text:
\section{Introduction}
\label{sec:intro}
\numberwithin{equation}{section}

This paper is a sequel to \cite{KLPSResolutionSchSing15}. In \cite {LascouxSyzDet78}, Lascoux constructed a minimal free resolution of the coordinate ring of the determinantal variety $D_k$ (consisting of $m \times n$
matrices (over a field $K$) of rank at most $k$, considered as a closed subvariety of the $mn$-dimensional affine space of all $m
\times n$ matrices), as a
module over the the $mn$-dimensional polynomial ring (the coordinate ring of the the $mn$-dimensional affine space).

In \cite{KLPSResolutionSchSing15}, the authors construct free resolutions for a larger class of
singularities, \textit{viz.},
Schubert singularities, i.e., the intersection of a singular Schubert
variety and the ``opposite big cell'' inside a Grassmannian.
The  technique adopted in \cite{KLPSResolutionSchSing15} is algebraic group-theoretic, and in this paper we follow this approach.

We now describe the results of this paper.
Let $n$ be a positive integer. Let $V=\mathbb{C}^{2n}$
together with a non-degenerate skew-symmetric bilinear form $(\cdot,\cdot)$.
Let $H=\mathrm{SL}(V)$ and

\noindent $G=\mathrm{Sp}(V)=$ $\left\{ Z\in \mathrm{SL}(V)\,\,|\,\, Z\mbox{ leaves the form }(\cdot,\cdot)\mbox{ invariant}\right\} $.
We take the matrix of the form, with respect to the standard bais of $V$,
to be

\begin{center}
$F=\left[\begin{array}{cc}
0 & \JDiag[n]\\
-\JDiag[n] & 0
\end{array}\right]$
\par\end{center}

\noindent where $\JDiag[n]$ is the anti-diagonal $(1,...,1)$, in this case of size $n$.
To simplify our notation we will normally omit specifying the size of $J$ as it will be obvious from the context. We may realize $\mathrm{Sp}(V)$ as the fixed point set
of the involution $\sigma\,:\, H\rightarrow H$ given by $\sigma(Z)=F(Z^{T})^{-1}F^{-1}$ (cf. \cite{SteinbergLectChev68}).

Denoting by $T_{H}$ and $B_{H}$ the maximal torus
in $H$ consisting of diagonal matrices and the Borel subgroup in
$H$ consisting upper triangular matrices, respectively, we have that $T_{H}$ and $B_{H}$ are stable under $\sigma$ and we
set $T_{G}=T_{H}^{\sigma}$, $B_{G}=B_{H}^{\sigma}$. It is easily
checked that $T_{G}$ is a maximal torus in $G$ and $B_{G}$ is a
Borel subgroup in $G$.

Thus we obtain $$W_G\hookrightarrow W_H$$ where $W_G, W_H$ denote the Weyl groups of $G,H$ respectively (with respect to $T_G,T_H$ respectively).
 Further, $\sigma$ induces an involution on $W_H$:
 $$w=(a_1,\cdots,a_{2n})\in W_H,
 \sigma(w)=(c_1,\cdots,c_{2n}),\,c_i=2n+1-a_{2n+1-i}$$ and
 $$W_G=W_H^\sigma$$Thus we obtain
$$W_G=\{(a_1 \cdots a_{2n}) \in S_{2n} \mid a_i=2n+1-a_{2n+1-i},\ 1 \leq i
\leq 2n \}.
$$ (here, $S_{2n}$ is the symmetric group on $2n$ letters). Thus $w=(a_1\cdots a_{2n}) \in W_G$ is known once $(a_1 \cdots
a_n)$ is known.  We shall denote an element $(a_1\cdots a_{2n})$
in $ W_G$ by just $(a_1\cdots a_n)$. Further, for $w\in W_G$, denoting by $X_G(w)$ (resp. $X_H(w)$), the the associated
Schubert variety in $G/B_G$ (resp. $H/B_H$), we have that under the canonical
inclusion $G/B_G\hookrightarrow H/B_H$,  $X_G(w)=X_H(w)\cap
G/B_G$,  scheme-theoretically.

Let $P=P_{\widehat{n}}$, the maximal parabolic subgroup of $G$ corresponding to omitting the simple root $\alpha_n$, the set of simple roots of $G$ being indexed as in \cite{BourbakuEMGAL68}. Let $1\leq k<r\leq n$ be positive integers, and let $w\in \mathcal{W}_{k,r}$ (cf. \eqref{notation:parabolicSubgroupsStr}).
Our main result (cf. Theorem \ref{theorem:birational}) is a description of the minimal free
resolution of the coordinate ring of $Y_P(w) := X_{P}(w) \cap
O^{-}_{G/{{P}}}$, the \emph{opposite cell} of $X_{{P}}(w)$, as a module
over the coordinate ring of $O^-_{G/P}$. For this, we use the Kempf-Lascoux-Weyman ``geometric technique'' of
constructing minimal free resolutions, as described below:
Suppose that we have a commutative diagram of varieties \begin{equation}
\label{equation:genericKLW}
\vcenter{\vbox{%
\xymatrix{%
Z \ar@{^(->}[r] \ar[d]^{q'} & \mathbb A  \times V \ar[d]^q \ar[r] & V \\
Y \ar@{^(->}[r] & \mathbb A }}}
\end{equation}
where $\mathbb A$ is an affine space, $Y$ a closed subvariety of $\mathbb A$ and $V$
a
projective variety.
The map $q$ is first projection, $q'$ is proper and
birational,  and the inclusion $Z \hookrightarrow \mathbb A  \times V$ is a
sub-bundle (over $V$) of the trivial bundle $\mathbb A  \times V$.
Let $\xi$ be the dual of the quotient bundle on $V$ corresponding to $Z$.
Then the derived direct image $\@ifnextchar[{\R@Der}{\ensuremath{\RDerChar}} q'_* \strSh_{Z}$ is quasi-isomorphic
to a minimal complex
$F_\bullet$ with
\[
F_i = \oplus_{j \geq 0} \homology^j(V, \bigwedge^{i+j} \xi)
\otimes_\mathbb{C} R(-i-j).
\]
Here $R$ is the coordinate ring of $\mathbb A$; it is a polynomial ring
and
$R(k)$ refers to twisting with respect to its natural grading.
If $q'$ is such that the natural map $\strSh_Y \longrightarrow \@ifnextchar[{\R@Der}{\ensuremath{\RDerChar}} q'_* \strSh_{Z}$
is a quasi-isomorphism,
(for example, if $q'$ is a desingularization of $Y$ and $Y$ has rational
singularities), then $F_\bullet$  is a minimal free resolution of
$\mathbb{C}[Y]$ over the polynomial ring $R$ .

In applying this technique in any given situation, there are two main steps involed: one must find a suitable $Z$ and a suitable morphism $q' : Z \longrightarrow Y$ such
that the map $\strSh_Y \longrightarrow \@ifnextchar[{\R@Der}{\ensuremath{\RDerChar}} q'_* \strSh_{Z}$
is a quasi-isomorphism and such that $Z$ is
a vector-bundle over a projective variety $V$; and, one must be
able to compute the necessary cohomology groups.
We carry this out for opposite cells $Y_P(w), w\in \mathcal{W}_{k,r}$.

As the first step, we establish the existence of a diagram as above, using the geometry of Schubert varieties. We now describe this briefly:

We take $\mathbb A = O^-_{G/P}$ and $Y = Y_P(w) $.  Let $\tilde{P}$ be the two-step parabolic
subgroup $P_{\widehat{r-k},\widehat{n}}$ of $G$, and
let $\tilde{w}$ the minimal representative of $w\tilde{P}$ in $W^{\tilde{P}}$. Let $w^{\prime}:=(k+1,..,r,n,..,r+1,k,..,1)\in S_n$, the Weyl group of $\mathrm{GL}_n$. Let $Z_{\tilde{P}}(\tilde{w}):= Y_{P}(w)\times_{X_{P}(w)}X_{\tilde{P}}(w) (=\,(O_{G/P}^{-}\times P/\tilde{P})\cap X_{\tilde{P}}(w)\,)$.
Then it turns out that $Z_{\tilde{P}}(\tilde{w})$ is smooth (cf. \ref{definition:ZPW}), and is a desingularization of $Y_{P}(w)$. Write $p$ for the composite map $Z_{\tilde{P}}(\tilde{w})\hookrightarrow O_{G/P}^{-}\times P/\tilde{P}\rightarrow P/\tilde{P}$
where the first map is the inclusion and the second map is the projection. We have (cf. Theorem \ref{theorem:birational})  that $p$ identifies
$Z_{\tilde{P}}(\tilde w)$ as a sub-bundle of  the trivial bundle
$O_{G/P}^{-}
\times X_{P'_{\widehat{r-k}}}(w^{\prime})$ over $X_{P'_{\widehat{r-k}}}(w^{\prime})$, which arises as the
restriction (to $X_{P'_{\widehat{r-k}}}(w^{\prime})$) of a certain homogeneous vector-bundle on $\mathrm{GL}_n/P'_{\widehat{r-k}}$. Denoting $V:=X_{P'_{\widehat{r-k}}}(w^{\prime})$,  we get:
\begin{equation}
\label{equation:KLWdiagram}
\vcenter{\vbox{%
\xymatrix{%
Z_{\tilde P}(\tilde w) \ar[d]^{q'} \ar@{^(->}[r] & O^-_{G/P} \times
V  \ar[d]^{q} \ar[r] & V\\
Y_{P}(w)  \ar@{^(->}[r] & O^-_{G/P}
}}}.
\end{equation}
In this diagram, $q'$ is a desingularization of $Y_P(w)$. Since it is known
that Schubert varieties have rational singularities, we have that the map
$\strSh_Y \longrightarrow \@ifnextchar[{\R@Der}{\ensuremath{\RDerChar}} q'_* \strSh_{Z}$ is a quasi-isomorphism, so
$F_\bullet$ is a minimal resolution.

As the second step, we need to determine the cohomology of the bundles $\wedge^t \xi$ over $V$. In the above situation, $V=X_{P'_{\widehat{r-k}}}(w^{\prime})\hookrightarrow \mathrm{GL}_n/P'_{\widehat{r-k}}$. As is easily seen, $X_{P'_{\widehat{r-k}}}(w^{\prime})$ is a Grassmannian, namely, $\mathrm{GL}_r/P''_{\widehat{r-k}}$; further, the bundles $\wedge^t \xi$ (on $\mathrm{GL}_r/P''_{\widehat{r-k}}$) are homogeneous, but are not of Bott-type, namely, they are not completely reducible (so one can not apply Bott-algorithm for computing the cohomology). This can be resolved in two ways. In \cite {OttavianiRubeiQuiversCohVB06} the authors determine the cohomology of general
homogeneous bundles on Hermitian symmetric spaces, and thus their results can be used to determine $H^{\bullet}(V,\wedge^t \xi)$. Alternatively, using a technique from \cite{WeymanCohVBSyz03}, we may compute the resolution of a related space (whose associated homogeneous vector bundle is of Bott-type) from which we retrieve the resolution of the coordinate ring of $Y_P(w)$ as a subresolution.

We hope to extend the results of this paper to Schubert varieties in the orthogonal Grassmannian. Details will appear in a subsequent paper.

The paper is organized as follows.
Section~\ref{sec:prelim} contains
notations and conventions and the necessary
background material on Schubert varieties in the flag variety(Section~\ref{sec:Preliminaries:NotationA}) and Schubert varieties in the symplectic flag variety
(Section~\ref{sec:Preliminaries:NotationC},Section~\ref{sec:Preliminaries:SchubertVarieitiesSP2n}) and
homogeneous bundles (Section~\ref{subsection:Preliminaries:homogVectorBundles}).
In Section~\ref{sec:desing}, we discuss properties of Schubert
desingularization, including the construction of
Diagram~\eqref{equation:KLWdiagram}.
Section~\ref{sec:freeresolutions} is devoted to a review
of the Kempf-Lascoux-Weyman technique and completes step one of the two part process of the geometric technique.
Section~\ref{sec:cohomologyHVBStepTwo} explains how the cohomologies of the homogeneous
bundles constructed in step one may be calculated.

\section*{Acknowledgements}
The authors thank Manoj Kummini for helpful discussions.

\section{Preliminaries}
\label{sec:prelim}
\numberwithin{equation}{subsection}

In this section we collect various results about Schubert varieties in the flag variety and symplectic flag variety, homogeneous vector bundles, and the Bott-algorithm.

\subsection{Notation and Conventions in Type A}
\label{sec:Preliminaries:NotationA}

We collect the symbols used and the conventions adopted in the rest of the
paper here.  For details on algebraic groups and Schubert varieties, the
reader may refer to ~\cite {BorelLinAlgGps91, JantzenReprAlgGps03, BilleyLakshmibaiSingularLoci00, SeshadriSMT07}.

Let $N$ be positive integer.
We denote by $\mathrm{GL}_N$ (respectively,
$B_N$, $B_N^{-}$) the group of all (respectively, upper-triangular,
lower-triangular) invertible
$N\times N$ matrices over $\mathbb{C}$.
The Weyl group $W$ of $\mathrm{GL}_N$ is isomorphic to the group $S_N$ of
permutations of $N$ symbols and is generated by the \emph{simple
reflections} $s_i, 1 \leq i \leq N-1$, which correspond to the
transpositions $(i,i+1)$.
For $w \in W$, its \emph{length} is the
smallest integer $l$ such that $w = s_{i_1}\cdots s_{i_l}$ as a
product of simple reflections.
For every $1 \leq i \leq N-1$, there is a
\emph{minimal parabolic subgroup} $P_i$ containing $s_i$ (thought of as
an element of $\mathrm{GL}_N$)
and a \emph{maximal parabolic subgroup} $P_{\widehat{i}}$ \emph{not
containing} $s_i$. Any parabolic subgroup can be written as
$P_{\widehat{A}} := \bigcap_{i \in A} P_{\widehat{i}}$ for some $A \subset
\{1, \ldots, N-1\}$. On the other hand, for $A \subseteq \{1, \ldots,
N-1\}$ write $P_A$ for the subgroup of $\mathrm{GL}_N$ generated by $P_i, i \in A$.
Then $P_A$
is a parabolic subgroup and $P_{\{1, \ldots, N-1\} \setminus A}=
P_{\widehat{A}}$.

We write the elements of $W$ in \emph{one-line} notation: $(a_1,
\ldots, a_N)$ is the permutation $i \mapsto a_i$.  For any $A \subseteq
\{1, \ldots, N-1\}$, define $W_{P_A}$ to be the
subgroup of $W$ generated by $\{s_i : i \in
A\}$. By $W^{P_A}$ we mean the subset of $W$ consisting of the minimal
representatives (under the Bruhat order) in $W$ of the elements of
$W/W_{P_A}$.
For $1 \leq i \leq N$, we represent the elements of
$W^{P_{\widehat{i}}}$ by sequences $(a_1, \ldots, a_i)$ with $1 \leq a_1 <
\cdots < a_i \leq N$ since under the action of the group
$W_{P_{\widehat{i}}}$, every element of $W$ can be represented minimally by
such a sequence.

We identify $\mathrm{GL}_N = \mathrm{GL}(V)$ for some $N$-dimensional vector-space $V$.
Let $A := \{i_1 < i_2 < \cdots < i_r\} \subseteq \{1, \ldots, N-1\}$.  Then
$\mathrm{GL}_N/P_{\widehat{A}}$ is the set of all flags $0 = V_0 \subsetneq V_{1}
\subsetneq V_2 \subsetneq  \cdots \subsetneq V_r \subsetneq V$ of subspaces
$V_j$ of dimension $i_j$ inside $V$. We call $\mathrm{GL}_N/P_{\widehat{A}}$ a
\emph{flag variety}. If $A = \{1, \ldots, N-1\}$ (i.e.  $P_{\widehat{A}}
= B_N$), then we call the the flag variety a \emph{full flag
variety}; otherwise, a \emph{partial flag variety}.
The \emph{Grassmannian} $Grass_{i,N}$ of $i$-dimensional subspaces of
$V$
is $\mathrm{GL}_N/P_{\widehat{i}}$.

Let ${\tilde{P}}$ be any parabolic subgroup containing $B_N$
and $\tau \in W$.  The \emph{Schubert
variety} $X_{{\tilde{P}}}(\tau)$ is the closure inside $\mathrm{GL}_N/{\tilde{P}}$ of $B_N\cdot e_w$
where $e_w$ is the coset $\tau{\tilde{P}}$, endowed with the canonical reduced
scheme structure.  Hereafter, when we write $X_{\tilde{P}}(\tau)$, we mean that $\tau$ is
the representative in $W^{\tilde{P}}$ of its coset.
The \emph{opposite big cell} $O^{-}_{\mathrm{GL}_N/{\tilde{P}}}$ in
$\mathrm{GL}_N/{\tilde{P}}$ is  the $B_N^{-}$-orbit of the coset
$(\mathrm{id}\cdot {\tilde{P}})$ in $\mathrm{GL}_N/{\tilde{P}}$.
Let $Y_{\tilde{P}}(\tau) := X_{\tilde{P}}(\tau) \cap
O^{-}_{\mathrm{GL}_N/{\tilde{P}}}$; we refer to $Y_{\tilde{P}}(\tau)$ as the
\emph{opposite cell} of $X_{\tilde{P}}(\tau)$.

We will write $R^+$, $R^{-}$, $R^+_{\tilde{P}}$,
$R^{-}_{\tilde{P}}$, to denote respectively, positive and negative roots for $\mathrm{GL}_N$
and for $\tilde{P}$.
We denote by $\epsilon_i$ the character that sends the invertible diagonal
matrix with $t_1, \ldots, t_n$ on the diagonal to $t_i$.

\subsection{Notation and Conventions in Type C}
\label{sec:Preliminaries:NotationC}

Below we review the properties of symplectic Schubert varieties relevant to this paper. For a more in depth introduction the reader may refer to \cite[Chapter 6]{LakRaghSMT08}.

Let $n$ be a positive integer. Let $V=\mathbb{C}^{2n}$
together with a non-degenerate skew-symmetric bilinear form $(\cdot,\cdot)$.Let $H=\mathrm{SL}(V)$ and $G=\mathrm{Sp}(V)=$ $\left\{ Z\in \mathrm{SL}(V)\,\,|\,\, Z\mbox{ leaves the form }(\cdot,\cdot)\mbox{ invariant}\right\} $.
We take the matrix of the form, w.r.t the standard bais of $V$,
to be

\begin{center}
$F=\left[\begin{array}{cc}
0 & \JDiag[n]\\
-\JDiag[n] & 0
\end{array}\right]$
\par\end{center}

\noindent where $\JDiag[n]$ is the anti-diagonal $(1,...,1)$, in this case of size $n$.
To simplify our notation we will normally omit specifying the size of $J$ as it will be obvious from the context. We may realize $\mathrm{Sp}(V)$ as the fixed point set
of the involution $\sigma\,:\, H\rightarrow H$ given by $\sigma(Z)=F(Z^{T})^{-1}F^{-1}$ (cf. \cite{SteinbergLectChev68}).
That is

\begin{center}
$\begin{alignedat}{2}G
& = \{Z\in \mathrm{SL}(V)\,\,|\,\, Z^{T}FZ=F\}\\
& = \{Z\in \mathrm{SL}(V)\,\,|\,\, F^{-1}(Z^{T})^{-1}F=Z\}\\
& = \{Z\in \mathrm{SL}(V)\,\,|\,\, F(Z^{T})^{-1}F^{-1}=Z\}\\
& = H^{\sigma} \end{alignedat} $
\par\end{center}

Denote by $T_{H}$ and $B_{H}$ the maximal torus
in $H$ consisting of diagonal matrices and the Borel subgroup in
$H$ consisting upper triangular matrices, respectively. It is easily
seen that $T_{H}$ and $B_{H}$ are stable under $\sigma$ and we
set $T_{G}=T_{H}^{\sigma}$, $B_{G}=B_{H}^{\sigma}$. It is easily
checked that $T_{G}$ is a maximal torus in $G$ and $B_{G}$ is a
Borel subgroup in $G$.

Thus we obtain $$W_G\hookrightarrow W_H$$ where $W_G, W_H$ denote the Weyl groups of $G,H$ respectively (with respect to $T_G,T_H$ respectively).
 Further, $\sigma$ induces an involution on $W_H$:
 $$w=(a_1,\cdots,a_{2n})\in W_H,
 \sigma(w)=(c_1,\cdots,c_{2n}),\,c_i=2n+1-a_{2n+1-i}$$ and
 $$W_G=W_H^\sigma$$Thus we obtain
$$W_G=\{(a_1 \cdots a_{2n}) \in S_{2n} \mid a_i=2n+1-a_{2n+1-i},\ 1 \leq i
\leq 2n \}.
$$ (here, $S_{2n}$ is the symmetric group on $2n$ letters). Thus $w=(a_1\cdots a_{2n}) \in W_G$ is known once $(a_1 \cdots
a_n)$ is known.  We shall denote an element $(a_1\cdots a_{2n})$
in $ W_G$ by just $(a_1\cdots a_n)$. For example, $(4231) \in
S_{4}$ represents $(42) \in W_{G},G=\mathrm{Sp}(4)$.

$\sigma$
induces an involution on $X(T_H)$, the character group of $T_H$:
$$\chi\in X(T_H),\,\sigma(\chi)(D)=\chi(\sigma(D)),\,D\in T_H$$
Let $\epsilon_i, 1\le i\le 2n$ be the character in $X(T_H)$,
$\epsilon_i(D)=d_i$, the $i$-th entry in $D(\in T_H)$. We have
$$\sigma(\epsilon_i)=-\epsilon_{2n+1-i}$$ Now it is easily seen
that the under the canonical surjective map
$$\varphi:X(T_H)\rightarrow X(T_G)$$ we have
$$\varphi(\epsilon_i)=-
\varphi(\epsilon_{2n+1-i}),\,1\le i\le 2n$$ Let
$R_H:=\{\epsilon_i-\epsilon_j,1\le i,j\le 2n\}$, the root system
of $H$ (relative to $T_H$), and
$R^+_H:=\{\epsilon_i-\epsilon_j,1\le i<j\le 2n\}$, the set of
positive roots (relative to $B_H$). We have the following:
\begin{enumerate}
\item $\sigma$ leaves $R_H$ (resp. $R^+_H$) stable. \item For
$\alpha,\beta\in R_H,\varphi(\alpha)=\varphi(\beta)\Leftrightarrow
\alpha=\sigma(\beta)$ \item $\varphi$ is equivariant for the
canonical action of $W_G$ on $X(T_H),X(T_G)$. \item
$R_H^\sigma=\{\pm(\epsilon_i-\epsilon_{2n+1-i}),1\le i\le n\}$
\end{enumerate}

Let $R_G$ (resp. $R_G^+$) the set of roots of $G$ with respect to
$T_G$ (resp. the set of positive roots with respect to $B_G$ ).
Using the above facts and the explicit nature of the adjoint
representation of $G$ on Lie$\,G$, we deduce that
$$R_G=\varphi(R_H),\,R^+_G=\varphi(R^+_H)$$ In particular, $R_G$ (resp. $R^+_G$) gets
identified with the orbit space of $R_H$ (resp. $R^+_H$) modulo
the action of $\sigma$. Thus we obtain the following
identification:
\begin{equation*}\label{e:roots.c}
R_G=\{ \pm (\epsilon_i \pm \epsilon_j),\ 1\leq i<j\leq n\} \cup
\{\pm 2\epsilon_i,\ i=1,...,n  \}
\end{equation*}
\begin{equation*}\label{e:pos.roots.c}
R_G^+ = \{  (\epsilon_i \pm \epsilon_j),\ 1\leq i<j\leq n\} \cup
\{ 2\epsilon_i,\ i=1,...,n  \}.
\end{equation*}
The set $S_G$ of simple roots in $R_G^+$ is given by
\begin{equation*}\label{e:sim.roots.c}
 S_G:=\{ \alpha_{i}=\epsilon_i - \epsilon_{i+1},\ 1 \leq i \leq n-1\}
\cup \{\alpha_{n}=2\epsilon_n  \}.
\end{equation*}
 Let us denote the simple reflections in $W_G$ by $\{ s_i,\  1
\leq
 i \leq n \}$, namely, $s_i$=  reflection with respect to
$\epsilon_i - \epsilon_ {i+1},\  1 \leq i \leq n-1$, and $s_n=$
reflection with respect to $2\epsilon_n.$ Then we have
\begin{equation}
\label{equation:simpleReflectionsSP2n}
s_i=\begin{cases}r_ir_{2n-i}, &\text{if}\ 1 \leq i \leq n-1\\
                 r_n,             &\text{if}\  i=n
\end{cases}
\end{equation}

where $r_i$ denotes the transposition $(i,i+1)$ in $S_{2n},\   1
\leq i \leq 2n-1.$

For $w\in W_G$, let us denote
$l(w,W_H)$ (resp. $l(w,W_G)$), the length of $w$ as an element of
$W_H$ (resp. $W_G$). For $w=(a_1,\cdots,a_{2n})\in W_H$, denote
$$m_w:=\#\{i\le n\,|\,a_i>n\}$$

\noindent Then for $w=(a_1,\cdots,a_{2n})\in W_G$, we have
$l(w,W_G)={\frac{1}{2}}(l(w,W_H)+m_w)$

\begin{proposition}
\label{proposition:inducedSchemeTheSchubert}
\cite[Proposition 6.2.5.1]{LakRaghSMT08}Let $w\in W_G$; let $X_G(w)$ (resp. $X_H(w)$) be the associated
Schubert variety in $G/B_G$ (resp. $H/B_H$). Under the canonical
inclusion $G/B_G\hookrightarrow H/B_H$, we have $X_G(w)=X_H(w)\cap
G/B_G$. Further, the intersection is scheme-theoretic.
\end{proposition}

\begin{notation}
\label{notation:parabolicSubgroupsPre}
For the remainder of the paper we fix the following notation. Let $1\leq k<r\leq n$ be positive integers. Let $Q=Q_{\hat{n}}$ to be the parabolic subgroup of $H$ corresponding to omitting the root $\alpha_{n}$ and $P=P_{\hat{n}}$
to be the parabolic subgroup of $G$ corresponding to omitting the
root $\alpha_{n}$. Let $\tilde{P}$ be the two-step parabolic
subgroup $P_{\widehat{r-k},\widehat{n}}$ of $G$. Let $\tilde{Q}$ be the three step parabolic subgroup $Q_{\widehat{r-k},\widehat{n},\widehat{2n-(r-k)}}$ in $H$. Note that $P=Q^{\sigma}$ and $\tilde{P}=\tilde{Q}^{\sigma}$. Finally, we will identify $P/\tilde{P}$ with $\mathrm{GL}_n/P'_{\widehat{r-k}}$ where $P'_{\widehat{r-k}}$ is the parabolic subgroup of $\mathrm{GL}_n$ omitting the root $\alpha_{r-k}$.
\end{notation}

\begin{definition}
\label{definition:persymmetric}
A square $m\times m$ matrix $X$ is persymmetric if $\JDiag[m]X=X^{T}\JDiag[m]$. Or equivalently if $\JDiag[m]X$ is symmetric.
\end{definition}

\begin{remark}
\label{remark:identificationOMinusGP}
Let $K$ be the subgroup of $H$ consisting of matrices
of the form

\begin{center}
$\left[\begin{array}{cc}
\mathrm{Id}_{n} & 0\\
Y & \mathrm{Id}_{n}
\end{array}\right],\, Y\in \mathrm{Mat}_{n}$.
\par\end{center}
The canonical morphism $H\rightarrow H/Q$ induces a morphism $\psi_H:K\rightarrow H/Q$. We have that $\psi_H$ is an open immersion, and
$\psi_H(K)$ gets identified with the \define{opposite big cell} $O^-_{H/Q}$ in $H/Q$.

\noindent $O_{H/Q}^{-}$ is $\sigma$-stable and we can identify the \define{opposite big cell} $O_{G/P}^{-}$ as
\begin{center}
$O_{G/P}^{-}= (O_{H/Q}^{-})^{\sigma}=\left\{ z\in K\,|\, \JDiag[n]Y^{T}\JDiag[n]=Y\right\}$\cite[Corollary 6.2.4.3]{LakRaghSMT08}
\end{center}
So $O_{G/P}^{-}$ is the subspace of $K$ with $Y$ persymmetric. Thus we can identify $O_{G/P}^{-}$ with the space of symmetric $n\times n$ matrices, $\mathrm{Sym}_n$, under the map $O_{G/P}^{-}\longrightarrow \mathrm{Sym}_n$ given by
\begin{center}
$\left[\begin{array}{cc}
\mathrm{Id}_{n} & 0\\
Y & \mathrm{Id}_{n}
\end{array}\right]\mapsto JY$.
\end{center}
\end{remark}

\subsection{Opposite Cells in Schubert Varieties in the Symplectic Flag Variety}
\label{sec:Preliminaries:SchubertVarieitiesSP2n}
A matrix $Z\in \mathrm{SL}(V)$ with $n\times n$ block form
\begin{center} $\left[\begin{array}{cc}
A_{n\times n} & C_{n\times n}\\
D_{n\times n} & E_{n\times n}
\end{array}\right]$
\end{center} \noindent is an element of $G$ if and only if $Z^{T}FZ=F$,
i.e. if and only if the following conditions hold on the $n\times n$
blocks.\begin{equation}
\label{equation:StructureSP2nAD}
A^{T}\JDiag[n]D=D^{T}\JDiag[n]A
\end{equation}\begin{equation}
\label{equation:StructureSP2nCE}
C^{T}\JDiag[n]E=E^{T}\JDiag[n]C
\end{equation}\begin{equation}
\label{equation:StructureSP2nADCE}
\JDiag[n]=(A^{T}\JDiag[n]E-D^{T}\JDiag[n]C)=(E^{T}\JDiag[n]A-C^{T}\JDiag[n]D)
\end{equation} The following proposition will prove useful throughout the rest of the paper.

\begin{proposition}
\label{proposition:structureOMinus}
Write $U_{P}^{-}$ for the negative unipotent radical of $P$.
\begin{asparaenum}
\item \label{proposition:structureOMinusUminus}
\noindent $O_{G/P}^{-}$ can be naturally
identified with $U_{P}^{-}P/P$

\item \label{proposition:structureOMinusinvertibleA}
\noindent For
\begin{center}
$z=\left[\begin{array}{cc}
A_{n\times n} & C_{n\times n}\\
D_{n\times n} & E_{n\times n}
\end{array}\right]\in G$
\par\end{center}

\noindent $zP\in O_{G/P}^{-}$ if and only if
$A$ is invertible.

\item \label{proposition:structureOMinusInverseImage}
\noindent The inverse image of $O_{G/P}^{-}$
under the natural map $G/\tilde{P}\rightarrow G/P$ is isomorphic
to $O_{G/P}^{-}\times P/\tilde{P}$ as schemes. Every element of
$O_{G/P}^{-}\times P/\tilde{P}$ is of the form

\begin{center}
$\left[\begin{array}{cc}
A_{n\times n} & 0\\
D_{n\times n} & \JDiag[n](A^{T})^{-1}\JDiag[n]
\end{array}\right]$ mod $\tilde{P}\in G/\tilde{P}$.
\par\end{center}

\noindent Moreover, two matrices
\[
\begin{bmatrix}
A_{n \times n} & 0_{n \times n} \\
D_{n \times n} & \JDiag[n](A^{T})^{-1}\JDiag[n]
\end{bmatrix} \;\text{and}\;
\begin{bmatrix}
A'_{n \times n} & 0_{n \times n} \\
D'_{n \times n} & \JDiag[n](A'^{T})^{-1}\JDiag[n]
\end{bmatrix}
\]
in $G$ represent the same element modulo $\tilde{P}$ if and only if
there exists a matrix $q \in P'_{\widehat{r-k}}$(cf. \eqref{notation:parabolicSubgroupsPre}) such that $A' = Aq$ and $D' = Dq$.

\item \label{proposition:structureOMinusSLnModQ}
\noindent $P/\tilde{P}$ is isomorphic
to $\mathrm{GL}_n/P'_{\widehat{r-k}}$. In particular, the projection map $O_{G/P}^{-}\times P/\tilde{P}\rightarrow P/\tilde{P}$
is given by

\begin{center}
$\left[\begin{array}{cc}
A_{n\times n} & 0\\
D_{n\times n} & \JDiag[n](A^{T})^{-1}\JDiag[n]
\end{array}\right]$ mod $\tilde{P}\longmapsto A$ mod $P'_{\widehat{r-k}}\in \mathrm{GL}_n/P'_{\widehat{r-k}}\cong P/\tilde{P}$
\par\end{center}
\end{asparaenum}
\end{proposition}

\begin{proof}
\noindent \eqref{proposition:structureOMinusUminus}: Note
that $U_{P}^{-}$ is the subgrioup of $G$ generated by the root subgroups
$U_{-\alpha},\alpha\in R^{+}\backslash R_{P}^{+}$. Under the canonical
projection $G\rightarrow G/P$, $g\mapsto gP$, $U_{P}^{-}$ is mapped
isomorphically onto its image $O_{G/P}^{-}$(c.f. {[}2-SingLoci{]} 4.4.4).
Thus we obtain the identification of $O_{G/P}^{-}$ with $U_{P}^{-}P/P$.

\noindent \eqref{proposition:structureOMinusinvertibleA}: Suppose that $zP\in O_{G/P}^{-}$. By ~\eqref{proposition:structureOMinusUminus} this means
that $\exists$ $n\times n$ matrices $A^{\prime},C^{\prime},D^{\prime},E^{\prime}$
such that

\begin{center}
$z_{1}=\left[\begin{array}{cc}
\mathrm{Id}_{n} & 0\\
D^{\prime} & \mathrm{Id}_{n}
\end{array}\right]\in U_{P}^{-}$ and $z_{2}=\left[\begin{array}{cc}
A^{\prime} & C^{\prime}\\
0 & E^{\prime}
\end{array}\right]\in P$ with $z=\left[\begin{array}{cc}
A & C\\
D & E
\end{array}\right]=z_{1}z_{2}$
\par\end{center}

\noindent Hence $A=A^{\prime}$, and $A^{\prime}$ inveritble
implies $A$ invertible.

\noindent Conversely, suppose $A$
is invertible. Let
\begin{center}
$z=\left[\begin{array}{cc}
A & C\\
D & E
\end{array}\right]\in G$
\end{center}
Then $A,C,D,\, E$ satisfy properties ~\eqref{equation:StructureSP2nAD}-\eqref{equation:StructureSP2nCE}. Since  $A$ is invertible we may write
\begin{center}
$z=z_{1}z_{2}$ where $z_{1}=\left[\begin{array}{cc}
\mathrm{Id}_{n} & 0\\
DA^{-1} & \mathrm{Id}_{n}
\end{array}\right]$, $z_{2}=\left[\begin{array}{cc}
A_{\,} & C\\
0 & E-DA^{-1}C_{\,}
\end{array}\right]$.
\par\end{center}

\noindent We shall now show that $z_1,z_2 \in G$.
We will need two important identities. For the first, ~\eqref{equation:StructureSP2nAD} implies that
\begin{equation}
\label{equation:identitySP2n1}
\JDiag[n](DA^{-1})=(DA^{-1})^{T}\JDiag[n]
\end{equation}
\noindent The second
\begin{equation}
\label{equation:identitySP2n2}
A^{T}\JDiag[n](E-DA^{-1}C)=\JDiag[n]
\end{equation}
follows from
\begin{center}
$\begin{array}{rlr}
A^{T}\JDiag[n](E-DA^{-1}C) & = A^{T}\JDiag[n]E-A^{T}\JDiag[n]DA^{-1}C & \\
\; & =A^{T}\JDiag[n]E-D^{T}\JDiag[n]AA^{-1}C & ~\eqref{equation:StructureSP2nAD} \\
\; & =\JDiag[n] & ~\eqref{equation:StructureSP2nADCE}\\
\end{array}$
\end{center}

\noindent Now ~\eqref{equation:identitySP2n1} shows that $z_{1}\in U_{P}^{-}$, and hence $z_{1}\in G$. Next we show that $z_{2}\in G$. Let $F$ be as in Section \ref{sec:Preliminaries:NotationC}, then

\begin{center}
$z_{2}^{T}Fz_{2}=\left[\begin{array}{cc}
0 & A^{T}\JDiag[n](E-DA^{-1}C)\\
-(E-DA^{-1}C)^{T}\JDiag[n]A & \,\,\,\,-(E-DA^{-1}C)^{T}\JDiag[n]C+C^{T}\JDiag[n](E-DA^{-1}C)
\end{array}\right]$
\par\end{center}

\noindent Then
\begin{center}
$\begin{array}{rlr}
-(E-DA^{-1}C)^{T}\JDiag[n]A & = -(E^{T}\JDiag[n]A-C^{T}(DA^{-1})^{T}\JDiag[n]A) & \\
\; & =-(E^{T}\JDiag[n]A-C^{T}\JDiag[n](DA^{-1})A) & ~\eqref{equation:identitySP2n1} \\
\; & =-(E^{T}\JDiag[n]A-C^{T}\JDiag[n]D) & \;\\
\; & =-\JDiag[n] & ~\eqref{equation:StructureSP2nADCE}\\
\end{array}$
\end{center}
\noindent and finally
\begin{center}
$\begin{array}{rlr}
-(E-DA^{-1}C)^{T}\JDiag[n]C & = -(E^{T}\JDiag[n]C-C^{T}(DA^{-1})^{T}\JDiag[n]C) & \\
\; & =-(C^{T}\JDiag[n]E-C^{T}\JDiag[n](DA^{-1})C) & ~\eqref{equation:identitySP2n1} \\
\; & =-C^{T}\JDiag[n](E-DA^{-1}C) & \;\\
\end{array}$
\end{center}

\noindent Combining these two results and ~\eqref{equation:identitySP2n2} we have $z_{2}^{T}Fz_{2}=F$, and thus $z_{2}\in G$. Further, since $A$ is invertible $z_{2}\in P$. Hence the coset $zP=z_{1}P$, which in view of the fact that $z_{1}\in U_{P}^{-}$, implies by part \eqref{proposition:structureOMinusUminus} that $zP \in O_{G/P}^{-}$.

\noindent \eqref{proposition:structureOMinusInverseImage}: Let $z\in U_{P}^{-}P\subset G$. Then we can write
$z=z_{1}z_{2}$ uniquely with $z_{1}\in U_{P}^{-}$, $z_{2}\in P$.
Suppose that

\begin{center}
$\left[\begin{array}{cc}
\mathrm{Id}_{n} & 0\\
D_{n\times n} & \mathrm{Id}_{n}
\end{array}\right]\left[\begin{array}{cc}
A_{n\times n} & C_{n\times n}\\
0_{n\times n} & E_{n\times n}
\end{array}\right]=\left[\begin{array}{cc}
\mathrm{Id}_{n} & 0\\
D_{n\times n}^{\prime} & \mathrm{Id}_{n}
\end{array}\right]\left[\begin{array}{cc}
A_{n\times n}^{\prime} & C_{n\times n}^{\prime}\\
0_{n\times n} & E_{n\times n}^{\prime}
\end{array}\right]$
\par\end{center}

\noindent then $A=A^{\prime}$, $C=C^{\prime}$, $DA=D^{\prime}A^{\prime}$
and $DC+E=D^{\prime}C^{\prime}+E^{\prime}$, which yields that $D^{\prime}=D$
(since $A=A^{\prime}$ is invertible), and then $E=E^{\prime}$. Hence
$U_{P}^{-}\times_{\mathbb{C}}P=U_{P}^{-}P$. Thus for any parabolic
subgroup $P^{\prime}\subseteq P$, $U_{P}^{-}\times_{\mathbb{C}}P/P^{\prime}=U_{P}^{-}P/P^{\prime}$.
The asserted isomorphism follows by part \eqref{proposition:structureOMinusUminus} from taking $P^{\prime}=\tilde{P}$.

\noindent To see the second assertion consider

\begin{center}
$z=\left[\begin{array}{cc}
A_{n\times n} & C_{n\times n}\\
D_{n\times n} & E_{n\times n}
\end{array}\right]\in G$
\par\end{center}

\noindent with $zP\in O_{G/P}^{-}$. Note that the $n\times n$ block matrices satisfy properties ~\eqref{equation:StructureSP2nAD}-\eqref{equation:StructureSP2nADCE} and by ~\eqref{proposition:structureOMinusinvertibleA} $A$ is invertible.

We have by the first part of ~\eqref{proposition:structureOMinusInverseImage} that the coset $zP$ is an element of $O_{G/P}^{-}\times P/\tilde{P}$, since $zP\in O_{G/P}^{-}$.

\noindent\textbf{Claim} We have a decomposition of $z$ in $G$,

\begin{center}
$\left[\begin{array}{cc}
A_{\,} & C\\
D & E_{\,}
\end{array}\right]=y_{1}y_{2}$ where $y_{1}=\left[\begin{array}{cc}
A_{\,} & 0\\
D & \JDiag[n](A^{T})^{-1}\JDiag[n]
\end{array}\right]\in G$, $y_{2}=\left[\begin{array}{cc}
\mathrm{Id}_{n} & A^{-1}C\\
0 & \mathrm{Id}_{n}
\end{array}\right]\in \tilde{P}$
\par\end{center}

\noindent We first check that $z=y_{1}y_{2}$. We first reformulate ~\ref{equation:identitySP2n2} into
\begin{equation}
\label{equation:identitySP2n3}
\JDiag[n]A^{T}\JDiag[n](E-DA^{-1}C)=\mathrm{Id}_n
\end{equation}
So that
\begin{center}
$\begin{array}{rlr}
DA^{-1}C+\JDiag[n](A^{T})^{-1}\JDiag[n] & = DA^{-1}C+\JDiag[n](A^{T})^{-1}\JDiag[n]\JDiag[n]A^{T}\JDiag[n](E-DA^{-1}C) & ~\eqref{equation:identitySP2n3}\\
\; & =DA^{-1}C+E-DA^{-1}C & \; \\
\; & =E & \;\\
\end{array}$
\end{center}

With this is easily verified that $z=y_{1}y_{2}$.\\
It is clear that $y_{1}\in G$. To show $y_{2}\in G$ we need to check that $\JDiag[n](A^{-1}C)^{T}\JDiag[n]=A^{-1}C$.
\begin{center}
$\begin{array}{rlr}
(A^{-1}C)^{T}\JDiag[n] & = (A^{-1}C)^{T}\JDiag[n]\JDiag[n]A^{T}\JDiag[n](E-DA^{-1}C) & ~\eqref{equation:identitySP2n3}\\
\; & =C^{T}\JDiag[n](E-DA^{-1}C) & \; \\
\; & =E^{T}\JDiag[n]C-C^{T}\JDiag[n]DA^{-1}C & ~\eqref{equation:StructureSP2nCE}\\
\; & =(E-DA^{-1}C)^{T}\JDiag[n]C & ~\eqref{equation:identitySP2n1}\\
\; & =(E-DA^{-1}C)^{T}\JDiag[n]A\JDiag[n]\JDiag[n]A^{-1}C & \; \\
\; & =(\JDiag[n]A^{T}\JDiag[n](E-DA^{-1}C))^{T}\JDiag[n](A^{-1}C) & \;\\
\; & =\JDiag[n](A^{-1}C) & ~\eqref{equation:identitySP2n3} \\
\end{array}$
\end{center}

\noindent Thus $y_{2}\in G$. It is clear additionally that
$y_{2}\in\tilde{P}$(in fact $y_{2}\in B_{G}$).

\noindent Hence our claim follows and we have

\begin{center}
$\left[\begin{array}{cc}
A_{\,} & C\\
D & E_{\,}
\end{array}\right]=\left[\begin{array}{cc}
A_{\,} & 0\\
D & \JDiag[n](A^{T})^{-1}\JDiag[n]
\end{array}\right]$ mod $\tilde{P}$
\par\end{center}

\noindent Finally,
\[
\begin{bmatrix}
A_{n \times n} & 0_{n \times n} \\
D_{n \times n} & \JDiag[n](A^{T})^{-1}\JDiag[n]
\end{bmatrix} \;=\;
\begin{bmatrix}
A'_{n \times n} & 0_{n \times n} \\
D'_{n \times n} & \JDiag[n](A'^{T})^{-1}\JDiag[n]
\end{bmatrix} \mod \tilde{P}
\]
\noindent if and only if there exist matrices $q \in P'_{\widehat{r-k}}$, and
$q' \in \mathrm{Mat}_n$ such that \[
\begin{bmatrix} A' & 0_{n \times n} \\ D' & \JDiag[n](A^{T})^{-1}\JDiag[n] \end{bmatrix} =
\begin{bmatrix} A & 0_{n \times n} \\ D & \JDiag[n](A'^{T})^{-1}\JDiag[n] \end{bmatrix} \begin{bmatrix} q & q' \\ 0_{n \times n} & \JDiag[n](q^{T})^{-1}\JDiag[n]
\end{bmatrix}, \]
which holds if and only if
$q'=0$, $A' = Aq$ and $D' = Dq$ (since
$A$ and $A'$ are invertible).

\noindent \eqref{proposition:structureOMinusSLnModQ}: There is a surjective morphism of $\mathbb{C}$-group
schemes $P\rightarrow \mathrm{GL}_{n}$

\begin{center}
$\left[\begin{array}{cc}
A_{\,} & C\\
0 & E_{\,}
\end{array}\right]\rightarrow A$
\par\end{center}

\noindent This induces the required isomorphism. The
element

\begin{center}
$\left[\begin{array}{cc}
A_{\,} & C\\
D & E_{\,}
\end{array}\right]$ mod $\tilde{P}\in O_{G/P}^{-}\times P/\tilde{P}$
\par\end{center}

\noindent decomposes uniquely as

\begin{center}
$\left[\begin{array}{cc}
\mathrm{Id}_{n} & 0\\
DA^{-1} & \mathrm{Id}_{n}
\end{array}\right]
\left(\left[\begin{array}{cc}
A_{\,} & C\\
0 & E-DA^{-1}C_{\,}
\end{array}\right]\mbox{ mod }\tilde{P}\right)$
\par\end{center}

\noindent and hence it is mapped to $A$ mod $P'_{\widehat{r-k}}$.
\end{proof}

\subsection{Homogeneous Bundles and Representations}
\label{subsection:Preliminaries:homogVectorBundles}

Let $Q$ be a parabolic subgroup of $\mathrm{GL}_n$. We collect here
some results about homogeneous vector-bundles on $\mathrm{GL}_n/Q$. Most of these
results are well-known, but for some of them, we could not find a
reference, so we give a proof here for the sake of completeness. Online
notes of G. Ottaviani~\cite{OttavianiRHV95} and of D. Snow~\cite {SnowHomogVB} discuss the details of many of these results.

Let $L_Q$ and $U_Q$  be respectively the Levi subgroup and the unipotent
radical of $Q$.

Let $E$ be a finite-dimensional vector-space on which $Q$ acts on the
right.

\begin{definition}
\label{definition:assbundle}
Define $\mathrm{GL}_n \times^Q E :=
(\mathrm{GL}_n \times E)/\sim$ where $\sim$ is the equivalence relation $(g,e) \sim
(gq,eq)$ for every $g \in \mathrm{GL}_n$, $q \in Q$ and $e \in E$.
Then $\pi_E : \mathrm{GL}_n \times^Q E \longrightarrow \mathrm{GL}_n/Q, (g,e)\mapsto gQ$, is a
vector-bundle called the
\emph{vector-bundle associated to} $E$ (and the principal $Q$-bundle
$\mathrm{GL}_n \longrightarrow \mathrm{GL}_n/Q$).  For $g\in \mathrm{GL}_n,e\in E$, we write $[g,e] \in \mathrm{GL}_n
\times^Q E$ for the equivalence class of
$(g,e) \in \mathrm{GL}_n \times E$ under $\sim$.
We say that a vector-bundle $\pi : \mathbf E \longrightarrow \mathrm{GL}_n/Q$ is
\emph{homogeneous} if  $\mathbf E$ has a $\mathrm{GL}_n$-action and $\pi$ is
$\mathrm{GL}_n$-equivariant, i.e, for every $y \in \mathbf E$, $\pi(g\cdot y) = g \cdot
\pi(y)$.
\end{definition}

\begin{remark}
There is a similar construction in the case when $E$ is a left $Q$-module.
\end{remark}

In this section, we abbreviate  $\mathrm{GL}_n \times^Q E$ as $\widetilde E$.  It
is known that $\mathbf E$ is homogeneous if and only if
$\mathbf E \simeq \widetilde E$ for some $Q$-module $E$. (If this is the case,
then $E$ is the fibre of
$\mathbf E$ over the coset $Q$.)
A homogeneous bundle $\widetilde E$ is said to be \emph{irreducible}
(respectively \emph{indecomposable},
\emph{completely reducible}) if $E$ is a
{irreducible} (respectively {indecomposable}, {completely reducible})
$Q$-module.
It is known that $E$ is completely reducible if and only if $U_Q$ acts
trivially and that $E$ is irreducible if and only if additionally it is
irreducible as a representation of $L_Q$.
See~\cite[Section~5]{SnowHomogVB} or~\cite[Section~10]{OttavianiRHV95} for the details.

\begin{discussionbox}
\label{discussionbox:taut}
For this paper we will primarily be interested in the case when $\mathrm{GL}_n/Q$ is a Grassmannian. Thus let $Q = P_{\widehat{m}}$,
with $1 \leq m \leq n-1$.
A weight $\lambda$ is said to be \define{$Q$-dominant} if and only if
when we express $\lambda$ as $\sum_{i=1}^n \lambda_i \epsilon_i$ (where $\epsilon_i$, $1\leq i\leq n$, is the character that sends a diagonal matrix in $T$ to its $i$th entry),
then $\lambda_{1} \geq \ldots \geq \lambda_{m}$ and $\lambda_{m+1} \geq \ldots \geq \lambda_{n}$.
We will write $\lambda = (\lambda_1, \ldots, \lambda_n)$
to mean that $\lambda = \sum_{i=1}^n \lambda_i \epsilon_i$.
Every finite-dimensional irreducible $Q$-module is of the form
$\homology^0(Q/B_n, L_\lambda)$ for a $Q$-dominant weight $\lambda$.
Hence the irreducible homogeneous vector-bundles on $\mathrm{GL}_n/Q$ are in correspondence with
$Q$-dominant weights. We describe them now. If $Q = P_{\widehat{n-i}}$,
then $\mathrm{GL}_n/Q = {\Grass_{i,n}}$. (Recall that, for us, the $\mathrm{GL}_n$-action
on
$\mathbb{C}^n$ is on the right.)
On $\Grass_{i,n}$, we have the \define{tautological sequence}
\begin{equation}
\label{equation:tautseq}
0 \longrightarrow \mathcal R_i \longrightarrow \mathbb{C}^{n} \otimes \strSh_{\Grass_{i,n}} \longrightarrow
\mathcal Q_{n-i} \longrightarrow 0
\end{equation}
of homogeneous vector-bundles. The bundle $\mathcal R_i$ is called the
\define{tautological sub-bundle} (of the trivial bundle $\mathbb{C}^n$) and
$\mathcal Q_{n-i}$ is called the \define{tautological quotient bundle}.  Every
irreducible homogeneous bundle on $\Grass_{i,n}$ is of the form
$\Schur_{(\lambda_{1}, \cdots, \lambda_{n-i})} \mathcal Q_{n-i}^* \otimes
\Schur_{(\lambda_{n-i+1}, \cdots, \lambda_{n})} \mathcal R_{i}^*$
for some $P_{\widehat{n-i}}$-dominant weight $\lambda$.
Here $\Schur_{\mu}$ denotes the \define{Schur functor} associated to the
partition $\mu$ (cf. \cite[\S 6.1]{FulHarRepTheory91}).

A $Q$-dominant weight is called \define{$(m)$-dominant}
in~\cite[p.~114]{WeymanCohVBSyz03}.  Although our definition looks like
Weyman's definition, we should keep in mind that our action is on the
right. We only have to be careful when we apply the Borel-Weil-Bott theorem
(more specifically, the Bott-algorithm). In this paper, our computations are
done only on Grassmannians. If $\mu$ and $\nu$ are partitions, then
$(\mu,\nu)$ will be $Q$-dominant (for a suitable $Q$), and will give us the
vector-bundle $\Schur_\mu \mathcal Q^* \otimes \Schur_\nu \mathcal R^*$ (this is
where the right-action of $Q$ becomes relevant) and to compute its
cohomology, we will have to apply the Bott-algorithm to the $Q$-dominant
weight $(\nu,\mu)$.  (In~\cite{WeymanCohVBSyz03}, one would get $\Schur_\mu
\mathcal R^* \otimes \Schur_\nu \mathcal Q^*$ and would apply the Bott-algorithm to $(\mu,\nu)$.)
\end{discussionbox}

We now give a brief description of the Bott-algorithm for computing the cohomology of irreducible homogeneous vector bundles~\cite[Remark 4.1.5]{WeymanCohVBSyz03}.

Let $\alpha=(\alpha_1,\ldots,\alpha_n)$ be a weight. The permutation $\sigma_i$ acts on the set of weights in the following way:
\begin{equation}
\label{equation:exchangeAction}
\sigma_i \alpha = ((\alpha_1,\ldots,\alpha_{i-1},\alpha_{i+1} - 1, \alpha_{i} + 1,\alpha_{i+2},\ldots,\alpha_n)
\end{equation}

The Bott-algorithm may be applied to our case as follows. Let $Q = P_{\widehat{m}}$,
with $1 \leq m \leq n-1$ and let $\lambda=(\lambda_1,\ldots,\lambda_n)$ be a $Q$-dominant weight with associated homogeneous vector bundle $V(\alpha)$. Let $\lambda'=(\lambda_{m+1},\ldots,\lambda_n,\lambda_1,\ldots,\lambda_m)$.

If $\lambda'$ is nonincreasing, then $\homology^{0}(\mathrm{GL}_n/Q,V(\lambda))=\Schur_{\lambda'}\mathbb{C}^{n}$ and $\homology^{i}(\mathrm{GL}_n/Q,V(\lambda))=0$ for $i>0$. Otherwise we start to apply the exchanges of type \eqref{equation:exchangeAction}, trying to move bigger numbers to the right past the smaller numbers. Two possibilities can occur:
\begin{enumerate}[label=(\arabic*)]
\item We apply an exchange of type \eqref{equation:exchangeAction} and it leaves the sequence unchanged. In this case  $\homology^{i}(\mathrm{GL}_n/Q,V(\lambda))=0$ for $i\geq0$.
\item After applying j exchanges, we transform $\lambda'$ into a nonincreasing
sequence $\beta$. Then we have $\homology^{i}(\mathrm{GL}_n/Q,V(\lambda))=0$ for $i\neq j$ and $\homology^{j}(\mathrm{GL}_n/Q,V(\lambda))=\Schur_{\beta}\mathbb{C}^{n}$.
\end{enumerate}

\section{Properties of Schubert Desingularization in Type C}
\label{sec:desing}
\numberwithin{equation}{section}

Recall the following result about the tangent space of a Schubert
variety, see \cite[Chapter 4]{BilleyLakshmibaiSingularLoci00} for details.

\begin{proposition}
\label{proposition:typeATangent}
Let $Q$ be a parabolic
subgroup of $\mathrm{SL}_{2n}$. Let $\tau\in W^{Q}$. Then the dimension of
the tangent space of $X_{Q}(\tau)$ at $e_{id}$ is $\#\{s_{\alpha}|\alpha\in R^{-}\backslash R_{Q}^{-}\,\, and\,\,\tau\geq s_{\alpha}\,\, in\,\, W/W_{Q}\}$.
In particular, $X_{Q}(\tau)$ is smooth if and only if dim$X_{Q}(\tau)=\#\{s_{\alpha}|\alpha\in R^{-}\backslash R_{Q}^{-}\,\, and\,\,\tau\geq s_{\alpha}\,\, in\,\, W/W_{Q}\}$.
\end{proposition}

\begin{notation}
\label{notation:parabolicSubgroupsStr}For an integer $i$ with $1\leq i\leq n$ we define $i^{\prime}=2n+1-i$. Let $1\leq k<r\leq n$. Then
\begin{center}
$\mathcal{W}_{k,r}=
\begin{cases}
(k+1,...,r,n^{\prime},...,(r+1)^{\prime},k^{\prime},...,1^{\prime})\in W^{P}&\text{if $r<n$}\\
(k+1,...,r,k^{\prime},...,1^{\prime})\in W^{P}&\text{if $r=n$}\\
\end{cases}$
\end{center}
Let $1\leq k<r\leq n$ be integers. Let $w \in \mathcal{W}_{k,r}$ with $\tilde{w}$ its minimal representative in $W^{\tilde{P}}$.
\end{notation}

\begin{proposition}
\label{proposition:schubertVarietySL2nSmooth}
The Schubert variety $X_{\tilde{Q}}(\tilde{w})$ in $H/\tilde{Q}$ is smooth.
\end{proposition}

\begin{proof}
Let $w_{max}\in W_{H}(=S_{2n})$ be the
maximal representative of $\tilde{w}$. Then
\begin{center}
$w_{max}=
\begin{cases}
([r,k+1][1^{\prime},k^{\prime}][(r+1)^{\prime},n^{\prime}][n,(r+1)][k,1][(k+1)^{\prime},r^{\prime}])&\text{if $r<n$}\\
([r,k+1][1^{\prime},k^{\prime}][k,1][(k+1)^{\prime},r^{\prime}])&\text{if $r=n$}\\
\end{cases}$
\end{center}
To see this we need to show that $X_{P_{\hat{i}}}(w_{max})=X_{P_{\hat{i}}}(\tilde{w})$
for $i=r-k,n,2n-(r-k)$ and that $w_{max}$ is the maximal element of $W_{H}$
with this property. But this follows from the fact that for $\tau=(c_{1},...,c_{2n})\in W_{H}$
and $1\leq i\leq2n$ we have that $X_{P_{\hat{i}}}(\tau)=X_{P_{\hat{i}}}(\tau^{\prime})$
where $\tau^{\prime}\in W^{P_{\hat{i}}}$ is the element with $c_{1},...c_{i}$
written in increasing order.

Thus $X_{B_{H}}(w_{max})$ is the inverse image of $X_{\tilde{Q}}(\tilde{w})$
under the natural morphism $H/B_{H}\rightarrow H/\tilde{Q}$. As $w_{max}$
is a $4231$ and $3142$ avoiding element of $W_{H}$ we have that
$X_{B_{H}}(w_{max})$ is nonsingular(see {[}BL00, 8.1.1{]}). Since
the morphism $H/B_{H}\rightarrow H/\tilde{Q}$ has nonsingular fibres
(namely $\tilde{Q}/B_{H}$) we have $X_{\tilde{Q}}(\tilde{w})$ must
be smooth.
\end{proof}

\begin{proposition}
\label{proposition:schubertVarietySP2nSmooth}
The Schubert variety $X_{\tilde{P}}(\tilde{w})$
in $G/\tilde{P}$ is smooth .
\end{proposition}

\begin{proof} Let $w_{max}$ be as in the proof of Proposition \ref{proposition:schubertVarietySL2nSmooth}. Then clearly  $w_{max}$ is in $W_G$ and $X_{B_{G}}(w_{max})$ is the inverse image of $X_{\tilde{P}}(\tilde{w})$
under the natural morphism $G/B_{G}\rightarrow G/\tilde{P}$.

\noindent\textbf{Claim} $X_{B_{G}}(w_{max})$ is smooth.

\noindent Note that the claim implies the required result (since, the canonical morphism $G/B_G\rightarrow G/\tilde{P}$ is a fibration with nonsingular fibers (namely, $\tilde{P}/B_G$)). To prove the claim, as seen in the proof of Proposition \ref{proposition:schubertVarietySL2nSmooth}, we have that  $X_{B_{H}}(w_{max})$ is smooth. We conclude the smoothness of $X_{B_{G}}(w_{max})$ using the following two formulas \cite[\S 3(VI), Remark 5.8]{LakshmibaiG/PVII87}: Let $\theta\in W_G$, say, $\theta=(a_1,\cdots a_n)$.
$$l_G(\theta)=\frac{1}{2}[l_H(\theta)+m(\theta)]\leqno{(1)}$$
where $m(\theta)=\#\{i,1\le i\le m|a_i>m\}$.
$$dim\ T_{id}(\theta,G)=\frac{1}{2}[dim\ T_{id}(\theta,H)+c(\theta)] \leqno{(2)}$$
 where $c(\theta)=\#\{1\le i\le m\,|\,\theta\ge s_{\epsilon_{2i}}\}$, and $T_{id}(\theta,G)$(resp $T_{id}(\theta,H)$) denotes the Zariski tangent space of $X_{B_G}(\theta)$(resp $X_{B_H}(\theta)$) at $e_{id}$. Note that $s_{\epsilon_{2i}}$ is just the transposition $(i,i')$(cf. ~\ref{equation:simpleReflectionsSP2n}). Now taking $\theta=w_{max}$,
we have, $c(w_{max})=m(w_{max})$. Hence we obtain from (1), (2) that dim$\, T_{id}(w_{max},G)=l_G(w_{max})$, proving that $X_{B_{G}}(w_{max})$ is smooth at $e_{id}$, and hence is nonsingular (note that for a Schubert variety X, the singular locus of $X$, $Sing(X)$, is $B$-stable implying $e_{id}\in Sing(X)$ if $Sing(X)\neq \emptyset$). Thus the claim  (and hence the required result) follows.
\end{proof}
\begin{remark}
We have that $X_{\tilde{P}}(\tilde{w})$  is the fixed
point set under an automorphism of  order two of the Schubert variety $X_{\tilde{Q}}(\tilde{w})$
and thus is smooth, provided char$\,K\ne 2$.\cite[Proposition 3.4]{EdixNeron92}
\end{remark}

\begin{discussionbox}
\label{discussionbox:OMinusPlucker}
To give a characterization of $Y_{\tilde{Q}}(\tilde{w})$ we first need a review of the structure of $O^{-}_{H/\tilde{Q}}$ and its Pl\"ucker coordinates.

Recall that for the Pl\"ucker embedding of the Grassmannian $\Grass_{d,n}$, the \define{Pl\"ucker coordinate} $p_{\underline{i}}(U)$, $U\in \Grass_{d,n}$ and $\underline{i}=(i_1,\ldots,i_d)$ with $1\leq i_1 < \ldots < i_d < n$, is just the $d\times d$ minor of the matrix $A_{n\times d}$ with row indices $(i_1,\ldots,i_d)$ (here the matrix $A_{n\times d}$ represents the $d$-dimensional subspace $U$ with respect to the standard basis).

$O^{-}_{H/\tilde{Q}}$ can be
identified with the affine space of lower-triangular matrices with possible
non-zero entries $x_{ij}$ at row $i$ and column $j$ where $(i,j)$ is such
that there exists a
$l \in \{r-k, n, 2n-(r-k)\}$ such that $j \leq l < i \leq N$.
To see this, note that we are interested in those
$(i,j)$ such that the root $\epsilon_i - \epsilon_j$ belongs to
$R^{-} \setminus R_{\tilde{Q}}^{-}$. Since
$R_{\tilde{Q}}^{-} = R_{Q_{\widehat{r-k}}}^{-} \bigcap R_{Q_{\widehat{n}}} \bigcap R_{Q_{\widehat{2n-(r-k)}}}$,
we see that we are looking for
$(i,j)$ such that $\epsilon_i - \epsilon_j \in
R^{-} \setminus R_{Q_{\widehat{l}}}^{-}$, for some $l \in
\{r-k, n, 2n-(r-k)\}$. For the maximal parabolic subgroup $P_{\widehat{l}}\,$, we have,
$R^{-} \setminus R_{Q_{\widehat{l}}}^{-} =
\{\epsilon_i-\epsilon_j \mid 1\le j\le l<i\le N\}$. We have $\dim O^{-}_{H/\tilde{Q}} = |R^{-}\setminus R^{-}_{\tilde{Q}}|$.

Thus we have the following identification \begin{equation}
\label{equation:StructureOMinusTQ}
O^{-}_{H/\tilde{Q}} = \left[\begin{array}{cccc}
\mathrm{Id}_{r-k} & 0 & 0 & 0 \\
A' & \mathrm{Id}_{n-(r-k)} & 0 & 0 \\
\mathcal{D}_{1} & \mathcal{D}_{2} & \mathrm{Id}_{n-(r-k)} & 0 \\
\mathcal{D}_{3} & \mathcal{D}_{4} & E' & \mathrm{Id}_{r-k} \\
\end{array}\right]
\end{equation}
\\ \noindent where the block matrices have possible non-zero entries $x_{ij}$ given by \\
\makebox[0.47\textwidth][l]{$A'=\left[\begin{array}{ccc} x_{(r-k)+1\;\;1}&\ldots&x_{(r-k)+1\;\;r-k} \\
\vdots&\;&\vdots \\
x_{n\;\;1}&\ldots&x_{n\;\;r-k} \\
\end{array}\right]$}
\makebox[0.5\textwidth][l]{$E'=\left[\begin{array}{ccc} x_{2n-(r-k)+1\;\;n+1} & \ldots & x_{2n-(r-k)+1\;\;2n-(r-k)} \\
\vdotswithin{\ldots} & \; & \vdotswithin{\ldots} \\
x_{2n\;\;n+1} & \ldots & x_{2n\;\;2n-(r-k)} \\
\end{array}\right]$} \par \noindent
\makebox[0.47\textwidth][l]{$\mathcal{D}_{1}=\left[\begin{array}{ccc}
x_{n+1\;\;1} & \ldots & x_{n+1\;\;r-k} \\
\vdotswithin{\ldots} & \; & \vdotswithin{\ldots} \\
x_{2n-(r-k)\;\;1} & \ldots & x_{2n-(r-k)\;\;r-k} \\
\end{array}\right]$}
\makebox[0.5\textwidth][l]{$\mathcal{D}_{2}=\left[\begin{array}{ccc}
x_{n+1\;\;(r-k)+1} & \ldots & x_{n+1\;\;n} \\
\vdotswithin{\ldots} & \; & \vdotswithin{\ldots} \\
x_{2n-(r-k)\;\;(r-k)+1} & \ldots & x_{2n-(r-k)\;\;n} \\
\end{array}\right]$} \par \noindent
\makebox[0.47\textwidth][l]{$\mathcal{D}_{3}=\left[\begin{array}{ccc}
x_{2n-(r-k)+1\;\;1} & \ldots & x_{2n-(r-k)+1\;\;r-k} \\
\vdotswithin{\ldots} & \; & \vdotswithin{\ldots} \\
x_{2n\;\;1} & \ldots & x_{2n\;\;r-k} \\
\end{array}\right]$}
\makebox[0.5\textwidth][l]{$\mathcal{D}_{4}=\left[\begin{array}{ccc}
x_{2n-(r-k)+1\;\;(r-k)+1} & \ldots & x_{2n-(r-k)+1\;\;n} \\
\vdotswithin{\ldots} & \; & \vdotswithin{\ldots} \\
x_{2n\;\;(r-k)+1} & \ldots & x_{2n\;\;n} \\
\end{array}\right]$} \par \noindent

For $i>r$, $j\leq r-k$ the Pl\"ucker coordinate $p_{(i,j)}^{(r-k)}$ on the Grassmannian $H/Q_{\widehat{r-k}}$ lifts to a regular function on $H/\tilde{Q}$. It's restriction to $O_{H/\tilde{Q}}^{-}$ is the $r-k\times r-k$ minor of ~\eqref{equation:StructureOMinusTQ} with column indices $\{1,2,\ldots,r-k\}$ and row indices $\{1,\ldots,j-1,j+1,\ldots,r-k,i\}$. This minor is the determinant of a $r-k\times r-k$ matrix with the top $(r-k)-1$ rows equal to $\mathrm{Id}_{r-k}$ omitting the $j$th row, and the bottom row equal to the first $r-k$ entries of the $i$th row of ~\eqref{equation:StructureOMinusTQ}. The determinant of this matrix is thus $(-1)^{(r-k)-j}x_{ij}$. Thus for $i>r$, $j\leq r-k$ \begin{equation}
\label{equation:PluckerRMK}
p_{(i,j)}^{(r-k)}\big{|}_{O_{H/\tilde{Q}}^{-}}=(-1)^{(r-k)-j}x_{ij}
\end{equation} For $i>2n-(r-k)$, $n<j\leq 2n-(r-k)$ the Pl\"ucker coordinate $p_{(i,j)}^{(2n-(r-k))}$ on the Grassmannian $H/Q_{\widehat{2n-(r-k)}}$ lifts to a regular function on $H/\tilde{Q}$. Its restriction to $O_{H/\tilde{Q}}^{-}$ is the $2n-(r-k)\times 2n-(r-k)$ minor of ~\eqref{equation:StructureOMinusTQ} with column indices $\{1,2,\ldots,2n-(r-k)\}$ and row indices $\{1,\ldots,j-1,j+1,\ldots,2n-(r-k),i\}$. This minor is the determinant of \begin{equation}
\label{equation:Plucker2nRMKMatrix}
\left[\begin{array}{ccc}
\mathrm{Id}_{r-k} & 0 & 0\\
A' & \mathrm{Id}_{n-(r-k)} & 0 \\
\hat{\mathcal{D}_{1}} & \hat{\mathcal{D}_{2}} & \hat{I_1}\\
\left[x_{i\;1}\ldots x_{i\;r-k}\right] & \left[x_{i\;(r-k)+1}\ldots x_{i\;n}\right] & \left[x_{i\;n+1}\ldots x_{i\;2n-(r-k)}\right]
\end{array}\right]
\end{equation} where $\hat{\mathcal{D}_{1}}, \hat{\mathcal{D}_{2}},$ and $\hat{I_1}$ are equal to, respectively,  $\mathcal{D}_{1}, \mathcal{D}_{2},$ and $\mathrm{Id}_{n-(r-k)}$ with their $(j-n)$th rows omitted. The determinant of ~\eqref{equation:Plucker2nRMKMatrix} is equal to the determinant of
\begin{center}
$\left[\begin{array}{c}
\hat{I_1} \\
\left[x_{i\;n+1}\ldots x_{i\;2n-(r-k)}\right] \\
\end{array}\right]$
\end{center}
As above this is just an identity matrix with a single row replaced and so its determinant is just equal $(-1)^{2n-(r-k)-j}x_{ij}$. Thus for $i>2n-(r-k)$, $n<j\leq 2n-(r-k)$\begin{equation}
\label{equation:Plucker2nRMK}
p_{(i,j)}^{(2n-(r-k))}\big{|}_{O_{H/\tilde{Q}}^{-}}=(-1)^{2n-(r-k)-j}x_{ij}
\end{equation} For $i>2n-(r-k)$, $r-k<j\leq n$ the Pl\"ucker coordinate $p_{(i,j)}^{(2n-(r-k))}$ on the Grassmannian $H/Q_{\widehat{2n-(r-k)}}$ lifts to a regular function on $H/\tilde{Q}$. Its restriction to $O_{H/\tilde{Q}}^{-}$ is the $2n-(r-k)\times 2n-(r-k)$ minor of ~\eqref{equation:StructureOMinusTQ} with column indices $\{1,2,\ldots,2n-(r-k)\}$ and row indices $\{1,\ldots,j-1,j+1,\ldots,2n-(r-k),i\}$. This minor is the determinant of\begin{equation}
\label{equation:Plucker2nRMKMatrix+}
\left[\begin{array}{ccc}
\mathrm{Id}_{r-k} & 0 & 0\\
\hat{A'} & \hat{I_2} & 0 \\
\mathcal{D}_{1} & \mathcal{D}_{2} & \mathrm{Id}_{n-(r-k)}\\
\left[x_{i\;1}\ldots x_{i\;r-k}\right] & \left[x_{i\;(r-k)+1}\ldots x_{i\;n}\right] & \left[x_{i\;n+1}\ldots x_{i\;2n-(r-k)}\right]
\end{array}\right]
\end{equation} where $\hat{A'}$ and $\hat{I_2}$ are equal to, respectively,  $A'$ and $\mathrm{Id}_{n-(r-k)}$ with their $j-(r-k)$th rows omitted. The determinant of ~\eqref{equation:Plucker2nRMKMatrix+} is equal to the determinant of \begin{equation}
\label{equation:Plucker2nRMKMatrix++}
\left[\begin{array}{cc}
\hat{I_2} & 0 \\
\mathcal{D}_{2} & \mathrm{Id}_{n-(r-k)}\\
\left[x_{i\;(r-k)+1}\ldots x_{i\;n}\right] & \left[x_{i\;n+1}\ldots x_{i\;2n-(r-k)}\right]
\end{array}\right]
\end{equation} To calculate this shift the bottom row so that it becomes the $j-(r-k)$th row of $\hat{I_2}$. Let $M=2n-(r-k)-j$. Then the determinant of ~\eqref{equation:Plucker2nRMKMatrix++} will be $(-1)^{M}$ times the determinant of\begin{equation}
\label{equation:Plucker2nRMKMatrix+++}
\left[\begin{array}{cc}
I_3 & Z \\
\mathcal{D}_{2} & \mathrm{Id}_{n-(r-k)}\\
\end{array}\right]
\end{equation} where $I_3$ is $\mathrm{Id}_{n-(r-k)}$ with the $j-(r-k)$th row replaced by $\left[x_{i\;(r-k)+1}\ldots x_{i\;n}\right]$ and $Z$ is the zero matrix with the $j-(r-k)$th row replaced by $\left[x_{i\;n+1}\ldots x_{i\;2n-(r-k)}\right]$. Since the lower right block matrix of ~\eqref{equation:Plucker2nRMKMatrix+++} commutes with its lower left block matrix we have that the determinant of ~\eqref{equation:Plucker2nRMKMatrix+++} is equal to the determinant of $I_3-Z\mathcal{D}_{2}$.
We have that $Z\mathcal{D}_{2}$ is equal to the zero matrix with its $j-(r-k)$th row replaced by
\begin{center}
$\left[x_{i\;(r-k)+1}\ldots x_{i\;n}\right]\mathcal{D}_{2}$
\end{center}
And thus $I_3-Z\mathcal{D}_{2}$ is equal to $\mathrm{Id}_{n-(r-k)}$ with the $j-(r-k)$th row replaced by
\begin{center}
$\left[x_{i\;(r-k)+1}\ldots x_{i\;n}\right] - \left[x_{i\;(r-k)+1}\ldots x_{i\;n}\right]\mathcal{D}_{2}$
\end{center}
And so the determinant of $I_3-Z\mathcal{D}_{2}$ is merely equal to the $j-(r-k)$th entry of $I_3-Z\mathcal{D}_{2}$ which is
\begin{center}
$x_{ij} - \left[x_{i\;(r-k)+1}\ldots x_{i\;n}\right]\left[x_{n+1\;j}\ldots x_{2n-(r-k)\;j}\right]^{T}$
\end{center}
Combining all our steps, we finally have that for $i>2n-(r-k)$, $r-k<j\leq n$ \begin{equation}
\label{equation:Plucker2nRMK+}
p_{(i,j)}^{(2n-(r-k))}\big{|}_{O_{H/\tilde{Q}}^{-}}=(-1)^{M}(x_{ij} - \left[x_{i\;(r-k)+1}\ldots x_{i\;n}\right]\left[x_{n+1\;j}\ldots x_{2n-(r-k)\;j}\right]^{T})
\end{equation}
\end{discussionbox}

\begin{theorem}
\label{theorem:schubertVarietySL2nMatrixForm}
The opposite cell $Y_{\tilde{Q}}(\tilde{w})$ can be identified with the subspace of $O_{H/\tilde{Q}}^{-}$ given by matrices of the form
\begin{center}
$\left[\begin{array}{cccc}
\mathrm{Id}_{r-k} & 0 & 0 & 0 \\
A' & \mathrm{Id}_{n-(r-k)} & 0 & 0 \\
0 & \mathcal{D}_{2} & \mathrm{Id}_{n-(r-k)} & 0 \\
0 & E'\mathcal{D}_{2} & E' & \mathrm{Id}_{r-k} \\
\end{array}\right]$
\end{center}
with $\mathcal{D}_{2} \in  \mathrm{Mat}_{n-(r-k)}$, $A'\in \mathrm{Mat}_{n-(r-k)\times r-k}$ with the bottom $n-r$ rows of $A'$ all zero, and $E'\in \mathrm{Mat}_{r-k\times n-(r-k)}$ with the left $n-r$ columns of $E'$ all zero.

\end{theorem}

\begin{proof}
For $j\leq r-k<i$ the reflection $(i,j)$
equals $(1,2,..,j-1,j+1,..,r-k,i)$ and $\tilde{w}$ equals $(k+1,...,r)$ in $W/W_{Q_{\widehat{r-k}}}$. Thus for $i>r$, and $j\leq r-k$
we have the reflection $(i,j)$ is not smaller than $\tilde{w}$ in
$W/W_{Q_{\widehat{r-k}}}$ so the Pl\"ucker coordinate $p_{(i,j)}^{(r-k)}$
vanishes on $X_{\tilde{Q}}(\tilde{w})$. We saw in ~\eqref{equation:PluckerRMK} that for such $(i,j)$ we have $p_{(i,j)}^{(r-k)}=(-1)^{(r-k)-j}x_{ij}$ and thus $x_{ij}\equiv0$
on $Y_{\tilde{Q}}(\tilde{w})$.

For $j\leq n<i$ the reflection $(i,j)$ equals $(1,2,..,j-1,j+1,..,n,i)$
and $\tilde{w}$ is equal to $(k+1,...,r,n^{\prime},...,(r+1)^{\prime},k^{\prime},...,1^{\prime})$ in $W/W_{Q_{\hat{n}}}$.
Thus there is no choice of $(i,j)$ such that $(i,j)$ is not smaller
than $\tilde{w}$ in $W/W_{Q_{\widehat{n}}}$.

For $j\leq 2n-(r-k)<i$ the reflection $(i,j)$ equals $(1,2,..,j-1,j+1,..,2n-(r-k),i)$
and $\tilde{w}$ equals $(1,...,n,n^{\prime},...,(r+1)^{\prime},k^{\prime},...,1^{\prime})$ in $W/W_{Q_{\hat{2n-(r-k)}}}$ .
Thus for $i>2n-(r-k)$, and $j\leq 2n-r$
we have the reflection $(i,j)$ is not smaller than $\tilde{w}$ in
$W/W_{Q_{\widehat{2n-(r-k)}}}$. We break these into two cases, ignoring those $j\leq r-k$ as we have already shown above that for $j\leq r-k$ and $i>2n-(r-k)$ we have $x_{ij}\equiv0$ on $Y_{\tilde{Q}}(\tilde{w})$.

The first case is for $(i,j)$ with $i>2n-(r-k)$, and $n<j\leq 2n-r$. The fact that $(i,j)$ is not smaller than $\tilde{w}$ in
$W/W_{Q_{\widehat{2n-(r-k)}}}$ implies that the Pl\"ucker coordinate $p_{(i,j)}^{(2n-(r-k))}$ vanishes on $X_{\tilde{Q}}(\tilde{w})$. We saw in ~\eqref{equation:Plucker2nRMK} that for such $(i,j)$ we have $p_{(i,j)}^{(2n-(r-k))}=(-1)^{2n-(r-k)-j}x_{ij}$ and thus $x_{ij}\equiv0$
on $Y_{\tilde{Q}}(\tilde{w})$.

The second case is for $(i,j)$ with $i>2n-(r-k)$ and $r-k<j\leq n$. $(i,j)$ is not smaller than $\tilde{w}$ in
$W/W_{Q_{\widehat{2n-(r-k)}}}$ implies that the Pl\"ucker coordinate $p_{(i,j)}^{(2n-(r-k))}$ vanishes on $X_{\tilde{Q}}(\tilde{w})$. We saw in ~\eqref{equation:Plucker2nRMK+} that for such $(i,j)$ we have $p_{(i,j)}^{(2n-(r-k))}=(-1)^{M}(x_{ij}-\left[x_{i\;(r-k)+1}\ldots x_{i\;n}\right]\left[x_{n+1\;j}\ldots x_{2n-(r-k)\;j}\right]^{T})$. Combining these two facts we get $x_{ij}=\left[x_{i\;(r-k)+1}\ldots x_{i\;n}\right]\left[x_{n+1\;j}\ldots x_{2n-(r-k)\;j}\right]^{T}$.

Noting that $\left[x_{i\;(r-k)+1}\ldots x_{i\;n}\right]$ is the $(2n-(r-k)-i)$th row of $E'$ and $\left[x_{n+1\;j}\ldots x_{2n-(r-k)\;j}\right]^{T}$ is the $(2n-(r-k)-j)$th column of $\mathcal{D}_{2}$ it is clear that $x_{ij}=(E'X)_{(2n-(r-k)-i)\;(2n-(r-k)-j)}$ on $Y_{\tilde{Q}}(\tilde{w})$.

On the other hand note that the reflections $(i,j)$ with $i>r$
and $j\leq r-k$, and $i>2n-(r-k)$ and $r-k<j\leq 2n-r$ are precisely the reflections $s_{\alpha}$ with
$\alpha\in R^{-}\backslash R_{\tilde{Q}}^{-}$ and $\tilde{w}\ngeq s_{\alpha}$
in $W/W_{\tilde{Q}}$. Since $X_{\tilde{Q}}(\tilde{w})$ is smooth
this implies by Proposition 3.1 that the codimension of $Y_{\tilde{Q}}(\tilde{w})$
in $O_{H/\tilde{Q}}^{-}$ equals $\#\{(i,j)|\, i>r\,\, and\,\, j\leq r-k,\,\,or\,\,i>2n-(r-k)\,\,and\,\,r-k<j\leq 2n-r\}$. Above we have shown that for each such $(i,j)$, $x_{ij}$ either vanishes, or is completely dependent on the entries of $E'X$. Thus $Y_{\tilde{Q}}(\tilde{w})$ is the subspace of $O_{H/\tilde{Q}}^{-}$
defined by the vanishing of $\{x_{ij}|\, i>r\,\, and\,\, j\leq r-k\,\,,or\,\,i>2n-(r-k)\,\,and\,\,n<j\leq 2n-r\}$ and $x_{ij}=(E'X)_{(2n-(r-k)-i)\;(2n-(r-k)-j)}$ for $i>2n-(r-k)$ and $r-k<j\leq n$.
\end{proof}

\begin{example}
Let $k=2$,$r=4$, and $n=5$. Then $\tilde{Q}=Q_{\hat{2},\hat{5},\hat{8}}$, $w=(3,4,6,9,10)$, and $\tilde{w}=(3,4,6,9,10,1,2,5)$. Then
\begin{center}
$O_{H/\tilde{Q}}^{-}=\left[\begin{array}{cccccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
x_{31} & x_{32} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
x_{41} & x_{42} & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
x_{51} & x_{52} & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
x_{61} & x_{62} & x_{63} & x_{64} & x_{65} & 1 & 0 & 0 & 0 & 0\\
x_{71} & x_{72} & x_{73} & x_{74} & x_{75} & 0 & 1 & 0 & 0 & 0\\
x_{81} & x_{82} & x_{83} & x_{84} & x_{85} & 0 & 0 & 1 & 0 & 0\\
x_{91} & x_{92} & x_{93} & x_{94} & x_{95} & x_{96} & x_{97} & x_{98} & 1 & 0\\
x_{101} & x_{102} & x_{103} & x_{104} & x_{105} & x_{106} & x_{107} & x_{108} & 0 & 1\\
\end{array}\right]$
\end{center}
And $Y_{\tilde{P}}(\tilde{w})$ will be the subspace of $O_{H/\tilde{Q}}^{-}$ given by
\begin{center}
$\left[\begin{array}{cccccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
x_{31} & x_{32} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
x_{41} & x_{42} & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & x_{63} & x_{64} & x_{65} & 1 & 0 & 0 & 0 & 0\\
0 & 0 & x_{73} & x_{74} & x_{75} & 0 & 1 & 0 & 0 & 0\\
0 & 0 & x_{83} & x_{84} & x_{85} & 0 & 0 & 1 & 0 & 0\\
0 & 0 & x_{97}x_{73}+x_{98}x_{83} & x_{97}x_{74}+x_{98}x_{84} & x_{97}x_{75}+x_{98}x_{85} & 0 & x_{97} & x_{98} & 1 & 0\\
0 & 0 & x_{107}x_{73}+x_{108}x_{83} & x_{107}x_{74}+x_{108}x_{84} & x_{107}x_{75}+x_{108} & 0 & x_{107} & x_{108} & 0 & 1\\
\end{array}\right]$
\end{center}
\end{example}

\begin{corollary}
\label{corollary:schubertVarietySP2nMatrixForm}
The opposite cell $Y_{\tilde{P}}(\tilde{w})$ can be identified with the subspace of $O_{G/\tilde{P}}^{-}$ given by matrices of the form
\begin{center}
$\left[\begin{array}{cccc}
\mathrm{Id}_{r-k} & 0 & 0 & 0 \\
A' & \mathrm{Id}_{n-(r-k)} & 0 & 0 \\
0 & \mathcal{D}_{2} & \mathrm{Id}_{n-(r-k)} & 0 \\
0 & -\JDiag[r-k](A')^{T}\JDiag[r-k]\mathcal{D}_{2} & -\JDiag[r-k](A')^{T}\JDiag[r-k] & \mathrm{Id}_{r-k} \\
\end{array}\right]$
\end{center}
with $\JDiag[n-(r-k)]\mathcal{D}_{2} \in  \mathrm{Sym}_{n-(r-k)}$ and $A'\in \mathrm{Mat}_{n-(r-k)\times r-k}$ with the bottom $n-r$ rows of $A'$ all zero.

\end{corollary}
\begin{proof}
Let $y \in Y_{\tilde{P}}(\tilde{w})=(Y_{\tilde{Q}}(\tilde{w}))^{\sigma}\subset Y_{\tilde{Q}}(\tilde{w})$. So $y$ is just an element of $Y_{\tilde{Q}}(\tilde{w})$ that is fixed under the involution $\sigma$. That is, an element which satisfies ~\eqref{equation:StructureSP2nAD}-\eqref{equation:StructureSP2nADCE}. Theorem \ref{theorem:schubertVarietySL2nMatrixForm} gives us that $y$ is of the form
\begin{center}
$\left[\begin{array}{cccc}
\mathrm{Id}_{r-k} & 0 & 0 & 0 \\
A' & \mathrm{Id}_{n-(r-k)} & 0 & 0 \\
0 & \mathcal{D}_{2} & \mathrm{Id}_{n-(r-k)} & 0 \\
0 & E'\mathcal{D}_{2} & E' & \mathrm{Id}_{r-k} \\
\end{array}\right]$
\end{center}
with $\mathcal{D}_{2} \in  \mathrm{Mat}_{n-(r-k)}$, $A'\in \mathrm{Mat}_{n-(r-k)\times r-k}$ with the bottom $n-r$ rows of $A'$ all zero, and $E'\in \mathrm{Mat}_{r-k\times n-(r-k)}$ with the left $n-r$ columns of $E'$ all zero.
We must now check what restrictions on $y$ are required for it to satisfy ~\eqref{equation:StructureSP2nAD}-\eqref{equation:StructureSP2nADCE}.
For $y$ to satisfy \eqref{equation:StructureSP2nADCE} we know that
\begin{center}
$\left[\begin{array}{cc}
\mathrm{Id}_{r-k} & 0 \\
A' & \mathrm{Id}_{n-(r-k)} \\
\end{array}\right]^{T}
\left[\begin{array}{cc}
0 & \JDiag[r-k] \\
\JDiag[n-(r-k)] & 0 \\
\end{array}\right]
\left[\begin{array}{cc}
\mathrm{Id}_{r-k} & 0 \\
E' & \mathrm{Id}_{n-(r-k)} \\
\end{array}\right] =\left[\begin{array}{cc}
(A')^{T}\JDiag[r-k]+\JDiag[r-k]E' & \JDiag[r-k] \\
\JDiag[n-(r-k)] & 0 \\
\end{array}\right]$
\end{center}
must equal
\begin{center}
$\left[\begin{array}{cc}
0 & \JDiag[r-k] \\
\JDiag[n-(r-k)] & 0 \\
\end{array}\right]$
\end{center}
which implies that $E'=-\JDiag[r-k](A')^{T}\JDiag[r-k]$.\\
Any $y$ clearly satisfies ~\eqref{equation:StructureSP2nCE}.\\
And finally for $y$ to satisfy ~\eqref{equation:StructureSP2nAD} we must have
\begin{center}
$\left[\begin{array}{cc}
0 & \mathcal{D}_{2} \\
0 & -\JDiag[r-k](A')^{T}\JDiag[r-k]\mathcal{D}_{2} \\
\end{array}\right]^{T}
\left[\begin{array}{cc}
0 & \JDiag[n-(r-k)] \\
\JDiag[r-k] & 0 \\
\end{array}\right]
\left[\begin{array}{cc}
\mathrm{Id}_{r-k} & 0 \\
A' & \mathrm{Id}_{n-(r-k)} \\
\end{array}\right] =\left[\begin{array}{cc}
0 & 0 \\
0 & \mathcal{D}_{2}^{T}\JDiag[n-(r-k)] \\
\end{array}\right]$
\end{center}
must equal
\begin{center}
$\left[\begin{array}{cc}
\mathrm{Id}_{r-k} & 0 \\
A' & \mathrm{Id}_{n-(r-k)} \\
\end{array}\right]^{T}
\left[\begin{array}{cc}
0 & \JDiag[r-k] \\
\JDiag[n-(r-k)] & 0 \\
\end{array}\right]
\left[\begin{array}{cc}
0 & \mathcal{D}_{2} \\
0 & -\JDiag[r-k](A')^{T}\JDiag[r-k]\mathcal{D}_{2} \\
\end{array}\right] =\left[\begin{array}{cc}
0 & 0 \\
0 & \JDiag[n-(r-k)]\mathcal{D}_{2} \\
\end{array}\right]$
\end{center}
which implies that $\JDiag[n-(r-k)]\mathcal{D}_{2}=\mathcal{D}_{2}^{T}\JDiag[n-(r-k)]$, or equivalently $\JDiag[n-(r-k)]\mathcal{D}_{2}\in \mathrm{Sym}_{n-(r-k)}$.
\end{proof}

\begin{remark}
\label{remark:identificationOMinusPTP}
We may identify $O_{P/\tilde{P}}^{-}$ with $O_{\mathrm{GL}_n/P'_{\widehat{r-k}}}^{-}$ under the map
\begin{center}
$\left[\begin{array}{cc}
\mathcal{A} & 0 \\
0 & \JDiag[n](\mathcal{A}^{T})^{-1}\JDiag[n]
\end{array}\right]\mapsto \mathcal{A}$
\end{center}
\end{remark}

\begin{remark}
\label{remark:identificationOfYPWasProduct}
Let $V_{w}$ be the linear subspace of $\mathrm{Sym}_n$ given by
$x_{ij}=0$ if $j\leq r-k$ or $i<n-(r-k)$.
And let $V_{w}^{\prime}$ is the linear subspace of $O_{\mathrm{GL}_n/P'_{\widehat{r-k}}}^{-}$
given by $x_{ij}=0$ if $i>r$ and $j\leq r-k$.

\noindent Consider the map $\delta:Y_{\tilde{P}}(\tilde{w})\hookrightarrow O_{G/\tilde{P}}^{-} = O_{G/P}^{-}\times O_{P/\tilde{P}}^{-} \cong O_{G/P}^{-}\times O_{\mathrm{GL}_n/P'_{\widehat{r-k}}}^{-}$, where the first map is inclusion, the second is simply the product decomposition, and the final map is from Remark \ref{remark:identificationOMinusPTP}. This map is given explicitly by
\begin{center}
$\left[\begin{array}{cccc}
\mathrm{Id}_{r-k} & 0 & 0 & 0 \\
A' & \mathrm{Id}_{n-(r-k)} & 0 & 0 \\
0 & \mathcal{D}_{2} & \mathrm{Id}_{n-(r-k)} & 0 \\
0 & -\JDiag[r-k](A')^{T}\JDiag[r-k]\mathcal{D}_{2} & -\JDiag[r-k](A')^{T}\JDiag[r-k] & \mathrm{Id}_{r-k} \\
\end{array}\right] \mapsto \left(\left[\begin{array}{cccc}
\mathrm{Id}_{r-k} & 0 & 0 & 0 \\
0 & \mathrm{Id}_{n-(r-k)} & 0 & 0 \\
-\mathcal{D}_{2}A' & \mathcal{D}_{2} & \mathrm{Id}_{n-(r-k)} & 0 \\
\JDiag[r-k](A')^{T}\JDiag[r-k]\mathcal{D}_{2}A' & -\JDiag[r-k](A')^{T}\JDiag[r-k]\mathcal{D}_{2} & 0 & \mathrm{Id}_{r-k} \\
\end{array}\right],\left[\begin{array}{cc}
\mathrm{Id}_{r-k} & 0 \\
A' & \mathrm{Id}_{n-(r-k)} \\
\end{array}\right]\right) $
\end{center}
\noindent Consider the isomorphism $\gamma:O_{G/P}^{-}\times O_{\mathrm{GL}_n/P'_{\widehat{r-k}}}^{-}\rightarrow \mathrm{Sym}_n\times O_{\mathrm{GL}_n/P'_{\widehat{r-k}}}^{-}$ (cf. Remark \eqref{remark:identificationOMinusGP}) given by
\begin{center}
$\left(\left[\begin{array}{cc}
\mathrm{Id}_{n} & 0 \\
L & \mathrm{Id}_{n)} \\
\end{array}\right],
\left[\begin{array}{cc}
\mathrm{Id}_{r-k} & 0 \\
N & \mathrm{Id}_{n-(r-k)} \\
\end{array}\right]\right) \mapsto \left((LN)^{T}\JDiag[n]N,\left[\begin{array}{cc}
\mathrm{Id}_{r-k} & 0 \\
N & \mathrm{Id}_{n-(r-k)} \\
\end{array}\right]\right)$
\end{center}
We have that under the map $\gamma \circ \delta$, $Y_{\tilde{P}}(\tilde{w})$ gets identified with $V_w \times V'_w$. This follows by a simple computation and Corollary \ref{corollary:schubertVarietySP2nMatrixForm}.
\end{remark}

\begin{definition}
\label{definition:ZPW}
Now let $Z_{\tilde{P}}(\tilde{w})\coloneqq Y_{P}(w)\times_{X_{P}(w)}X_{\tilde{P}}(w)$.
Then $Z_{\tilde{P}}(\tilde{w})=(O_{G/P}^{-}\times P/\tilde{P})\cap X_{\tilde{P}}(w)$.
Hence $Z_{\tilde{P}}(\tilde{w})$ is smooth, being open in the smooth $X_{\tilde{P}}(\tilde{w})$ (cf. Proposition \ref{proposition:schubertVarietySL2nSmooth}).
\end{definition}

Write $p$ for the composite map $Z_{\tilde{P}}(\tilde{w})\rightarrow O_{G/P}^{-}\times P/\tilde{P}\rightarrow P/\tilde{P}\;(\cong \mathrm{GL}_n/P'_{\widehat{r-k}})$
where the first map is the inclusion and the second map is the projection.
Using Proposition ~\ref{proposition:structureOMinus}\eqref{proposition:structureOMinusInverseImage} and ~\eqref{proposition:structureOMinusSLnModQ} we see that
\begin{center}
$p(\left[\begin{array}{cc}
A_{\,} & 0\\
D_{\,} & \JDiag[n](A^{T})^{-1}\JDiag[n]
\end{array}\right](\mbox{mod}\tilde{P}))=A(\mbox{mod}P'_{\widehat{r-k}})$
\end{center}
Note that $A$ is invertible by ~\ref{proposition:structureOMinus}\eqref{proposition:structureOMinusinvertibleA}.

Using the injective map
\begin{center}
$A\in B_n\longmapsto\left[\begin{array}{cc}
A & 0_{n\times n}\\
0_{n\times n} & \JDiag[n](A^{T})^{-1}\JDiag[n]
\end{array}\right]\in B_{G}$
\end{center}
$B_{n}$ can be thought of as a subgroup of $B_{G}$. With this identification
we have the following Proposition.

\begin{proposition}
\label{proposition:ZBnEquivariant}
$Z_{\tilde{P}}(\tilde{w})$
is $B_{n}$-stable for the action on the left by multiplication. Further
$p$ is $B_{n}$ equivariant.
\end{proposition}

\begin{proof}
Let $z\in \mathrm{Sp}_{2n}$ such that $z\tilde{P}\in Z_{\tilde{P}}(\tilde{w})$. Then by Proposition \ref{proposition:structureOMinus}\eqref{proposition:structureOMinusInverseImage} we may write
\begin{center}
$z=\left[\begin{array}{cc}
A_{\,} & 0\\
D_{\,} & \JDiag[n](A^{T})^{-1}\JDiag[n]
\end{array}\right]mod\tilde{P}$
\end{center}
be such that $z\tilde{P}\in Z_{\tilde{P}}(\tilde{w})$. Since $X_{B_{G}}(\tilde{w})\rightarrow X_{\tilde{P}}(\tilde{w})$
is surjective, we may assume that $z(\mbox{mod}B_{G})\in X_{B_{G}}(\tilde{w})$,
i.e., $z\in\overline{B_{G}\tilde{w}B_{G}}$. Then for every $A^{\prime}\in B_{n}$
\begin{center}
$\left[\begin{array}{cc}
A^{\prime} & 0_{n\times n}\\
0_{n\times n} & \JDiag[n](A^{\prime T})^{-1}\JDiag[n]
\end{array}\right]z=\left[\begin{array}{cc}
A^{\prime}A & 0_{\,}\\
\JDiag[n](A^{\prime T})^{-1}\JDiag[n]D\,\,\,\,\, & \JDiag[n](A^{\prime T})^{-1}(A^{\prime T})^{-1}\JDiag[n]
\end{array}\right]=$: $z^{\prime}$.
\end{center}
Then $z^{\prime}\in\overline{B_{G}\tilde{w}B_{G}}$, so $z^{\prime}(\mbox{mod}\tilde{P})\in X_{\tilde{P}}(\tilde{w})$.
By Proposition ~\ref{proposition:structureOMinus}\eqref{proposition:structureOMinusinvertibleA}, we have that $A$ is invertible, and hence
$AA^{\prime}$. This implies again by Proposition ~\ref{proposition:structureOMinus}\eqref{proposition:structureOMinusinvertibleA} that $z^{\prime}(\mbox{mod}\tilde{P})\in Z_{\tilde{P}}(\tilde{w})$.
Thus $Z_{\tilde{P}}(\tilde{w})$ is $B_{n}$ stable. Also $p(A^{\prime}z)=p(z^{\prime})=A^{\prime}A=A^{\prime}p(z)$.
Hence $p$ is $B_{n}$-equivariant.
\end{proof}

\begin{theorem}
\label{theorem:birational}
With notation as above, and $w^{\prime}:=(k+1,..,r,n,..,r+1,k,..,1)\in S_n$, the Weyl group of $\mathrm{GL}_n$,
\begin{enumerate}

\item \label{enum:birationalMap}
The natural map $X_{\tilde P}({\tilde{w}}) \longrightarrow X_P(w)$
is proper and birational. In particular, the map $Z_{\tilde{P}}(\tilde w)
\longrightarrow Y_P(w)$ is proper and birational. In particular, $Z_{\tilde{P}}(\tilde w)$ is a desingularization of $Y_P(w)$.

\item \label{enum:birationalFibre}
$X_{P'_{\widehat{r-k}}}(w')$ is the fibre of the natural map
$Z_{\tilde P}({\tilde{w}}) \longrightarrow Y_P(w)$ at $e_{id} \in  Y_P(w)$.

\item \label{enum:birationalFibration}
$X_{P'_{\widehat{r-k}}}(w')$ is the image of $p$. Further, $p$ is a fibration with fibre
isomorphic to $V_w$.

\item \label{enum:birationalVB}
$p$ identifies
$Z_{\tilde{P}}(\tilde w)$ as a sub-bundle of  the trivial bundle
$O_{G/P}^{-}
\times X_{P'_{\widehat{r-k}}}(w^{\prime})$, which arises as the
restriction of the vector-bundle on $\mathrm{GL}_n/P'_{\widehat{r-k}}$ associated to the
$P'_{\widehat{r-k}}$-module $V_w$ (which, in turn, is a $P'_{\widehat{r-k}}$-submodule of
$O_{G/P}^{-}$).
\end{enumerate}
\end{theorem}

\begin{proof}
~\eqref{enum:birationalMap}: The map $X_{\tilde{P}}(\tilde{w})\hookrightarrow G/\tilde{P} \rightarrow G/P$ is proper and its (scheme-theoretic) image is $X_{P}(w)$ hence $X_{\tilde{P}}(\tilde{w})\rightarrow X_{P}(w)$ is proper. Birationality follows from the fact that $\tilde{w}$ is the minimal representative of the coset $w\tilde{P}$.

~\eqref{enum:birationalFibre}: The fibre at $e_{id} \in Y_P(w)$ of the map $Y_{\tilde P}({\tilde{w}}) \longrightarrow Y_P(w)$ is ${0}\times V_{w}^{\prime}(\subseteq V_w\times V_{w}^{\prime}=Y_{\tilde P}({\tilde{w}}))$. Since $Z_{\tilde P}({\tilde{w}})$ is the closure of $Y_{\tilde P}({\tilde{w}})$ inside $O_{G/P}^{-} \times P/\tilde{P}$ and $X_{P'_{\widehat{r-k}}}(w^{\prime})$ is the closure of $V_{w}^{\prime}$ inside $P/\tilde{P}$(note that as a subvariety of $O_{P/\tilde{P}}^{-}$, $Y_{P'_{\widehat{r-k}}}(w^{\prime})$ is identified with $V_{w}^{\prime}$), we see that the fibre at  $e_{id}$ (belonging to $Y_P(w)$) of $Z_{\tilde P}({\tilde{w}})\longrightarrow Y_P(w)$ is $X_{P'_{\widehat{r-k}}}(w^{\prime})$.

~\eqref{enum:birationalFibration}: From Remark \ref{remark:identificationOfYPWasProduct} we have $p(Y_{\tilde P}({\tilde{w}}))=V_{w}^{\prime}\subseteq X_{P'_{\widehat{r-k}}}(w^{\prime})$. Since $Y_{\tilde P}({\tilde{w}})$ is dense inside $Z_{\tilde P}({\tilde{w}})$ and $X_{P'_{\widehat{r-k}}}(w^{\prime})$ is closed in $\mathrm{GL}_n/P'_{\widehat{r-k}}$ we see that $p(Z_{\tilde P}({\tilde{w}}))\subseteq X_{P'_{\widehat{r-k}}}(w^{\prime})$. The other inclusion $X_{P'_{\widehat{r-k}}}(w^{\prime})\subseteq p(Z_{\tilde P}({\tilde{w}}))$ follows from ~\eqref{enum:birationalFibre}. Hence, $p(Z_{\tilde P}({\tilde{w}}))= X_{P'_{\widehat{r-k}}}(w^{\prime})$.
To prove the second assertion of ~\eqref{enum:birationalFibration} we shall show that for every $A\in \mathrm{GL}_n$ with $A$ mod $P'_{\widehat{r-k}}\in X_{P'_{\widehat{r-k}}}(w^{\prime})$, we have $p^{-1}(A\;\text{mod}P'_{\widehat{r-k}})$ is isomorphic to $V_w$.

To prove this we first observe that $p^{-1}(e_{id})$ is isomorphic to $V_w$ in view of Remark ~\ref{remark:identificationOfYPWasProduct}. Next observe that every $B_n$-orbit inside $X_{P'_{\widehat{r-k}}}(w^{\prime})$ meets $V_{w}^{\prime}(=Y_{P'_{\widehat{r-k}}}(w^{\prime}))$; further $p$ is $B_n$-equivariant by Proposition ~\ref{proposition:ZBnEquivariant} and hence every fibre is isomorphic to the fibre at $e_{id}$, i.e. isomorphic to $V_w$.

~\eqref{enum:birationalVB}: Define a right action of $\mathrm{GL}_n$ on $O_{G/P}^{-}$(identified with $\mathrm{Sym}_n$ as in Remark ~\ref{remark:identificationOMinusGP}) as $g\circ v=g^{T}vg$ for $g\in \mathrm{GL}_n,\;v\in \mathrm{Sym}_n$. This induces an action of $P'_{\widehat{r-k}}$ on $O_{G/P}^{-}$ under which $V_w$ is stable. Thus we get the homogeneous bundle $\mathrm{GL}_n \times ^{P'_{\widehat{r-k}}} V_w \longrightarrow \mathrm{GL}_n/P'_{\widehat{r-k}}$. Now to prove the assertion about
$Z_{\tilde{P}}(\tilde w))$ being a
vector-bundle over $X_{P'_{\widehat{r-k}}}(w^{\prime})$, we will show that there is a
commutative
diagram given as below, with $\psi$ an isomorphism:
\[
\xymatrix{%
Z_{\tilde{P}}(\tilde{w}) \ar[rrrd]^{\phi} \ar[rrdd]_p \ar[rrd]_\psi\\
&&(\mathrm{GL}_{n}\times^{P'_{\widehat{r-k}}} V_w)|_{X_{P'_{\widehat{r-k}}}(w^{\prime})} \ar[d] \ar[r]&
\mathrm{GL}_{n}\times^{P'_{\widehat{r-k}}} V_w \ar[d]^\alpha \\
&&{X_{P'_{\widehat{r-k}}}(w^{\prime})} \ar[r]^\beta& \mathrm{GL}_n/P'_{\widehat{r-k}}
}
\]
The map $\alpha$ is the homogeneous bundle map
and $\beta$ is the inclusion map.
Define $\phi$ by \[
\phi :  {{\begin{bmatrix}
A & 0_{n \times n} \\
D & \JDiag[n](A^{T})^{-1}\JDiag[n]
\end{bmatrix}} \mod \tilde P}
 \longmapsto (A,D^T\JDiag[n]A)/\sim.
\]
Using Proposition~\ref{proposition:structureOMinus}\eqref%
{proposition:structureOMinusInverseImage} and Remark~\ref{remark:identificationOfYPWasProduct}
we conclude the following:
$\phi$ is well-defined and injective;
$\beta \cdot p = \alpha \cdot \phi$; hence, by the universal property of
products, the map $\psi$ exists; and, finally, the injective map $\psi$ is
in fact an isomorphism (by dimension considerations).
\end{proof}

\noindent As an immediate consequence of Theorem \ref{theorem:birational} we have

\begin{corollary}
\label{corollary:geometricTechniqueRealizationSp2n1}
We have the following realization of the
diagram in ~\eqref{equation:KLWdiagram}:
\[
\vcenter{\vbox{%
\xymatrix{%
Z_{\tilde P}(\tilde{w}) \ar[d]^{q^{\prime}} \ar@{^(->}[r] &
O_{G/P}^{-} \times
X_{P'_{\widehat{r-k}}}(w^{\prime})  \ar[d]^{q} \ar[r] & X_{P'_{\widehat{r-k}}}(w^{\prime})\\
Y_{P}(w)  \ar@{^(->}[r] & O_{G/P}^{-}
}}}.
\]
$\,$
$\,$\par
$\,$\par
\end{corollary}

\begin{proposition}
\label{proposition:smallerGrass}
$\;$

\begin{enumerate}
\item \label{proposition:smallerGrass:Ident}
The Schubert variety $X_{P'_{\widehat{r-k}}}(w^{\prime})$ is isomorphic to the Grassmannian $\mathrm{GL}_r/P''_{\widehat{r-k}}$, where $P''_{\widehat{r-k}}$ is the parabolic subgroup in $\mathrm{GL}_r$ omitting $\alpha_{r-k}$.
\item \label{proposition:smallerGrass:VB}
$(\mathrm{GL}_{n}\times^{P'_{\widehat{r-k}}} V_w)|_{X_{P'_{\widehat{r-k}}}(w^{\prime})} \cong (\mathrm{GL}_{n}\times^{P'_{\widehat{r-k}}} V_w)|_{\mathrm{GL}_r/P''_{\widehat{r-k}}} \cong \mathrm{GL}_{r}\times^{P''_{\widehat{r-k}}} V_w$ as homogenous vector bundles.
\end{enumerate}
\end{proposition}

\begin{proof}
$\;$

\noindent ~\eqref{proposition:smallerGrass:Ident}: This is clear.

\noindent ~\eqref{proposition:smallerGrass:VB}: Consider the embedding $i:\mathrm{GL}_r \xhookrightarrow{} \mathrm{GL}_n$ given by
\begin{center}
$R\mapsto \left[\begin{array}{cc}
R & 0\\
0 & \mathrm{Id}_{n-r}
\end{array}\right]$
\end{center}
Define the action of $\mathrm{GL}_r$ on $\mathrm{Sym}_n$ as the action induced by this embedding. This induces an action of $P''_{\widehat{r-k}}$ on $\mathrm{Sym}_n$. As $i(P''_{\widehat{r-k}})\subset P'_{\widehat{r-k}}$, the $P'_{\widehat{r-k}}$ stability of $V_w$ implies the $P''_{\widehat{r-k}}$ stability of $V_w$. Hence our result follows.
\end{proof}

\begin{corollary}
\label{corollary:geometricTechniqueRealizationSp2n2}
We have the following realization of the
diagram in ~\eqref{equation:KLWdiagram}:
\[
\vcenter{\vbox{%
\xymatrix{%
Z_{\tilde P}(\tilde{w}) \ar[d]^{q^{\prime}} \ar@{^(->}[r] &
O_{G/P}^{-} \times
\mathrm{GL}_r/P''_{\widehat{r-k}}  \ar[d]^{q} \ar[r] &\mathrm{GL}_r/P''_{\widehat{r-k}}\\
Y_{P}(w)  \ar@{^(->}[r] & O_{G/P}^{-}
}}}.
\]
\end{corollary}

\section{Free Resolutions}
\label{sec:freeresolutions}

\subsection*{Kempf-Lascoux-Weyman geometric technique}%

We summarize the geometric technique of computing free resolutions,
following \cite[Chapter~5]{WeymanCohVBSyz03}.

Consider Diagram~\eqref{equation:genericKLW}. There is a natural map $f : V \longrightarrow \Grass_{r, d}$ (where $r = \rank_V\! Z$
and $d=\dim \mathbb A$)
such that the inclusion $Z \subseteq \mathbb A \times V$ is
the pull-back of the tautological sequence~\eqref{equation:tautseq};
here $\rank_V Z$ denotes the rank of $Z$ as a vector-bundle  over $V$,
i.e., $\rank_V Z = \dim Z - \dim V$.
Let $\xi = (f^* \mathcal Q)^*$. Write $R$ for the polynomial ring
$\mathbb{C}[\mathbb A]$ and ${\mathfrak m}$ for its homogeneous maximal ideal.  (The
grading on $R$ arises as follows. In
Diagram~\eqref{equation:genericKLW}, $\mathbb A$ is thought of as the fibre of a
trivial vector-bundle, so it has a distinguished point, its origin. Now,
being a sub-bundle, $Z$ is defined by linear equations in each fibre; i.e.,
for each $v \in V$, there exist $s := (\dim \mathbb A - \rank_V Z)$ linearly
independent linear polynomials $\ell_{v, 1}, \ldots, \ell_{v, s}$ that
vanish along $Z$ and define it. Now $Y = \{y \in \mathbb A :
\text{there exists $v \in V$ such that}\; \ell_{v,1}(y) = \cdots =
\ell_{v,s}(y)=0\}$. Hence $Y$ is defined by homogeneous polynomials. This
explains why the resolution obtained below is graded.) Let ${\mathfrak m}$ be the
homogeneous maximal ideal, i.e., the ideal defining the origin in $\mathbb A$.
Then:
\begin{theorem}[\protect{\cite[Basic Theorem~5.1.2]{WeymanCohVBSyz03}}]
\label{theorem:geometrictechnique}
With notation as above, there is a finite complex $(F_\bullet,
\partial_\bullet)$ of finitely generated graded free $R$-modules that is
quasi-isomorphic to $\@ifnextchar[{\R@Der}{\ensuremath{\RDerChar}} q'_* \strSh_{Z}$, with \[
F_i = \oplus_{j \geq 0} \homology^j(V, \bigwedge^{i+j} \xi)
\otimes_\mathbb{C} R(-i-j),
\]
and $\partial_i(F_i) \subseteq {\mathfrak m} F_{i-1}$.  Furthermore, the following
are equivalent:
\begin{asparaenum}
\item $Y$ has rational singularities, i.e.  $\@ifnextchar[{\R@Der}{\ensuremath{\RDerChar}} q'_* \strSh_{Z}$ is
quasi-isomorphic to $\strSh_{Y}$;
\item $F_\bullet$ is a minimal $R$-free resolution of $\mathbb{C}[Y]$, i.e.,
$F_0 \simeq R$ and $F_{-i} = 0$ for every $i > 0$.
\end{asparaenum}
\end{theorem}

We give a sketch of the proof because one direction of the
equivalence is only implicit in the proof of~\cite[5.1.3]{WeymanCohVBSyz03}.

\begin{proof}[Sketch of the proof] One constructs a suitable $q_*$-acyclic
resolution $\mathscr I^\bullet$ of the Koszul complex that resolves $\strSh_{Z}$
as an $\strSh_{\mathbb A \times V}$-module so that the
terms in $q_*\mathscr I^\bullet$ are \emph{finitely generated free} graded
$R$-modules. One places the Koszul complex on the negative horizontal axis
and thinks of $\mathscr I^\bullet$ as a second-quadrant double complex, thus to
obtain a complex $G_\bullet$ of finitely generated free $R$-modules whose
homology at the $i$th position is $R^{-i}q_* \strSh_Z$. Then,
using standard homological considerations, one constructs a subcomplex
$(F_\bullet, \partial_\bullet)$ of $G_\bullet$ that is quasi-isomorphic to
$G_\bullet$ with $\partial_i(F_i) \subseteq {\mathfrak m} F_{i-1}$ (we say that
$F_\bullet$ is \define{minimal} if this happens), and since
$\homology_i(G_\bullet) = 0$ for every $|i| \gg 0$, $F_i = 0$ for every $|i|
\gg 0$. Now using the minimality of $F_\bullet$, we see that
$R^iq_*\strSh_Z = 0$ for every $i \geq 1$ if and only if $F_{-i} = 0$ for every
$i \geq 1$. When one of these  conditions
holds, then $F_\bullet$ becomes a minimal free resolution of
$q_*\strSh_Z$ which is a finitely generated
$\strSh_Y$-module, and therefore $q_*\strSh_Z = \strSh_Y$ if and only if
$q_*\strSh_Z$ is generated by one element as an
$\strSh_Y$-module if and only if $q_*\strSh_Z$ is a generated by one
element as an $R$-module
if and only if $F_0$ is a free $R$-module of rank one
if and only if $F_0 = R(0)$ since $\homology^0(V, \bigwedge^0 \xi) \otimes
R$ is a summand of $F_0$.
\end{proof}

\subsection*{Our situation}

We now apply Theorem~\ref{theorem:geometrictechnique} to our situation.
We keep the notation of Theorem~\ref{theorem:birational}.
Theorem~\ref{theorem:geometrictechnique} and
Corollary~\ref{corollary:geometricTechniqueRealizationSp2n2} yield the following result:

\begin{theorem}
\label{theorem:stepone}
Write $\xi$ for the homogeneous vector bundle on $\mathrm{GL}_r/P''_{\widehat{r-k}}$
associated to the $P''_{\widehat{r-k}}$-module $\left(O^\mhyphen_{G/P}/ V_w\right)^*$
(this is the dual of the quotient of $O^\mhyphen_{G/P} \times \mathrm{GL}_r/P''_{\widehat{r-k}}$ by $Z_{\tilde{P}}(\tilde w)$).
Then we have a minimal $R$-free resolution $(F_\bullet, \partial_\bullet)$
of
$\mathbb{C}[Y_P(w)]$  with
\[
F_i = \oplus_{j \geq 0} \homology^j(\mathrm{GL}_r/P''_{\widehat{r-k}}, \bigwedge^{i+j} \xi)
\otimes_\mathbb{C} R(-i-j).
\]
\end{theorem}

Computing the cohomology groups required in Theorem~\ref{theorem:stepone}
in the general situation is a difficult problem. Techniques for computing them in our specific case are discussed in the following section.
\section{Cohomology of Homogeneous Vector-Bundles}
\label{sec:cohomologyHVBStepTwo}

We have shown in Theorem ~\ref{theorem:stepone} that the calculation of a minimal $R$-free resolution of $\mathbb{C}[Y_P(w)]$ comes down to the computation of the cohomology of certain homogeneous bundles over $\mathrm{GL}_r/P''_{\widehat{r-k}}$. In particular we need to calculate
\begin{equation}
\label{equation:cohomologyVB}
\homology^{\bullet}(\mathrm{GL}_r/P''_{\widehat{r-k}}, \bigwedge^{t} \xi)
\end{equation}
for arbitrary $t$.

The $P''_{\widehat{r-k}}$-module $\left(O^\mhyphen_{G/P}/ V_w\right)^*$ is not completely reducible (the unipotent radical of $P''_{\widehat{r-k}}$ does not act trivially), and thus we can not use the Bott-algorithm to compute its cohomology. In \cite{OttavianiRubeiQuiversCohVB06} the authors determine the cohomology of general homogeneous bundles on Hermitian symmetric spaces. As $\mathrm{GL}_r/P''_{\widehat{r-k}}$ is such a space their results could be used to determine \eqref{equation:cohomologyVB}. In practice proceeding along these lines is possible though extremely complicated.

Another approach to the calculation of these cohomologies comes from using a technique employed in \cite[Chapter 6.3]{WeymanCohVBSyz03}. There the minimal $R$-free resolution of a related space is computed and the minimal $R$-free resolution of $\mathbb{C}[Y_P(w)]$ can be seen as a sub-resolution. In \cite{WeymanCohVBSyz03} this method is used for the case when $n=r$. That is, the case where $Y_P(w)$ is the symmetric determinental variety. In this case the authors assume that $k=2u$(the odd case can be reduced to this even case). They look at the subspace $T_w$ of $\mathrm{Sym}_n$ given by symmetric matrices of block form
\begin{center}
$\left[\begin{array}{cc}
0_{n-u \times n-u} & R\\
R^{T} & S_{u \times u}
\end{array}\right]$
\end{center}
Let $P'_{\widehat{n-u}}$ be the parabolic subgroup of $\mathrm{GL}_n$ omitting the root $\alpha_{n-u}$, then $T_w$ is a $P'_{\widehat{n-u}}$-module under the same action. If $Z_w$ is the homogeneous vector bundle associated with $T_w$ we have the following diagram
\begin{center}
$\vcenter{\vbox{%
\xymatrix{%
Z_w \ar@{^(->}[r] \ar[d]^{q'} & \mathrm{Sym}_n  \times \mathrm{GL}_n/P'_{\widehat{n-u}} \ar[d]^q \ar[r] & \mathrm{GL}_n/P'_{\widehat{n-u}} \\
Y \ar@{^(->}[r] & \mathrm{Sym}_n }}}$
\end{center}
They show that the resolution of $\mathbb{C}[Y_P(w)]$ can be realized as a sub-resolution of the resolution of $\mathbb{C}[Y]$. In this case, the $P'_{\widehat{n-u}}$-module $(\mathrm{Sym}_n/T_w)^{*}$(this is the dual of the quotient of $\mathrm{Sym}_n  \times \mathrm{GL}_n/P'_{\widehat{n-u}}$ by $Z_w$) is completely reducible and thus the cohomology of the corresponding homogeneous vector bundles $\bigwedge^{t} \xi$  may be computed using the Bott-algorithm, leading to
\begin{theorem}
\cite[Theorem 6.3.1(c)]{WeymanCohVBSyz03} The $i$th term $G_i$ of the minimal free resolution of $\mathbb{C}[Y_P(w)]$ as an $R$ module is given by the formula
\begin{center}
$G_i=\bigoplus\limits_{\substack{\lambda \in Q_{k-1}(2t) \\ \mathrm{rank}\lambda\;\mathrm{even} \\ i=t-k\frac{1}{2}\mathrm{rank}\lambda}} \Schur_{\chk{\lambda}}\mathbb{C}^{n} \otimes_{\mathbb{C}} R$
\end{center}

\end{theorem}
\noindent Here $Q_{k-1}(2t)$ is the set of partitions $\lambda$ of $2t$ which in hook notation can be written as $\lambda = (a_1, . . . , a_{s} |b_1, . . . , b_{s} )$, where $s$ is a positive integer, and for each $j$ we have $a_j = b_j + (k-1)$. And $\chk{\lambda}$  is the conjugate (or dual) partition of $\lambda$. And finally, rank$\lambda$ is defined as being equal to $l$, where the largest square fitting inside $\lambda$ is of size $l \times l$.

Similar methods may be used to compute a closed form formula for the minimal free resolution of $\mathbb{C}[Y_P(w)]$ as an $R$ module in the case $r\neq n$.

redpajama_set_name: RedPajamaArXiv


Unlike the average photo that you take for your school year book, you want to be prepared as you can for your senior photos. In some cases, they will be documented in your final high school year book and exchanged with friends and peers for remembrance. They also serve as a reflection of who you were in the moments you prepared yourself to end one meaningful chapter in your life and embark into your new chapter, adulthood.
Below are a list of instructions on how to prepare yourself for your senior photo session.
Select two to three outfits, or dress in layers to change up your look throughout the shoot. Go for one casual and one dress that is definitely "YOU" which says something about your personality. Choose suitable jewelry, accessories, socks, stocking, shoes and underwear for each. Avoid bright colours and busy patterns- these can be a distraction in photos along with sleeveless tops, which can make your arms look big. Iron in advance, bring all attire on hangers and bring an iron for emergency touch-ups.
Choose props like fresh-cut flowers, stuffed animals, musical instruments, skateboards, or other hobby related equipment. You can even put your friend or pet in the photo if you want.
Plan your hairstyle or haircut and schedule a hair appointment a week before the shoot. Being photographed too soon after getting a fresh style can prevent you from looking like yourself. Or you can contact us to refer you to a hair and make-up artist that we use for our photo shoots.
If you want to wear glasses, wear glasses that don't have lenses in them or borrow a pair of similar frames without lenses. Even glare-resistant lenses light, and removing distortions is costly when possible at all.
Don't be afraid to brainstorm with your photographer regarding locations. Choose places that better defines YOU and your personality.
Perfect your face. Ladies- wear make-up, even if you don't usually. Keep coverage natural, decide your look before the day of your shoot, and bring make-up along for touch-ups. A great idea is visit your local make-up counter and get ideas from make-up artists. Have them teach you the tricks to bring out your best features like your eyes, cheek bones, etc. Stay away from blinging your face with glitter- which may show up as white spots- and SPF products, which can make your skin look shiny. Guys- you should also consider wearing powder to keep your skin from looking shiny. Give yourself a clean shave, neatly trim any facial hair or bring a razor to shave before formal pictures if you want facial hair in the others.
Have a manicure on the day of or before your photo shoot. Chipped polish will show up in your photos, so clear polish or French manicures are ideal.
Avoid over-exposure to the sun. Redness can be difficult and expensive to retouch when the photographer does post-production on your photos.
Get everything together no later than the day before your photo shoot. Plan a music plaits to set the mood for a fun shoot, and invite a friend or two to join you at your shoot. We have a bluetooth wireless speaker that can link up to your iPod or iPhone so you can listen to the music that gets you in that move in front of the camera.
Last but not least, get a good nights' sleep for at least the two days before your shoot to look well-rested and stress-free in your photos. Having bags under your eyes can be expensive to touch up.
Your photo shoot is a resemblance of you. Think about what makes you "YOU", and how you want to be remembered by the friends you're leaving behind. Look at senior portraits from previous years- of older friends or in your photographer's portfolio for ideas and inspiration to personalize your shoot. Make your photos look different from everyone else. Remember, there are plenty of photographers that take the same type of photos just because they own a DSLR camera. Your photographer is the pro and he'll know the type of lighting that you'll need to create that magic photo. When you get your photos, make your friends ENVIOUS of you and don't be afraid to send them to AfallaStudios if they want to make their friends envious of their photos. Its not common when a senior retakes their photos by a more experienced photographer.


text:
package org.opensextant.solrtexttagger;

import org.apache.lucene.analysis.TokenFilter;
import org.apache.lucene.analysis.TokenStream;
import org.apache.lucene.util.Attribute;

/**
 * Attribute used by the {@link Tagger} to decide if a token can start a
 * new {@link TagLL tag}.
 *

 * By default this Attribute will return true, but it might be
 * reset by some {@link TokenFilter} added to the {@link TokenStream} used
 * to analyze the parsed text. Typically this will be done based on NLP
 * processing results (e.g. to only lookup Named Entities).
 *

 * NOTE: that all Tokens are used to advance existing {@link TagLL tags}.
 *
 * @author Rupert Westenthaler
 */
public interface TaggingAttribute extends Attribute {

  /**
   * By default this Attribute will be initialised with true.
   * This ensures that all tokens are taggable by default (especially if
   * the {@link TaggingAttribute} is not set by any component in the configured
   * {@link TokenStream}
   */
  public static final boolean DEFAULT_TAGGABLE = true;

  /**
   * Getter for the taggable state of the current Token
   *
   * @return the state
   */
  public boolean isTaggable();

  /**
   * Setter for the taggable state. Typically called by code within
   * {@link TokenFilter#incrementToken()}.
   *
   * @param lookup the state
   */
  public void setTaggable(boolean lookup);

}

redpajama_set_name: RedPajamaGithub


text:
Evelyn Montone
Parkland Softball
Class: Junior
2021-2022 13 3 4 1
Career Totals: 13 3 4 1
03/29/2022 @ Northampton L 2-12 — — — 1 — — — — — — —
04/02/2022 vs. William Allen W 15-0 — — — — — — 1 — — — —
04/09/2022 @ Tunkhannock L 3-4 — — — 1 — — — 1 — — —
04/11/2022 @ East Stroudsburg North W 15-0 — — — 1 — 1 — — — — —
04/14/2022 @ Freedom W 11-9 — — — 1 — — — — — — —
04/21/2022 @ Bethlehem Catholic W 6-1 — — — — — 1 — — — — —
04/28/2022 @ William Allen W 16-3 — — — 1 — — — — — — —
04/29/2022 vs. Nazareth W 6-3 — — — 2 — — — — — — —
05/03/2022 vs. Northampton L 2-5 — — — — — — 1 — — — —
05/05/2022 @ Emmaus W 8-3 — — — 1 — — — — — — —
05/12/2022 vs. Freedom W 10-0 — — — 2 — — 1 — — — —
05/19/2022 @ Northampton W 3-1 — — — 2 — 1 1 — — — —
05/26/2022 vs. Pleasant Valley W 11-0 — — — 1 — — — — — — —
Season Totals: — — — 13 — 3 4 1 — — —
Parkland softball gets revenge against Northampton to take back EPC gold
Parkland softball starts fast, rolls to EPC playoff win over Freedom

redpajama_set_name: RedPajamaCommonCrawl


it is exciting to watch kids level up or get their badges at the COAs!
A great and worth while activity for our kids. I loved girl scouts when I was young.
Awesome! A good leader is a must. Glad Josh has that again.


text:
Kolmoskanava (also known as TV3 and informally as Kolmonen) was a Finnish free-to-air television station owned and operated by Oy Kolmostelevisio Ab, a joint venture between MTV Oy, Yle and Nokia. The channel was launched on December 1, 1986, and closed on the New Year's Eve night of December 31, 1992 with MTV3 starting its broadcasts simultaneously. Being the first nationwide commercial channel in Finland, Kolmoskanava did not produce any of its programmes themselves, instead relying on imported programmes (mostly TV shows and movies from North America), sports and a small number of Finnish programmes, most notably the bingo game show Megavisa, which aired from 1991 until 1992 on Kolmoskanava, and from 1993 until 1995 on its successor channel MTV3.

External links
Kolmoskanava archive

Defunct television channels in Finland
Television channels and stations established in 1986
Television channels and stations disestablished in 1992

redpajama_set_name: RedPajamaWikipedia


Which Flower Hall Friend Are You?
What's your favorite drunk food?
What TV show can you quote backwards and forwards?
What movie can you quote backwards and forwards?
What's your go-to alcoholic drink?
Which article of clothing defines you the most?
What's your favorite song to sing at the top of your lungs?
What's your favorite part about Miami?
Which nickname do you relate to the most?
Why do you get made fun of?
"No one makes fun of me"
What's your favorite bar at Miami?