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\begin{document}

\centerline{\large \bf DEGENERACY SCHEMES, QUIVER SCHEMES}
\centerline{\large \bf AND SCHUBERT VARIETIES}
\vspace{1em}
\centerline{\sc \large v.~lakshmibai and peter magyar
%
\footnote
{Both authors partially supported by the
National Science Foundation.}
%
}
\vspace{1em}
\centerline{November 1997, revised March 1998}
\vspace{2em}

\noindent
{\small {\sc abstract.}  A result of Zelevinsky states that
an orbit closure in the space of representations of the
equioriented quiver of type $A_h$ is in bijection with the
opposite cell in a Schubert variety of a partial flag
variety $SL(n)/P$.  We prove that Zelevinsky's bijection is
a scheme-theoretic isomorphism,
which shows that the
degeneracy schemes of Fulton and Buch are reduced and
Cohen-Macaulay in arbitrary characteristic.
}
\\[2em]
Among all algebraic varieties, the best understood are the
flag varieties and their Schubert subvarieties.
They first appear as interesting examples, but acquire a
general importance in the theory of characteristic
classes of vector bundles.

Fulton \cite{F} and Buch-Fulton \cite{BF}
have recently given a theory  of
``universal degeneracy loci'', characteristic
classes associated to maps among vector bundles,
in which the role of Schubert varieties is taken by
certain degeneracy schemes.  The underlying varieties
of these schemes
arise in the theory of quivers: they are the
orbit-closures in the space of representations
of the equioriented quiver
$A_h$.  Many other classical varieties also appear as such
quiver varieties, such as determinantal varieties and
the variety of complexes (cf. \S1.4).
The same quiver varieties also arise in Deligne-Langlands theory for
the $p$-adic general linear group \cite{Z1}:
the intersection homology of these varieties gives the $p$-adic
analog of Kazhdan-Lusztig polynomials (which, by Zelevinsky's
result below, become identical to ordinary Kazhdan-Lusztig
polynomials).

%Much effort has gone into giving a
%geometric theory for these varieties
%parallel to that for Schubert varieties.

It turns out that a separate theory is  not necessary
to understand these spaces (for this particular quiver).
By a remarkable but little-known result of Zelevinsky
\cite{Z}, all the above quiver varieties can be identified
set-theoretically with open subsets
of Schubert varieties.
In this paper, we prove a scheme-theoretic strengthening
of Zelevinksy's identification:
the ``naive'' determinantal conditions defining each
quiver variety generate the same ideal as
the Plucker equations defining
the corresponding Schubert variety.
Since the latter ideal is well understood via
Standard Monomial Theory, we conclude that
the corresponding quiver schemes
are reduced and their singularities are identical
to those of Schubert varieties.
In particular, the quiver varieties in arbitrary characteristic
are normal, Cohen-Macaulay, etc.
These properties give a more concrete interpretation
to the intersection theory in Fulton and Buch's
work.

Our results extend early work by Hochster-Eagon \cite{HE},
Kempf \cite{K}, and Deconcini-Strickland \cite{DS}.
Musili-Seshadri \cite{MS},
proved the above scheme-theoretic identification for
the variety of complexes.
Some of the consequences of our
identification were known for more general quiver varieties by
work of Abeasis, Del Fra, and Kraft
\cite{AFK},\cite{A}: that the quiver varieties are Cohen-Macaulay
with rational singularities over a field of characteristic zero,
and that the determinantal conditions generate
the reduced ideals of the quiver varieties of codimension one.
Our methods are similar to those of Gonciulea and Lakshmibai
\cite{GL}.

\section{Zelevinsky's bijection}

In this section we establish the set-theoretic
identification between quiver varieties and
Schubert varieties.
In \S1.4, we give several examples,
including Fulton's degeneracy schemes.

\subsection{Quiver  varieties}

For the basic results below on quivers,
we follow Abeasis-del Fra \cite{AF} and Zelevinsky \cite{Z1}.
Fix an $h$-tuple of non-negative integers
$\nn = (n_1,\ldots,n_h)$
and a list of vector spaces $V_1,\ldots, V_h$
over an arbitrary field $\kk$
with respective dimensions $n_1,\ldots,n_h$.
Define $Z$, the {\it variety of quiver representations}
(of dimension $\nn$, of the equioriented quiver of type
$A_h$) to be the affine space of all
$(h\!-\!1)$-tuples of linear maps $(f_1,\ldots,f_{h\!-\!1}):$
$$
V_1 \stackrel{f_1}{\to} V_2
 \stackrel{f_2}{\to} \cdots \stackrel{f_{h\!-\!2}}{\to} V_{h\!-\!1}
 \stackrel{f_{h\!-\!1}}{\to} V_h \ .
$$
If we endow each $V_i$ with a basis, we get $V_i \cong \kk^{n_i}$
and
$$
Z \cong M(n_2 \!\times\! n_1) \!\times\! \cdots
\times M(n_{h} \!\times\! n_{h\!-\!1}) ,
$$
where $M(l\!\times\! m)$ denotes the affine space of matrices
over $\kk$ with $l$ rows and $m$ columns.
The group
$$
G_{\nn} = GL(n_1) \times \cdots \times GL(n_h)
$$
acts on $Z$ by
$$
(g_1,g_2,\cdots,g_h) \cdot (f_1,f_2,\cdots,f_{h\!-\!1})
= (g_2 f_1 g_1^{-1}, g_3 f_2 g_2^{-1},\cdots,
g_{h}f_{h\!-\!1} g_{h\!-\!1}^{-1}),
$$
corresponding to change of basis in the $V_i$.

Now, let $\rr = (r_{ij})_{1 \leq i \leq j \leq h}$
be an array of non-negative integers with $r_{ii} = n_i$,
and define $r_{ij} = 0$ for any indices other than
$1\leq i\leq j \leq h$.  Define the set
$$
Z^{\circ}(\rr) = \{(f_1,\cdots,f_{h\!-\!1}) \in Z
\ \mid\  \forall\, i\!<\!j,\ \rank (f_{j\!-\!1} \cdots f_i : V_i \to V_j)
 = r_{ij} \}.
$$
(This set might be empty for a bad choice of $\rr$.)
\\[1em]
{\bf Proposition.} {\it The $G_{\nn}$-orbits of
$Z$ are exactly the sets $Z^{\circ}(\rr)$
for $\rr=(r_{ij})$ with
$$
r_{ij}-r_{i,j\!+\!1}-r_{i\!-\!1,j}+r_{i\!-\!1,j\!+\!1}
\geq 0,\quad
\forall\ 1\! \leq\! i\! \leq\! j\! \leq\! h.
$$
}

\noindent
{\bf Proof.} This is a standard result of algebraic
quiver theory \cite{BGP}, \cite{ARS}, first stated in this form by
Abeasis-del Fra and Zelevinsky.
Since this theory is not well known among geometers,
we recall it here.

Consider the abelian category
$\RR$ of quiver representations defined as follows.
An object of $\RR$ is a sequence of linear maps
$(V_1 \stackrel{f_1}{\to} \cdots \stackrel{f_{h\!-\!1}}{\to} V_h)$,
where the $V_i$ are {\it any} vector spaces of
{\it arbitrary} dimension.
A morphism of $\RR$ from the object
$(V_1 \stackrel{f_1}{\to} \cdots \stackrel{f_{h\!-\!1}}{\to} V_h)$
to the object
$(V'_1 \stackrel{f'_1}{\to} \cdots \stackrel{f'_{h\!-\!1}}{\to} V'_h)$
is defined to be an $h$-tuple of linear maps $(\phi_i:V_i \to V'_i)$
such that each of the following squares commutes:
$$
\begin{array}{ccc}
V_i & \stackrel{f_i}{\to} & V_{i+1} \\
\mbox{\tiny $\phi_i$} \downarrow &&
 \downarrow \mbox{\tiny $\phi_{i+1}$} \\
V'_i & \stackrel{f'_i}{\to} & V'_{i+1}
\end{array}
$$


Direct sum of objects is defined componentwise, and it is
known by the Krull-Schmidt Theorem \cite{ARS}
that any object $R \in
\RR$ can be written uniquely as a direct sum of
indecomposable objects.
By elementary linear algebra,
these indecomposables are seen to be
$$
\begin{array}{c@{\!}c@{\!}c@{\!}c@{\!}c}
R_{ij} =
(0 \to \cdots \to 0\to &\kk& \eqto \cdots \eqto &\kk&\to 0 \to \cdots \to 0) \\
&V_i&&V_j &
\end{array}
$$
for $1 \leq i\leq j \leq h$ (corresponding to the
positive roots of the root system $A_h$).
That is, there are unique multiplicities $m_{ij} \in \ZZ^+$ with
$$
R \cong \bigoplus_{1 \leq i\leq j \leq h} m_{ij} R_{ij}.
$$


Our variety $Z$ consists of representations
with fixed $(V_i)$ and all possible $(f_i)$.
Two points of $Z$ are in the same $G_{\nn}$-orbit
exactly if they are isomorphic as objects in $\RR$.
So the orbits correspond to arrays
 $(m_{ij})_{1 \leq i\leq j \leq h}$ with $m_{ij} \in \ZZ^+$
and $n_i = \sum_{k\leq i\leq l} m_{kl}$.

We can compute the rank numbers $\rr=(r_{ij})$ from
the multiplicities $\mm=(m_{ij})$:
$$
r_{ij} = \sum_{k\leq i\leq j\leq l} m_{kl},
$$
and conversely
$$
m_{ij}=
r_{ij}-r_{i,j\!+\!1}-r_{i\!-\!1,j}+r_{i\!-\!1,j\!+\!1}
$$
Hence the arrays $(r_{ij})$ with the stated conditions
classify the $G_{\nn}$-orbits on $Z$. $\bullet$
\\[1em]
We define the {\it quiver variety} as the algebraic set
$$
Z(\rr)
=\{(f_1,\cdots,f_{h\!-\!1}) \in Z
\mid  \forall i,j,\ \rank (f_{j\!-\!1} \cdots f_i : V_i \to V_j)
\leq r_{ij}\}.
$$
It will follow from Zelevinsky's theorem (\S1.3)
that $Z(\rr)$ is an irreducible
variety and is the Zariski closure of $Z^{\circ}(\rr)$\
(provided the base field $\kk$ is infinite).


\subsection{Schubert varieties}

Given $\nn=(n_1,\cdots,n_h)$, for $1 \leq i \leq h$ let
$$
a_i = n_1 + n_2 + \cdots +n_i,
\qquad \mbox{and} \qquad
n = n_1  + \cdots + n_h \ .
$$
For positive integers $i \leq j$, we shall frequently use
the notations
$$
[i,j] = \{ i, i+1, \ldots, j\}, \qquad [i] = [1,i],
\qquad [0] = \{\} \ .
$$

Let $\kk^n \cong V_1 \oplus \cdots \oplus V_h$
have basis $e_1,\ldots,e_n$ compatible
with the $V_i$.  Consider its general
linear group $GL(n)$, the subgroup $B$ of upper-triangular
matrices, and the parabolic subgroup $P$ of block upper-triangular
matrices
$$
P = \{ (a_{ij}) \in GL(n) \mid a_{ij}=0 \
\mbox{whenever}\ j\leq a_k <i
\ \mbox{for some}\ k \}\ .
$$
A {\it partial flag of type $(a_1<a_2<\cdots <a_h=n)$ }
(or simply a {\it flag}) is a sequence of subspaces
$U\bdot = (U_1 \subset U_2 \subset \cdots \subset U_h = \kk^n)$
with $\dim U_i = a_i$.
Let $E_i = V_1\oplus\cdots\oplus V_i
= \langle e_{1},\ldots,e_{a_i}\rangle$,
and $E'_i = V_{i\!+\!1} \oplus \cdots \oplus V_{h}
=\langle e_{a_i+1},\ldots,e_n\rangle$, so that
$E_i \oplus E'_i = \kk^n$.
%Call $E\bdot = (E_1 \subset E_2 \subset \cdots)$
%the {\it standard flag}.
%$E'. = (\kk^n \supset E'_1 \supset \cdots\supset E'_{h-1} \supset
%E'_h = 0)$ the
%{\it opposite standard flag}.
The {\it flag variety} $\Fl$ is the set of all flags $U\bdot$ as above.

$\Fl$ has a transitive $GL(n)$-action induced from
$\kk^n$, and $P = \Stab_{GL(n)}( E\bdot)$, so we may identify
$\Fl \cong GL(n)/P$, \ $g\!\! \cdot\!\! E\bdot \leftrightarrow gP$\ .
The {\it Schubert varieties} are the closures of $B$-orbits
on $\Fl$.  Such orbits are usually indexed by certain
permutations of $[n]$, but we prefer to use
{\it flags of subsets} of $[n]$, of the form
$$
\tau = (\tau_1 \subset \tau_2 \subset\cdots \subset \tau_h = [n]),
\qquad \#\tau_i=a_i\ .
$$
(A permutation $w: [n]\to[n]$
corresponds to the subset-flag with
$\tau_i = w[a_i] = \{w(1),w(2),\ldots,w(a_i)\}$.
This gives a one-to-one correspondence between cosets
of the symmetric group $W=S_n$ modulo the Young subgroup
$W_{\nn}=S_{n_1} \times \cdots \times S_{n_h}$,
and subset-flags.)

Given such $\tau$, let
$E_i(\tau) = \langle e_j \mid j \in \tau_i \rangle$
be a coordinate subspace of $\kk^n$, and
$E\bdot(\tau) = (E_1(\tau) \subset E_2(\tau) \subset \cdots) \in \Fl$.
Then we may define the {\it Schubert cell}
$$
\begin{array}{rcl}
X^{\circ}(\tau) &= &B\cdot E(\tau)\\
&=& \left\{(U_1\subset U_2\subset\cdots)\in \Fl\ \ \left|\
\begin{array}{c}
\dim U_i \cap \kk^j = \#\, \tau_i \cap [j]\\[.2em]
1\leq i \leq h,\ 1\leq j \leq n
\end{array}
\right.\right\}
\end{array}
$$
and the {\it Schubert variety}
$$
\begin{array}{rcl}
X(\tau) &= &\overline{X^{\circ}(\tau)}\\
&=& \left\{(U_1\subset U_2\subset\cdots)\in \Fl\ \ \left|\
\begin{array}{c}
\dim U_i \cap \kk^j \geq \#\, \tau_i \cap [j]\\[.2em]
1\leq i \leq h,\ 1\leq j \leq n
\end{array}
\right.\right\}
\end{array}
$$
where $\kk^j = \langle e_1,\ldots,e_j\rangle \subset \kk^n$.

We define the {\it opposite cell} $\OO \subset \Fl$
to be the set of flags in general position with respect
to the spaces
$E'_1 \supset \cdots \supset E'_{h-1}$:
$$
\OO = \{(U_1\subset U_2\subset\cdots)\in \Fl\ \mid\
U_i \cap E'_{i}=0\}.
$$
In fact, $\OO = B_{-}\cdot E\bdot$, the orbit
of the standard flag in $\Fl$ under
the group $B_{-}$ of lower triangular matrices.
We also define $Y(\tau) = X(\tau) \cap \OO$, an open subset of
$X(\tau)$ and $Y^{\circ}(\tau) = X^{\circ}(\tau) \cap \OO$.
By abuse of language, we call $Y(\tau)$ the
{\it opposite cell} of $X(\tau)$,
even though it is not a cell.


\subsection{The bijection $\zeta$}

We define a special subset-flag
$\taum = (\taum_1 \subset \cdots \subset\taum_h = [n])$
corresponding
to $\nn = (n_1,\ldots,n_h)$.
We want each $\taum_i$
to contain numbers as large as possible
given the constraints $[a_{j\!-\!1}]\subset \taum_j$ for
all $j$ (here $a_0=0$).
Namely, we define $\taum_i$ recursively by
$$
\taum_h = [n];\quad \taum_{i} = [a_{i\!-\!1}]
\cup \{ \mbox{largest $n_i$ elements of $\taum_{i+1}$}\}.
$$
Furthermore, given $\rr = (r_{ij})_{1\leq i\leq j\leq h}$
indexing a quiver variety, define a subset-flag $\taur$ to
contain numbers as large as possible given the
constraints
$$
\#\, \taur_i\cap [a_j] =
\left\{ \begin{array}{cl}
a_i -r_{i,j+1} & \mbox{for}\ i\leq j \\
a_j& \mbox{for}\ i> j \\
\end{array} \right.
$$
Namely,
$$
\taur_i = \{\,
\underbrace{1\ldots a_{i\!-\!1}}_
{\mbox{\small $a_{i\!-\!1}$}}
\ \underbrace{. \ldots\ldots a_{i}}_
{\mbox{\small $r_{ii}\!-\!r_{i,i+1}$}}
\ \underbrace{.\ldots\ldots a_{i+1}}_
{\mbox{\small $r_{i,i+1}\!-\!r_{i,i+2}$}}
\ \underbrace{.\ldots\ldots a_{i+2}}_
{\mbox{\small $r_{i,i+2}\!-\!r_{i,i+3}$}}\ \ldots\
\ \underbrace{.\ldots\ldots n_{\mbox{}}}_
{\mbox{\small $r_{i,h}$}}
\}
$$
where we use the visual notation
$$
\underbrace{\cdots\cdots a}_{\mbox{\small $b$}} =
[a\!-\!b\!+\!1,a].
$$
Recall that $a_j = a_{j-1}+n_j$
and $0\leq r_{ij} -r_{i,j+1} \leq n_j$,
so that each $\taur_i$ is an increasing list of integers.
Also $r_{ij}-r_{i,j+1}\leq r_{i+1,j}-r_{i+1,j+1}$,
so that $\taur_i \subset \taur_{i+1}$.  Thus
$\taur$ is indeed a subset-flag.
See \S1.4 for examples.

Now define the Zelevinsky map
$$
\begin{array}{rccc}
\zeta: & Z & \to &\Fl \\
& (f_1,\ldots,f_{h\!-\!1}) & \mapsto &(U_1\subset U_2 \subset \cdots)
\end{array}
$$
where
$$
U_i = \{ (v_1,\ldots,v_h)\in
V_1\!\oplus\! \cdots \!\oplus\! V_h = \kk^n \mid
\forall\, j\geq i,\ v_{j+1} = f_j(v_j)\}.
$$
In terms of coordinates, if we identify the linear maps
$(f_1,\ldots,f_{h\!-\!1})$ with the matrices $(A_1,\ldots,A_{h\!-\!1})$,
and identify $\Fl \cong GL(n)/P$,
we have
$$
\zeta(A_1,\ldots,A_{h-1}) =
\left(
\begin{array}{ccccc}
I_1 & 0 & 0 & 0& \cdots \\
A_1 & I_2 &0 & 0& \cdots \\
A_2 A_1 & A_2 & I_3 & 0& \cdots \\
A_3 A_2 A_1 & A_3 A_2 &A_3 & I_4 & \cdots  \\[-.4em]
\vdots & \vdots & \vdots &  \vdots&
%A_{h\!-\!1} \cdots A_1 & A_{h\!-\!1}\cdots A_{2} &
%A_{h\!-\!1} \cdots A_3 & \cdots &I_{h\!-\!1}
\end{array}
\right) \ \
\mod \ \ P
$$
where $I_i$ is an identity matrix of size $n_i$.
\pagebreak

\noindent
{\bf Theorem.}  {\it (Zelevinsky \cite{Z})\\
(i) $\zeta$ is a bijection of $Z$ onto its image $Y(\taum)$:
\quad $\zeta : Z \eqto Y(\taum)$.\\
Also, \\[-1em]
$$
\mbox{}\hspace{-2em} (*)\qquad
Y(\taum)=
\{(U_1\subset U_2 \subset \cdots) \ \mid \
\forall\ i,\ \ E_{i-1} \subset U_i,
\ \ U_i \cap E'_{i} = 0 \}.
$$
(ii) $\zeta$ restricts to a bijection from $Z(\rr)$
onto $Y(\taur)$:\quad $\zeta : Z(\rr) \eqto Y(\taur)$.\\
Also, \\[-1em]
$$
\mbox{} \hspace{-1em} (**) \quad
Y(\taur)=\left\{(U_1\subset U_2 \subset \cdots) \ \left| \
\begin{array}{c}
 \forall\ i\leq j,\quad \dim \, U_i \cap E_j \geq a_i-r_{i,j\!+\!1}
,\\[.3em]
 E_{i-1} \subset U_i,
\quad U_i \cap E'_{i} = 0\end{array}
\right.\right\}\ .
$$
}


\noindent{\bf Proof.}  Obviously $\zeta$ is injective.
To prove (i), we first show equation $(*)$.  The inclusion
$\subset$ is clear.  For the inclusion $\supset$, consider
a flag $U\bdot$ with $E_{i-1} \subset U_i$ for all $i$.
Since acting by $B$ does not change $\dim U_i\cap \kk^j$,
we may suppose $U\bdot = E\bdot(\mu)$ for some
$\mu=(\mu_1\subset\cdots\subset \mu_{h}=[n])$
with $[a_{i\!-\!1}]\subset \mu_i$ for all $i$.
By definition $\# \taum_i \cap [j]$ is as small as
possible given $[a_{i\!-\!1}]\subset \taum_i$, so
$$
\dim U_i \cap \kk^j = \# \mu_i \cap [j]
\geq  \# \taum_i \cap [j],
$$
which shows $(*)$.

Now we show that $\zeta(Z)$ is equal to the right
hand side of $(*)$.  The inclusion
$\zeta(Z) \subset \mbox{RHS}(*)$
is clear, so we show $\zeta(Z) \supset \mbox{RHS}(*)$.
Each $U_i$ is tansverse to $E'_i$, so $U_i$ is the graph
of a linear map
$$
(f_{i,i\!+\!1},\ldots,f_{i,h}): E_i \to
E'_i = V_{i\!+\!1} \oplus \cdots \oplus V_h.
$$
Since $E_{i-1} \subset U_i$,
we have $f_{ij}(E_{i\!-\!1}) =0$,
and we may consider $f_{ij}:V_i \cong E_i/E_{i\!-\!1}\to V_j$.
Any element of $U_j$ can be written
$(v_1,\ldots,v_j,f_{j,j\!+\!1}(v_j),\ldots)$.
Let $i<j$.  Any $(v_1,\ldots,v_h) \in U_i$ is also
an element of $U_j$, so $v_{j\!+\!1} = f_{j,j\!+\!1}(v_j)$.
Taking $f_i = f_{i,i\!+\!1}$,
we find $U\bdot = \zeta(f_1,\ldots,f_{h\!-\!1})$.

The proof of (ii) is similar.
Equation $(**)$ follows just as before.
Now consider a flag
$U\bdot = \zeta(f_1\ldots f_{h\mo})
\in \zeta(Z) = Y(\taum)$.
Then
$$
\begin{array}{rcl}
\dim\, U_i \cap E_j &=&
\dim E_{i\!-\!1} + \dim \mbox{Ker}(f_j f_{j\!-\!1} \cdots f_i)\\
&=& \dim E_{i\!-\!1} + \dim V_i - \rank( f_j f_{\!j-\!1}\cdots f_i)\\
&=& a_i -\rank( f_j f_{j\!-\!1}\cdots f_i).
\end{array}
$$
Hence $U\bdot \in \zeta(Z(\rr))\ \Leftrightarrow\
U\bdot \in \mbox{RHS}(**) = Y(\taur)$. $\bullet$
\\[1em]
{\bf Corollary.} {(\cite{AF}, \cite{Z1}).
For each $\rr$,\ $Z^{\circ}(\rr)$ is
an open dense $G_{\nn}$-orbit in $Z(r)$. }
\\[.5em]
{\bf Proof.}  Arguing as in the proof of (ii) above,
we find that $\zeta(Z^{\circ}(\rr)) \supset Y^{\circ}(\taur)$.
But it is known that $Y^{\circ}(\taur)$ is Zariski open
and dense in $Y(\taur)$.
 $\bullet$
\\[1em]
{\bf Remarks.} (i) The map $\zeta$
is an algebraic isomorphism onto its image, since
it is clear from the coordinate definition that
$\zeta$ is injective on points and on tangent vectors.\\
(ii) For each $\rr$, \ $X^{\circ}(\taur)$
is an orbit of $P$.  If we embed $G_{\nn}$ into $P$
as block-diagonal matrices, then $\zeta$ is a
$G_{\nn}$-equivariant map.
Now, clearly $\zeta(Z^{\circ}(\rr))\subset Y^{\circ}(\taur)$.
Also the $Z^{\circ}(\rr)$ are a complete list
of $G_{\nn}$-orbits on $Z$ and the $Y^{\circ}(\taur)$
are disjoint subsets of $Y(\taum)\cong \zeta(Z)$.
We conclude that $\zeta(Z^{\circ}(\rr))=Y^{\circ}(\taur)$.



\subsection{Examples}

{\bf Example.} A small generic case.
Let $h=4$,\ $\nn = (2,3,2,2)$,\
$$
\rr=
\begin{array}{|cccc|}\hline
2&2&0&0\\
&3&1&1\\
&&2&2\\
&&&2\\
\hline
\end{array}
\qquad
\mm=
\begin{array}{|cccc|}\hline
0&2&0&0\\
&0&0&1\\
&&0&1\\
&&&0\\
\hline
\end{array}
$$
where $r_{ij}$ and $m_{ij}$ are written
in the usual matrix positions.  Note that $r_{ij}$
is obtained by summing the entries in $\mm$
weakly above and to the right of $m_{ij}$.

Then we get $(a_1,a_2,a_3,a_4) = (2,5,7,9)$,\ $n=9$,
and
$$
\taum = (89\subset 12589 \subset 1234589 \subset [9]),
\qquad
\taur = (45\subset 12459 \subset 1234589 \subset [9]),
$$
which correspond to the cosets in $W/W_{\nn}$
$$
w^{\mbox{\tiny max}} = 89|125|34|67,
\qquad
w^{\rr} = 45|129|38|67.
$$
(The minimal-length representatives of these cosets
are the permutations as written; the other elements
are obtained by permuting numbers within each block.)
The partial flag variety is
$\Fl = \{ U_1 \subset U_2 \subset U_3
\subset \kk^9 \mid \dim U_i = a_i \}$, and
the Schubert varieties are:
$$
X(\taum)=\left\{
U\bdot \left|
\begin{array}{c}
\kk^2 \subset U_2\\
\kk^5 \subset U_3
\end{array}
\right. \right\},
\quad
X(\taur)=\left\{
U\bdot \left|
\begin{array}{c}
 U_1\!\subset\! \kk^5\!\subset\! U_3,\
\kk^2\! \subset\! U_2\\
 \dim U_2 \cap \kk^5 \geq 4
\end{array}
\right. \right\}.
$$
The opposite cells $Y(\tau)$ are defined by the extra
conditions $U_i \cap E'_i = 0$.
\\[1em]
%
{\bf Example.} Fulton's universal
degeneracy schemes \cite{F}.
Given $m>0$, let $Z$ be the affine space associated
to the quiver data $h=2m$,\ $\nn = (1,2,\ldots,m,m,\ldots,2,1)$.
For each $w \in S_{m\po}$, Fulton defines a
``degeneracy scheme'' $\Om_{w} = Z(\rr)$ as follows.
(Here $\Om_{w}=Z(\rr)$ is a variety.  We will define
scheme structures for quiver varieties in \S2.)
Denote $\bi = 2m+1-i$, and define $\rr = \rr(w) = (r_{ij})$
and $\mm = (m_{ij})$ by:
$$
\begin{array}{c}
r_{ij} = r_{\bj\bi} = i \\[.2em]
r_{i\bj} = \#\,[i]\cap w[j]
\end{array}
\qquad
m_{ij} = \left\{ \begin{array}{cl}
1, & (i,j) = (w(k),\overline{k}),\ \exists k \leq m\\
0, & \mbox{otherwise}
\end{array}
\right.
$$
for $1\leq i,j\leq m$.
The associated Schubert varieties $Y(\taur)$ are
given by  $\taur = (\taur_1\subset \cdots  \subset
\taur_{\overline{1}})$
or by cosets $\tw = \tw_1 |\cdots |\tw_{\overline 1}
\in W/W_{\nn}$
$$
\mbox{\hspace{-1em}}
\begin{array}{c}
\taur_i = [a_{i\mo}] \cup
\{ a_{\overline{w^{\mo}(1)}}\,, a_{\overline{w^{\mo}(2)}}\,,
\ldots, a_{\overline{w^{\mo}(i)}} \},\\[.5em]
\taur_{\bi} = [a_{\bi}-\!1] \cup
\{ a_{\overline{1}}, a_{\overline{2}},
\ldots, a_{\overline{m}} \}
\end{array}
\quad
\begin{array}{c}
\tw_i = [a_{i\!-\!2}+\!1,a_{i\mo}] \cup
\{ a_{\overline{ w^{\mo}(i)}} \}
\\[.2em]
\tw_{\overline{m}} = [a_{m\!-\!1}\po,a_{m}\mo] \cup
\{ a_{\overline{ w^{\mo}(m\po) }}\}
\\[.2em]
\tw_{\bj} = [a_{\bj\!-\!2}\po,a_{\bj\mo}]
\end{array}
$$
for $1\leq i\leq  m$,\ \ $1\leq j\leq  m\mo$.
Furthermore $\taum = \tau^{\rr(w)}$
and $\tw^{\mbox{\tiny max}} = \tw^{\rr(w)}$ for
$w = e \in S_{m\po}$, the identity permutation.
\\[1em]
%
{\bf Example.}  For a given $h$ and $\nn$, the
{\it variety of complexes} is defined as the union
${\cal C} = \cup_{\rr} Z(\rr)$ over all
$\rr = (r_{ij})$ with $r_{i,i\!+\!2}=0$ for each $i$.
The subvarieties $Z(\rr)$ correspond to the
multiplicity matrices $\mm = (m_{ij})$ with
$m_{ij}=0$ for all $i+2 \leq j$, and $m_{ii}+m_{i\mo,i}+
m_{i,i\po} = n_i$ for all $i$.
Musili-Seshadri \cite{MS} find the irreducible
components (maximal subvarieties)  of ${\cal C}$,
and show that each component is isomorphic
to the opposite cell of some Schubert variety.
\\[1em]
%
{\bf Example.} The classical {\it determinantal variety}
of $k \times l$ matrices of rank $\leq m$ (where $m\leq k,l$) is
${\cal D} = Z(\rr)$ for
$\rr=$
{\footnotesize
$\left( \begin{array}{@{\!}cc@{\!}}l&m\\0&k\end{array}\right)$
},
and
$\mm=$
{\footnotesize
$\left( \begin{array}{@{\!}cc@{\!}}l\!-\!m&m\\
0&k\!-\!m\end{array}\right)$
}.
Also $n=k+l$,\
$$
\taum = ([k+1,n]\subset [n]),\quad
\taur = ([m+1,l]\cup [n-m+1,n]\subset [n])
$$
$$
X(\taum) = \Fl = \Gr(l,\kk^n), \quad\
X(\taur) \cong
\{ U\in \Gr(l,\kk^n)\mid \dim U\cap \kk^l \geq l-m \},
$$
$$
{\cal D} = Z(\rr) \cong Y(\taur) =
\{U\in \Gr(l,\kk^n)\mid \dim U\cap \kk^l\geq l-m, \ U\cap E'=0\},
$$
where $E'=\langle e_{l\po},e_{l\!+\!2},\ldots,e_n\rangle$.


\section{Plucker coordinates and determinantal ideals}

In this section we prove the scheme-theoretic version
of Zelevinsky's bijection.
From now on, we assume our field $\kk$ is infinite.

\subsection{Coordinates on the opposite big cell}

Consider the opposite
cell $\OO \subset GL(n)/P$.  It is easily seen that
$\OO$ consists of those cosets which have a unique
representative $A$ of the form
$$
A = (a_{kl}) =
\left( \begin{array}{ccccc}
I_1& 0 & 0& \cdots & 0\\
A_{21} & I_2 & 0 & \cdots &0\\
A_{31}& A_{32} & I_3 &\cdots & 0\\
\vdots&\vdots&\vdots &&\vdots\\
A_{h1}& A_{h2}& A_{h3}&\cdots& I_{h}
\end{array} \right)\  \mod \   P,
$$
where $I_i$ is the identity matrix of size $n_i$,
and $A_{ij}$ is an arbitrary matrix of size $n_i \times n_j$.
That is, $\OO$ is an affine space with coordinates
$a_{kl}$ for those positions $(k,l)$
with $1 \leq l \leq a_i <k \leq n$ for some $i$.
Its coordinate ring is the polynomial ring
$$
\kk[\OO] = \kk[a_{kl}].
$$

For a matrix $M \in M(k\times l)$ and subsets
$\lam\subset [k]$, $\mu \subset [l]$, let
$\det M_{\lam\times \mu}$ be the minor with row indices $\lam$
and column indices $\mu$.
Now let $\sig \subset [n]$ be a subset of size
$\#\sig = a_i$ for some $i$.  Define the {\it Plucker
coordinate} $p_{\sig} \in \kk[\OO]$ to be the
$a_i$-minor of our matrix $A$ with row indices $\sig$ and
column indices the interval $[a_i]$:
$$
 p_{\sig}=p_{\sig}(A)=\det A_{\sig \times [a_i]}.
$$
Define a partial order on Plucker coordinates by:
$$
\sig \leq \sig'
\quad \Longleftrightarrow \quad
\begin{array}{c}
\sig = \{\sig(1)<\sig(2)<\cdots<\sig(a_i)\},\\
\sig' = \{\sig'(1)<\sig'(2)<\cdots<\sig'(a_i)\},\\
\sig(1)\leq \sig'(1),\ \sig(2) \leq \sig'(2),
\cdots, \sig(a_i) \leq \sig'(a_i).
\end{array}
$$
This is a version of the Bruhat order.
\\[1em]
{\bf Proposition.} {\it
Let $\tau = (\tau_1 \subset \cdots \subset \tau_{h} = [n])$
be a subset-flag and $Y(\tau)$ the intersection of the
Schubert variety $X(\tau)$ with the opposite cell $\OO$.
Then the (reduced) vanishing ideal
$\II(\tau) \subset \kk[\OO]$ of
$Y (\tau) \subset \OO$ is generated by
those Plucker coordinates $p_{\sig}$
which are incomparable with one of the $p_{\tau_i}$:
$$
\II(\tau) = \langle p_{\sig} \mid
\exists\, i,\ \#\sig = a_i,
\ \sig \not\leq \tau_i \rangle.
$$
}

\noindent
{\bf Proof.}  This follows from well-known results of
Lakshmibai-Musili-Seshadri in
Standard Monomial Theory (see e.g.~\cite{MS},\cite{LS}).

\subsection{The main theorem}

Denote a generic element of the quiver space
$ Z = M(n_2\times n_1) \times \cdots
\times M(n_{h}\times n_{h\!-\!1})$
by $(A_1,\ldots,A_{h-1})$, so that the coordinate ring
of $Z$ is the polynomial ring in the entries of all the matrices
$A_i$.  Let $\rr = (r_{ij})$ index the quiver variety
$Z(\rr) = \{(A_1,\ldots,A_{h-1}) \mid
\rank\, A_{j-1}\cdots A_i \leq r_{ij}\}$.

Let $\JJ(\rr) \subset \kk[Z]$ be the ideal generated by
the determinantal conditions implied by the definition
of $Z(\rr)$:
$$
\JJ(\rr) = \left\langle \det(A_{j-1} A_{j-2} \cdots A_i)_
{\lam\times\mu}
\ \left| \
\begin{array}{c}
j>i,\ \lam \subset [n_j],\ \mu \subset [n_i] \\[.2em]
\#\lam = \#\mu = r_{ij}+1
\end{array}
\right.
\right\rangle\ .
$$
Clearly $\JJ(\rr)$ defines $Z(\rr)$ set-theoretically.
\\[1em]
{\bf Theorem.} {\it
$\JJ(\rr)$ is a prime ideal and is the vanishing ideal
of $Z(\rr)\subset Z$.  There are isomorphisms of
reduced schemes
$$
Z(\rr) = \mbox{Spec}(\kk[Z]\,/\,\JJ(\rr)) \cong
\mbox{Spec}(\kk[\OO]\,/\,\II(\taur))
= Y(\taur).
$$
That is, the quiver scheme $Z(\rr)$ defined by $\JJ(\rr)$ is
isomorphic to the reduced variety $Y(\taur)$,
the opposite cell of a Schubert variety.
}
\\[1em]
{\bf Corollary.} {\it For a ring $R$, consider the polynomial ring
$R[\OO] = R[a_{kj}]$ and the ideal $\JJ(\rr)_R \subset R[\OO]$
generated by the same determinants as above.
Define the scheme $Z(\rr)_R = {\rm Spec}(R[\OO]/\JJ(\rr)_R)$.\\
(i) If $\kk$ is an arbitrary field,
then $Z(\rr)_{\kk}$ is reduced, irreducible, Cohen-Macaulay, normal, and
has rational singularities. \\
(ii) If $R$ is a noetherian ring, then $Z(\rr)_R$
is reduced (resp.~irreducible, Cohen-Macaulay,  normal)
exactly when ${\rm Spec}(R)$
is reduced (irreducible, Cohen-Macaulay, normal).
}
\\[1em]
{\bf Proof of Corollary.}
(i) Let $\overline{\kk}$ be the algebraic closure of $\kk$.
Then the desired properties of $Z(\rr)_{\overline{\kk}}$
follow from the corresponding properties of
Schubert varieties (see e.g.~\cite{J}, \cite{R1}, \cite{R2}).
But this implies these properties for
$Z(\rr)_{\kk}$ as well,
since $Z(\rr)_{\overline{\kk}} \to Z(\rr)_{\kk}$ is a
faithfully flat morphism.  (See \cite{Mm}, \S21.E.) \\
(ii) By \cite{R1}, the morphism
$Z(\rr)_{\ZZ} \to{\rm Spec}(\ZZ)$ is faithfully flat,
so $Z(\rr)_R \to{\rm Spec}(R)$ is as well (\cite{Mm}, \S 3.C).
Now, the fibers of this latter morphism
are reduced, Cohen-Macaulay, and normal by (i), so the corresponding
properties hold for the total space $Z(\rr)_R$ exactly when
they hold for the base (\cite{Mm}, \S21.E).
Finally, $Z(\rr)_R \to{\rm Spec}(R)$ is a closed surjective
morphism with irreducible fibers of the same dimension,
and it is elementary that the total space
is irreducible exactly when the
base is irreducible (see \cite{Sh}, \S I.6.3).
\\[1em]
{\bf Proof of Theorem.}
The map of \S1.3,
$\zeta: Z \eqto Y(\taum) \subset \OO$
is an algebraic isomorphism onto its image,
so the restriction homomorphism
$\zeta^*:\kk[\OO] \to \kk[Z]$ is surjective.

Now, $\zeta$ maps $Z(\rr)$ isomorphically onto $Y(\taur)$,
so we have
$$
\begin{array}{rcl}
Z(\rr) = \mbox{Spec}(\kk[Z]\,/\,\tJ(\rr)) &\cong&
\mbox{Spec}(\kk[\OO]\,/\,\zi\tJ(\rr))\\
&=&\mbox{Spec}(\kk[\OO]\,/\,\II(\taur))
\ = \ Y(\taur).
\end{array}
$$
where $\tJ(\rr) \subset \kk[Z]$ denotes the (reduced)
vanishing ideal of $Y(\taur)$.

We must show $\JJ(\rr) = \tJ(\rr)$.  Since
clearly $\JJ(\rr) \subset \tJ(\rr)$, we are left
with the other inclusion, which is equivalent to
$$
\zi\JJ(\rr) \supset \zi \tJ(\rr) = \II(\taur).
$$
We prove this in the next section.


\subsection{Proof of the main theorem}

Let $A \in \OO$ be the generic matrix of \S2.1.
We define ideals $\II_0, \II_1, \II_2 \subset \kk[\OO]$
generated by certain minors of $A$:
$$
\II_0 =\zi\JJ(\rr) \hspace{3.2in} \mbox{}
$$
\vspace{-1.8em}
$$
= \zi
\left\langle \det(A_{j-1} A_{j-2} \cdots A_i)_
{\lam\times\mu}
\ \left| \
\begin{array}{c}
j>i,\ \lam \subset [n_j],\ \mu \subset [n_i] \\[.2em]
\#\lam = \#\mu = r_{ij}+1
\end{array}
\right.
\right\rangle\ .
$$
\vspace{.4em}
$$
\II_1\ =
\left\langle \det A_{\lam\times\mu}\ \left|\
\begin{array}{c}
i\leq j,\ \
\lam \subset [a_j\!+\!1,n],\ \ \mu\subset [a_i]\\[.2em]
\# \lam = \#\mu = r_{ij}\!+\!1
\end{array}
\right. \right\rangle
$$
\vspace{.4em}
$$
\II_2\ =\  \II(\taur)\ =\
\left\langle \det A_{\sig\times [a_i]}\ \left|\
\begin{array}{c}
1\leq i \leq h\!-\!1,\ \ \sig \subset [n]\\[.2em]
\# \sig = a_i,\ \ \sig \not \leq \taur_i
\end{array}
\right. \right\rangle
$$
To finish the proof of Theorem 2.2, we will show
$$
\II_0 \supset \II_1 \supset \II_2\ .
$$

\noindent
{\bf Lemma 1.} {\it
Let $X = (x_{ij})$ and $Y= (y_{kl})$ be matrices of
variables $x_{ij}$, $y_{kl}$ generating a polynomial ring.
Let $\JJ_{X}$  (resp.~$\JJ_{Y}$) be the ideal generated by all
$r\!+\!1$-minors of $X$ (resp.~$Y$).  Then $\JJ_X$ and $\JJ_Y$
both contain all $r\!+\!1$-minors of the product $XY$.
}
\vspace{.5em}

\noindent {\bf Proof.}
$$
\det (XY)_{\lam\times\mu}
= \sum_{\nu} \det X_{\lam\times \nu}\, \det Y_{\nu\times\mu}.
\quad \bullet
$$

\noindent {\bf Lemma 2.} {\it
Let $(A_1,\ldots,A_{h-1})$ be a generic element of $Z$, and
for $j>i$
let $\JJ_{ji}$ be the ideal generated by
all $r+1$-minors of the $n_j \times n_i$ product matrix
$A_{j\mo}\cdots A_i$.
Then $\JJ_{ji}$ contains all $r\!+\!1$-minors of the
$(n\!-\!a_{j\mo}) \times a_i$ matrix
$$
\widetilde{A}^{(ji)} =
\left( \begin{array}{cccc}
A_{j\mo}\!\! \cdots\! A_{1} & A_{j\mo}\!\!\cdots\! A_2& \cdots& A_{j\mo}\!\! \cdots\! A_i \\
A_{j}\!\! \cdots\! A_{1} & A_{j}\!\!\cdots \!A_2&
 \cdots& A_{j}\!\! \cdots\! A_i \\
\vdots&\vdots&&\vdots\\
A_{h\!-\!1}\!\! \cdots\! A_{1} & A_{h\!-\!1}\!\!\cdots\! A_2&
\cdots& A_{h\!-\!1}\!\! \cdots A_i\!
\end{array} \right)
$$
}
\noindent {\bf Proof.} Note that we can factor the matrix
$$
\widetilde{A}^{(ji)} = \left(\!\!\!\begin{array}{c}
I_{j\mo} \\ A_{j} \\ \vdots \\ A_{h\!-\!1}\! \cdots\! A_{j}
\end{array} \!\!\!\right)
 \cdot\, A_{j\mo}\!\!\cdots\!\! A_i\, \cdot\
( A_{i\!-\!1}\!\!\cdots\!\! A_1,\
A_{i\!-\!1}\!\!\cdots\!\! A_2,\ \cdots\ ,\ A_{i\!-\!1},\ I_i).
$$
Now apply Lemma 1 twice. $\bullet$
\\[1em]
%
{\bf Lemma 3.}\qquad
$ \II_0 \supset \II_1\ .$
\\[.5em]
{\bf Proof.}
For generic elements $A \in \OO$ and $(A_1,\ldots,A_{h\mo})
\in Z$, we have by definition
$\zeta^*(f(A)) = f(\zeta(A_1,\ldots,A_{h\mo}))$
for any polynomial $f$ in the matrix entries.
Now let $\lam \subset [a_{j\mo}\!+\!1,n]$,\
$\mu \subset [a_i]$, $\#\lam = \#\mu = r_{ij}+1$,
and consider a generator $\det A_{\lam\times\mu}$
of $\II_1$.  Then
$$
\zeta^*( \det A_{\lam\times\mu} ) =
\det \zeta(A_1,\ldots,A_{h\mo})_{\lam\times\mu}
=\det \tA^{(ji)}_{\lam'\times\mu}
$$
where $\lam' \subset [n-a_{j\mo}]$ is a translate of $\lam$.
By Lemma 2, $\det \tA^{(ji)}_{\lam'\times\mu} \in \JJ(\rr)$,
so $\II_1=\langle\det A_{\lam\times\mu}\rangle
\subset \zi \JJ(\rr) =\II_0$. $\bullet$
\\[1em]
{\bf Lemma 4.} {\it (Gonciulea-Lakshmibai)\
Let $A$ be a generic element of $\OO$.
Let $1 \leq t\leq a_i$, \ $1\leq s \leq n$, and
$\sig = \{ \sig(1)<\sig(2)<\cdots<\sig(a_i)\} \subset [n]$
with $\sig(a_i-t+1) \geq s$.  Then $p_{\sig}(A)$
belongs to the ideal of $\kk[\OO]$ generated by $t$-minors
of $A$ with row indices $\geq s$ and column indices $\leq a_i$.
}
\vspace{.5em}

\noindent{\bf Proof.}  Choose $\sig'\subset [s,n] \cap \sig$
with $\#\sig' =t$, and let $\sig'' = \sig \!\setminus\! \sig'$.
Then the Laplace expansion of $p_{\sig}(A)$ with respect
to the rows $\sig'$, $\sig''$, gives
$$
p_{\sig}(A) = \det A_{\sig\times[a_i]}
= \sum_{\lam' \cup \lam''= [a_i]}
\!\!\!\pm \det A_{\sig'\times \lam'}
\det A_{\sig''\times\lam''},
$$
where the sum is over all partitions of the interval $[a_i]$.
The first factor of each term in the sum
is of the form required. $\bullet$
\\[1em]
{\bf Lemma 5.}\qquad  $\II_1 \supset \II_2\ .$
\\[.5em]
{\bf Proof.}
Let $\sig \subset [n]$ with $\# \sig = a_i$,\
$\sig \not \leq \taur_i$ for some $i$,\ $1 \leq i\leq h\!-\!1$.
Now, $\taur_i$ has the largest
possible entries such that
$$
\taur_i(a_i-r_{i,j+1})\leq a_j,\qquad \forall \,j\geq i ,
$$
so $\sig \not \leq \taur_i$ must violate this condition for
some $j$:
$$
\sig(a_i-r_{i,j+1}) \geq a_j+1,\qquad \exists\, j\geq i.
$$
Hence by Lemma 4, $p_{\sig}(A)$ is in $\II_1$. $\bullet$
\\[1em]
The Main Theorem  2.2 is therefore proved.



\small

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\end{document}

[5] 86j:16028 Abeasis, S.; Del Fra, A. Degenerations for the representations of a quiver of type ${\scr A}\sb m$. J.
Algebra 93 (1985), no. 2, 376--412. (Reviewer: Sheila Brenner) 16A64 (14D25)

[6] 86h:14038 Abeasis, S. Codimension $1$ orbits and semi-invariants for the representations of an equioriented
graph of type $D\sb{n}$. Trans. Amer. Math. Soc. 286 (1984), no. 1, 91--123. (Reviewer: V. L. Popov) 14L30
(14D25 16A64)

[7] 86h:14037 Abeasis, S. Codimension $1$ orbits and semi-invariants for the representations of an oriented graph
of type ${\cal A}\sb{n}$. Trans. Amer. Math. Soc. 282 (1984), no. 2, 463--485. (Reviewer: V. L. Popov) 14L30
(14D25 16A64)

[8] 86g:16039 Abeasis, S.; Del Fra, A. Degenerations for the representations of an equioriented quiver of type
$D\sb{m}$. Adv. in Math. 52 (1984), no. 2, 81--172. (Reviewer: Idun Reiten) 16A64 (14L30)

[9] 84e:16019 Abeasis, S.; Del Fra, A. Degenerations for the representations of an equioriented quiver of type
$A\sb{m}$. Boll. Un. Mat. Ital. Suppl. 1980, no. 2, 157--171. (Reviewer: Sheila Brenner) 16A64 (16A58)

[10] 84d:16036 Abeasis, S. On the ring of semi-invariants of the representations of an equioriented quiver of type
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[11] 83h:14038 Abeasis, S.; Del Fra, A.; Kraft, H. The geometry of representations of $A\sb{m}$. Math. Ann.
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