GENERATING THE IDEALS DEFINING UNIONS OF SCHUBERT VARIETIES

arXiv:1405.2945v1 [math.AG] 12 May 2014

ANNA BERTIGER

A BSTRACT. This note computes a Gröbner basis for the ideal defining a union of Schubert varieties.
More precisely, it computes a Gröbner basis for unions of schemes given by northwest rank conditions on the space of all matrices of a fixed size. Schemes given by northwest rank conditions include
classical determinantal varieties and matrix Schubert varieties–closures of Schubert varieties lifted
from the flag manifold to the space of matrices.

1. I NTRODUCTION
We compute a Gröbner basis, and hence ideal generating set, for the ideal defining a union of
schemes each given by northwest rank conditions with respect to an “antidiagonal term order.”
A scheme defined by northwest rank conditions is any scheme whose defining equations are of
the form “all k × k minors in the northwest i × j sub-matrix of a matrix of variables,” where i, j,
and k can take varying values. These schemes represent a generalization of classical determinantal
varieties–those varieties with defining equations all (r+1)×(r+1) minors of a matrix of variables.
One geometrically important collection of schemes defined by northwest rank conditions is the set
of matrix Schubert varieties. Matrix Schubert varieties are closures of the lift of Schubert varieties
from the complete flag manifold to matrix space [Ful92]. In general, a matrix Schubert variety
for a partial permutation π is the subvariety of matrix space given by the rank conditions that
the northwest i × j sub-matrix must have rank at most the number of 1s in the northwest i × j
sub-matrix of the partial permutation matrix for π. Notice that the set of matrix Schubert varieties
contains the set of classical determinantal varieties, which are the zero locus of all minors of a
fixed size on the space of all matrices of fixed size.
Matrix Schubert varieties associated to honest, that is non-partial, permutations are the closures
of the lifts of the corresponding Schubert varieties in the flag manifold, B− \GLn . If Xπ is the matrix
Schubert variety for an honest permutation π the projection
{full rank matrices}  B− \ GLn C = F`Cn
sends Xπ ∩ GLn C onto the Schubert variety Xπ ⊆ F`Cn . Schubert varieties, orbits of B+ , stratify
F`Cn and give a basis for H∗ (F`Cn ). It is this application that led to the introduction of matrix
Schubert varieties in [Ful92]. Knutson and Miller showed that matrix Schubert varieties have a
rich algebro-geometric structure corresponding to beautiful combinatorics [KM05]. Fulton’s generators are a Gröbner basis with respect to any antidiagonal term order and their initial ideal is
the Stanley-Reisner ideal of the “pipe dream complex.” Further, Knutson and Miller show that
the pipe dream complex is shellable, hence the original ideal is Cohen-Macaulay. Pipe dreams,
the elements of the pipe dream complex, were originally called RC graphs and were developed
by Bergeron and Billey [BB93] to describe the monomials in polynomial representatives for the
classes corresponding to Schubert varieties in H∗ (F`Cn ).
The importance of Schubert varieties, and hence matrix Schubert varieties, to other areas of
geometry has become increasing evident. For example, Zelevinsky [Zel85] showed that certain
quiver varieties, sequences of vector space maps with fixed rank conditions, are isomorphic to
Date: May 14, 2014.
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Schubert varieties. Knutson, Miller and Shimozono, [KMS06] produce combinatorial formulae for
quiver varieties using many combinatorial tools reminiscent of those for Schubert varieties.
1.1. Notation and Background. Much of the background surveyed here can be found in [MS05].
Let B− (respectively B+ ) denote the group of invertible lower triangular (respectively upper triangular) n × n matrices. Let M = (mi,j ) be a matrix of variables. In what follows π will be a possibly
partial permutation, written in one-line notation π(1) . . . π(n), with entries for π(i) undefined are
written ?. We shall write permutation even when we mean partial permutation in cases where
there is no confusion. A matrix Schubert variety Xπ is the closure B− πB+ in the affine space of all
matrices, where π is a permutation matrix and B− and B+ act by downward row and rightward
column operations respectively. Notice that for π an honest permutation Xπ is the closure of the
lift of Xπ = B− \B− πB+ ⊆ B− \GLn C to the space of n × n matrices.
The Rothe diagram of a permutation is found by looking at the permutation matrix and crossing
out all of the cells weakly below, and the cells weakly to the right of, each cell containing a 1. The
remaining empty boxes form the Rothe diagram. The essential boxes [Ful92] of a permutation are
those boxes in the Rothe diagram that do not have any boxes of the diagram immediately south
or east of them. The Rothe diagrams for 2143 and 15432 are given in Figure 1.1. In both cases the
essential boxes are marked with the letter e.
1
e

e

1
e

1
e

e

1

1

1

1

1

1

F IGURE 1.1. The Rothe diagrams and essential sets of 2143 (left) and 15432 (right).
The rank matrix of a permutation π, denoted r(π), gives in each cell r(π)ij the rank of the i × j
northwest-justified sub-matrix of the permutation matrix for π. For example, the rank matrix of
15432 is


1 1 1 1 1
 1 1 1 1 2 


 1 1 1 2 3 .


 1 1 2 3 4 
1 2 3 4 5
Theorem 1.1 ( [Ful92]). Matrix Schubert varieties have radical ideal I(Xπ ) = Iπ given by determinants
representing conditions given in the rank matrix r(π), that is, the (r(π)ij + 1) × (r(π)ij + 1) determinants
of the northwest i × j sub-matrix of a matrix of variables. In fact, it is sufficient to impose only those rank
conditions r(π)ij such that (i, j) is an essential box for π.
Hereafter we call the determinants corresponding the to essential rank conditions, or the analogous determinants for any ideal generated by northwest rank conditions, the Fulton generators.
One special form of ideal generating set is a Gröbner basis. To define a Gröbner basis we set
a total ordering on the monomials in a polynomial ring such that 1 ≤ m and m < n implies
mp < np for all monomials m, n and p. Let init f denote the largest monomial that appears in the
polynomial f. A Gröbner basis for the ideal I is a set {f1 , . . . fr } ⊆ I such that init I := hinit f : f ∈
Ii = hinit f1 , . . . init fr i. Notice that a Gröbner basis for I is necessarily a generating set for I.
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The antidiagonal of a matrix is the diagonal series of cells in the matrix running from the most
northeast to the most southwest cell. The antidiagonal term (or antidiagonal) of a determinant

is the product of the entries in the antidiagonal. For example, the antidiagonal of ac db is the
cells occupied by b and c, and correspondingly, in the determinant ad − bc the antidiagonal term
is bc. Term orders that select antidiagonal terms from a determinant, called antidiagonal term
orders have proven especially useful in understanding ideals of matrix Schubert varieties. There
are several possible implementations of an antidiagonal term order on an n×n matrix of variables,
any of which would suit the purposes of this paper. One example is weighting the top right entry
highest and decreasing along the top row before starting deceasing again at the right of the next
row; monomials are then ordered by their total weight.
Theorem 1.2 ( [KM05]). The Fulton generators for Iπ form a Gröbner basis under any antidiagonal term
order.
Typically we will denote the cells of a matrix that form antidiagonals by A or B. In what follows if A is the antidiagonal of a sub-matrix of M we will use the notation det(A) to denote the
determinant of this sub-matrix. We shall be fairly liberal in exchanging antidiagonal cells and the
corresponding antidiagonal terms, thus, for any antidiagonal term order, A = init det(A).
1.2. Statement of Result. Let I1 , . . . Ir be ideals defined by northwest rank conditions. We will
produce a Gröbner basis, and hence ideal generating set, for I1 ∩ · · · ∩ Ir . For each list of antidiagonals A1 , . . . , Ar , where Ai is the antidiagonal of a Fulton generator of Ii , we will produce a
Gröbner basis element gA1 ,...,Ar for ∩Ii . The generators gA1 ,...,Ar will be products of determinants,
though not simply the product of the r determinants corresponding to the Ai . For a fixed list of
antidiagonals A1 , . . . , Ar , build the generator gA1 ,...,Ar by:
(1) Begin with gA1 ,...,Ar = 1
(2) Draw a diagram with a dot of color i in each box of Ai and connect the consecutive dots of
color i with a line segment of color i.
(3) Break the diagram into connected components. Two dots are connected if they are either
connected by lines or are connected by lines to dots that occupy the same box.
(4) For each connected component, remove the longest series of boxes B such that there is
exactly one box in each row and column and the boxes are all in the same connected component. If there is a tie use the most northwest of the longest series of boxes. Note that B
need not be any of A1 , . . . , Ar . Multiply gA1 ,...,Ar by det(B). Remove this antidiagonal from
the diagram of the connected component, break the remaining diagram into components
and repeat.
Theorem 1.3. {gA1 ...Ar : Ai is an antidiagonal of a Fulton generator of Ii , 1 ≤ i ≤ r} form a Gröbner
basis, and hence a generating set, for ∩ri=1 Ii .
1.3. Acknowledgements. This work constitutes a portion of my PhD thesis completed at Cornell
University under the direction of Allen Knutson. I wish to thank Allen for his help, advice and
encouragement in completing this project. Thanks also go to Jenna Rajchgot for helpful discussions in the early stages of this work. I’d also like to thank the authors of computer algebra system
Macaulay2, [GS] which powered the computational experiments nessecary to do this work. I’m especially grateful to Mike Stillman who patiently answered many of my Macaulay2 questions over
the course of this work. Kevin Purbhoo gave very helpful comments on drafts of this manuscript
for which I cannot thank him enough.
2. E XAMPLES
We delay the proof of Theorem 1.3 to Section 3 and first give some examples of the generators
produced for given sets of antidiagonals. These examples are given by pictures of the antidiagonals on the left and corresponding determinantal equations on the right. Note that we only give
3

particular generators, rather than entire generating sets, which might be quite large. We then give
entire ideal generating sets for two smaller intersections.
If r = 1 then for each Fulton generator with antidiagonal A the algorithm produces the generator gA = det(A). Therefore, if we intersect only one ideal the algorithm returns the original set of
Fulton generators. The generator for the antidiagonal shown is exactly the determinant of the one
antidiagonal pictured:

m1,1 m1,2 m1,4
m3,1 m3,2 m3,4
m4,1 m4,2 m4,4

.

The generator for two disjoint antidiagonals is the product of the determinants corresponding
to the two disjoint antidiagonals:

m1,1 m1,2
m2,1 m3,2

m1,1 m1,2 m1,4
m3,1 m3,2 m3,4
m4,1 m4,2 m4,4

.

In general, if A1 , . . . Ar are disjoint antidiagonals then the then the algorithm looks at each Ai separately as they are part of separate components and the result is that gA1 ,...Ar = det(A1 ) · · · det(Ar ).
If A1 , . . . Ar overlap to form one antidiagonal X then the last step of the algorithm will occur
only once and will produce gA1 ,...Ar = det(X). For example,

m1,1
m2,1
m3,1
m4,1

m1,2
m2,2
m3,2
m4,2

m1,3
m2,3
m3,3
m4,3

m1,4
m2,4
m3,4
m4,4

.

In this example, there are two longest possible antidiagonals, the three cells occupied by the
green dots and the three cells occupied by the red dots. The ones occupied by the green dots are
more northwest, hence the generator for the three antidiagonals shown below is
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m1,1 m1,2
m2,1 m2,2

m1,2 m1,3 m1,3
m2,2 m2,3 m2,3
m3,2 m3,3 m3,4

m4,2 m4,3
m5,2 m2,2

.

In the picture below, the longest possible anti diagonal uses all of the cells in the green anti
diagonal but only some of the cells in the red antidiagonal, however, there is only one possible
longest antidiagonal. Thus the generator is

m1,1 m1,2
m2,1 m2,2

m1,1
m2,1
m3,1
m4,1

m1,2
m2,2
m3,2
m4,2

m1,4
m2,4
m3,4
m4,4

m1,5
m2,5
m3,5
m4,5

m5,1

.

We now give two examples where the complete ideals are comparatively small. Firstly, we
calculate I(X231 ∪ X312 ) = I(X231 ) ∩ I(X312 ). I(X231 ) = hm1,1 , m2,1 i and I(X312 ) = hm1,1 , m1,2 i. The
antidiagonals and corresponding generators are shown below with antidiagonals from generators
of I(X231 ) shown in red and antidiagonals of generators of I(X312 ) shown in blue. Note that the
antidiagonals are only one cell each in this case.

m1,1

m1,1 m1,2

m1,1 m2,1

m1,2 m2,1

Theorem 1.3 results in
I(X231 ∪ X312 ) = I(X231 ) ∩ I(X312 ) = hm1,1 , m1,1 m1,2 , m1,1 m2,1 , m1,2 m2,1 i.
As a slightly larger example, consider I(X1423 ∪ X1342 ) = I(X1423 ) ∩ I(X1342 ). These generators
are given below in the order that the antidiagonals are displayed reading left to right and top to
bottom. The antidiagonals for I(X1423 are shown in red while the antidigaonals I(X1342 ) are shown
in blue. for Note that the full 4 × 4 grid is not displayed, but only the northwest 3 × 3 portion
where antidiagonals for these two ideals may lie.
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Here Theorem 1.3 produces

m1,1
m1,1 m1,2
,
m2,1
m2,1 m2,2

m1,1 m1,2
m1,1
+
m3,1 m2,2

m1,1 m1,2
m1,2
+
m2,1 m3,2
m2,2

m1,2
m2,2

m1,1

m1,1 m1,3
,
m2,1 m2,3
m1,3
m2,2

m2,1 m2,2
m3,1 m2,2

m3,1 m1,2
,
m2,1 m2,2
m1,1 m1,3
m3,1 m3,2

m1,1 m1,2
,
m3,1 m2,3

m1,1 m1,3
,
m3,1 m2,3

m1,2 m1,3
m2,2 m2,2




m2,1 m2,2
m3,1 m2,3

m1,1 m1,2 m1,3
, m2,1 m2,2 m2,3
m3,1 m3,2 m3,4



3. P ROOF OF T HEOREM 1.3
We now prove the main result of this paper, Theorem 1.3, which states that the gA1 ,...,Ar generate
I1 ∩ · · · ∩ Ir .
We begin with a few fairly general statements:
Theorem 3.1 ( [Knu]). If {Ii : i ∈ S} are ideals generated by northwest rank conditions then init(∩i∈S Ii ) =
∩i∈S (init Ii ).
Lemma 3.2 ( [KM05]). If J ⊆ K are homogeneous ideals in a polynomial ring such that init J = init K
then J = K.
Lemma 3.3. Let IA and IB be ideals that define schemes of northwest rank conditions and let det(A) ∈ IA
and det(B) ∈ IB be determinants with antidiagonals A and B respectively such that A ∪ B = X and
A ∩ B 6= ∅. Then det(X) is in IA ∩ IB .
Proof. Let VX = V(det(X)), VA = V(IA ) and VB = V(IB ) be the varieties corresponding to the ideals
hdet(X)i, IA and IB . It is enough to show that VA ⊆ VX and VB ⊆ VX .
We will show that given a matrix with antidiagonal X with a sub-matrix with antidiagonal
A ⊆ X where the sub-matrix northwest of the cells occupied by A has rank at most length(A) − 1
then the full matrix has rank at most length(X) − 1. The corresponding statement for sub-matrix
with antidiagonal B can be proven by replacing A with B everywhere.
The basic idea of this proof is that we know the rank conditions on the rows and columns northwest of those occupied by A. The rank conditions given by A then imply other rank conditions as
adding either a row or a column to a sub-matrix can increase its rank by at most one.
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column t + 1 column c
rank at most l
northwest of row k − t column c

rank at most l + k − c ≤ k − t − 2
northwest of column k row k − t

rank at most k − 2

row k − t

northwest of column k row k

row k
F IGURE 3.1. The proof of Lemma 3.3. The antidiagonal cells in A are marked in
black and the antidiagonal cells in X − A ⊆ B are marked in white.
Let k be the number of rows, also the number of columns in the antidiagonal X. Let the length
of A be l + 1, so the rank condition on all rows and columns northwest of those occupied by A
is at most l. Assume that the rightmost column of A is c and the leftmost column of A is t + 1.
Notice that this implies that the bottom row occupied by A is k − t, as the antidiagonal element in
column t + 1 is in row k − t. Thus, the northwest (k − t) × c of matrices in VA has rank at most l.
Notice c ≥ (t+1)+(l+1), with equality if A occupies a continuous set of columns, so matrices in
VA have rank at most l in the northwest (k−t)×(t+l+2). Adding k−c ≤ k−(n−t−l−2) columns to
this sub-matrix gives a (k−t)×k sub-matrix with rank at most l+k−c ≤ r+(k−t−l−2) = k−t−2.
Further, by the same principle, moving down t rows, the northwest k × k, i.e. the whole matrix
with antidiagonal X, has rank at most k − t − 2 + t = k − 2, hence has rank at most k − 1 and so is
in VX .

For a visual explanation of the proof of Lemma 3.3 see Figure 3.1.
Lemma 3.4. gA1 ,...,Ar ∈ Ii for 1 ≤ i ≤ r and hence
hgA1 ,...,Ar : Ai ranges over all antidiagonals for Fulton generators of Ii i ⊆ ∩ri=1 Ii .
Proof. Fix i. Let S be the first antidiagonal containing a box occupied by a box contained in Ai
added to gA1 ,...,Ar . We shall show that det(S) is in Ii and hence gA1 ,...,Ar ∈ Ii as it is a multiple of
det(S). If Ai ⊆ S then det(S) ∈ Ii either because S = Ai or S ( Ai in which case we apply Lemma
3.3. Otherwise, |S| ≥ |Ai | and S is weakly to the northwest of Ai . Therefore, there is a subset B
of S such that |B| = |Ai |, and B is weakly northwest of Ai . Hence, B is an antidiagonal for some
determinant in Ii , and again by Lemma 3.3 det(S) ∈ Ii .

Lemma 3.5. init gA1 ,...,Ar = A1 ∪ · · · ∪ Ar under any antidiagonal term order.
Proof. init gA1 ,...,Ar is a product of determinants, with collective antidiagonals A1 ∪ · · · ∪ Ar .



When we combine Lemma 3.5 and Theorem 3.1 we see that inithgA1 ,...Ar i = init(∩Ii ). Then,
Lemmas 3.2 and 3.4 combine to complete the proof of Theorem 1.3.
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Note that Theorem 1.3 may produce an oversupply of generators. For example, if I1 = I2 , then
inputting the same set of p Fulton generators twice results in a Gröbner basis of p2 polynomials
for I1 ∩ I2 = I1 = I2 .
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