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\begin{document}
\title[Cohomology of smooth Schubert varieties]
{Cohomology of smooth Schubert varieties in partial flag manifolds}
\author{Vesselin Gasharov}
\address{Department of Mathematics\\
Cornell University\\
Ithaca, NY 14853\\
USA}
\email[V. Gasharov]{vesko@math.cornell.edu}
\author{Victor Reiner}
\address{School of Mathematics\\
University of Minnesota\\
Minneapolis, MN 55455\\
USA}
\email[V. Reiner]{reiner@math.umn.edu}
\subjclass{05E, 14M15}
\thanks{Second author partially supported by NSF grant
DMS-9877047.}
\begin{abstract}
We use the fact that smooth Schubert varieties in partial flag manifolds
are iterated fiber bundles over Grassmannians to give
a simple presentation for their integral cohomology ring,
generalizing Borel's presentation for the cohomology of the
partial flag manifold itself. More generally, such a presentation is shown
to hold for a larger class of subvarieties of the partial flag manifolds
(which we call {\it subvarieties defined by inclusions}). The Schubert
varieties
which lie within this larger class are characterized combinatorially by
a pattern avoidance condition.
\end{abstract}
\maketitle
\section{Introduction}
This paper concerns smooth Schubert subvarieties within the
partial flag varieties
$$
G/P_J: =
\{0 \subset V_{d_1} \subset V_{d_2} \subset \cdots \subset V_{d_{s}}
\subset \complexes^n \}.
$$
Here $G=GL_n(\complexes)$, and $P_J$ is the subgroup that
stabilizes the standard partial flag having $V_{d_i}$ spanned
by the first $d_i$ standard basis vectors in $\complexes^n$.
When $J=\emptyset$, $P_J=B$ is the Borel group of upper triangular
matrices and $G/B$ is usual (complete) flag variety.
The Schubert varieties $\{X_{wW_J}\}$ in $G/P_J$ are subvarieties
indexed by cosets $w{W_J}$ of the parabolic subgroup $W_J$ of the
Weyl group $W= S_n$. The equations which define
$X_{wW_J}$ as a subvariety of $G/P_J$ are a
conjunction of certain conditions having the general form
\begin{equation}
\label{Schubert-conditions}
\dim V_{d_i} \cap \complexes^e \geq r.
\end{equation}
Smooth Schubert varieties have several special properties,
which make them almost as well-behaved as the partial flag manifolds
themselves. We review some of these properties here
to provide a context for our results.
Note that condition (\ref{Schubert-conditions}) says
\begin{equation}
\label{inclusion-conditions}
\begin{aligned}
V_{d_i} \subset \complexes^e &\text{ when } r=d_i \\
\complexes^e \subset V_{d_i} &\text{ when } r = e.
\end{aligned}
\end{equation}
\noindent
Any subvariety $X$ of $G/P_J$ defined by a conjunction of
conditions of the special forms in (\ref{inclusion-conditions})
will be called a subvariety of $G/P_J$ {\it defined by inclusions}.
Say that a pair of such inclusion conditions of the form
$$
\begin{aligned}
V_{d_i} &\subset \complexes^{e} \\
\complexes^{e'} &\subset V_{d_j}
\end{aligned}
$$
are {\it crossing} if $d_i < d_j$ and $e > e'$. If $X$ is defined
by inclusions but contains no crossing pair, we
say it is {\it defined by non-crossing inclusions}. By results of
Ryan \cite{Ryan} and Wolper \cite{Wolper}, this property characterizes
the smooth Schubert varieties. Theorem~\ref{paraphrase} below
summarizes some of their work, along with a result due independently to
Lakshmibai and Sandhya \cite{LakshmibaiSandhya} and Haiman \cite{Haiman}.
For the purposes of stating it, we introduce two pieces of terminology.
Say that a (complex) algebraic variety $X$ is {\it fibered by Grassmannians}
if it lies within the smallest class of algebraic varieties that contains
all Grassmannians $\G(k,\complexes^n)$ and has the property that the
total space of an algebraic fiber bundle is in the class whenever both
its base and fiber are in the class.
Given a coset $wW_J$ in $S_n$ and permutation $u=u_1 \cdots u_k$ in $S_k$
for some $k \leq n$, say that $wW_J$ {\it avoids the pattern} $u$ if the
longest coset representative $w$ for $wW_J$ contains no subsequence
$w_{i_1} \cdots w_{i_k}$ whose values are in the same relative order
as $u_1 \cdots u_k$ (i.e. the permutation matrix representing $w$ when
restricted to columns $\{i_1,\ldots,i_k\}$ and rows
$\{w_{i_1},\ldots,w_{i_k}\}$
gives exactly the permutation matrix representing $u$).
\begin{theorem}
\label{paraphrase}
The following are equivalent for a subvariety $X$ of $G/P_J$:
\begin{enumerate}
\item[(i)] $X$ is smooth and $B$-stable.
\item[(ii)] $X$ is smooth and defined by inclusions.
\item[(iii)] $X$ is defined by non-crossing inclusions.
\item[(iv)] $X$ is $B$-stable and fibered by Grassmannians.
\item[(v)] $X$ is a smooth Schubert variety $X_{wW_J}$.
\item[(vi)] $X$ is a Schubert variety $X_{wW_J}$
for some $w$ that avoids both patterns $4231$, $3412$.
\end{enumerate}
\end{theorem}
After reviewing combinatorial and topological facts about
partial flag manifolds and Schubert varieties in Section 2,
in Section 3, we use some very special cases of the fiber bundles from
(iv) above to prove a presentation (Theorem \ref{main-result}) for the
integral cohomology ring of a smooth Schubert variety.
This presentation is nearly as simple as Borel's picture for
the cohomology of $G/P_J$.
For example, the Schubert variety $X_{352164}$ in the flag variety
$G=GL_6(\complexes)/B$ is defined by the following inclusions
$$
\{0 \subset V_{1} \subset V_{2} \subset V_{3} \subset V_{4} \subset V_{5}
\subset \complexes^6: V_1 \subset \complexes^3 \subset V_4 \subset
\complexes^5\}
$$
and has cohomology ring
$$
\cohom(X_{352164},\integers) \cong
\integers[x_1,x_2,x_3,x_4,x_5,x_6]/I
$$
where $I$ is the ideal having generators which are elementary symmetric
functions in consecutive variable sets $x_i,x_{i+1},\ldots,x_j$
of the form $e_r(i,j):=e_r(x_i,x_{i+1},\ldots,x_j)$,
coming from the various inclusion conditions that define $X_{352164}$:
$$
\begin{matrix}
e_1(1,6),e_2(1,6),e_3(1,6),e_4(1,6),e_5(1,6),e_6(1,6)
&\text{ from } V_i \subset \complexes^6\\
e_3(2,6),e_4(2,6),e_5(2,6) &\text{ from } V_1 \subset \complexes^3\\
e_2(5,6) &\text{ from } V_4 \subset \complexes^5\\
e_2(1,4),e_3(1,4),e_4(1,4)) &\text{ from } \complexes^3 \subset V_4.\\
\end{matrix}
$$
In fact, Theorem~\ref{main-result} proves such a
presentation more generally for all subvarieties of $G/P_J$ defined by
inclusions.
In Section 4, the Schubert varieties $X_{wW_J}$ defined by inclusions are
characterized combinatorially (Theorem~\ref{inclusions-characterization})
as those with $wW_J$ avoiding the four patterns $4231, 35142,42513, 351624$.
\section{Preliminaries}
We start by defining partial flag manifolds and their Schubert
varieties. A good reference for some of this
material is \cite[Chapter 10]{Fulton}.
Let $G=GL_n(\complexes)$ be the general linear group,
$B$ the Borel subgroup of upper triangular matrices in $G$,
and $W=S_n$ the symmetric group on $n$ letters $[n]:=\{1,2,\ldots,n\}$,
which we may also think of as the subgroup of permutation matrices in $G$.
The group $W$ has a set of generators $S=\{s_1,\ldots,s_{n-1}\}$, where $s_i$
is the transposition swapping $i, i+1$ and keeping all other
numbers fixed, which makes $(W,S)$ a Coxeter system (see \cite{Humphreys}).
For any $J =\{s_{i_1},\ldots,s_{i_p}\} \subseteq S$,
let $W_J$ denote the {\it Young} subgroup of $W$ generated by $J$,
let $P_J$ denote the {\it standard parabolic} subgroup of $G$ generated
by $B$ and $J$, and let
$$
D(J) = \{0\} \cup \{ t\in [n-1]: s_t \not\in J\} \cup\{n\}
=\{0 < d_1 < d_2 < \cdots < d_s < n\}.
$$
$G$ acts transitively on the set of complete flags
$$
0 \subset V_1 \subset V_2 \subset \cdots \subset V_{n-1} \subset \complexes^n
$$
where $V_i$ is a subspace of dimension $i$ for each $i$.
If one fixes the {\it standard flag} having $V_i = \complexes^i$,
where $\complexes^i$ is the subspace of $\complexes^n$ spanned by
the first $i$ standard basis vectors $\{e_1, e_2, \ldots, e_i\}$,
then $B$ is exactly the subgroup which stabilizes this standard
flag. Hence one may identify the coset space $G/B$ with
the set of complete flags. More generally, for each $J \subseteq S$
with $D(J)$ defined as above, the stabilizer of the partial flag
$$
0 \subset \complexes^{d_1} \subset \complexes^{d_2} \subset \cdots
\subset \complexes^{d_{s}}\subset\complexes^n
$$
is $P_J$, hence one can identify the set of partial flags of the form
$$
0 \subset V_{d_1} \subset V_{d_2} \subset \cdots \subset V_{d_{s}} \subset
\complexes^n
$$
with $G/P_J$
(the complete flags and $G/B$ are the special case $J =\emptyset$).
This homogeneous space $G/P_J$ can be embedded as a closed subvariety in
a suitable complex projective space $\Proj(\complexes^N)$, and is a smooth
variety called the {\it partial flag manifold} $G/P_J$.
In the special case $J=\emptyset$ where $P_J=B$, it is
called the {\it flag manifold} $G/B$. In the special case
where $|J|=|S|-1$ so $D(J)=\{0 w_j \}|.
$$
If $D(J)=\{0,d_1,\ldots,d_s,n\}$, then the freedom
in choosing a coset representative for $wW_J$ is exactly
the choice of ordering within each block
$w_{d_{i}+1} w_{d_{i}+2} \cdots w_{d_{i+1}}$.
The length function allows one to distinguish two coset
representatives $\min wW_J$ and $\max wW_J$
for $wW_J$, which are unique in achieving the minimum and maximum length,
respectively. More explicitly, $\min wW_J, \max wW_J$ are the
representatives for which each block
$w_{d_{i}+1} w_{d_{i}+2} \cdots w_{d_{i+1}}$ is in increasing, decreasing
order respectively.
The {\it Bruhat order}
is a partial order on the cosets $W/W_J$ defined as follows:
$uW_J \leq vW_J$ if $X^o_{uW_J} \subset X_{vW_J}$.
The open Schubert cell $X^o_{wW_J}$
is isomorphic to a complex affine space of dimension $l(\min wW_J)$.
Thus the cell decomposition (\ref{full-cell-decomposition}) has only cells
in even (real) dimensions. Note also that since the open Schubert cells
are the $B$-orbits of $G/P_J$, any $B$-stable subvariety $X$, such as
a Schubert variety itself, will also be a union
\begin{equation}
\label{cell-decomposition}
X = \coprod_{wW_J \in \I} X^o_{wW_J}
\end{equation}
of open Schubert cells for some subset $\I$ of $W/W_J$.
Since $X$ is a closed subvariety, this subset $\I$ must form an order ideal
in the Bruhat order on $W/W_J$, and therefore the above decomposition
is again a cell decomposition. The fact that the Schubert cells
have even real dimensions have several topological consequences,
which we collect in the following proposition.
\begin{proposition}
\label{even-cells}
Let $Y$ be a finite CW-complex with cells $\{\sigma_j\}_{j\in J}$
occurring only in even dimensions, and exactly one $0$-cell.
Let $X$ be any subcomplex, with cells $\{\sigma_i\}_{i \in I}$ for some subset
$I \subset J$.
(In particular, this applies when $Y$ is any $B$-stable subvariety of
$G/P_J$, and $X$ is any $B$-stable subvariety of $Y$).
Then
\begin{enumerate}
\item[(i)] $X$ is simply-connected.
\item[(ii)] The integral homology $H_*(X;\integers)$
and cohomology $\cohom(X;\integers)$ are free abelian groups
which are non-vanishing only in even dimensions.
They form dual lattices under the Kronecker pairing,
having dual $\integers$-bases given by the cellular homology
classes $\{[\sigma_i]\}_{i \in I}$ and their (Kronecker)
dual cohomology classes $\{[\sigma_i]\}^*_{i \in I}$, respectively.
\item[(iii)] Consequently,
$$
\begin{aligned}
\Poin(X,q) &:= \sum_{k \geq 0} \rank_\integers H_k(X;\integers) \,\, q^i\\
&= \sum_{i \in I} q^{\dim \sigma_i}.
\end{aligned}
$$
\item[(iv)] The maps $i_*, i^*$ induced on homology and cohomology
$$
\begin{aligned}
H_*(X;\integers) &\overset{i_*}{\rightarrow} H_*(Y;\integers)\\
\cohom(Y;\integers) &\overset{i^*}{\rightarrow} \cohom(X;\integers)
\end{aligned}
$$
by the inclusion $X \overset{i}{\hookrightarrow}Y$
are injective and surjective, respectively, with
$$
\begin{aligned}
i_*([\sigma_i]) &= [\sigma_i] \text{ for all }i\text{ in }I\\
i^*([\sigma_j]^*) &=
\begin{cases}
[\sigma_j]^* & \text{ if }j \in I\\
0 & \text{ if }j \in J-I
\end{cases}
\end{aligned}
$$
\item[(v)] (cf. \cite[Cor. 4.4]{Carrell1})
Consequently, the kernel of the surjection $i^*$ is
generated by $\{[\sigma_j]^*\}_{j \in J - I}$,
both additively and as an ideal in $\cohom(X;\integers)$. $\qed$
\end{enumerate}
\end{proposition}
\begin{proof}
Since $X$ has only one $0$-cell, it is connected. Since it has
no $1$-cells, it is simply connected. For the rest, note
that the absence of odd-dimensional cells implies that all
boundary and coboundary maps are zero in the cellular chain and
cochain complexes computing $H_*(X;\integers),\cohom(X;\integers)$.
\end{proof}
We have seen the geometric significance of the coset representative
$\min wW_J$, whose length gives the dimension of $X_{wW_J}$.
The other distinguished coset representative $\max wW_J$ has
the following geometric interpretation.
\begin{proposition}
\label{parabolic-fibration}
Let $w = \max wW_J$. Then the Schubert conditions
on flags in $X_{w}$ inside $G/B$ are all implied by the
conditions on $\dim V_{d} \cap \complexes^e$ with
$d \in D(J)=\{0,d_1,\ldots,d_s,n\}$.
Consequently, the canonical surjection
$$
\begin{matrix}
G/B &\,\,{\overset{p}\longrightarrow} &\,\,G/P_J \\
0 \subset V_{1} \subset \cdots \subset V_{n-1} \subset \complexes^n
&\longmapsto &
0 \subset V_{d_1} \subset \cdots \subset V_{d_s} \subset \complexes^n
\end{matrix}
$$
is an algebraic fiber bundle with
fiber isomorphic to a product of flag manifolds,
and restricts to a fiber bundle map
$X_{\max{wW_J}} {\overset{p}\rightarrow} X_{wW_J}$ with the
same fiber.
\end{proposition}
\begin{proof}
See Proposition \ref{parabolic-essential}.
\end{proof}
Such fiber bundle maps (and others that will appear later)
have strong combinatorial and geometric consequences,
via the following proposition.
\begin{proposition}
\label{Leray-Serre}
Let $F, E, B$ be complex algebraic varieties, and
$E {\overset{p}\rightarrow} B$ an algebraic fiber bundle map
with fiber isomorphic to $F$.
%\begin{enumerate}
%\item[(i)] If any two of the three varieties $F,E,B$ are smooth,
%then so is the third.
%\item[(ii)]
If $F,B$ are simply connected, have free abelian
integral homology, and have homology occurring only
in even dimensions, then the same is true for $E$. Furthermore
$$
\Poin(E,q) = \Poin(F,q) \, \Poin(B,q).
$$
%\end{enumerate}
\end{proposition}
\begin{proof}
%For assertion (i), note that smoothness of a variety $X$
%is a local property defined by equality in all of the
%inequalities
%$$
%\dim_{\complexes} T_x X \geq \dim X
%$$
%where $T_x X$ is the Zariski tangent space to $X$ at a point $x$,
%and $\dim X$ is the (global) dimension of $X$ as a variety.
%Since fiber bundles are locally products, assertion (i) follows.
%
Note that Proposition \ref{even-cells} (ii)
implies that the maps $d^i$ in the $E^i$ page of the
Leray-Serre spectral sequence associated
to $E {\overset{p}\rightarrow} B$ are all $0$ for $i \geq 1$, so that
it degenerates at the $E^1$ page. Also
$$
E^1_{p,q} = H^p(B) \otimes_\integers H^q(F)
$$
because $B$ is simply-connected, and the homology of $F$ is free abelian.
The assertion follows.
\end{proof}
%The following proposition follows immediately from
%Propositions \ref{parabolic-fibration} and \ref{Leray-Serre} (i).
%
%\begin{proposition} \label{partial-complete}
%A Schubert variety $X_{wW_J}$ in $G/P_J$ is smooth (resp.
%defined by non-crossing inclusions) if and only
%if $X_{\max wW_J}$ in $G/B$ is smooth (resp.
%defined by non-crossing inclusions). $\qed$
%\end{proposition}
\section{A presentation for the cohomology ring of subvarieties
of $G/P_J$ defined by inclusions}
We begin by establishing notation for subvarieties of $G/P_J$
defined by inclusions. Let $J \subseteq S$ index a partial flag
manifold $G/P_J$, so that if $D=D(J)$, we can define
$$
X_D:=G/P_J=\{0 \subset V_{d_1} \subset V_{d_2} \subset \cdots \subset V_{d_{s}}
\subset \complexes^n \}.
$$
Given a triple $(D,L,U)$ where
\begin{enumerate}
\item[$\bullet$]
$D=D(J)=\{0 d_i-l_i\}\\
&\quad \quad \quad \cup \{ e_k(d_i+1,n):
(d_i,u_i) \in U, \,\,k > u_i-d_i\})\\
R_{(D,L,U)}&:=A_D/I_{(D,L,U)}
\end{aligned}
$$
Recall from Proposition \ref{even-cells}
that for any $B$-stable subvariety $X$ of $X_D$,
the cohomology $\cohom(X)$ is free abelian, and is the
image of the surjective restriction map
$\cohom(X_D) \overset{i^*}{\longrightarrow} \cohom(X)$.
Consequently, for any $X_{(D,L,U)}$, the composite ring homomorphism
$$
\phi: A_D \longrightarrow A_D/I_n \cong
\cohom(X_D) \overset{i^*}{\longrightarrow} \cohom(X_{(D,L,U)})
$$
will be surjective.
\begin{theorem}
\label{main-result}
Let $X_{(D,L,U)}$ be a subvariety of $X_D$ defined
by inclusions. Then the composite $\phi$ defined above has
kernel $I_{(D,L,U)}$, and therefore induces a (grade-doubling)
ring isomorphism
$$
\overline{\phi}:R_{(D,L,U)} \cong \cohom(X_{(D,L,U)}).
$$
\end{theorem}
The proof of this theorem occupies the remainder of this section.
We begin by showing that $I_{(D,L,U)} \subset \ker \phi$.
Note that if $(l_i,d_i)$ is in $L$, then the fact that
$\complexes^{l_i} \subset V_{d_i}$ on $X_{(D,L,U)}$
implies by the splitting principle
that the vector bundle $V_{d_i}$ of rank $d_i$ has the
same Chern classes as the direct sum
$\complexes^{l_i} \oplus V_{d_i}/\complexes^{l_i}$,
where $\complexes^{l_i}$ here represents a trivial bundle
of rank $l_i$ over $X_{(D,L,U)}$. The Chern classes of the
bundle $V_{d_i}$ on $X_{(D,L,U)}$ therefore coincide with
those of $V_{d_i}/\complexes^{l_i}$, a bundle of rank $d_i-l_i$,
so $c_k(V_{d_i})$ will vanish for $k > d_i - l_i$.
Hence the $k^{th}$ elementary symmetric functions in the
Chern roots $x_1,\ldots,x_{d_i}$ of the bundle $V_{d_i}$
must vanish in $\cohom(X_{(D,L,U)})$ for $k > d_i - l_i$.
Similarly, if $(d_i,u_i)$ is in $U$, then
$V_{d_i} \subset \complexes^{u_i}$ on $X_{(D,L,U)}$
implies the bundle $\complexes^{n}/V_{d_i}$ of rank $n-d_i$ has the
same Chern classes as the direct sum
$\complexes^{n}/\complexes^{u_i} \oplus \complexes^{u_i}/V_{d_i}$,
where $\complexes^{n_i}/\complexes^{u_i}$ is a trivial bundle on
$X_{(D,L,U)}$. Consequently, $c_k(\complexes^n/V_{d_i})$ vanishes
for $k > u_i - d_i$, implying vanishing in $\cohom(X_{(D,L,U)})$ of
the $k^{th}$ elementary symmetric function in the Chern roots
$x_{d_i+1},\ldots,x_{n}$ of the bundle $\complexes^n/V_{d_i}$
for $k > u_i - d_i$.
Thus $I_{(D,L,U)} \subset \ker \phi$, so that $\phi$
induces a (grade-doubling) surjective ring homomorphism
$$
\overline{\phi}:R_{(D,L,U)} \longrightarrow \cohom(X_{(D,L,U)}).
$$
For the rest of the
theorem, we must show that this surjection
$\overline{\phi}$ is an isomorphism. We observe that because
$\overline{\phi}$ is
a surjection, it will be an isomorphism if we can show either of
two other equivalent conditions:
\begin{enumerate}
\item[(C1)] If $R_{(D,L,U)}$ has a $\integers$-spanning set of
cardinality equal to the rank $r$
of the free abelian group $\cohom(X_{(D,L,U)})$,
then we would have a composite surjection of abelian groups
$$
\integers^{r} \rightarrow
R_{(D,L,U)} \overset{\overline{\phi}}{\rightarrow}
\cohom(X_{(D,L,U)}) \cong \integers^{r}
$$
forcing $\overline{\phi}$ to be an isomorphism.
\item[(C2)] Given the decomposition
$$
X_{(D,L,U)} =X = \coprod_{wW_J \in \I} X^o_{wW_J}
$$
we know from Proposition \ref{even-cells} (v) that
the cohomology classes $[X_{wW_J}]^*$ with $wW_J$ not in $\I$
generate the kernel of the map
$$
A_D/I_n \cong \cohom(G/P_J) \rightarrow \cohom(X_{(D,L,U)}).
$$
Consequently, $\overline{\phi}$ will be an isomorphism
if we can show that each $[X_{wW_J}]^*$ with
$wW_J$ not in $\I$, when considered as an element of
$A_D/I_n \cong \cohom(G/P_J)$, lies in the ideal $I_{(D,L,U)}/I_n$.
\end{enumerate}
Our strategy consists of three steps:
\noindent {\bf Step 1.} Use equivalent condition (C2) to reduce
to the case where $|L|+|U|=1$, \hglue 1.5cm that is, when $X_{(D,L,U)}$ is
defined by a {\it single} inclusion condition.\hfil\break
{\bf Step 2.} Do an independent reduction that
uses equivalent condition (C1) to
reduce \hfil\break \hglue 1.4cm to the case of complete flags, i.e.
$D=D_0:=\{0,1,2,\ldots,n-1,n\}$.
\hfil\break
{\bf Step 3.} Deal with the remaining very special case where
$D=\{0,1,2,\ldots,n-1,n\}$ \hfil\break \hglue 1.4cm and $|L|+|U|=1$, using
equivalent condition (C1) and one of the fibrations \hfil\break \hglue
1.5cm from Theorem~\ref{paraphrase} (iv).
\vskip .1in
\noindent
{\bf Step 1.}
We assume that $\overline{\phi}$ is an isomorphism whenever $|L|+|U|=1$,
that is, when there is only one inclusion condition on the flags
in $X_{(D,L,U)}$. Given an arbitrary $(D,L,U)$, to use condition (C2)
we assume $wW_J$ is not in $\I$, so that at least one inclusion condition
parametrized by $L$ or $U$ is not satisfied by all flags in $X_{wW_J}$.
Let $X_{(D,L_0,U_0)}$ be the subvariety defined by this single inclusion
condition, so that $|L_0|+|D_0|=1$. By our assumption,
condition (C2) is satisfied for $X_{(D,L_0,U_0)}$, so that
$[X_{wW_J}]^*$ lies in the ideal $I_{(D,L_0,U_0)}/I_n$ when we consider it
as an element of $A_D/I_n$. Since $L_0\subseteq L$ and $U_0\subseteq U$
implies $I_{(D,L_0,U_0)}\subseteq I_{(D,L,U)}$, this implies
$[X_{wW_J}]^*$ lies in the ideal $I_{(D,L,U)}/I_n$, as desired.
\vskip .1in
\noindent
{\bf Step 2.}
We assume that $\overline{\phi}$ is an isomorphism whenever $D=D_0$.
Given $X_{(D,L,U)}$, let $J \subset S$ satisfy $D=D(J)$.
Then a fiber bundle as in Proposition \ref{parabolic-fibration},
along with Proposition \ref{Leray-Serre} imply that, as free abelian groups,
\begin{equation}
\label{rank-equation}
\rank_\integers \cohom(X_{(D_0,L,U)}) = r \cdot
\rank_\integers \cohom(X_{(D,L,U)})
\end{equation}
where $r$ is the rank of the cohomology of
$\prod_{i=1}^{s+1} GL_{d_i-d_{i-1}}(\complexes)/B$, that is
$$
r = \prod_{i=1}^{s+1}{(d_i-d_{i-1})!} = |W_J|.
$$
In light of (\ref{rank-equation}), and
since we have assumed $\overline{\phi}$ is an
isomorphism when $D=D_0$, it would suffice to show
that $R_{(D_0,L,U)}$ is a free $R_{(D,L,U)}$-module
of rank $r$, since this would imply that $R_{(D,L,U)}$ must be free
abelian and have the correct rank.
To this end, it is known that the inclusion
$$
A_D= \integers[\x]^{W_J} \overset{\iota}{\hookrightarrow}
\integers[\x] = A_{D_0}
$$
makes $A_{D_0}$ a free $A_D$-module of rank equal to the
same number $r = |W_J|$. Furthermore, there is a $A_{D}$-splitting
$$
\pi_{w_0(J)} : A_{D_0} \rightarrow A_D
$$
of this inclusion, where $w_0(J)$ is the longest permutation in $W_J$,
and $\pi_{w_0(J)}$ is the {\it total $W_J$-symmetrizer} (or {\it
isobaric divided difference}, {\it Demazure operator}) corresponding
to $w_0(J)$ (see \cite{Macdonald}).
Since by definition $I_{(D_0,L,U)}$ is exactly the ideal in
$A_{D_0}$ generated by $\iota(I_{(D,L,U)})$, the following lemma
(whose straightforward proof we omit) completes Step 2:
\begin{lemma}
Let $A \overset{\iota}{\hookrightarrow} B$ be an inclusion of rings
which makes $B$ a free $A$-module of rank $r$, and which has an
$A$-module splitting $\sigma:B \rightarrow A$, i.e. $\sigma \circ \iota =
id_A$.
Then for any ideal $I$ in $A$, and $J$ the ideal in $B$ generated
by $i(I)$, the induced map $A/I \overset{\iota}{\hookrightarrow} B/J$
is injective and makes $B/J$ into a free $A/I$-module of rank $r$. $\qed$
\end{lemma}
\vskip .1in
\noindent
{\bf Step 3.}
We must show that $\overline{\phi}$ is an isomorphism when $D=D_0$ and
$|L|+|U|=1$.
Assume without loss of generality that
$L =\emptyset$ and $U=\{(d,u)\}$ is a singleton; the case
where $L$ is a singleton and $U = \emptyset$ follows via the
symmetry $x_i \longleftrightarrow x_{n+1-i}$ on $\integers[\x]$.
We will use condition (C1). A special case of the fiber bundles referred
to in Theorem~\ref{paraphrase} (iv) is the map
$$
X_{(D_0,\emptyset,{(d,u)})} \rightarrow
X_{(\{0,1,2,\ldots,d,u\},\emptyset,\emptyset)}
$$
which sends a complete flag
$0 \subset V_1 \subset \cdots \subset V_{n-1} \subset \complexes^n$
satisfying $V_d \subset \complexes^u$ to the partial flag
$0 \subset V_1 \subset \cdots \subset V_{d} \subset \complexes^u$,
and has fiber $GL_{n-d}(\complexes)/B$.
Since the fiber and base are partial flag manifolds whose
cohomology rings are free abelian of rank $(n-d)!$ and
$u (u-1) \cdots (u-d+1)$, respectively, one concludes using
Proposition \ref{Leray-Serre} that
$\cohom(X_{(D_0,\emptyset,{(d,u)})})$ is free abelian of rank
$$
r:= u (u-1) \cdots (u-d+1) \cdot (n-d)!.
$$
To show that $R_{(D_0,\emptyset,(d,u))}$ has a spanning set of
this cardinality $r$, we borrow a trick from \cite[p. 163]{Fulton}.
Recall $I_n = ( e_i(1,n): i=1,2,\ldots,n)$, so that we have the
following identities in $R_{D_0}[t] = A_{D_0}/I_n[t]$:
$$
\begin{aligned}
\prod_{i=1}^n (1-x_i t) &= \sum_{s \geq 0} (-1)^s e_s(1,n) t^s \\
&= 1 \\
\prod_{i=1}^m (1-x_i t)^{-1}&=\prod_{i=m+1}^n (1-x_i t)
\end{aligned}
$$
and consequently
\begin{equation}
\label{Fulton-trick}
\sum_{r \geq 0} h_r(1,m) t^r =
\sum_{s=0}^{n-m} (-1)^s e_s(m+1,n) t^s
\end{equation}
where $h_r(i,j)$ is the ({\it complete}) {\it homogeneous symmetric function}
of degree $r$ in the variables $x_i,x_{i+1},\ldots,x_j$,
i.e. the sum of all monomials of degree $r$ in these variables.
We conclude that in $R_{D_0}$
(and hence also in the further quotient $R_{(D_0,\emptyset,{(d,u)})}$),
we have for any $m$ that
\begin{equation}
\label{first-GB-elements}
h_r(1,m) = 0 \text{ for }r>n-m.
\end{equation}
But we can say more in $R_{(D_0,\emptyset,{(d,u)})}$, since
$I_{(D_0,\emptyset,{(d,u)})}$ contains also the elements
$e_s(d+1,n)$ for
$s \geq u-d+1$. Using the fact that
$$
e_s(m,n) = e_s (m+1,n)
+ x_m e_{s-1}(m+1,n)
$$
we conclude that
$I_{(D_0,\emptyset,{(d,u)})}$ also contains
$e_s(m+1,n)$ for every $m \leq d$ and $s \geq u-m+1$.
Consequently, using (\ref{Fulton-trick}) we conclude that in
$R_{(D_0,\emptyset,{(d,u)})}$
\begin{equation}
\label{second-GB-elements}
h_r(1,m) = 0 \text{ for }m \leq d \text{ and }r \geq u-m+1.
\end{equation}
Since $h_r(1,m)$ equals $x_m^r$ plus a sum of monomials
which are lower in a lexicographic ordering with $x_n > \cdots > x_1$,
one can use such relations to systematically rewrite polynomials that use
powers $x_m^s$ with $s>r$ in terms of monomials involving
lower powers of $x_m$ and arbitrary powers of $x_1,\ldots,x_{m-1}$.
Therefore the relations (\ref{first-GB-elements}), (\ref{second-GB-elements})
imply that $R_{(D_0,\emptyset,{(d,u)})}$ is spanned by all monomials
$$
x_1^{i_1} x_2^{i_2} \cdots x_d^{i_d} x_{d+1}^{i_{d+1}}
x_{d+2}^{i_{d+2}} \cdots x_n^{i_n}
$$
satisfying $i_m \geq 0$ for each $m$ and
\[
\begin{aligned}
i&_1 \leq u-1,\,\, i_2 \leq u-2,\,\, \cdots,\,\, i_d \leq u-d \\
i&_{d+1} \leq n-d-1,\,\, i_{d+2} \leq n-d-2,\,\, \cdots,
\,\,i_{n-1} \leq 1,\,\,i_n \leq 0.
\end{aligned}
\]
Since this spanning set has cardinality
$r=u (u-1) \cdots (u-d+1) \cdot (n-d)!$,
we are done with Step 3, and with the proof of
Theorem \ref{main-result}.
\begin{remark} \rm \ \\
In the special case where
\begin{enumerate}
\item[] $D = \{0,1,2\ldots,l,n\},$
\item[] $L = \emptyset$,
\item[] $U = \{(1,\lambda_1),\ldots, (l,\lambda_l)\}$
\end{enumerate}
for some number partition $\lambda = (\lambda_1 \leq \cdots \leq \lambda_l)$,
the varieties $X_{(D,L,U)}$ are all smooth Schubert subvarieties of $X_D$,
which were studied by K. Ding \cite{Ding} under the name
of {\it partition varieties}. Among other things, he
observed that their Schubert cell decomposition is indexed by rook
placements on the Ferrers board corresponding to the partition $\lambda$.
In this special case, the proof of Step 3 in Theorem \ref{main-result}
actually showed that
\begin{equation}
\label{Groebner-basis-presentation}
R_{(D,L,U)} \cong \integers[x_1,\ldots,x_l]/
( h_{\lambda_i - i + 1}(1,i): i=1,2,\ldots,l )
\end{equation}
and that $\{ h_{\lambda_i - i + 1}(1,i): i=1,2,\ldots,l \}$
forms a Gr\"obner basis for the ideal that it generates
with respect to a lexicographic ordering of the variables in
which $x_1 < \cdots j$,
and also some $\times$ located in $(i'',w_{i''})$ with $i'' > i$ and
$w_{i''}