COHOMOLOGY OF SMOOTH SCHUBERT VARIETIES IN PARTIAL FLAG MANIFOLDS

VESSELIN GASHAROV AND VICTOR REINER

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 subvarieties defined by inclusions). The Schubert varieties which lie within this larger class are characterized combinatorially by a pattern avoidance condition.

1. Introduction

This paper concerns smooth Schubert subvarieties within the partial flag varieties

G/P_{J} := {0 ⊂ V_{d_{1}} ⊂ V_{d_{2}} ⊂ ⋯ ⊂ V_{d_{s}} ⊂ ℂ^{n}}.

Here G = GL_{n}(ℂ), 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 ℂ^{n}. When J = ∅, 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 wW_{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

(1)  dim V_{d_{i}} ∩ ℂ^{e} ≥ r.

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 (1) says

(2)

V_{d_{i}} ⊂ ℂ^{e} when r = d_{i}
ℂ^{e} ⊂ V_{d_{i}} when r = e.

Any subvariety X of G/P_{J} defined by a conjunction of conditions of the special forms in (2) will be called a subvariety of G/P_{J} defined by inclusions. Say that a pair of such inclusion conditions of the form

\begin{aligned}V_{d_{i}} & ⊂ ℂ^{e} \\ ℂ^{e′} & ⊂ V_{d_{j}} \end{aligned}

1991 Mathematics Subject Classification. 05E, 14M15.

Second author partially supported by NSF grant DMS-9877047.

are crossing if d_{i} < d_{j} and e > e′. If X is defined by inclusions but contains no crossing pair, we say it is defined by non-crossing inclusions. By results of Ryan [9] and Wolper [11], this property characterizes the smooth Schubert varieties. Theorem 1.1 below summarizes some of their work, along with a result due independently to Lakshmibai and Sandhya [7] and Haiman [5]. For the purposes of stating it, we introduce two pieces of terminology.

Say that a (complex) algebraic variety X is fibered by Grassmannians if it lies within the smallest class of algebraic varieties that contains all Grassmannians 𝔾r(k, ℂ^{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} ⋯ u_{k} in S_{k} for some k ≤ n, say that wW_{J} avoids the pattern u if the longest coset representative w for wW_{J} contains no subsequence w_{i_{1}} ⋯ w_{i_{k}} whose values are in the same relative order as u_{1} ⋯ u_{k} (i.e. the permutation matrix representing w when restricted to columns {i_{1}, …, i_{k}} and rows {w_{i_{1}}, …, w_{i_{k}}} gives exactly the permutation matrix representing u).

Theorem 1.1. The following are equivalent for a subvariety X of G/P_{J}:

(i) X is smooth and B-stable.

(ii) X is smooth and defined by inclusions.

(iii) X is defined by non-crossing inclusions.

(iv) X is B-stable and fibered by Grassmannians.

(v) X is a smooth Schubert variety X_{wW_{J}}.

(vi) X is a Schubert variety X_{wW_{J}} for some w that avoids both patterns 4231, 3412.

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 3.1) 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}(ℂ)/B is defined by the following inclusions

{0 ⊂ V_{1} ⊂ V_{2} ⊂ V_{3} ⊂ V_{4} ⊂ V_{5} ⊂ ℂ^{6} : V_{1} ⊂ ℂ^{3} ⊂ V_{4} ⊂ ℂ^{5}}

and has cohomology ring

H^{∗}(X_{352164}, ℤ) ≅ ℤ[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}, …, x_{j} of the form e_{r}(i, j) := e_{r}(x_{i}, x_{i+1}, …, x_{j}), coming from the various inclusion conditions that define X_{352164}:

e_{1}(1, 6), e_{2}(1, 6), e_{3}(1, 6), e_{4}(1, 6), e_{5}(1, 6), e_{6}(1, 6) from V_{i} ⊂ ℂ^{6}
e_{3}(2, 6), e_{4}(2, 6), e_{5}(2, 6)  from V_{1} ⊂ ℂ^{3}
e_{2}(5, 6)  from V_{4} ⊂ ℂ^{5}
e_{2}(1, 4), e_{3}(1, 4), e_{4}(1, 4))  from ℂ^{3} ⊂ V_{4}.

In fact, Theorem 3.1 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 4.2) as those with wW_{J} avoiding the four patterns 4231, 35142, 42513, 351624.

2. Preliminaries

We start by defining partial flag manifolds and their Schubert varieties. A good reference for some of this material is [3, Chapter 10].

Let G = GL_{n}(ℂ) 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, …, 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}, …, 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 [6]).

For any J = {s_{i_{1}}, …, s_{i_{p}}} ⊆ S, let W_{J} denote the Young subgroup of W generated by J, let P_{J} denote the standard parabolic subgroup of G generated by B and J, and let

D(J) = {0} ∪ {t ∈ [n − 1] : s_{t} ∉ J} ∪ {n} = {0 < d_{1} < d_{2} < ⋯ < d_{s} < n}.

G acts transitively on the set of complete flags

0 ⊂ V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{n−1} ⊂ ℂ^{n}

where V_{i} is a subspace of dimension i for each i. If one fixes the standard flag having V_{i} = ℂ^{i}, where ℂ^{i} is the subspace of ℂ^{n} spanned by the first i standard basis vectors {e_{1}, e_{2}, …, 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 ⊆ S with D(J) defined as above, the stabilizer of the partial flag

0 ⊂ ℂ^{d_{1}} ⊂ ℂ^{d_{2}} ⊂ ⋯ ⊂ ℂ^{d_{s}} ⊂ ℂ^{n}

is P_{J}, hence one can identify the set of partial flags of the form

0 ⊂ V_{d_{1}} ⊂ V_{d_{2}} ⊂ ⋯ ⊂ V_{d_{s}} ⊂ ℂ^{n}

with G/P_{J} (the complete flags and G/B are the special case J = ∅). This homogeneous space G/P_{J} can be embedded as a closed subvariety in a suitable complex projective space ℙ(ℂ^{N}), and is a smooth variety called the partial flag manifold G/P_{J}. In the special case J = ∅ where P_{J} = B, it is called the flag manifold G/B. In the special case where |J| = |S| − 1 so D(J) = {0 < d < n} for some d, then G/P_{J} is called the Grassmannian or Grassman manifold 𝔾r(d, ℂ^{n}).

The partial flag manifold has a cellular decomposition

(3)  G/P_{J} = ∐_{wW_{J} ∈W/W_{J}} X_{wW_{J}}^{o}

where

X_{wW_{J}}^{o} = BwP_{J}/P_{J}

are the orbits under the left-translation action of B in G, called Schubert cells and indexed by cosets wW_{J}. The Schubert cell X_{wW_{J}}^{o} has the following description as a collection of flags: if w = w_{1}w_{2} ⋯ w_{n} is a permutation in one-line notation, that is, w(i) = w_{i}, and D(J) = {0 < d_{1} < d_{2} < ⋯ < d_{s} < n} is defined as above, then

(4)

X_{wW_{J}}^{o} = {0 ⊂ V_{d_{1}} ⊂ ⋯ ⊂ V_{d_{s}} ⊂ ℂ^{n} :
dim V_{d_{i}} ∩ ℂ^{e} = #{k ≤ d_{i} : w_{k} ≤ e}
for 1 ≤ i ≤ s and 1 ≤ e ≤ n}

One can check that this is independent of the choice of w ∈ wW_{J}. The Zariski closure X_{wW_{J}} := \overline{X_{wW_{J}}^{o}} is called a Schubert variety in G/P_{J}, and is the subset of partial flags satisfying the same intersection conditions as X_{wW_{J}}^{o} except replacing the “ = ” by “ ≥ ”. We will call these conditions Schubert conditions for X_{wW_{J}}.

Using the one-line notation w = w_{1}w_{2} ⋯ w_{n} for w in W = S_{n}, the usual Coxeter group length function l : W → ℕ, is in this case the number of inversions

l(w) = |{(i, j) : 1 ≤ i < j ≤ n, w_{i} > w_{j}}|.

If D(J) = {0, d_{1}, …, 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} ⋯ 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} ⋯ w_{d_{i+1}} is in increasing, decreasing order respectively.

The Bruhat order is a partial order on the cosets W/W_{J} defined as follows: uW_{J} ≤ vW_{J} if X_{uW_{J}}^{o} ⊂ X_{vW_{J}}. The open Schubert cell X_{wW_{J}}^{o} is isomorphic to a complex affine space of dimension l(min wW_{J}). Thus the cell decomposition (3) 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

(5)  X = ∐_{wW_{J} ∈I} X_{wW_{J}}^{o}

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.

Proposition 2.1. Let Y be a finite CW-complex with cells {σ_{j}}_{j∈J} occurring only in even dimensions, and exactly one 0-cell. Let X be any subcomplex, with cells {σ_{i}}_{i∈I} for some subset I ⊂ 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

(i) X is simply-connected.

(ii) The integral homology H_{∗}(X; ℤ) and cohomology H^{∗}(X; ℤ) are free abelian groups which are non-vanishing only in even dimensions. They form dual lattices under the Kronecker pairing, having dual ℤ-bases given by the cellular homology classes {[σ_{i}]}_{i∈I} and their (Kronecker) dual cohomology classes {[σ_{i}]}_{i∈I}^{∗}, respectively.

(iii) Consequently,

Poin(X, q) := ∑_{k≥0} rank_{ℤ}H_{k}(X; ℤ) q^{i}

= ∑_{i∈I} q^{dim σ_{i}}.

(iv) The maps i_{∗}, i^{∗} induced on homology and cohomology

\begin{matrix}H_{∗}(X; ℤ) & →^{i_{∗}} H_{∗}(Y; ℤ) \\ H^{∗}(Y; ℤ) & →^{i^{∗}} H^{∗}(X; ℤ) \end{matrix}

by the inclusion X ↪^{i} Y are injective and surjective, respectively, with

i_{∗}([σ_{i}]) = [σ_{i}] for all i in I
i^{∗}([σ_{j}]^{∗}) = \begin{cases}[σ_{j}]^{∗} & if j ∈ I \\ 0 & if j ∈ J − I \end{cases}

(v) (cf. [1, Cor. 4.4]) Consequently, the kernel of the surjection i^{∗} is generated by {[σ_{j}]^{∗}}_{j∈J−I}, both additively and as an ideal in H^{∗}(X; ℤ). □

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; ℤ), H^{∗}(X; ℤ). □

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. Proposition 2.2. 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} ∩ ℂ^{e} with d ∈ D(J) = {0, d_{1}, …, d_{s}, n}.

Consequently, the canonical surjection

\begin{matrix}G/B & ⟶^{p} & G/P_{J} \\ 0 ⊂ V_{1} ⊂ ⋯ ⊂ V_{n−1} ⊂ ℂ^{n} & ⟼ & 0 ⊂ V_{d_{1}} ⊂ ⋯ ⊂ V_{d_{s}} ⊂ ℂ^{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}} →^{p}X_{wW_{J}} with the same fiber.

Proof. See Proposition 4.1. □

Such fiber bundle maps (and others that will appear later) have strong combinatorial and geometric consequences, via the following proposition.

Proposition 2.3. Let F, E, B be complex algebraic varieties, and E→^{p}B an algebraic fiber bundle map with fiber isomorphic to F.

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).

Proof. Note that Proposition 2.1 (ii) implies that the maps d^{i} in the E^{i} page of the Leray-Serre spectral sequence associated to E→^{p}B are all 0 for i ≥ 1, so that it degenerates at the E^{1} page. Also

E_{p,q}^{1} = H^{p}(B) ⊗_{ℤ} H^{q}(F)

because B is simply-connected, and the homology of F is free abelian. The assertion follows. □

3. 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 ⊆ S index a partial flag manifold G/P_{J}, so that if D = D(J), we can define

X_{D} := G/P_{J} = {0 ⊂ V_{d_{1}} ⊂ V_{d_{2}} ⊂ ⋯ ⊂ V_{d_{s}} ⊂ ℂ^{n}}.

Given a triple (D, L, U) where • D = D(J) = {0 < d_{1} < d_{2} < ⋯ < d_{s} < n},

• L is a set of ordered pairs of the form (l_{i}, d_{i}) with l_{i} ≤ d_{i}, and
• U is a set of ordered pairs of the form (d_{i}, u_{i}) with d_{i} ≤ u_{i},

define a subvariety X_{(D,L,U)} of X_{D} by

X_{(D,L,U)} = {0 ⊂ V_{d_{1}} ⊂ V_{d_{2}} ⊂ ⋯ ⊂ V_{d_{s}} ⊂ ℂ^{n} :
ℂ^{l_{i}} ⊂ V_{d_{i}} if (l_{i}, d_{i}) ∈ L
V_{d_{i}} ⊂ ℂ^{u_{i}} if (d_{i}, u_{i}) ∈ U}

Clearly every subvariety X of G/P_{J} defined by inclusions has this form. Note that since the Borel subgroup stabilizes the subspaces ℂ^{i}, the subvarieties X_{(D,L,U)} are always B-stable, and consequently Proposition 2.1 applies to them.

We review here Borel’s presentation for H^{∗}(G/P_{J}) (see [3, §10.3, 10.6]). For J = ∅, so D = D(J) = {1, 2, …, n − 1} and X_{D} = G/B, Borel showed

ℤ[x_{1}, …, x_{n}]/(e_{1}(1, n), …, e_{n}(1, n)) ≅ H^{∗}(G/B, ℤ)

where we recall from the Introduction that e_{k}(i, j) := e_{k}(x_{i}, x_{i+1}, …, x_{j}) is the k^{th} elementary symmetric function in the variables x_{i}, x_{i+1}, …, x_{j}. The isomorphism sends x_{i} to −c_{1}(L_{i}) where L_{i} denotes the line bundle on G/B whose fiber over a complete flag

0 = V_{0} ⊂ V_{1} ⊂ ⋯ V_{n−1} ⊂ V_{n} = ℂ^{n}

is the space V_{i}/V_{i−1}, and c_{1}(L_{i}) is its first Chern class. He also showed that for D = D(J) arbitrary, the surjection π : G/B → G/P_{J} = X_{D} induces an inclusion

π^{∗} : H^{∗}(G/P_{J}) ↪ H^{∗}(G/B)

sending the k^{th} Chern class c_{k}(V_{d_{j}}/V_{d_{j−1}}) of the vector bundle V_{d_{j}}/V_{d_{j−1}} to the elementary symmetric function e_{k}(d_{j−1} + 1, d_{j}). The image of this inclusion is exactly the invariant subring

ℤ[x_{1}, …, x_{n}]^{W_{J}} /(e_{1}(1, n), …, e_{n}(1, n)).

Define for a triple (D, L, U) as above

A_{D} := ℤ[x_{1}, …, x_{n}]^{W_{J}}
I_{n} = I_{d_{s+1}} := (e_{1}(1, n), …, e_{n}(1, n))
R_{D} := A_{D}/I_{n} ≅ H^{∗}(G/P_{J})
I_{(D,L,U)} := I_{n} + ({e_{k}(1, d_{i}) : (l_{i}, d_{i}) ∈ L, k > d_{i} − l_{i}}
∪ {e_{k}(d_{i} + 1, n) : (d_{i}, u_{i}) ∈ U, k > u_{i} − d_{i}})
R_{(D,L,U)} := A_{D}/I_{(D,L,U)}

Recall from Proposition 2.1 that for any B-stable subvariety X of X_{D}, the cohomology H^{∗}(X) is free abelian, and is the image of the surjective restriction map H^{∗}(X_{D}) ⟶^{i^{∗}} H^{∗}(X). Consequently, for any X_{(D,L,U)}, the composite ring homomorphism

ϕ : A_{D} ⟶ A_{D}/I_{n} ≅ H^{∗}(X_{D}) ⟶^{i^{∗}} H^{∗}(X_{(D,L,U)})

will be surjective. Theorem 3.1. Let X_{(D,L,U)} be a subvariety of X_{D} defined by inclusions. Then the composite ϕ defined above has kernel I_{(D,L,U)}, and therefore induces a (gradedoubling) ring isomorphism

\overline{ϕ} : R_{(D,L,U)} ≅ H^{∗}(X_{(D,L,U)}).

The proof of this theorem occupies the remainder of this section.

We begin by showing that I_{(D,L,U)} ⊂ ker ϕ. Note that if (l_{i}, d_{i}) is in L, then the fact that ℂ^{l_{i}} ⊂ 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 ℂ^{l_{i}} ⊕ V_{d_{i}}/ℂ^{l_{i}}, where ℂ^{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}}/ℂ^{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}, …, x_{d_{i}} of the bundle V_{d_{i}} must vanish in H^{∗}(X_{(D,L,U)}) for k > d_{i} − l_{i}.

Similarly, if (d_{i}, u_{i}) is in U, then V_{d_{i}} ⊂ ℂ^{u_{i}} on X_{(D,L,U)} implies the bundle ℂ^{n}/V_{d_{i}} of rank n − d_{i} has the same Chern classes as the direct sum ℂ^{n}/ℂ^{u_{i}} ⊕ ℂ^{u_{i}}/V_{d_{i}}, where ℂ^{n_{i}}/ℂ^{u_{i}} is a trivial bundle on X_{(D,L,U)}. Consequently, c_{k}(ℂ^{n}/V_{d_{i}}) vanishes for k > u_{i}−d_{i}, implying vanishing in H^{∗}(X_{(D,L,U)}) of the k^{th} elementary symmetric function in the Chern roots x_{d_{i}+1}, …, x_{n} of the bundle ℂ^{n}/V_{d_{i}} for k > u_{i} − d_{i}.

Thus I_{(D,L,U)} ⊂ ker ϕ, so that ϕ induces a (grade-doubling) surjective ring homomorphism

\overline{ϕ} : R_{(D,L,U)} ⟶ H^{∗}(X_{(D,L,U)}).

For the rest of the theorem, we must show that this surjection \overline{ϕ} is an isomorphism. We observe that because \overline{ϕ} is a surjection, it will be an isomorphism if we can show either of two other equivalent conditions:

(C1) If R_{(D,L,U)} has a ℤ-spanning set of cardinality equal to the rank r of the free abelian group H^{∗}(X_{(D,L,U)}), then we would have a composite surjection of abelian groups

ℤ^{r} → R_{(D,L,U)} →^{\overline{ϕ}} H^{∗}(X_{(D,L,U)}) ≅ ℤ^{r}

forcing \overline{ϕ} to be an isomorphism. (C2) Given the decomposition

X_{(D,L,U)} = X = ∐_{wW_{J} ∈I} X_{wW_{J}}^{o}

we know from Proposition 2.1 (v) that the cohomology classes [X_{wW_{J}}]^{∗} with wW_{J} not in I generate the kernel of the map

A_{D}/I_{n} ≅ H^{∗}(G/P_{J}) → H^{∗}(X_{(D,L,U)}).

Consequently, \overline{ϕ} 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} ≅ H^{∗}(G/P_{J}), lies in the ideal I_{(D,L,U)}/I_{n}.

Our strategy consists of three steps: Step 1. Use equivalent condition (C2) to reduce to the case where |L| + |U| = 1,

that is, when X_{(D,L,U)} is defined by a single inclusion condition.

Step 2. Do an independent reduction that uses equivalent condition (C1) to reduce

to the case of complete flags, i.e. D = D_{0} := {0, 1, 2, …, n − 1, n}. Step 3. Deal with the remaining very special case where D = {0, 1, 2, …, n−1, n}

and |L|+|U| = 1, using equivalent condition (C1) and one of the fibrations from Theorem 1.1 (iv).

Step 1.

We assume that \overline{ϕ} 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} ⊆ L and U_{0} ⊆ U implies I_{(D,L_{0},U_{0})} ⊆ I_{(D,L,U)}, this implies [X_{wW_{J}}]^{∗} lies in the ideal I_{(D,L,U)}/I_{n}, as desired.

Step 2.

We assume that \overline{ϕ} is an isomorphism whenever D = D_{0}.

Given X_{(D,L,U)}, let J ⊂ S satisfy D = D(J). Then a fiber bundle as in Proposition 2.2, along with Proposition 2.3 imply that, as free abelian groups,

(6)  rank_{ℤ}H^{∗}(X_{(D_{0},L,U)}) = r · rank_{ℤ}H^{∗}(X_{(D,L,U)})

where r is the rank of the cohomology of ∏_{i=1}^{s+1} GL_{d_{i}−d_{i−1}}(ℂ)/B, that is

r = ∏_{i=1}^{s+1} (d_{i} − d_{i−1})! = |W_{J}|.

In light of (6), and since we have assumed \overline{ϕ} 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} = ℤ[x]^{W_{J}} ↪^{ι} ℤ[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

π_{w_{0}(J)} : A_{D_{0}} → A_{D}

of this inclusion, where w_{0}(J) is the longest permutation in W_{J}, and π_{w_{0}(J)} is the total W_{J}-symmetrizer (or isobaric divided difference, Demazure operator) corresponding to w_{0}(J) (see [8]). Since by definition I_{(D_{0},L,U)} is exactly the ideal in A_{D_{0}} generated by ι(I_{(D,L,U)}), the following lemma (whose straightforward proof we omit) completes Step 2:

Lemma 3.2. Let A ↪^{ι} B be an inclusion of rings which makes B a free A-module of rank r, and which has an A-module splitting σ : B → A, i.e. σ ∘ ι = id_{A}.

Then for any ideal I in A, and J the ideal in B generated by i(I), the induced map A/I ↪^{ι} B/J is injective and makes B/J into a free A/I-module of rank r. □

Step 3.

We must show that \overline{ϕ} is an isomorphism when D = D_{0} and |L| + |U| = 1.

Assume without loss of generality that L = ∅ and U = {(d, u)} is a singleton; the case where L is a singleton and U = ∅ follows via the symmetry x_{i} ⟷ x_{n+1−i} on ℤ[x].

We will use condition (C1). A special case of the fiber bundles referred to in Theorem 1.1 (iv) is the map

X_{(D_{0},∅,(d,u))} → X_{({0,1,2,...,d,u},∅,∅)}

which sends a complete flag 0 ⊂ V_{1} ⊂ ⋯ ⊂ V_{n−1} ⊂ ℂ^{n} satisfying V_{d} ⊂ ℂ^{u} to the partial flag 0 ⊂ V_{1} ⊂ ⋯ ⊂ V_{d} ⊂ ℂ^{u}, and has fiber GL_{n−d}(ℂ)/B. Since the fiber and base are partial flag manifolds whose cohomology rings are free abelian of rank (n − d)! and u(u − 1) ⋯ (u − d + 1), respectively, one concludes using Proposition 2.3 that H^{∗}(X_{(D_{0},∅,(d,u))}) is free abelian of rank

r := u(u − 1) ⋯ (u − d + 1) · (n − d)!.

To show that R_{(D_{0},∅,(d,u))} has a spanning set of this cardinality r, we borrow a trick from [3, p. 163]. Recall I_{n} = (e_{i}(1, n) : i = 1, 2, …, n), so that we have the following identities in R_{D_{0}}[t] = A_{D_{0}}/I_{n}[t]:

∏_{i=1}^{n}(1 − x_{i}t) = ∑_{s≥0}(−1)^{s}e_{s}(1, n)t^{s}

= 1

∏_{i=1}^{m}(1 − x_{i}t)^{−1} = ∏_{i=m+1}^{n}(1 − x_{i}t)

and consequently

(7)  ∑_{r≥0} h_{r}(1, m)t^{r} = ∑_{s=0}^{n−m}(−1)^{s}e_{s}(m + 1, n)t^{s}

where h_{r}(i, j) is the (complete) homogeneous symmetric function of degree r in the variables x_{i}, x_{i+1}, …, 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},∅,(d,u))}), we have for any m that

(8)  h_{r}(1, m) = 0 for r > n − m.

But we can say more in R_{(D_{0},∅,(d,u))}, since I_{(D_{0},∅,(d,u))} contains also the elements e_{s}(d + 1, n) for s ≥ 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},∅,(d,u))} also contains e_{s}(m + 1, n) for every m ≤ d and s ≥ u − m + 1. Consequently, using (7) we conclude that in R_{(D_{0},∅,(d,u))}

(9)  h_{r}(1, m) = 0 for m ≤ d and r ≥ u − m + 1.

Since h_{r}(1, m) equals x_{m}^{r} plus a sum of monomials which are lower in a lexicographic ordering with x_{n} > ⋯ > 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}, …, x_{m−1}. Therefore the relations (8), (9) imply that R_{(D_{0},∅,(d,u))} is spanned by all monomials

x_{1}^{i_{1}}x_{2}^{i_{2}} ⋯ x_{d}^{i_{d}}x_{d+1}^{i_{d+1}}x_{d+2}^{i_{d+2}} ⋯ x_{n}^{i_{n}}

satisfying i_{m} ≥ 0 for each m and

\begin{matrix}i & _{1} ≤ u − 1, i_{2} ≤ u − 2, ⋯, i_{d} ≤ u − d \\ i & _{d+1} ≤ n − d − 1, i_{d+2} ≤ n − d − 2, ⋯, i_{n−1} ≤ 1, i_{n} ≤ 0. \end{matrix}

Since this spanning set has cardinality r = u(u − 1) ⋯ (u − d + 1) · (n − d)!, we are done with Step 3, and with the proof of Theorem 3.1. Remark 3.3. In the special case where

D = {0, 1, 2 …, l, n},

L = ∅,
U = {(1, λ_{1}), …, (l, λ_{l})}

for some number partition λ = (λ_{1} ≤ ⋯ ≤ λ_{l}), the varieties X_{(D,L,U)} are all smooth Schubert subvarieties of X_{D}, which were studied by K. Ding [2] under the name of 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 λ.

In this special case, the proof of Step 3 in Theorem 3.1 actually showed that

(10)  R_{(D,L,U)} ≅ ℤ[x_{1}, …, x_{l}]/(h_{λ_{i}−i+1}(1, i) : i = 1, 2, …, l)

and that {h_{λ_{i}−i+1}(1, i) : i = 1, 2, …, l} forms a Gröbner basis for the ideal that it generates with respect to a lexicographic ordering of the variables in which x_{1} < ⋯ < x_{l} (cf. [10, §1.2]).

4. A pattern avoidance characterization for Schubert varieties defined by inclusions

In this section we give a characterization of the Schubert varieties defined by inclusions (Theorem 4.2), making use of Fulton’s notion of the essential set of a permutation w.

The essential set was introduced to parametrize a minimal set of Schubert conditions which imply all of the other Schubert conditions on X_{wW_{J}}. We present here a version slightly different from his original definition that is adapted to our conventions, and which may be inferred from the discussion in [4, pp. 20-21]. Given w = w_{1}w_{2} ⋯ w_{n}, we first describe an adapted version of its diagram: First draw an n-by-n matrix of bubbles. For each i, place an × in the (i, w_{i}) entry, and erase all bubbles weakly to its left in the same row, and all bubbles weakly below it in the same column. The bubbles remaining define the diagram.

Note that the number of bubbles in this diagram is not the length l(w), but rather l(w_{0}w) where w_{0} is the longest element in W = S_{n}. This corresponds to the fact that this diagram is actually a mirror image of the usual diagram D(w_{0}w) associated to w_{0}w [4, p. 11], [8, p. 8].

The essential set is defined to be the set of bubbles (i, j) for which there is neither a bubble in (i + 1, j) nor in (i, j − 1). Assuming that w = max wW_{J}, for each such bubble there is a Schubert condition on the flags in X_{wW_{J}} asserting

(11)  dim V_{i} ∩ ℂ^{j−1} ≥ r

where r is the number of bubbles in column j lying weakly above the bubble (i, j). We will call such Schubert conditions essential. The essential Schubert conditions imply all of the other Schubert conditions on flags in X_{wW_{J}} and no essential Schubert condition is implied by the other Schubert conditions. For example, if w = 56238417, then the diagram of w is:

1 2 3 4 5 6 7 8
1  × • ∘ ∘
2  × ∘ ∘
3  × • ∘  ∘ ∘
4  × •  • ∘
5  ×
6 × ∘
7 ×  •
8  ×

Here the bubbles in the essential set are shown darkened and the essential Schubert conditions on X_{wW_{J}} are:

dim V_{1} ∩ ℂ^{5} ≥ 1, i.e. V_{1} ⊂ ℂ^{5}
dim V_{3} ∩ ℂ^{2} ≥ 1
dim V_{4} ∩ ℂ^{6} ≥ 4, i.e. V_{4} ⊂ ℂ^{6}
dim V_{4} ∩ ℂ^{3} ≥ 2
dim V_{7} ∩ ℂ^{6} ≥ 6, i.e. ℂ^{6} ⊂ V_{7}.

The following observation is immediate from the construction of the diagram of w. Proposition 4.1. The essential set of w contains an element in row i if and only if w_{i} < w_{i+1}.

In particular, if w = max wW_{J}, then the essential set of w is contained entirely in the rows indexed by D(J). □

Note that the second assertion of Proposition 4.1 implies that the Schubert conditions on complete flags in X_{w} are essentially the same as those on partial flags in X_{wW_{J}}, proving Proposition 2.2.

The following theorem gives a combinatorial characterization of the Schubert varieties X_{wW_{J}} defined by inclusions. Theorem 4.2. Let w = max wW_{J}. Then X_{wW_{J}} is defined by inclusions if and only if w does not contain a subsequence with the same relative order as

(12)  4231, 35142, 42513, or 351624.

Proof. We begin with two facts about an essential Schubert condition (11). Both follow immediately from the way the diagram is constructed.

Fact 1: r ≤ i, with strict inequality if and only if there exists some × in the diagram strictly to the northeast of (i, j). Fact 2: r ≤ j − 1, with strict inequality if and only if there exists some × in the diagram strictly to the southwest of (i, j).

Given these facts, we now prove the theorem. Assume X_{wW_{J}} is not defined by inclusions, so there is an essential bubble (i, j) in the diagram of w that does not correspond to an inclusion condition. Consequently the conditions in Facts 1 and 2 must hold, i.e. there must exist some × located in (i′, w_{i′}) with i′ < i and w_{i′} > j, and also some × located in (i′′, w_{i′′}) with i′′ > i and w_{i′′} < j −1. Call these two ×’s witnesses. We will further assume, without loss of generality, that the southwest witness located in (i′′, w_{i′′}) is chosen with i′′ minimal.

We show that at least one of the four sequences in (12) will then appear, depending upon whether the rectangular box in the diagram of w bounded by rows i′, i′′ and columns w_{i′′}, w_{i′} contains an × strictly to the left of column j, and an × strictly below row i. If one can find

- both, then combining these with the two witnesses produces a subsequence in w with the same relative order as 4231,

- the former, but not the latter, then combining the two witnesses with the former and with the two ×’s in (w^{−1}(j), j), (i + 1, w_{i+1}) produces a subsequence in w with the same relative order as 42513,

- the latter, but not the former, then combining the two witnesses with the latter and with the two ×’s in (w^{−1}(j − 1), j − 1), (i, w_{i}) produces a subsequence in w with the same relative order as 35142,

- neither, then combining the two witnesses with the four ×’s in

(w^{−1}(j − 1), j − 1), (w^{−1}(j), j), (i, w_{i}), (i + 1, w_{i+1})

produces a subsequence in w with the same relative order as 351624. The converse is easier to see. First suppose that u is one of the four permutations in (12). Then it is easy to exhibit a (unique!) essential bubble that corresponds to a non-inclusion Schubert condition in X_{u}: it is the bubble at position

- (2, 3) for u = 4231,

- (3, 4) for u = 35142,
- (2, 3) for u = 42513,
- (3, 4) for u = 351624.

Then if w contains a subsequence with the same relative order as u, there will be a “region” containing some bubbles within the diagram for w that corresponds to the above essential bubble in u. By Facts 1 and 2, it is easy to see that any essential bubble in that region will give rise to a non-inclusion Schubert condition on X_{wW_{J}}. □

5. Acknowledgements

The authors thank Conan Leung, Sorin Popescu, Nathan Reading, and Andrei Zelevinsky for helpful comments. They also thank an anonymous referee for pointing out references [9, 11].

References

[1] J. Carrell, Some remarks on regular Weyl groups orbits and the cohomology of Schubert varieties, Contemporary Mathematics 139 (1992).

[2] K. Ding, Rook placements and cellular decomposition of partition varieties, Discrete Math. 170 (1997), 107–151

[3] W. Fulton, Young Tableaux, Cambridge University Press, Cambridge, 1997.

[4] W. Fulton and P. Pragacz, Schubert varieties and degeneracy loci, Springer Lecture Notes in Mathematics 1689, 1998.

[5] M. Haiman, unpublished.

[6] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge studies in advanced mathematics 29 Cambridge Univ. Press, 1990.

[7] V. Lakshmibai and B. Sandhya, Criterion for smoothness of Schubert varieties in SL(n)/B, Proc. Indian Acad. Sci. (Math. Sci.), 100(1) (1990), 45–52.

[8] I. G. Macdonald, Notes on Schubert polynomials, LACIM Montreal, Canada 1991.

[9] K. M. Ryan, On Schubert varieties in the flag manifold of Sl(n, C), Math. Ann. 276 (1987), 205–224.

[10] B. Sturmfels, Algorithms in invariant theory, Springer-Verlag, New York 1993.

[11] J. S. Wolper, A combinatorial approach to the singularities of Schubert varieties. Adv. Math. 76 (1989), 184–193

Department of Mathematics, Cornell University, Ithaca, NY 14853, USA E-mail address, V. Gasharov: vesko@math.cornell.edu

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA E-mail address, V. Reiner: reiner@math.umn.edu