i+1,\dots,n\}=\{r_1,\dots,r_t\}$.

Assume now that the assertion is true for all matrices with row indices
$r_1<\dots<r_t$ and column indices $d_1<\dots <d_t$  such that
$(d_1,\dots,d_t)>(c_1,\dots,c_t)$ (i.e. such that
$d_j\ge c_j$ for all $1\le j\le t$ and $(d_1,\dots,d_t)\ne (c_1,\dots,c_t)$). We shall now
prove it for the matrix $M$ with row indices  $r_1<\dots <r_t$ and column indices
$c_1<\dots<c_t$. Consider a typical
$N_{c_1',\dots,c_t'}$ in $(*)$. If there exists a $j$ such that $c_j'<c_j$, then the column with
index
$c_j$ is replacing the column with index $c_j'$ while obtaining $N_{c_1',\dots,c_t'}$ from
$N_{c_1,\dots,c_t}$; hence
$N_{c_1',\dots,c_t'}$ is still lower triangular, but the diagonal entry in the column with index
$c_j$ is $0$, which implies that $\det N_{c_1',\dots,c_t'}=0$. Consequently we
obtain
$$\det T=\pm\det M +\sum\pm\det N_{c_1',\dots,c_t'}\det M_{c_1',\dots,c_t'},$$
and hence
$$\det M =\pm p_\t\res_{O^-}+\sum\pm\det N_{c_1',\dots,c_t'}\det M_{c_1',\dots,c_t'},$$
the sum being taken over all  $(c_1,\dots c_t')\in I_{t,a_i}$ such that
$(c_1',\dots,c_t')>(c_1,\dots,c_t)$. The required result now follows by induction hypothesis.
\end{pf}

\begin{lem}\label{two}
Let $1\le t\le a\le a_i$,  $1\le s\le n$ and $\t\in W^{a_i}$ such that $\t(a-t+1)\ge s$.
 Then $p_\t\res_{O^-}$ belongs to the ideal of $k[H]$ generated by $t$-minors in $X^-$ with
row indices  $\ge s$ and column indices  $\le a$.
\end{lem}
\begin{pf}

Let $T$ be the $a_i\times a_i$ submatrix of $X^-$ with row indices $\{\t(1),\dots,\t(a_i)\}$ and
column indices $\{1,\dots,a_i\}$. Then $p_\t\res_{O^-}=\det T$. Using Laplace expansion with
respect to the first $a$ columns, we have $\det T= \sum_p \det A_p\det B_p$, where $A_p$ (resp.
$B_p$) is an $a\times a$ (resp. $(a_i-a)\times (a_i-a)$) matrix. Clearly, all the column indices
of a typical $A_p$ are $\le a$, and since $\t(a-t+1)\ge s$, at least $t$ of the row indices of
$A_p$ are
$\ge s$. Using Laplace expansion for $A_p$ with respect to $t$ rows with indices $\ge s$, we
obtain
$\det A_p=\sum_q \det C_q\det D_q$, where $C_q$ (resp. $D_q$) is a $t\times t$ (resp.
$(a-t)\times (a-t)$) matrix, the row indices of $C_q$ are $\ge s$, and column indices of
$C_q$ are $\le a$. The required result follows from this.
\end{pf}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Ladder determinantal varieties and Schubert varieties}\label{5}

Let $L\subset X$ be an one-sided ladder  in $X$ defined by the outside corners $(b_i,a_i)$, $1\le
i\le h$, $1\le b_1<\dots<b_h< n$, $1< a_1<\dots<a_h\le n$ where $X$ is a generic
$n\times n$  matrix $X=(x_{ba})$, with $n$ large enough such that $L$ is situated below the main
diagonal, i.e.
$b_i\ge a_i+1$, $1\le i\le h$. Let $G=SL(n)$, $Q=P_{a_1}\cap\dots\cap P_{a_h}$. Let
$O^-$ be the opposite big cell in $G/Q$. Let $H$ be the one-sided ladder defined by the outside
corners $(a_i+1,a_i)$, $1\le i\le h$.
Let $\us,\ut\in\Bbb{Z}_+^l$ satisfying (L1), (L2) and (L3), as in Section \ref{1}, with $m=n$.
Let notations be as in Section \ref{1}. Let $Z$ be the variety in
$\Bbb{A}(H)\simeq O^-$ defined by the vanishing of the $t_i$-minors in
$L(i)$, $1\le i\le l$. Note that $Z\simeq D_{\us,\ut}(L)\times\Bbb{A}(H\setminus L)\simeq
D_{\us,\ut}(L)\times\Bbb{A}^r$, where $r=\text{dim\,} SL(n)/Q-|L|$.

 We shall now define an element $w\in W_Q^{\text{min}}$, such that  the variety $Z$
identifies with the opposite cell  in the the Schubert variety $X(w)$ in $G/Q$. We define $w\in
W_Q^{\text{min}}$ by specifying
$w^{(a_i)}\in W^{a_i}$ $1\le i\le h$, where $\pi_i(X(w))=X(w^{(a_i)})$ under the projection
$\pi_i:G/Q\to G/P_{a_i}$.

Define $w^{(a_i)}$, $1\le i\le h$, inductively, as  the (unique) maximal element in
$W^{a_i}$ such that

$(1)$ $w^{(a_i)}(a_i-t_j+1)=s_j-1$ for all $j\in\{1,\dots,l\}$ such that $s_j\ge b_i$, and
$t_j\ne t_{j-1}$ if $j>1$.

$(2)$ 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 $W_Q^{\text{min}}$.



\subsection{} Let us denote the
distinct elements in $\{t_1,\dots,t_l\}$ by $t_1=t_{i_1}>t_{i_2}>\dots >t_{i_m}=t_l$, where
$t_{i_k-1}>t_{i_k}$ for $2\le k\le m$. For
$2\le k\le m$, let $I_k=[e_{i_k},s_{i_k}-1]$, where $e_{i_k}=s_{i_k}-(t_{i_{k-1}}-t_{i_k})$. Let
$I_1=[b_1-(a_1-t_1+1),b_1-1]$, $I_{m+1}=[n-t_l+2,n]$ (here for $p,q\in\Bbb{Z}$, $p<q$, $[p,q]$
denotes the set $\{p,p+1,\dots,q\}$).

\begin{rem}\label{5.2}
Fix $j$, $1\le j\le h$. Let $b_j=s_c$, for some $c$, $1\le c\le l$. Let $i_k$ be the smallest
such that $s_{i_k}> b_j$. Then in $w^{(a_j)}$, $b_j-1$ appears at the $(a_i-t_c+1)$-th place,
and is followed by the blocks $I_k,I_{k+1},\dots,I_{m+1}$.
\end{rem}
\begin{lem}\label{5.3}
We have

$(1)$ $w^{(a_1)}=I_1\cup I_2\cup\dots\cup I_{m+1}$.

$(2)$ $I_j\subset w^{(a_i)}$, $1\le j\le m+1$, $1\le i\le h$.

$(3)$ The entries in $w^{(a_i)}\setminus w^{(a_{i-1})}$ are $\le b_i-1$, $1\le i\le h$.


\end{lem}
All the assertions are clear from the definition of $w$.
\begin{lem}\label{5.4}
Fix $j$, $1\le j\le l$.

$(1)$ We have $s_j\not\in I_r$, $1\le r\le m+1$.

$(2)$  Let $t_j=t_{i_{k-1}}$, for some $k$, $2\le k\le m+1$. Then $e_{i_k}>s_j$

\noindent (here, $e_{i_{m+1}}=n-t_l+2$).
\end{lem}
\begin{pf}
If $k=m+1$, then $t_j=t_l$, $e_{i_{m+1}}=n-t_l+2>s_j$ (since $t_j<n-s_j+1$). Further, $s_j\ge
s_{i_m}$, and hence $s_j\not\in I_r$ for any $1\le r\le m+1$. Let then $k\le m$. We have
 $s_{i_k}-s_j>t_j-t_{i_k}=t_{i_{k-1}}-t_{i_{k}}$. This implies $e_{i_k}>s_j$. Hence $s_j\not\in
I_r$, $r\ge k$. Also the fact that $s_j\ge s_{i_{k-1}}$ implies that $s_j\not\in I_r$, $r\le
k-1$.
\end{pf}

\begin{rem}
Consider a block of consecutive integers in $w^{(a_i)}$, $1\le i\le h$, ending with $s_j-1$ at
the $(a_k-t_j+1)$-th place, for some $k\le i$. Then either $k=i$, or $k=j^*$; in other
words, $k$ is the largest integer in $\{1,\dots,i\}$ such that $b_k\le s_i$. In particular, if
$j^*\le i$, then $k=j^*$.
\end{rem}


\begin{thm}\label{w}
The  variety $Z$ $(=D_{\us,\ut}(L)\times\Bbb{A}^r)$ identifies with the opposite cell
in $X(w)$, i.e. $Z=X(w)\cap O^-$ (scheme theoretically).
\end{thm}
\begin{pf}
Let $f=\det M$ , where $M$ is a $t_i\times t_i$ matrix contained in $L(i)$ for some $1\le i\le
l$, be a generator of $I(Z)$. Let $k=i^*$, i.e. $k$ is  the largest integer such that $b_k\le
s_i$. Then
$M$ is contained in $H_k$. By Lemma \ref{one}, $f$ can be written in the form $f=\sum g_\f
p_\f\res_{O^-}$, with $\f\in W^{a_k}$ such that
$\{\f(1),\dots,\f(a_k)\}\cap\{a_k+1,\dots,n\}=\{ r_1,\dots,r_{t_i}\}$, and $g_\f\in k[H]$ (here
$r_1,\dots,r_{t_i}$ are the row indices of $M$). In particular, we have $\f(a_k-t_i+1)=r_1$.
Since $M$ is contained in $L(i)$, we have
$r_1\ge s_i$, and hence $\f(a_k-t_i+1)\ge s_i$. We have
$w^{(a_k)}(a_k-t_i+1)=s_i-1$, and hence
$\f(a_k-t_i+1)>w^{(a_k)}(a_k-t_i+1)$. This shows that $\f\not\le w^{(a_k)}$, and therefore
$p_\f\in I(X(w)\cap O^-)$. Thus $f\in I(X(w)\cap O^-)$.

Let now $g$ be a generator of the ideal $I(X(w)\cap O^-)$, i.e. $g=p_\t\res_{O^-}$, with
$\t\in W^{a_i}$ for some $i$, $1\le i\le h$, such that $\t\not\le w^{(a_i)}$. Since $w^{(a_i)}$
consists of several blocks of consecutive integers ending with  $s_m-1$ at the $(a_k-t_m+1)$-th
place, for some  $m\in\{1,\dots,l\}$, where $k\in\{1,\dots,i\}$ is the largest such that
$b_k\le s_m$, and a last block ending with $n$ at the $a_i$-th place, it follows that
 $\t(a_k-t_m+1)\ge s_m$ for some $m$, where $k\in\{1,\dots,i\}$ is the largest such that  $s_m\ge
b_k$. Using Lemma
\ref{two}, we deduce that $p_\t\res_{O^-}$ belongs to the ideal of $k[H]$ generated by
$t_m$-minors in $L$ with row indices $\ge s_m$, and column indices $\le a_k$. Thus
$p_\t\res_{O^-}$ belongs to the ideal generated by $t_m$-minors contained in $L(m)$, which shows
that $g\in I(Z)$.
\end{pf}

Since the  Schubert varieties are irreducible, normal,
Cohen-Macaulay, and have rational singularities (cf. \cite{KR}, \cite{RR}, \cite{R1},
\cite{R2}), as a consequence of Theorem
\ref{w} we obtain
\begin{thm}\label{normal}
The variety $D_{\us,\ut}(L)$ is irreducible, normal, Cohen-Macaulay, and has rational
singularities.
\end{thm}



%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The dimension of $D_{\us,\ut}(L)$}\label{6}

Let $X=(x_{ba})$, $1\le b\le m$, $1\le a\le n$ be a
$m\times n$ matrix of indeterminates.

\subsection{The partial order among minors}
 We shall denote the  determinant of the $r\times r$ submatrix of $X$ whose row indices are
$i_1<\dots<i_r$ and column indices are $j_1<\dots<j_r$  by
$[i_1,\dots,i_r|j_1,\dots,j_r]$.
 We introduce a partial order on the set of all minors of
$X$ as follows:
$[i_1,\dots,i_r|j_1,\dots,j_r]\le [i_1',\dots,i_s'|j_1',\dots,j_s']$ if
$$r\ge s\text{ and }i_r\ge i_s',i_{r-1}\ge i_{s-1}',\dots,i_{r-s+1}\ge i_1',
\ j_1\le j_1',j_2\le j_2',\dots,j_s\le j_s'.$$

We say that an ideal $I$ of $k[X]$ is {\em cogenerated} by a given minor $M$ if $I$ is
generated by the minors in the set $\{M'\mid M'\text{ a minor of }X\text{ such that }
M'\not\ge M\}$.

\subsection{The monomial order $\prec$ and Gr\" obner bases}\label{ord}
 We introduce a total order on the variables as follows:
$$x_{m1}>x_{m2}>\dots>x_{mn}>x_{m-1\, 1}>x_{m-1\, 2}>\dots>x_{m-1\, n}>\dots>
x_{11}>x_{12}>\dots>x_{1n}.$$

This induces a total order, namely the lexocographic order, on the set of monomials in
$k[X]=k[x_{11},\dots,x_{mn}]$, denoted by
$\prec$. The largest monomial (with respect to $\prec$)  present in a polynomial $f\in k[X]$
is called the {\em initial term} of  $f$, and is denoted by $\text{in}(f)$.
 Note that the initial term  (with respect to $\prec$) of a minor of $X$ is equal to the product
of its elements on the skew diagonal.

Given an ideal $I\subset k[X]$, a set $G\subset I$ is called a {\em Gr\" obner basis } of $I$
(with respect to the monomial order $\prec$) if the ideal $\text{in}(I)$ generated by the initial
terms of the elements in $I$ is generated by the initial terms of the elements in $G$. Note that
a Gr\" obner basis of $I$ generates  $I$ as an ideal.


We recall the following (see \cite{HeTr})
\begin{thm}\label{HT-GB}
Let $M=[i_1,\dots,i_r|j_1,\dots,j_r]$ be a minor of $X$, and $I$ the ideal of $k[X]$
cogenerated by $M$. For $1\le t\le r+1$, let $G_t$ be the set of all $t$-minors
$[i_1',\dots,i_r'|j_1',\dots,j_r']$  satisfying the conditions

\begin{alignat}{1}
          i_t'\le i_r,i_{t-1}'\le i_{r-1},\dots, i_2'&\le i_{r-t+2},\tag{1}\\
                j_{t-1}'\ge j_{t-1},\dots,j_2'&\ge j_2,j_1'\ge j_1\notag\\
    \text{if }t\le r,\text{ then } i_1'>i_{r-t+1}\text{ or } j_t'<j_t.\tag{2}
\end{alignat}
Then the set $G=\cup_{i=1}^{r+1}G_i$ is a Gr\" obner basis for the ideal $I$ with respect to
the monomial order $\prec$.
\end{thm}

\subsection{The ideal $I_{\us,\ut}(X)$ and the set $\cal{G}$}
The matrix $X$ can be viewed as an one-side ladder with a unique outside corner, namely $(1,n)$.
Let $\us,\ut\in\Bbb{Z}_+^l$ satisfying (L1), as in Section \ref{1} (where $b_1=1$).  Let
$I_{\us,\ut}(X)$ be as in Section \ref{1}, for $L=X$. In other words, $I_{\us,\ut}(X)$ is
the ideal of $k[X]$ generated by the $t_i$-minors in $X_i=\{x_{ba}\mid s_i\le b\le m\}$,
$1\le i\le l$. For $1\le i<l$, let  $\cal{G}_i$ be the set
consisting of the $t_i$ minors in
$X_i$ such that the number of   rows   contained in
$X_j$  is less than $t_j$, for all $j$, $i<j\le l$, and
$\cal{G}_l$  the set consisting of the $t_l$ minors in $X_l$. Let
$\cal{G}=\cup_{i=1}^l\cal{G}_i$. Clearly, $I_{\us,\ut}(X)$ is generated by $\cal{G}$.


\begin{prop}\label{GB}
Let $\us,\ut\in\Bbb{Z}_+^l$ satisfy (L1), and let $\cal{G}$ be as above. Then
$\cal{G}$  is a Gr\" obner basis of $I_{\us,\ut}(X)$, with respect to the monomial order
$\prec$.
\end{prop}
\begin{pf}

 Let
$M_{\us,\ut}$ be the minor of $X$ of size $t_1-1$ given by the last $t_i-t_{i+1}$ rows of
$X_i\setminus X_{i+1}$, $1\le i< l$ and the last $t_l-1$ rows of $X_l$, and the first $t_1-1$
columns of $X$.
 First we show that the ideal $I_{\us,\ut}(X)$ is
cogenerated by $M_{\us,\ut}$. Let
$M_{\us,\ut}=[i_1,\dots,i_{t_1-1}|j_1,\dots,j_{t_1-1}]$, and $\cal{F}=\{M'\mid M'\not\ge
M_{\us,\ut}\}$. Note that $M'\ge M_{\us,\ut}$ if and only if $M'$ contains at most
$t_i-1$ rows in $X_i$,
$1\le i\le l$. Thus $\cal{F}=\cup_{i=1}^l\cal{F}_i$, where $\cal{F}_i=\{M'\mid M'\text{
contains at least  }t_i\text{ rows in }X_i\}$. Now $\cal{F}_i\subset I_{\us,\ut}(X)$, $1\le
i\le l$, and hence
$\langle\cal{F}\rangle\subset I_{\us,\ut}(X)$. On the other hand, $\cal{G}_i\subset\cal{F}_i$,
$1\le i\le l$, and
$\langle\cal{G}\rangle=I_{\us,\ut}(X)$. Therefore $I_{\us,\ut}(X)=\langle\cal{F}\rangle$, i.e.
$I_{\us,\ut}(X)$ is cogenerated by $M_{\us,\ut}$.

The inequalities regarding $j$'s in condition $(1)$  of Theorem \ref{HT-GB} are
redundant in our case (since $j_t=t$, $1\le t\le t_1-1$); also, condition $(2)$ reduces to
the condition that if $t\le r$, then $i_1'>i_{r-t+1}$ (since $j_t=t$, and hence $j_t'\ge j_t$
for all $t$, $1\le t\le t_1-1$). Therefore, in our case the conditions $(1)$ and $(2)$ are
equivalent to
$$ i_t'\le i_{t_1-1},i_{t-1}'\le i_{t_1-2},\dots, i_2'\le i_{t_1-t+1},
    \text{ and if }t\le t_1-1,\text{ then } i_1'>i_{t_1-t}.$$
Note that   the above inequalities
imply $i_{t_1-t+1}\ge i_2'>i_1'>i_{t_1-t}$; now, if $t\not\in\{t_1,\dots,t_l\}$, then this is
not possible, since
$i_{t_1-t+1}=i_{t_1-t}+1$. Hence $G_t=\emptyset$ for
$t\in\{1,\dots,t_1\}\setminus\{t_1,\dots,t_l\}$. It is easily seen that $G_{t_i}=\cal{G}_i$, for
$1\le i\le l$. Therefore Theorem \ref{HT-GB} implies that $\cal{G}$ is a Gr\" obner basis for
$I_{\us,\ut}(X)$ with respect to the  monomial order $\prec$.
\end{pf}


We recall the following well-known
\begin{lem}\label{above}
Let $k[X]$ be the polynomial ring in the set of indeterminates $X$, $I$ an ideal of $k[X]$, and
$G$ a Gr\" obner basis of $I$ with respect to a certain monomial order. Let $L\subset X$
such that
$$\text{if }f\in G\text{ and }\text{in}(f)\in k[L],\text{ then }f\in k[L].$$
Then the set $G\cap k[L]$ is a Gr\" obner basis of the ideal $I\cap k[L]$.
\end{lem}
\begin{pf}
Let $g\in I\cap k[L]$. Since $G$ is a Gr\" obner basis of $I$, there exists $f\in G$
such that $\text{in}(g)=\langle\text{in}(f)\rangle$. Since
$g\in k[L]$, we have  $\text{in}(g)\in k[L]$, and hence $\text{in}(f)\in k[L]$. By hypothesis,
$f\in k[L]$, and hence $f\in G\cap k[L]$. Therefore, the initial terms of the elements of $G\cap
k[L]$ generate the ideal
$\text{in}(I\cap k[L])$.
\end{pf}

As a
direct consequence, we obtain the following
\begin{prop}\label{GBL}
Let $L\subset X$ be an one-sided ladder and $\us,\ut\in \Bbb{Z}_+^l$ satisfying (L1). Then
$I_{\us,\ut}(L)=I_{\us,\ut}(X)\cap k[L]$, and $\cal{G}_L=\cal{G}\cap k[L]$ is a Gr\"
obner basis of $I_{\us,\ut}(L)$ with respect to the monomial order $\prec$.
\end{prop}
\begin{pf}
By Proposition \ref{GB}, $\cal{G}$ is a Gr\" obner basis of $I_{\us,\ut}(X)$. By Lemma
\ref{above}, $\cal{G}_L$ is a Gr\" obner basis of the ideal $I_{\us,\ut}(X)\cap k[L]$. On the
other hand it easily seen that $\cal{G}_L$ generates $I_{\us,\ut}(L)$, and the result follows.
\end{pf}



\subsection{The set $\cal{C}$}\label{c}
 We construct
a set $\cal{C}_{\us,\ut}(X)\subset X$ as follows. Let $\cal{C}_l(X)$ be the submatrix obtained
from
$X_l$ by  deleting the first $t_l-1$ columns and the last $t_l-1$ rows. For $i<l$, let
$\cal{C}_i(X)$ be the  matrix obtained from $\tilde{X}_i=X_i\setminus X_{i+1}$ by deleting the
first $t_i-1$ columns and the last $t_i-t_{i+1}$ rows. Now let
$\cal{C}_{\us,\ut}(X)=\cup_{i=1}^l\cal{C}_i(X)$.

For an one-sided ladder $L\subset X$, and $\us,\ut\in\Bbb{Z}_+^l$ satisfying (L1), we define
$\cal{C}_i(L)=\cal{C}_i(X)\cap L$,
$\cal{C}_{\us,\ut}(L)=\cal{C}_{\us,\ut}(X)\cap L$.

Note that in a solid minor in $\cal{G}_L$ (i.e. a minor with consecutive row indices and
consecutive column indices), the smallest (for the order in \ref{ord}) element belongs to
$\cal{C}_{\us,\ut}(L)$, and conversely, an element $\a\in\cal{C}_{\us,\ut}(L)$ determines
uniquely a solid minor in $\cal{G}_L$ having $\a$ as the smallest element. Hence the number of
elements in $\cal{C}_{\us,\ut}(L)$ is equal to the number of solid minors in the set
$\cal{G}_L$.

The following is a generalization of Proposition 8 in \cite{GS}.
\begin{prop}\label{dim}
Let  $L\subset X$ an   one-sided ladder, and $\us,\ut\in\Bbb{Z}_+^l$ satisfying (L1). Then
$$\text{codim\,}_{\Bbb{A}(L)}D_{\us,\ut}(L)=|\cal{C}_{\us,\ut}(L)|.$$
\end{prop}
\begin{pf}
By Proposition \ref{GBL}, the ideal $I_{\us,\ut}(L)$ and the ideal $J_{\us,\ut}(L)$ of its
initial terms determine graded
quotient rings of $k[L]$ having the same Hilbert series, and hence  the codimension of the
variety
$D_{\us,\ut}(L)$ is equal to the height  of the monomial ideal $J_{\us,\ut}(L)$. In general,
the height  of a monomial ideal $J$ in a polynomial ring $k[x_1,\dots,x_N]$ is equal to the
minimal cardinality of a set $\cal{C}\subset \{x_1,\dots,x_N\}$ of variables such that
\begin{equation}
\begin{aligned}
&\text{each monomial in a set of monomial generators for }J\\
&\text{contains a variable from }\cal{C}.
\end{aligned}\tag{$\ast$}
\end{equation}
Let $J=J_{\us,\ut}(L)$  and
$\cal{C}=\cal{C}_{\us,\ut}(L)$. Then it is easy to see that $\cal{C}$ satisfies ($\ast$), the
set of monomial generators being the set of the initial terms of all the $t_i$-minors in $L_i$,
$1\le i\le l$. Let us denote
$\D_k=\{x_{ba}\in L\mid b+a=k+1\}$, $k\ge 1$. Then $L=\dot{\cup}_{k\ge 1}\D_k$, and
$\cal{C}=\dot{\cup}_{k\ge 1}(\cal{C}\cap \D_k)$.

Let now $\cal{C}'\subset\{x_{ba}\mid x_{ba}\in L\}$ be a set  such that
$|\cal{C}'|<|\cal{C}|$. Then there exists a $k$ such that $|\cal{C}'\cap \D_k|<|\cal{C}\cap
\D_k|$ (in particular $\cal{C}\cap\D_k\ne\emptyset$).
Let  $i\in\{1,\dots,l\}$ be the largest such that  $\D_k\cap\cal{C}\subset L_i$. Then
$$
|\cal{C}'\cap (\D_k\cap L_{i})|\le|\cal{C}'\cap \D_k|<|\cal{C}\cap \D_k|=|\D_k\cap L_{i}|
-(t_{i}-1).
$$
Therefore there exist $t_{i}$ distinct variables in $(\D_k\cap L_{i})\setminus\cal{C}'$.
Thus the initial term of the $t_{i}$-minor in $L_{i}$ having these elements on the
skew diagonal does not contain any variable in $\cal{C}'$, and hence $\cal{C}'$ does not
satisfy ($\ast$).

Therefore $\cal{C}$ is a set of minimal cardinality among the sets satisfying ($\ast$), and the
required result follows.
\end{pf}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The singular locus of $D_{\us,\ut}(L)$}\label{7}

Let $X=(x_{ba})$, $1\le b< m$, $1< a\le n$ be a $m\times n$ matrix of indeterminates.
Let   $L\subset X$ be an one-sided ladder defined by the outside
corners $\o_i=x_{b_ia_i}$, $1\le i\le h$,  $1\le b_1<\dots<b_h\le m$, $1\le a_1<\dots
<a_h\le n$. Let $\us,\,\ut\in\Bbb{Z}_+^l$ satisfy (L1), (L2) and (L3) of Section \ref{1}.
We preserve the notations of Section \ref{1}. Let   $V=D_{\us,\ut}(L)$,
$\cal{C}=\cal{C}_{\us,\ut}(L)$.


 For $1\le i\le l$, let $V_i\subset\Bbb{A}(L)$ be the variety
defined by the vanishing of the $t_j$-minors in $L(j)$, with $j\in\{1,\dots,l\}\setminus\{i\}$,
and the $(t_i-1)$-minors in $L(i)$.


\begin{thm}\label{sing}
With notations as above, we have
$$\text{Sing\,}V=\cup_{i=1}^lV_i.$$
\end{thm}
\begin{pf}

For  simplicity of notation, we identify the variable $x_{ba}$ with the element
$(b,a)$.

First, we prove that $V_i\subset\text{Sing\,}V$, for all $1\le i\le l$. Let $x\in V_i$ for some
$1\le i\le l$. Let $\cal{J}$ be the jacobian matrix associated to the variety
$V\subset\Bbb{A}(L)$, evaluated at
$x$. Then the rows of
$\cal{J}$ are indexed by $t_j$-minors in $L(j)$, $1\le j\le l$, and the columns are indexed by
the elements
$\a\in L$. The
$(M,\a)$-th  entry in $\cal{J}$ is equal to $\pm (\det M')(x)$, where $M'$ is the matrix
obtained from $M$ by  deleting the row and the column containing $\a$, if $\a$ appears in $M$,
and $0$ otherwise.

We distinguish two cases.

(I) $s_i\in\{b_1,\dots,b_h\}$

Let $s_i=b_j$, for some $1\le j\le h$. It is easily seen that
$$\o_j\in\cal{C}_{\us,\ut}(L)$$
(since $s_{i+1}-s_i>t_i-t_{i+1}$ and $a_j\ge t_i$).
Now consider the one-sided ladder  $L'$ obtained from $L$ by deleting the element $\o_j$,
i.e. the one-sided ladder defined by the outside corners
\begin{align}
&\o_1=(b_1,a_1),\dots,\o_{j-1}=(b_{j-1},a_{j-1}),\o_{j^-}=(b_j,a_j-1),\notag\\
&\o_{j^+}=(b_j+1,a_j),\o_{j+1}=(b_{j+1},a_{j+1}),\dots,\o_l=(b_l,a_l),\notag
\end{align}
where $\o_{j^-}$ is present only if $a_j-1>a_{j-1}$, and $\o_{j^+}$ is present only if
$b_j+1<b_{j+1}$.

Since $x\in V_i$, a row of $\cal{J}$ indexed by a $t_i$-minor involving $\o_j=x_{b_ja_j}$ is $0$.
Also, the column of $\cal{J}$ indexed by $\o_j$ is $0$.
Let $\cal{J}'$ be the matrix obtained from $\cal{J}$ by deleting the column indexed by $\o_j$
and the rows indexed by $t_i$-minors containing $\o_j$. Then
$$\text{rank\,} \cal{J}=\text{rank\,}\cal{J}',$$
since $\cal{J}'$ is obtained from $\cal{J}$ by deleting zero rows and columns.
 Let $x'=(x_\a)_{\a\in L'}$. Then $x'\in D_{\us,\ut}(L')$, and
$\cal{J}'$ is the jacobian matrix associated to the variety $D_{\us,\ut}(L')\subset
\Bbb{A}(L')$, evaluated at
$x'$.  Thus
$$\text{rank\,}\cal{J}'\le\text{codim\,}_{\Bbb{A}(L')} D_{\us,\ut}(L').$$
Now,  using Proposition \ref{dim} we obtain
$$\text{codim\,}_{\Bbb{A}(L')}D_{\us,\ut}(L')=|\cal{C}_{\us,\ut}(L')|=|\cal{C}_{\us,\ut}
(L)\setminus\{\o_j\}| <|\cal{C}_{\us,\ut} (L)|=\text{codim\,}_{\Bbb{A}(L)}D_{\us,\ut}(L).$$
Hence
$\text{rank\,}\cal{J}'<\text{codim\,}_{\Bbb{A}(L)}V$,
which implies
$\text{rank\,}\cal{J}<\text{codim\,}_{\Bbb{A}(L)}V$,
i.e. $x\in \text{Sing\,}V$.

(II) $s_i\not\in\{b_1,\dots,b_h\}$

We have $i>1$ and $t_{i-1}>t_i$. Let $k=i^*$, i.e. $k$ is the largest integer such that
$b_k<s_i$.
 Define
$\us'=(s_1,\dots,s_{i-1},\widehat{s_i},s_{i+1},\dots,s_l)$,
$\ut'=(t_1,\dots,t_{i-1},\widehat{t_i},t_{i+1},\dots,t_l)$.
Let $\cal{C}=\cal{C}_{\us,\ut}(L)$,
$\cal{C}'=\cal{C}_{\us',\ut'}(L)$, and
$$\cal{C}=\bigcup_{j\in\{1,\dots,l\}}\cal{C}_j,\quad
\cal{C}'=\bigcup_{j\in\{1,\dots,l\}\setminus\{i\}}\cal{C}_j',$$ as defined in
\S \ref{c}. Then
$\cal{C}_j=\cal{C}_j'$ for $j\not\in \{i-1,i\}$, and
$$
\begin{aligned}
|\cal{C}|-|\cal{C}'|=&|\cal{C}_{i-1}|+|\cal{C}_{i}|-|\cal{C}_{i-1}'|=\\
&[(s_i-s_{i-1})-(t_{i-1}-t_i)][a_k-(t_{i-1}-1)]+\\
&[(s_{i+1}-s_i)-(t_i-t_{i+1})][a_k-(t_i-1)]-\\
&[(s_{i+1}-s_{i-1})-(t_{i-1}-t_{i+1})][a_k-(t_{i-1}-1)]=\\
&[(s_{i+1}-s_i)-(t_i-t_{i+1})](t_{i-1}-t_i)>0\notag
\end{aligned}
$$
(here $s_{i+1}=m+1$, $t_{i+1}=1$, if $i=l$).
Therefore $$|\cal{C}_{\us',\ut'}(L)| <|\cal{C}_{\us,\ut}(L)|.$$

Since
$x\in V_i$, a row indexed by a $t_i$-minor contained in $L(i)$ is $0$. Let $\cal{J}'$ be the
matrix obtained from
$\cal{J}$ by deleting the rows indexed by $t_i$-minors contained in $L(i)$. Then
$$\text{rank\,} \cal{J}=\text{rank\,}\cal{J}'.$$

Now, $x\in D_{\us',\ut'}(L)$, and
$\cal{J}'$ is the Jacobian matrix associated to the variety $D_{\us',\ut'}(L)\subset\Bbb{A}(L)$,
evaluated at
$x$.  Thus
$$\text{rank\,}\cal{J}'\le\text{codim\,}_{\Bbb{A}(L)} D_{\us',\ut'}(L).$$


Now,  using Proposition \ref{dim} we obtain
$$\text{codim\,}_{\Bbb{A}(L)}D_{\us',\ut'}(L)=|\cal{C}_{\us',\ut'}(L)| <|\cal{C}_{\us,\ut}(L)|=
\text{codim\,}_{\Bbb{A}(L)}D_{\us,\ut}(L).$$ Hence
$\text{rank\,}\cal{J}'<\text{codim\,}_{\Bbb{A}(L)}V$,
which implies
$\text{rank\,}\cal{J}<\text{codim\,}_{\Bbb{A}(L)}V$,
i.e. $x\in \text{Sing\,}V$.

Now we prove that $\text{Sing\,}V\subset \cup_{i=1}^l V_i$. Let $\cal{C}=\cal{C}_{\us,\ut}(L)$,
$\cal{C}=\cup_{i=1}^l\cal{C}_i$, as defined in \S \ref{c}.

We introduce a total order on the set of minors of $L$ of size $r$, with $r\ge 1$ fixed, as
follows:
$[i_1,\dots,i_r|j_1,\dots,j_r]<[i_1',\dots,i_r'|j_1',\dots,j_r']$ if there exists $1\le k\le
r$ such that
 $$
\begin{aligned}
\text{either }&i_1=i_1',\dots,i_{k-1}=i_{k-1}',i_k<i_k',\notag\\
\text{ or }&i_1=i_1',\dots,i_r=i_r',j_1=j_1',\dots,j_{k-1}=j_{k-1}',j_k<j_k'\notag
\end{aligned}
$$
(this is simply the lexicographic order on $\{i_1,\dots,i_r,j_1,\dots,j_r\}$).
Let $x\in V\setminus\cup_{i=1}^l V_i$.
 For each $1\le i\le l$, let $M_i$ be the largest $(t_i-1)$-minor in $L(i)$ such that $(\det
M_i)(x)\ne 0$. Let
$\cal{T}_l$ be the set of elements in $L_l$ not in the rows or the columns given by the rows
and the columns of
$M_l$. Clearly,
$|\cal{T}_l|=|\cal{C}_l|$. By (decreasing) induction on $i$, suppose that, for some $i$,
$1<i\le l$, the sets $\cal{T}_i,\dots,\cal{T}_l$ have been constructed, such that

$(1)_i$ $\cal{T}_j\subset L(j)$, $i\le j\le l$,

$(2)_i$ the sets $\cal{T}_i,\dots,\cal{T}_l$ are pairwise disjoint,

$(3)_i$ $|\cal{T}_j|=|\cal{C}_j|$, $i\le j\le l$,

$(4)_i$ $\cal{T}_j$ contains no elements appearing in the  rows or in the columns of $L$ given
by the rows and the columns  of $M_j$, $i\le j\le l$,

$(5)_i$ there exist $t_i-1$ rows in $L(i)$ not containing any element from
$\cal{T}_i\cup\dots\cup\cal{T}_l$.

We define the set $\cal{T}_{i-1}$ as follows. Let $r$ be the number of the rows of $M_{i-1}$
contained in $\tilde{L}(i-1)=L(i-1)\setminus L(i)$. We distinguish two cases.

(I) $t_{i-1}-t_i\ge r$

In this case $\cal{T}_{i-1}$ is obtained from $\tilde{L}(i-1)$ by deleting the rows
given by the rows  of $M_{i-1}$, and  $t_{i-1}-t_i-r$ other rows, followed by
the deletion of the $t_{i-1}-1$ columns given by the columns of $M_{i-1}$. Then properties
$(1)_{i-1}-(4)_{i-1}$ are obvious; the $t_{i-1}-t_i$ rows of
$\tilde{L}(i-1)$ which were deleted while defining $\cal{T}_{i-1}$, and the $t_i-1$ rows of
$L_(i)$ in $(5)_i$, intersested with $L(i-1)$, give
$t_{i-1}-1$ rows of $L(i-1)$ not containing any elements in
$\cal{T}_{i-1}\cup\cal{T}_i\cup\dots\cup\cal{T}_l$, so that we have $(5)_{i-1}$.

(II) $t_{i-1}-t_i<r$

In this case $\cal{T}_{i-1}$ is obtained from $\tilde{L}(i-1)$ by deleting the $r$ rows
given by the rows  of $M_{i-1}$, then adding $r-t_{i-1}+t_i$ rows from the $t_i-1$ rows of
$L(i)$ in $(5)_i$ which are not rows of $M_{i-1}$, intersected with $L(i-1)$
(this is possible, since there are $t_{i-1}-1-r$ rows of $M_{i-1}$  in $L(i)$, and hence at
least
$(t_i-1)-(t_{i-1}-1-r)=r-t_{i-1}+t_i$ rows from the $t_i-1$ rows of $L(i)$ in $(5)_i$ are not
rows of $M_{i-1}$), followed by the deletion of the $t_{i-1}-1$ columns given by the columns
of $M_{i-1}$. Again, the properties $(1)_{i-1}-(4)_{i-1}$ are obvoius; the $r$ rows of
$M_{i-1}$ which were deleted from $\tilde{L}(i-1)$, and the $(t_i-1)-(r-t_{i-1}+t_i)$ rows
from the $t_i-1$ rows in $(5)_i$ which were not used while defining
$\cal{T}_{i-1}$, intersected with $L(i-1)$,  give $t_{i-1}-1$ rows of $L(i-1)$ not containing any
elements in
$\cal{T}_{i-1}\cup\cal{T}_i\cup\dots\cup\cal{T}_l$, so that we have $(5)_{i-1}$.

 Thus, using induction, we obtain the
disjoint sets $\cal{T}_j\subset L(j)$, $1\le j\le l$, such that  $|\cal{T}_j|=|\cal{C}_j|$, and
$\cal{T}_j$ contains no elements in the rows or columns of $L$ given by the rows and
columns of $M_j$.


For $\t\in\cal{T}_i\subset\cal{T}$, $1\le i\le l$, let $M^\t$ be the $t_i$-minor
obtained from $M_i$ by adding the row and the column containing $\t$. Obviously,
$M^\t\ne M^{\t '}$ for $\t$, $\t '\in\cal{T}$, with $\t\ne\t '$.

We now take a total order on $\cal{T}$, namely $(b,a)>(b',a')$ if either $b>b'$, or $b=b'$ and
$a>a'$.

  Let us fix $\t\in\cal{T}$, say $\t\in\cal{T}_i$ for some $i$, $1\le i\le l$. Then the
$(M^\t,\t)$-th entry   in $\cal{J}$ is equal to $\pm (\det M_i)(x)$, so it is nonzero. Let now
$\s\in\cal{T}$,
$\s<\t$. If
$\s$ is not an entry of $M^\t$, then the $(M^\t,\s)$-th entry of $\cal{J}$ is equal to $0$.
Assume now that $\s$ is the $(r,s)$-th entry of $M^\t$. Then the $(M^\t,\s)$-th entry of
$\cal{J}$ is equal to $\pm (\det M')(x)$, where $M'$ is the $(t_i-1)\times (t_i-1)$ matrix
obtained from $M^\t$ by deleting the $r$-th row and the $s$-th column. Let $\t=(b,a)$,
$\s=(b',a')$. If $b'<b$, then the indices of the first $r-1$ rows of $M'$ and $M_i$ are the
same, while the index of the $r$-th row of $M'$ is $>b'$, which is the index of the $r$-th row
of $M_i$. Thus, $M'>M_i$, and by the maximality of $M_i$, we obtain
$(\det M')(x)=0$. If $b'=b$, then $a'<a$. The indices of all the rows and those of the first
$s-1$  columns are the same, while the index of the $s$-th column in $M'$ is $>a'$, which is the
index of the $s$-th column of $M_i$. Thus
$M'>M_i$, and the maximality of $M_i$ implies that $(\det M')(x)=0$. Thus, for $\s<\t$, the
$(M^\t,\s)$-th entry in $\cal{J}$ is $0$.

Let $\cal{J}'$ be the submatrix of $\cal{J}$given by the rows indexed by $M^\t$'s and the columns
indexed by $\t$'s, with $\t\in\cal{T}$. We suppose that both rows and columns of $\cal{J}'$
are indexed   by the elements in $\cal{T}$, and we arrange them increasingly, with respect to
the total order on $\cal{T}$ defined above. Then $\cal{J}'$ is upper
triangular, and all the diagonal entries are nonzero. Thus $\det\cal{J}'\ne 0$, and  this
implies that
$$\text{rank\,}\cal{J}'=|\cal{T}|=|\cal{C}|=\text{codim\,}_{\Bbb{A}(L)}D_{\us,\ut}(L).$$
Consequently
$\text{rank\,}\cal{J}=\text{codim\,}_{\Bbb{A}(L)}V$,
i.e. $x\not\in\text{Sing\,}V$.
\end{pf}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The irreducible components of $Sing\,V$ and $Sing\,X(w)$}\label{8}


We preserve the notations of Section \ref{5}.

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 $X(\te_j)$ 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 $i<r$), let
$\te_j^{(a_i)}=w^{(a_i)}\setminus\{v_i\}\cup\{s_j-1\}$.

If $s_j-1\in w^{(a_i)}$ and $u_j\not\in w^{(a_i)}$, then
$\te_j^{(a_i)}=w^{(a_i)}\setminus\{v_i\}\cup\{u_j\}$.

If $s_j-1$ and $u_j\in w^{(a_i)}$, then $\te_j^{(a_i)}=w^{(a_i)}$ (note that in this case
$i>r$).

 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 $t_{i_k}<t_j\le t_{i_{k-1}}$.

(I) If $j\not\in\{i_1,\dots,i_m\}$ (i.e. $j>1$ and $t_{j-1}=t_j$), then

for $i<r$,  $\te_j^{(a_i)}=w_j^{(a_i)}\setminus\{e_{i_k}\}\cup\{s_j-1\}$;

for $i=r$,  $\te_j^{(a_r)}=w_j^{(a_r)}\setminus\{e_{i_k}\}\cup\{u_j\}$, 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
$w^{(a_i)}\setminus\te_j^{(a_{i-1})}$.

(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{te}
The subvariety $Z_j\subset Z$ identifies with the opposite cell in $X(\te_j)$, i.e.
$Z_j=X(\te_j)\cap O^-$ (scheme theoretically).
\end{thm}
\begin{pf}
Let $f=\det M$, $M$ being either a $t_i$-minor contained in $L(i)$, $i\in\{1,\dots,h\}\setminus
\{j\}$, or a $(t_j-1)$-minor contained in $L(j)$ be a generator of $I(Z_j)$. In the former case
we have $f\in I(Z)$, and Theorem \ref{w} implies that $f\in I(X(w)\cap O^-)\subset I(X(\te_j)\cap
O^-)$. In the latter case,
$M$ is contained in
$H_k$, where $k\in\{1,\dots,h\}$ is the largest such that $b_k\le s_j$.  By Lemma
\ref{one}, $f$ can be  written in the form $f=\sum g_\f p_\f\res_{O^-}$, with $\f\in W^{a_k}$
such that
$\{\f(1),\dots,\f(a_k)\}\cap\{a_k+1,\dots,n\}=\{ r_1,\dots,r_{t_j-1}\}$, and $g_\f\in k[H]$
(here $r_1,\dots,r_{t_j-1}$ are the row indices of $M$). In particular we have
$\f(a_k-t_j+2)=r_1$. Since
$M$ is contained in $L(j)$, we deduce that $r_1\ge s_j$, and hence $\f(a_k-t_j+2)\ge s_j$. We
have
$\te_j{}^{(a_k)}(a_k-t_j+2)=s_j-1$, and hence
$\f(a_k-t_j+2)>\te_j{}^{(a_k)}(a_k-t_j+2)$. This shows that $\f\not\le \te_j{}^{(a_k)}$, and
therefore $p_\f\in I(X(\te)\cap O^-)$. Thus
$f\in I(X(\te)\cap O^-)$.

Let now  $g=p_\t\res_{O^-}$, with $\t\in W^{a_i}$ for  some $i$, $1\le i\le h$, such that
$\t\not\le
\te^{(a_i)}$, be a generator of the ideal $I(X(\te_j)\cap O^-)$.
 Since $\te_j{}^{(a_i)}$
consists of several blocks of consecutive integers ending with  $s_m-1$ at the $(a_k-t_m+1)$-th
place, for some  $m\in\{1,\dots,l\}\setminus\{j\}$, where
$k\in\{1,\dots,i\}$ is the largest such that $b_k\le s_m$, a possible block ending with
$s_j-1$ at the $(a_k-t_j+2)$-th place, where $k\in\{1,\dots,i\}$ is the largest such that
$b_k\le s_j$, and a last block ending with $n$ at the
$a_i$-th place, it follows that either  $\t(a_k-t_m+1)\ge s_m$, for some $m\ne j$,
where $k\in\{1,\dots,i\}$ is the largest such that $s_m\ge b_k$,  or
$\t(a_k-t_j+2)\ge s_j$, where $k\in\{1,\dots,i\}$ is the largest such that  $s_j\ge b_k$.
In the first case we have
$\t\not\le w$, and hence $p_\t\res_{O^-}\in I(X(w)\cap O^-)=I(Z)\subset I(Z_j)$.
Suppose now that
$\t(a_k-t_j+2)\ge s_j$, $k\in\{1,\dots,i\}$ being the largest such that  $s_j\ge b_k$. Using
Lemma
\ref{two}, we deduce that
$p_\t\res_{O^-}$ belongs to the ideal of $k[H]$ generated by $(t_j-1)$-minors with row indices
$\ge s_j$, and   column indices  $\le a_k$. Thus $p_\t\res_{O^-}$ belongs to the ideal
generated by $(t_j-1)$-minors contained in $L(j)$, which implies that $g\in I(Z_j)$.
\end{pf}
\begin{thm}\label{comp}
The irreducible components of $\text{Sing\,}D_{\us,\ut}(L)$ are precisely the $V_j$'s, $1\le j\le l$.
\end{thm}
\begin{pf}
In view of Theorem \ref{te}, we obtain that $V_j$, $1\le j\le l$, is irreducible, and the
required result follows from Theorem \ref{sing}.
\end{pf}

 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 $X(w)$ (resp. $X(\te_j)$, $1\le j\le l$) under the canonical projection
$\pi :SL(n)/B\to SL(n)/Q$.
Then using Theorems \ref{sing}, \ref{w} and \ref{te}, we obtain
\begin{thm}\label{8.4}
The irreducible components of $\text{Sing\,}X(w^{\text{max}})$ are precisely
$X(\te_j^{\text{max}})$, $1\le j\le l$.
\end{thm}
%%%%%%%%%%%%%%%%%%%%%%%%
%%% conjecture

\section{A conjecture on the irreducible components of a Schubert variety in $SL(n)/B$}\label{9}
Let $G=SL(n)$. In this section we state a conjecture which is a refinement of  the conjecture in
\cite{LS} on the irreducible components of the singular locus of a  Schubert variety, and prove
the conjecture  for a certain class of Schubert varieties, namely the pull-backs
$\pi^{-1}(X_Q(w))$ under
$\pi :G/B\to G/Q$, where $w$ and $Q$ are as in Section \ref{5}.

For $\t\in W$, let $P_\t$ (resp. $Q_\t$) be the maximal element of the set of parabolic
subgroups which leave $\overline{B\t B}$ (in $G$) stable under multiplication on the left
(resp. right).


We recall the following two well-known results (for a proof, see \cite{LMS} for example).
\begin{lem}
Let $\a$ be a simple root, and let $P_\a$ be the rank $1$ parabolic subgroup with
$S_{P_\a}=\{\a\}$. Let $\t\in W$. Then $\overline{B\t B}$ is stable under multimplication on
the right (resp. left) by $P_\a$ if and only if $\t(\a)\in R^-$ (resp. $\t^{-1}(\a)\in R^-$).
\end{lem}
\begin{cor}
With notations as in \ref{W_Q}, we have

\begin{align}
S_{P_\t}&=\{\a\in S\mid\t^{-1}(\a)\in R^-\},\notag\\
S_{Q_\t}&=\{\a\in S\mid\t(\a)\in R^-\}.\notag
\end{align}

\end{cor}

\begin{defn}
Given parabolic subgroups $P$, $Q$, we say that $\overline{B\t B}$ is $P$-$Q$ stable if $P\subset
P_\t$ and $Q\subset Q_\t$.
\end{defn}

\begin{lem}\label{9.4}
Let $G=SL(n)$. Let $\t\in\cal{S}_n$, say $\t=(a_1,\dots,a_n)$. Let $\a=\e_i-\e_{i+1}$. Then

$(1)$ $\t(\a)\in R^-$ if and only if $a_i>a_{i+1}$.

$(2)$ $\t^{-1}(\a)\in R^-$ if and only if $i+1$ occurs before $i$ in $\t$.
\end{lem}
\begin{pf}
We have $\t(\a)=\e_{a_i}-\e_{a_{i+1}}$ and $\t^{-1}(\a)=\e_j-\e_k$, where $a_j=i$ and $a_k=i+1$.
The results follow from this.
\end{pf}



Let $\eta\in W$. We shall denote $X_B(\eta)$ by just $X(\eta)$.
 We first recall
the criterion given in \cite{LS} for $X(\eta)$ to be singular.
\begin{thm}
Let $\eta=(a_1\dots a_n)\in\cal{S}_n$. Then $X(\eta)$ is singular if and only if there exist
$i,j,k,m$, $1\le i<j<k<m\le n$ such that
$$\text{either }a_k<a_m<a_i<a_j\,\text{ or }a_m<a_j<a_k<a_i\, .$$
\end{thm}

\subsection{The set $F_{\eta}$}\label{9.6}


Let $\eta=(a_1\dots a_n)\in\cal{S}_n$. Let $E_{\eta}$ be the set of all $\t'\le\eta$ such that
either $1)$ or $2)$ below holds.

\noindent $1)$ There exist $i,j,k,m$, $1\le i<j<k<m\le n$, such that

$(a)$ $a_k<a_m<a_i<a_j$

$(b)$ if $\t'=(b_1\dots b_n)$, then there exist  $i',j',k',m'$, $1\le i'<j'<k'<m'\le n$ such
that $b_{i'}=a_k$, $b_{j'}=a_i$, $b_{k'}=a_m$, $b_{m'}=a_j$

$(c)$ if $\t$ (resp. $\eta'$) is the element obtained from $\eta$ (resp. $\t'$) by replacing
$a_i,a_j,a_k,a_m$ respectively by $a_k,a_i,a_m,a_j$ (resp. $b_{i'},b_{j'},b_{k'},b_{m'}$
respectively by $b_{j'},b_{m'},b_{i'},b_{k'}$), then $\t'\ge\t$ and $\eta'\le\eta$.

\noindent $2)$ There exist $i,j,k,m$, $1\le i<j<k<m\le n$, such that

$(a)$ $a_m<a_j<a_k<a_i$

$(b)$ if $\t'=(b_1\dots b_n)$, then there exist  $i',j',k',m'$, $1\le i'<j'<k'<m'\le n$ such
that $b_{i'}=a_j$, $b_{j'}=a_m$, $b_{k'}=a_i$, $b_{m'}=a_k$

$(c)$ if $\t$ (resp. $\eta'$) is the element obtained from $\eta$ (resp. $\t'$) by replacing
$a_i,a_j,a_k,a_m$ respectively by $a_j,a_m,a_i,a_k$ (resp. $b_{i'},b_{j'},b_{k'},b_{m'}$
respectively by $b_{k'},b_{i'},b_{m'},b_{j'}$), then $\t'\ge\t$ and $\eta'\le\eta$.

Let $F_\eta=\{\t\in E_\eta\mid\overline{B\t B}\text{ is }P_\eta\text{-}Q_\eta\text{ stable}\}$.
\begin{conj}
The singular locus of $X(\eta)$ is equal to $\cup_{\l}X(\l)$, where $\l$ runs over the
maximal (under the Bruhat order) elements of $F_{\eta}$.
\end{conj}



\subsection{}
Let $\eta=(a_1\dots a_n)\in\cal{S}_n$. Let $\text{Sing\,}X(\eta)\ne \emptyset$. Let $(a,b,c,d)$
be four distinct entries in $\{1,\dots,n\}$ such that $a<b<c<d$. An occurence in
$\eta$ of the form $d,b,c,a$, where $d=a_i$, $b=a_j$, $c=a_k$, $a=a_m$, $i<j<k<m$, will be
referred to as a {\em Type I bad occurance in } $\eta$. An occurance in $\eta$ of the form
$(c,d,a,b)$, where
$c=a_i$, $d=a_j$, $a=a_k$, $b=a_m$, $i<j<k<m$, will be referred to as a {\em Type II bad
occurance in} $\eta$. Let $(d,b,c,a)$ (resp. $(c',d',a',b')$) be a bad occurance of Type  I
(resp. Type II), where $a<b<c<d$ (resp. $a'<b'<c'<d'$). Let $\te$, $\te'$ be both $\le w$.
Further, let $b,a,d,c$ (resp. $a',c',b',d'$) appear in that order in $\te$ (resp. $\te'$). By
abuse of language, we shall refer to $(b,a,d,c)$ (resp. $(a',c',b',d')$) as a bad occurance in
$\te$ (resp. $\te'$) corresponding to the bad occurance $(d,b,c,a)$ (resp. $(c',d',a',b')$) in
$\eta$.


Let $\t\in W_Q^{\text{min}}$. We have $\pi
^{-1}(X_Q(\t))=X_B(\t^{\text{max}})$, where $\t^{\text{max}}$, as a permutation, is given by
$\t^{(a_1)}$ arranged in descending order, followed by $\t^{(a_2)}\setminus \t^{(a_1)}$ arranged
in descending order, etc.. We shall refer to the set
$\t^{(a_i)}\setminus \t^{(a_{i-1})}$, $1\le i\le l+1$, arranged in descending order, as the
$i$-th block in $\t^{\text{max}}$ (here, $\t^{(a_0)}=\emptyset$, and $\t^{(a_{l+1})}$ is the set
$\{1,\dots,n\}\setminus \t^{(a_l)}$ arranged in descending order).

For the rest of this section, $w$ and $Q$ will be as in Section \ref{5}.

\begin{rem} \label{9.8}
 Set $b_{h+1}-1=n-t_l+1$. All of the entries in the $i$-th block in $w^{\text{max}}$ are $\le
b_i-1$, $2\le i\le h+1$. In particular, for $1\le j\le l$, $s_j$ occurs after $s_j-1$ in
$w^{\text{max}}$ (in view of lemma \ref{5.4}).
\end{rem}

\begin{lem}\label{9.9}
We have

$(1)$ $Q_{w^{\text{max}}}=Q$.

$(2)$ Let $I_{w^{\text{max}}}=\{\e_i-\e_{i+1}\mid i=s_j-1,1\le j\le l\}$. Then
$S_{P_{w^{\text{max}}}}=S\setminus I_{w^{\text{max}}}$.
\end{lem}
The assertions are clear from the description of $w^{\text{max}}$ in view of Lemma \ref{9.4}
and Remark \ref{9.8}.

\begin{lem}\label{9.10}
Let $P=P_{w^{\text{max}}}$, $Q=Q_{w^{\text{max}}}$. Then $\overline{B\te_j^{\text{max}}B}$ is
$P$-$Q$ stable.
\end{lem}
\begin{pf}
The $Q$-stability of $\overline{B\te_j^{\text{max}}B}$ on the right is obvious. Regarding the
$P$-stability of $\overline{B\te_j^{\text{max}}B}$ on the left, let $x$ denote either
$e_{i_k}$ or $v_i$, where $i>j$, $u_j\not\in w^{(a_i)}$ (notations being as in Section
\ref{8}). Then $x-1$ occurs after $x$ in $w^{\text{max}}$. It is clear from the definition
of $\te_j^{\text{max}}$ that $x-1$ occurs after $x$ in $\te_j^{\text{max}}$ also. For any
other entry $y\ne x$, $s_j-1$, if $y-1$ occurs after $y$ in $w^{\text{max}}$, then it does so
in $\te_j^{\text{max}}$ also. The result now follows from this.
\end{pf}

\begin{lem}\label{new}
Fix $j$, $1\le j\le h$. Let $C$ be a block of consecutive integers in $w^{(a_j)}$ ending with
$s_k-1$ at the $(a_j-t_k+1)$-th place (for some $k$) and beginning with $x_k$. Let the block
preceding $C$ end with $s_i-1$ for some $i$. Suppose $k^*\le j$. Then for $\a=\e_y-\e_{y+1}$,
where $y\in[s_i,x_k]$, the rank $1$ parabolic subgroup $P_\a$ is contained in
$P$($=P_{w^{\text{max}}}$).
\end{lem}
\begin{pf}
The result follows (in view of Lemma \ref{9.9}) from the fact that $[s_i,x_k]$ does not contain
$s_t-1$ for any $t$, $1\le t\le l$.
\end{pf}

We first show the above conjecture to be true for $X(w^{\text{max}})$ for the case $t_1=\dots
=t_l$, since the exposition in this case is much neater (and simpler) than the general case.
Let then $t_1=\dots =t_l=t$ say. In this case, we have $b_i-1\in w^{(a_i)}\setminus
w^{(a_{i-1})}$, $2\le i\le l$. Also, $h=l$, and $\{s_j,1\le j\le l\}=\{b_i,1\le i\le h\}$.




\begin{lem}\label{9.11}
Any bad occurance in $w^{\text{max}}$ is of Type I.
\end{lem}
\begin{pf}
Let $w^{\text{max}}=(a_1\dots a_n)$. Assume that $(c,d,a,b)$ is a bad occurance of Type II in
$w^{\text{max}}$, where $a<b<c<d$. Clearly, $c$ and $d$ (resp. $a$ and $b$) cannot both
appear in the same block , in view of the description of $w^{\text{max}}$. Let then
$c,d,a,b$ appear in the $r$-th, $i$-th, $j$-th, $k$-th blocks respectively, where $r<i\le j<k$.
This implies that $a<b<c<d\le b_i-1$ (cf. Remark \ref{9.8}). But now, $a$ and $b$ are both
$<b_i-1$, and they both appear after $b_i-1$; further, $a$ appears before $b$ in
$w^{\text{max}}$, which is not possible by the construction of $w^{\text{max}}$ (note that
$a<b$). The required result follows from this.
\end{pf}
\begin{rem}\label{9.12}
Of course, there are several bad occurances in $w^{\text{max}}$ of Type I. For example, fix some
$j$, $1\le j\le h$. Observe that $b_j$ appears
after $b_j-1$ (cf. Remark \ref{9.8}), and $u_j$ appears after $b_j$ in $w^{\text{max}}$
(notations being as in Section \ref{8}).  Take $d$, to be any entry in $\{n-t+2,\dots,n\}$,
$b=b_j-1$, $c=b_j$, $a=u_j$. Then
$d,b,c,a$ occur in the $1$-st, $j$-th, $k$-th, $m$-th blocks respectively, where $m\ge k>j$.
This provides an example of a Type I bad occurance in $w^{\text{max}}$.
\end{rem}
\begin{lem}\label{9.13}
Let $d,b,c,a$ be a Type I bad occurance in $w^{\text{max}}$, where $a<b<c<d$. Assume that $b$
belongs to the $i$-th block, for some $i$ (note that $i\le h$, since $b<c$). Then

$(1)$ $c<n-t+2$

$(2)$ $b\le b_i-1$

$(3)$ $d\ge n-t+2$
\end{lem}
\begin{pf}
Let $d,b,c,a$ occur in the $r$-th, $i$-th, $j$-th, $k$-th blocks respectively in
$w^{\text{max}}$, where $r\le i<j\le k$. The hypothesis that $b<c$ implies that $j>1$. Hence we
obtain $c\le b_j-1$ (cf. Remark
\ref{9.8}), and $(1)$ follows. Now, if $i\ge 2$, then the assertion $(2)$ follows from Remark
\ref{9.8}. If $i=1$, then the assertion $(2)$ follows from the fact that $b<c<n-t+2$.
\begin{claim}
$d>b_i-1$.
\end{claim}
\begin{pf}
Assume that $d\le b_i-1$. Then assumption implies $c<b_i-1$ (since $c<d$). Now both $c$ and
$b$ are $<b_i-1$, and $b$ belongs to the $i$-th block  in $w^{\text{max}}$. This implies that
$c$ should occur before $b$, which is not possible.
Hence our assumption is wrong, and the claim follows.
\end{pf}
Note that the Claim and Remark \ref{9.8} imply that $d\ge n-t+2$, and $d$ appears in the
first block.
\end{pf}
\begin{lem}\label{9.14}
Fix $j$, $1\le j\le h$. Then $\te_j^{\text{max}}$ is the unique maximal element of the set
$\{\t\in W\mid \t\le w^{\text{max}},\t^{(a_j)}(a_j-t+2)\le b_j-1\}$.
\end{lem}
The proof is clear from the definition of $\te_j^{\text{max}}$.
\begin{prop}\label{9.15}
The maximal elements in $F_{w^{\text{max}}}$ are precisely $\te_j^{\text{max}}$, $1\le i\le h$
(here $F_{w^{\text{max}}}$ is as in \S \ref{9.6}).
\end{prop}
\begin{pf}
We first observe that $\te_j^{\text{max}}\in F_{w^{\text{max}}}$; for, corresponding to the
bad occurance $d=n-t+2$, $b=b_j-1$, $c=b_j$, $a=u_j$ (cf. Remark \ref{9.12}), we have the bad
occurance
$(b,a,d,c)$ (note that $b,a,d,c$ occur in that order in $\te_j^{\text{max}}$). Let us denote
$\te_j^{\text{max}}$ by $\t'$. Let $w'$ (resp. $\t$) be the element of $\cal{S}_n$ obtained from
$\t'$ (resp. $w$) by replacing $b,a,d,c$ (resp. $d,b,c,a$) respectively by $d,b,c,a$ (resp.
$b,a,d,c$). Then clearly $\t\le\t'$, and $w'\le w$. Further, $\overline{B\te_j^{\text{max}}B}$ is
$P$-$Q$ stable (cf. Lemma \ref{9.10}). Thus $\te_j^{\text{max}}\in F_{w^{\text{max}}}$

Let now $\t'\in F_{w^{\text{max}}}$. In particular, we have $\t'\in W_Q^{\text{max}}$.

We have a bad occurrance in $\t'$ which has to be of the form $(b,a,d,c)$, $a<b<c<d$,
corresponding to the occurrance $(d,b,c,a)$ in $w^{\text{max}}$ (cf. Lemma \ref{9.11}). Let
$b,a,d,c$ occur in the $p$-th, $q$-th, $r$-th, $s$-th blocks respectively in $\t'$, where
$p\le q<r\le s$ (note that $\t'\in W_Q^{\text{max}}$).

We have
$$w'{}^{(a_q)}(a_q-t+1)\le w^{(a_q)}(a_q-t+1)= b_q-1$$
(here $w'$ is as in \S \ref{9.6}). Further, $\t'{}^{(a_q)}$ is obtained from $w'{}^{(a_q)}$
by replacing $d$ by $a$, where $a(<b)<n-t+2\le d$ (cf. Lemma \ref{9.13}). Hence we obtain $a\le
b_q-1$ (since $\t'{}^{(a_q)}\le w^{(a_q)}$), and $$\t'{}^{(a_q)}(a_q-t+2)\le
w'{}^{(a_q)}(a_q-t+1)\le b_q-1.$$
This implies $\t'\le \te_q^{\text{max}}$ (cf. Lemma \ref{9.14})
\end{pf}
\begin{thm}\label{9.16}
The conjecture \ref{9.6} holds for $X(w^{\text{max}})$.
\end{thm}
\begin{pf}
In view of Theorem \ref{8.4}, $X(\te_j^{\text{max}})$,
$1\le j\le h$ are precisely  the irreducible components of $X(w^{\text{max}})$. On the other
hand, we have (cf. Proposition \ref{9.15}) that the maximal elements in $F_{w^{\text{max}}}$ are
precisely $\te_j^{\text{max}}$, $1\le j\le h$. Hence the irreducible components of
$\text{Sing\,}X(w^{\text{max}})$ are precisely
$\{X(\te)\mid\te\text{ a maximal element of } F_{w^{\text{max}}}\}$. Thus the conjecture holds
for
$X(w^{\text{max}})$.
\end{pf}

Now we prove the conjecture for $X(w^{\text{max}})$ in the general case.


\begin{lem}\label{9.17}
Fix $j$, $1\le j\le l$. Let $j^*=r$. Then $\te_j^{\text{max}}$ is the unique maximal
element of the set $\{\t\in W\mid \t\le w^{\text{max}}, \t^{(a_{r})}(a_r-t_j+2)\le s_j-1\}$.
\end{lem}
The proof is clear from the definition of $\te_j$.

\begin{lem}
A bad occurrance in $w^{\text{max}}$ has to be of Type I.
\end{lem}
\begin{pf}
If possible, let $c,d,a,b$, where $a<b<c<d$, occur in the $i$-th, $j$-th, $k$-th, $p$-th
blocks respectively in $w^{\text{max}}$. Now $c<d$ implies that $i<j$. Hence $j>1$. Hence
$d\le b_j-1$ (cf. Remark \ref{9.8}), and this implies that $b<d\le b_j-1\le b_k-1$. But then $a$
cannot appear before $b$ (by definition of $w^{\text{max}}$).
\end{pf}
\begin{rem}\label{9.19}
Of course, there are several Type I bad occurrances. For example, take $j$, $1\le j\le l$. Let
$j^*=r$. With notations as in Lemma \ref{5.4}, let $d=e_{i_k}$. We have (cf. Lemma \ref{5.4})
$d>s_j$. Also, in view of  Remark \ref{9.8}, $s_j$ is not an entry
in
$w^{(a_i)}$, $i\le r$, and $s_j$ appears after $s_{j}-1$ in $w^{\text{max}}$. From the definition
of $w^{\text{max}}$, it is clear that $u_j$ appears after $s_j$ in $w^{\text{max}}$
(notations being as in Section \ref{8}). Take
$d=e_{i_k}$, $b=s_j-1$, $c=s_j$, $a=u_j$.
\end{rem}

\begin{lem}\label{9.20}
 Let $d,b,c,a$ be a Type I bad occurrance in $w^{\text{max}}$. Then

$(1)$ $d\in I_r$, for some $r$, $1\le r\le m+1$.

$(2)$ $a,c\not\in I_r$, for any $r$, $1\le r\le m+1$.
\end{lem}
\begin{pf}
Let $d,b,c,a$ belong to the $i$-th, $j$-th, $k$-th, $p$-th blocks respectively in
$w^{\text{max}}$. Assertion $(2)$ is immediate, since $p,k>1$. Note that assertion $(1)$ is
equivalent to the assertion that $i=1$. If $j=1$, then $i=1$, and
$(1)$ follows (cf. Lemma \ref{5.3}). Let then $j>1$. This implies $b\le b_j-1<c$. Suppose
$i>1$. Then we would obtain that $d\le b_i-1\le b_j-1<c$, which is not possible. Hence $i=1$,
and $(1)$ follows.
\end{pf}

\begin{rem}\label{rema}
With notations as in Lemma \ref{9.20}, we have in fact $d\in I_r$ for some $r\ge 2$. This is
clear if $j\ge 2$ (since $b\le b_j-1<c<d$). If $j=1$, then we have $b_1-1<c<d$. Thus we get
that $r\ge 2$.
\end{rem}


\begin{prop}\label{9.21}
The maximal elements of $F_{w^{\text{max}}}$ are precisely $\te_j^{\text{max}}$.
\end{prop}
\begin{pf}
Let us denote $j^*$ by $r$. Then with $d,b,c,a$ as in Remark \ref{9.19}, we have that
$b,a,d,c$ occur in that order in $\te_j^{\text{max}}$. Let us denote
$\te_j^{\text{max}}$ by $\t'$. Let $w'$ (resp. $\t$) be the element of $\cal{S}_n$ obtained from
$\t'$ (resp. $w$) by replacing $b,a,d,c$ (resp. $d,b,c,a$) respectively by $d,b,c,a$ (resp.
$b,a,d,c$). Then clearly $\t\le\t'$, and $w'\le w$. Further, $\overline{B\te_j^{\text{max}}B}$ is
$P$-$Q$ stable (cf. Lemma \ref{9.10}). Thus
$\te_j^{\text{max}}\in F_{w^{\text{max}}}$. Let now
$\t'\in F_{w^{\text{max}}}$. Let $b,a,d,c$ be a bad occurance in $\t'$. Further, let $b,a,d,c$
appear in the $p$-th, $q$-th, $r$-th, $s$-th blocks respectively in $\t'$ (note that $\t'\in
W_Q^{\text{max}}$). Let $b_q=s_z$ for some $z$, $1\le z\le l$. If $a\le b_q-1$, and $d>b_q-1$, as
in the proof of Proposition
\ref{9.15}, we obtain $\t'{}^{(a_q)}(a_q-t_z+2)\le b_q-1(=s_z-1)$. This implies  $\t'\le
\te_z^{\text{max}}$ (note that $z^*=q$).

We now distinguish the following two cases

 {\sl Case 1:}\quad $d\le b_q-1$

Let $d\in I_k$($=[e_{i_k},s_{i_k}-1]$) for some $k\ge 2$ (cf. Remark \ref{rema}). Let
$j=i_k^*$. We first observe that $j\le q$. For, if $i_k=i_k^*(=j)$, then $j\le q$ (since $d\le
b_q-1$ ). If $i_k>i_k^*$, then again in view of Lemma \ref{5.4}, we
have $s_{i_k^*}<d\le b_q-1$, and hence $b_j-1<b_q-1$ (note that $s_{i_k^*}=b_j$). Hence we get
$j<q$. Thus in either case we have $j\le q$.

We further divide this case into the following two subcases.

 {\sl Subcase 1 (a)} \quad $j<i_k$

Now, $I_k$ appears in $w^{(a_j)}$ as a block of consecutive integers (cf. Remark \ref{5.2}), and
$s_{i_k}-1$ appears at the $(a_j-t_{i_k}+1)$-th place. Let the block in $w^{(a_j)}$ preceding
this block end with
$s_i-1$ at the $(a_u-t_i+1)$-th place, for some $u$ and $i$. Then $u=j$ necessarily (since
$j<i_k$), and hence $i^*=u=j$. Now, in view of Lemmas \ref{9.9} and \ref{new} for
$\a=\e_y-\e_{y+1}$, where
$y\in [s_i,d-1]$, the rank $1$ parabolic subgroup $P_\a$ is contained in
$P(=P_{w^{\text{max}}})$. This, together with the fact that $d\not\in\t'{}^{(a_j)}$, implies that
$[s_i,d]\cap\t'{}^{(a_j)}\ne\emptyset$ (in view of the $P$-stability on the left of
$X(\t')$(cf. lemma \ref{9.4})). Hence we obtain that $\t'{}^{(a_j)}(a_j-t_i+2)\le s_i-1$, where
$i^*=j$. This implies
$\t'\le
\te_i^{\text{max}}$ (cf. Lemma \ref{9.17}).

 {\sl Subcase 1 (b)} \quad $j=i_k$

Note that $j>1$ (cf. Remark \ref{rema}). Consider $w^{(a_{j-1})}$. Now $I_k$ appears in
$w^{(a_{j-1})}$ as a block (cf. Remark \ref{5.2}, since $i_k^*>j-1$), and
$d$ belongs to this block. Further, $s_{i_k}-1$ appears at the $(a_{j-1}-t_{i_k}+1)$-th place.
Let the block in
$w^{(a_{j-1})}$ preceding this block end with $s_i-1$ at the $(a_{j-1}-t_i+1)$-th place for
some $i$. Then $i^*=j-1$, necessarily (since $j=i_k$). Further, for
$\a=\e_y-\e_{y+1}$, where $y\in [s_i,d-1]$, the rank $1$ parabolic subgroup $P_\a$ is contained
in
$P$ (in view of Lemma \ref{9.9}, since $[s_i,d-1]$ does not contain $s_t-1$ for any $t$, $1\le
t\le l$). Now, the fact that
$d\not\in\t'{}^{(a_q)}$ implies that
$\t'{}^{(a_{j-1})}\cap [s_i,d]=\emptyset$ (in view of $P$-stability on the left of $X(\t')$).
 Hence we obtain $\t'{}^{(a_{j-1})}(a_{j-1}-t_i+2)\le s_i-1$, where $i^*=j-1$. This implies
$\t'\le\te_i^{\text{max}}$ (cf. Lemma \ref{9.17}).

{\sl Case 2:} \quad $a>b_q-1$

Let $d,b,c,a$ appear in the $i$-th, $j$-th, $k$-th, $x$-th blocks respectively in
$w^{\text{max}}$, where $i\le j<k\le x$. Let $u$ be the smallest such that $a\le s_u-1$. We have
$q\le u^*$ (since $q>u^*$ would imply $a\le s_u-1<b_q-1$, which is not true).

\begin{claim} $x>u^*$.
\end{claim}
If $j\ge 2$, then we have $b_q-1<a<b\le b_j-1$ (cf. Remark \ref{9.8}). Hence we obtain $u^*\le j$
from which the Claim follows (since $x>j$).

If $j=1$, let $b\in I_v$ for some $v\ge 2$ (cf. Lemma \ref{5.3}; note that $b_q-1<a<b$ implies
$b>b_1-1$). We have $b_q-1<a<b\le s_{i_v}-1$. Hence we obtain $s_u-1\le s_{i_v}-1$, and $u^*\le
i_v^*$.  Now, we have
$b_k-1\ge c>s_{i_v}-1\ge s_{i_v^*}-1$ (by the definition of $w^{\text{max}}$). This implies
$c\not\in w^{(a_{i_v^*})}$, and hence
$k>i_v^*\ge u^*$. The Claim now follows from this (since $x\ge k$). Thus we obtain $q\le u^*<x$.
Now the fact that
$a\in\t'{}^{(a_q)}$ implies
$a\in\t'{}^{(a_{u^*})}$. This, together with the $P$-stability on the left of $X(\t')$, implies
that $[a,s_u-1]\subset\t'{}^{(a_{u^*})}$ (note that $s_j-1\not\in [a,s_u-1]$, for any $j\ne u$,
and hence for $\a=\e_y-\e_{y+1}$, where $y\in [a,s_u-2]$, the rank $1$ parabolic subgroup
$P_\a$ is contained in $P$). From this, we obtain $\t'{}^{(a_{u^*})}(a_{u^*}-t_u+2)\le s_u-1$
(since $\t'{}^{(a_{u^*})}\le w^{(a_{u^*})}$, and $a\not\in w^{(a_{u^*})}$ (note that $x>u^*$)).
This implies $\t'\le \te_u^{\text{max}}$ (cf. Lemma \ref{9.17}).
 \end{pf}
\begin{thm}\label{last}
The Conjecture \ref{9.6} holds for $X(w^{\text{max}})$.
\end{thm}
\begin{pf}
As in the proof of Theorem \ref{9.16}, the result follows from Theorem \ref{8.4} and Proposition
\ref{9.21}.
\end{pf}





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\end{document}