On the Intersection Ring of the Grassmannian of Complex Structures

Haibao Duan and Piotr Pragacz

Contents

1 Introduction 1

2 Preliminaries, notation, and conventions. 3

3 Vanishing in nil Hecke ring of type D. 5

4 A group-theoretic approach to Schubert calculus for CS_{n}. 18

5 Schubert cycles of complex structures on R^{2n}. 25

References 33

1 Introduction

The Grassmannian of complex structures parametrizes orthogonal automorphisms of the Euclidean space R^{2n} whose square is the minus identity. Equivalently, it parametrizes minimal geodesics from the identity to the minus identity in the orthogonal group SO(2n, R) [Mi]. This space is usually denoted by CS_{n}. It played a significant role in several important achievements in topology:

• in the investigation of orthonormal vector fields on spheres by Hurewicz and Adams;

• in the study of the existence of complex structures on even dimensional spheres by Borel and Serre;

• in the Bott’s discovery of the eight-periodicity of homotopy groups of the stable real orthogonal groups.

Also, CS_{n} serves as the classifying space of all complex bundles whose real reduction is trivial, by a result of the first author [Du1].

The goal of the present paper is to develop in a systematic way a Schubert calculus for CS_{n}. We hope that it will be useful also for topologists.

The space CS_{n} has two connected components, each isomorphic to the homogeneous space

SO(2n, C)/U(n) or SO(2n, C)/P ,

where P is the maximal parabolic corresponding to omitting the “right end root”. This space is a connected component of the Grassmannian of all isotropic subspaces of C^{2n} w.r.t. to the orthogonal form induced by the scalar product, and as such, it is also known as the orthogonal Grassmannian.

With the help of the group-theoretic description, we can use the characteristic map of Borel [Bo], and - via the theory of Bernstein-Gelfand-Gelfand [BGG] and Demazure [De] - divided differences of type D to study the intersection theory on the space in question. In order to make the work with the characteristic map efficient, one needs a proper family of “invariant” polynomials that are well suited to divided differences, and also to geometry/topology at the same time.

A result of the second author [P1] identified Schubert classes in the homogeneous space SO(2n, C)/U(n) with suitable Schur P -polynomials. In loc. cit. this identification used a geometric argument, namely an isomorphism SO(2n, C)/U(n) ≃ SO(2n − 1, C)/U(n − 1), and an identification of the Schubert classes for the latter Grassmannian with Schur P -functions. (This last identification was based on comparison of the Pieri-type formulas from [HB] and [Mo].)

In the present paper we revisit the identification for the Schubert classes for SO(2n, C)/U(n) with Schur P -functions via a direct group-theoretic argument based on the calculus of divided differences of type D. More precisely, we give a group-theoretic proof of a Pieri-type formula that is based on some vanishing results for operators composed of divided differences of type D and simple reflections from the Weyl group of type D (i.e. products of elements in the nil Hecke ring of type D). These last results form the most technical part of the present work. (We recommend to omit them by the first reading for the reader not having a sufficient combinatorial background.) Our proof of the Pieri-type formula follows a strategy for deriving similar formulas for various homogeneous spaces worked out by Ratajski and the second autor in a series of papers summarized in [P2]. This particular proof was promised in [PR2] – a paper that is now under revision. The proof uses esentially an iteration of the Leibniz-type formula for a simple divided difference applied to the product of two functions.

Combining the Pieri formula with a combinatorial lemma of Schur for the projective characters of the symmetric groups, we get a formula for the degree of Schubert varieties in SO(2n, C)/U(n). (Occasionally, we also sketch another proof of the lemma of Schur.)

We remark that there exists now a refinement of Schur P -functions that seems to be even better adapted for some aspects of geometry. These are the so called P̃-functions of [PR1], which are modeled on Schur P -functions. In [LP], A. Lascoux and the second author worked out a connection of orthogonal divided differences to P̃-functions using vertex operators. This has led to orthogonal Schubert polynomials that are useful in various cohomological computations (cf. [KT] and [T2]).

After presenting the Schubert varieties in a group-theoretic way, we also describe them via Schubert-type conditions relative to some flag of linear subspaces, and finally we study them in terms of complex structures. To this end, we are guided by the Mahowald-Vassiljev-type formula ([DV],[V]):

H_{i}(CS_{n}) = ⊕_{k=0}^{n}H_{i−k(k−1)}(G_{k}(C^{n})) ,

where G_{k}(C^{n}) is the Grassmannian of all complex k-planes through zero in C^{n}. This is done by some vector bundle argument.

We end the paper by illustrating how the Schubert calculus developed here can be used to solve problems about enumeration of complex structures which satisfy some natural conditions of “partial overlaping” with a certain number of complex structures in general position in R^{2n}. One of the applications leads to an interesting algebraic conjecture about homomorphisms between the cohomology ring of CS_{n}^{+} and that of the Grassmannian G_{k}(C^{n}).

Acknowledgements. This work was partially supported by grant No.5031 of French-Polish cooperation C.N.R.S. – P.A.N. (the “French connection” of this grant being A. Lascoux whose comments clarified our Remark 4.11).

The second author is grateful to the Max-Planck-Institut für Mathematik in Bonn for excellent working conditions during the preparation of the first draft of this paper.

2 Preliminaries, notation, and conventions.

We start with some algebraic preliminaries on even orthogonal groups. We fix a positive integer n. Suppose that H = SO(2n, C) is the orthogonal group (of type D_{n}) over the field of complex numbers. Let us use the following notation:

B - a fixed Borel subgroup of H, T ⊂ B - a fixed maximal torus. R - the root system of H associated with T, Σ - a set of simple roots of R associated with B, W - the Weyl group of (H, T),

In a standard Bourbaki [Bu] realization, we have:

R = {±e_{i} ± e_{j} : 1 ≤ i < j ≤ n} ⊂ R^{n} = ⊕_{i=1}^{n}Re_{i},
Σ = {e_{1} − e_{2}, …, e_{n−1} − e_{n}, e_{n−1} + e_{n}},
W = S_{n}|× Z_{2}^{n−1} .

A typical element of W can be written as a pair (τ, ϵ), where τ ∈ S_{n} and ϵ = (ϵ_{1}, …, ϵ_{n}) is a sequence of elements of Z_{2} = {−1, 1} such that #{i : ϵ_{i} = −1} is even. Multiplication in W is given by

(τ, ϵ) · (τ′, ϵ′) = (τ ∘ τ′, δ),

where “∘” denotes the composition of permutations and δ_{i} = ϵ_{τ′(i)} · ϵ′_{i}. The following lemma can be easily verified (and is pretty well-known). For w ∈ W, let l(w) denote the length of w takem w.r.t. to the above Σ.

Lemma 2.1 For any w ∈ W, l(w) is

∑_{i=1}^{n} #{j | j > i & w(j) < w(i)} + ∑_{ϵ_{p}=−1} 2#{q | q > p & w(q) > w(p)} .

We will use the “barred-permutation” notation, indicating by a bar a place in the permutation where ϵ_{i} = −1.

The poset W^{∗} of the minimal length left coset representatives of S_{n} in W consists o f elements of the form

[y_{1}, y_{2}, …, y_{n−k}, \overline{z}_{k}, \overline{z}_{k−1}, …, \overline{z}_{1}]

where k runs over the set of even integers between 1 and n, y_{1} < ⋯ < y_{n−k}, and z_{k} > ⋯ > z_{1}. The following lemma, that is easy to prove, gives us the lengths of some barred permutations important for this paper.

Lemma 2.2 Let, as above, y_{1} < ⋯ < y_{n−k} and z_{k} > ⋯ > z_{1} be sequences of integers that are complementary in {1, …, n}. Assume that k is even. Then in W we have

l([y_{1}, y_{2}, …, y_{n−k}, \overline{z}_{k}, \overline{z}_{k−1}, …, \overline{z}_{1}]) = ∑_{j=1}^{k}(n − z_{j}) .

We set ρ(k) := (k, k − 1, …, 1), a “triangular partition” of length k. Given a strict partition α ⊂ ρ(n − 1), we set

α^{+} := (α_{1} + 1, α_{2} + 1, …, α_{l} + 1)

if l = l(α) is even, and

α^{+} := (α_{1} + 1, α_{2} + 1, …, α_{l} + 1, 1)

if l = l(α) is odd. Note that α^{+} is of even length.

Given a strict partition μ ⊂ ρ(n) of even length k, we associate with it the following element w_{μ} of W^{∗}. We set

w_{μ} := [y_{1}, y_{2}, …, y_{n−k}, \overline{n + 1 − μ_{1}}, \overline{n + 1 − μ_{2}}, …, \overline{n + 1 − μ_{k}}] .

Note that l(w_{μ}) = |μ| − k.

Setting for a strict partition α ⊂ ρ(n − 1), λ := α^{+}, we have l(w_{λ}) = |α|. Finally, we adopt a convention that all homology or cohomology groups

in the present paper are taken with integer coefficients.

3 Vanishing in nil Hecke ring of type D.

We define simple divided differences of type D_{n} which are operators ∂_{i} : Z[X] → Z[X], i = 1, …, n, of degree −1 acting on the ring of polynomials Z[X] where X is a fixed set of indeterminates X = {x_{1}, x_{2}, …, x_{n}}. To this end, we denote by s_{i}, 1 ≤ i ≤ n − 1, the transposition

[1, …, i − 1, i + 1, i, i + 2, …, n] ∈ S_{n} ⊂ W ,

acting on X by interchanging x_{i} and x_{i+1}. Moreover, let

s_{n} = [1, …, n − 2, \overline{n}, \overline{n − 1}]

be the reflection which transposes x_{n−1} with x_{n} and changes the signs of both the variables. The remaining variables are invariant under the action of these transpositions. This action is extended multiplicatively to the ring Z[X]. Note that s_{n} commutes with s_{i}, i ≠ n − 2, and

s_{n−2} · s_{n} · s_{n−2} = s_{n} · s_{n−2} · s_{n} .

Simple divided differences of type D are defined as follows:

∂_{i}(f) = (f − s_{i}f)/(x_{i} − x_{i+1}), i = 1, …, n − 1;

∂_{n}(f) = (f − s_{n}f)/(x_{n−1} + x_{n}).

The algebra generated by the divided diferences ∂_{i}, 1 ≤ i ≤ n − 1 that goes back to Newton, has been recently extensively studied by Lascoux and Schutzenberger [LSc]. Here we investigate the effect of adding one extra divided difference ∂_{n} (which changes the picture in a drastic way).

For every f, g ∈ Z[X] and any i, we have:

∂_{i}(f · g) = f · (∂_{i}g) + (∂_{i}f) · (s_{i}g)  (1)

(a Leibniz-type formula).

For a given a = (a_{n}, a_{n−1}, …, a_{2}, a_{1}) ∈ {−1, 0, 1}^{n}, we define the generating function:

E_{a} = ∏_{i=1}^{n}(1 + a_{i}x_{i}).  (2)

In particular, for a = (1, …, 1), the resulting generating function, denoted by E, is the generating function for the elementary symmetric polynomials e_{i}(X) = e_{i}(x_{1}, …, x_{n}), i = 1, …, n.

Lemma 3.1 a) We have s_{i}(E_{a}) = E_{a′}, where

a′ = \{ \begin{matrix}(a_{n}, …, a_{i+2}, a_{i}, a_{i+1}, a_{i−1}, …, a_{1}) & i < n, \\ (−a_{n−1}, −a_{n}, a_{n−2}, …, a_{1}) & i = n. \end{matrix}

b) For i = 1, 2, …, n − 1,

∂_{i}(E_{a}) = d · E_{a′}  if a_{i} = a_{i+1} + d  (d = −2, −1, 0, 1, 2),

where a′ = (a_{n}, …, 0, 0, …, a_{1}) is the sequence a with a_{i+1}, a_{i} replaced by zeros.

c) ∂_{n}(E_{a}) = (a_{n} + a_{n−1}) · E_{(0,0,a_{n−2},...,a_{1})}. In particular if Δ is a composition of some s- and ∂-operations, then for every a, Δ(E_{a}) = (scalar) · E_{a′}, where a′ is uniquely determined if this scalar is not zero.

Proof. We prove e.g. c). We have, with a′ = (0, 0, a_{n−2}, …, a_{1}),

∂_{n}(E_{a}) = \frac{(1 + a_{n−1}x_{n−1})(1 + a_{n}x_{n}) − (1 − a_{n−1}x_{n})(1 − a_{n}x_{n−1})}{x_{n−1} + x_{n}} · E_{a′}

= \frac{(a_{n−1} + a_{n})(x_{n−1} + x_{n})}{x_{n−1} + x_{n}} · E_{a′} = (a_{n} + a_{n−1}) · E_{a′} ,

as desired. □

We now recall the following fact from [BGG] and [De]. For any w ∈ W and any reduced decomposition w = s_{i_{1}} ⋯ s_{i_{l}} one can define ∂_{w} = ∂_{i_{1}} ∘ ⋯ ∘ ∂_{i_{l}} – an operator on Z[X] of degree −l(w). In fact, since divided differences satisfy braid relations, ∂_{w} does not depend on the chosen reduced decomposition of w.

Suppose a strict partition μ with even l(μ) is given. Let us use the following coordinates for boxes in the Ferrers’ diagram D_{μ} of μ:

n⋮1  n n − 1 … 2 1

We associate with μ a certain distinguished reduced decomposition of w_{μ} ∈ W. To this end, let us modify the diagram D_{μ} in the following way. Remove one box from each row of D_{μ} : from rows with even numbers remove the box in the n-th column, and from rows with odd numbers remove the box in the (n − 1)-th column.

We display the removed boxes in the picture using the symbol × and denote the so obtained diagram by D^{∘}_{μ}. For example, D^{∘}_{(8,7,4,2)} is:

×× ××

Assume now, that a subset D ⊂ D^{∘}_{μ} is given. A box belonging to D will be called a D-box and a box from the difference D_{μ} \ D will be called a ∼D-box.

Definition 3.2 Read D^{∘}_{μ} row by row from left to right and from top to bottom. Every D-box (resp. ∼ D-box) in the i-th column gives us s_{i} (resp. ∂_{i}). Then ∂_{μ}^{D} is the composition of the resulting s_{i}’s and ∂_{i}’s (the composition written from right to left).

Definition 3.3 Read D^{∘}_{μ}. Every D-box in the i-th column gives us s_{i}. ∼Dboxes give no contribution. Then, r_{D} is the word obtained by writing the resulting s_{i}’s from right to left. ( In other words, one obtains r_{D} by erasing all the ∂_{i}’s from ∂_{μ}^{D}.)

For example, for n = 9 and μ = (8, 7, 4, 2),

×× ×× • • • • • • • • •^{9} ^{8} ^{7} ^{6} ^{5} ^{4} ^{3} ^{2} ^{1}

∂_{μ}^{D} = ∂_{8} ∘ s_{6} ∘ ∂_{7} ∘ ∂_{9} ∘ ∂_{3} ∘ ∂_{4} ∘ s_{5} ∘ ∂_{6} ∘ s_{7} ∘ s_{8} ∘ ∂_{2} ∘ ∂_{3} ∘ s_{4} ∘ ∂_{5} ∘ s_{6} ∘ s_{7} ∘ s_{9}

r_{D} = s_{6} · s_{5} · s_{7} · s_{8} · s_{4} · s_{6} · s_{7} · s_{9} .

One can easily prove that for D = D^{∘}_{μ}, we have r_{D} ∈ R(w_{μ}) - the set of reduced decompositions of w_{μ}. This is our distinguished reduced decomposition of w_{μ}. For example, for n = 9 and μ = (8, 7, 4, 2),

×× ×× • • • • • • • • • • • • • • • • •^{9} ^{8} ^{7} ^{6} ^{5} ^{4} ^{3} ^{2} ^{1}

w_{μ} = s_{8} · s_{6} · s_{7} · s_{9} · s_{3} · s_{4} · s_{5} · s_{6} · s_{7} · s_{8} · s_{2} · s_{3} · s_{4} · s_{5} · s_{6} · s_{7} · s_{9} .

Let now μ be a strict partition with even length. We will examine subsets D ⊂ D^{∘}_{μ} for which ∂_{μ}^{D}(E) = 0 (we say: “ D causes the vanishing”).

In many computations in this section, we will apply compositions of the operators of boxes of D_{μ} to the generating functions E_{a}. With the following example we illustrate how such operators act.

Example 3.4 Let n = 9. We apply the operators of boxes from left to right to E_{a}, where a = (a_{9}, a_{8}, a_{7}, a_{6}, a_{5}, a_{4}, a_{3}, a_{2}, a_{1}) , and obtain E_{a′}; “•” denotes a D-box and empty boxes are ∼D-boxes. We give 2 examples of the action of the operators associated with a row in D_{μ}:

× • • • • • • •^{9} ^{8} ^{7} ^{6} ^{5} ^{4} ^{3} ^{2} ^{1}  a′ = (−a^{8}, a^{7}, a^{6}, a^{5}, a^{4}, a^{3}, a^{2}, 0, 0)

× • • • • • •^{9} ^{8} ^{7} ^{6} ^{5} ^{4} ^{3} ^{2} ^{1}  a′ = (a^{8}, a^{7}, a^{6}, 0, a^{4}, a^{3}, 0, a^{1}, 0)

Lemma 3.5 The following configurations of ∼D-boxes (noted by “∘”) in D_{μ} give the vanishing:

1. (A three ∼D-box configuration)

∘ ? ⋯ ? ∘ ? ⋯ ? ∘

2. (A three ∼D-box configuration)

× ? ⋯ ? ∘ ? ⋯ ? ∘ ∘ ×

3. (A three ∼D-box configuration)

? × ? ⋯ ? ∘ ? ⋯ ? ∘ × ∘  also:  ∘ × ? ,  ⋯ ? ∘ × ∘

4.

× ∘ ⋯ ∘ ⋯ ∘| ∘  in particular: × ∘ ⋯ ∘| ∘

∘ × ∘ ⋯ ∘ ⋯ ∘| ∘  in particular: ∘ × ∘ ⋯ ∘| ∘

(Above, “?” can be ∘, •, or ×; in 4. and 5., the symbol “|” denotes the end of a row; the skew directions are all parallel to the antidiagonal.) Proof. Direct calculation using Lemma 3.1. □

We state this lemma in a stronger form than is needed in the present paper. This stronger form will be used in the revised version of [PR2].

Let us fix an element w = [y_{1}, y_{2}, …, y_{n−k}, \overline{z}_{k}, \overline{z}_{k−1}, …, \overline{z}_{1}] ∈ W^{∗}. Recall that k is even. We treat a given reduced decomposition w = s_{i_{1}} · s_{i_{2}} · . . . · s_{i_{l}} as a sequence of simple transposition operations, which produces the element in question from the identity permutation:

[y_{1}, y_{2}, …, y_{n−k}, \overline{z}_{k}, \overline{z}_{k−1}, …, \overline{z}_{1}] = (⋯ ([1, 2, …, n] · s_{i_{1}}) ⋯) · s_{i_{l}}.

In the following, the simple transpositions involved will be called “the s_{i_{h}}- operations” (h = 1, …, l).

Proposition 3.6 We have the following two possibilities for the action of s_{i_{h}}-operations on the z’s:

1. If i_{h} = n then this operation is:

[…, z, z′] → […, \overline{z}′, \overline{z}]

2. If i_{h} < n, then this operation is:

[…, z, x, …] → […, x, z, …]

where x ≠ z_{j} for j = 1, …, k.

Proof. The most economical way (i.e. the one that requires the minimal number of s_{i_{h}}-operations) to supply all the z’s with bars, is the following one. We move the pair […, z_{k−1}, … z_{k}, …] towards the end of the permutation (i.e. towards the n − 1-th and n-th places), and then the s_{n}-operation transforms […, z_{k−1}, z_{k}] to […, \overline{z}_{k}, \overline{z}_{k−1}]. This requires at least

(n − z_{k}) + (n − 1 − z_{k−1}) + 1 = (n − z_{k}) + (n − z_{k−1})

s_{i_{h}}-operations. Then, we do the same with the pairs:

[…, z_{k−3}, …, z_{k−2} …] , […, z_{k−5}, …, z_{k−4}, …] , …

up to the last pair […, z_{1}, …, z_{2}, …]. We see that l(w) (which is the number of s_{i_{h}}-operations) is greater than or equal to

∑_{j=1}^{k}(n − z_{j}) .

But by Lemma 2.2, this last number is actually equal to l(w). This means that no s_{i_{h}}-operation, for i_{h} < n, interchanges two (bar-free) z’s (on their way towards the end of the permutation), and that exactly k/2 s_{i_{h}}-operations with i_{h} = n appear. This implies immediately both assertions of the proposition. □

Corollary 3.7 1. No s_{i_{h}}-operation for i_{h} < n interchanges two (bar-free) z’s. (Also, it does not interchange two barred z’s.)

2. If i_{h} = n then the s_{i_{h}}-operation interchanges two (bar-free) z’s on the n-th and n−1-th places (more precisely, these are z’s od the form: z_{2i−1} and z_{2i}), and supplies them with bars.

Now we assume that λ is another strict partition, that D ⊂ D^{∘}_{μ}, and that r_{D} ∈ R(w_{λ}). Suppose that a D-box appears in the i-th column where i < n. We define the mark of this box to be p, if the corresponding s_{i_{h}}-operation acts on the i-th and i + 1-th places as follows:

[…, z_{p}, x, …] → […, x, z_{p}, …] ,

where x ≠ z_{j} , j = 1, …, k. A D − box in the n-th column has mark p if the corresponding (s_{i_{h}} = s_{n})-operation acts via

[…, z_{p−1}, z_{p}] → […, \overline{z}_{p}, \overline{z}_{p−1}] .

In particular, boxes in the n-th column have only odd marks. In the following lemma, we collect some simple properties of marks.

Lemma 3.8 1. (Conectedness) The D-boxes with a fixed mark in one row form a connected set. (The boxes marked by × are treated as empty boxes).

2. (Separation) In a fixed row, the two sets of D-boxes equipped with different marks are disconnected (i.e. there is at least one ∼ D-box between them; a box marked by × is not treated as as ∼D-box).

3. The sequence of boxes with odd mark p is of the form:

(t_{n}, n), (t_{n−2}, n − 2), …, (t_{z_{p}}, z_{p}) ,

where p ≤ t_{n} ≤ t_{n−2} ≤ ⋯ ≤ t_{z_{p}} .

4. The sequence of boxes with even mark p is of the form:

(t_{n−1}, n − 1), (t_{n−2}, n − 2), …, (t_{z_{p}}, z_{p}) ,

where p ≤ t_{n−1} ≤ t_{n−2} ≤ ⋯ ≤ t_{z_{p}} .

5. The marks of boxes in a fixed column (strictly) increase from top to bottom.

6. The marks of boxes in a fixed row (weakly) decrease from left to right.

Definition 3.9 The set of D-boxes with mark p is called the ribbon of mark p.

We have two basic operations of deforming ribbons.

• (“Push down”) Let i be odd and suppose that the boxes

(i, n), (i, n − 2), …, (i, j)

form an entire ribbon. The operation transforms them to

(i − 2, n), (i − 2, n − 2), …, (i − 2, j) .

Let i be even and suppose that the boxes

(i, n − 1), (i, n − 2), …, (i, j)

form an entire ribbon. The operation transforms them to

(i − 2, n − 1), (i − 2, n − 2), …, (i − 2, j) .

For example, the ribbons

×× ×  × × × • • • • • • •  • • • • • • •

can be pushed down to

[figure: ×; ×; ×; ×; ×; ×; •; •; •; •; •; •; •; •; •; •; •; •; •; •]

• (“Breaking a ribbon”) Let j ≤ n − 2. The operation transforms a final

segment

(i, j), (i, j − 1), …, (i, h)

of a ribbon to

(i − 1, j), (i − 1, j − 1), (i − 1, h) ,

provided (i − 1, j + 1) is a ∼ D-box, or it is × and (i − 1, j + 2) is a ∼ D-box. The box (i, j) (before the operation) is called the breaking box.

For example, for a breaking box a, b a ∼ D-box, or b = × and c a ∼D-box:

? ^{?} ^{?} ^{a}^{?}  • • • • • • •^{c} ^{b}

can be broken at a and transformed to

? ^{?} ^{?}^{?}  • • • • • • •^{c} ^{b} ^{a}

Using the braid relations in W one easily shows that if r_{D} ∈ R(w_{λ}), then, after breaking a ribbon in D, we get D′ such that r_{D′} ∈ R(w_{λ}). In the case of the push down operation, it is clear that we get D′ with r_{D} = r_{D′}. Note that any configuration of boxes D ⊂ D_{μ} such that r_{D} ∈ R(w_{λ}) can be obtained from D^{∘}_{λ} ⊂ D^{∘}_{μ} by a sequence of operations of the above described type.

Definition 3.10 (“ Maximal deformation” of D^{∘}_{λ} ⊂ D^{∘}_{μ})

• Pick the lowest ribbon. Push it down as many times as possible. Then choose the leftmost breaking box on this ribbon (if it exists) and break the ribbon.

• Pick a ribbon and suppose that lower ribbons in D^{∘}_{λ} have been already deformed. Push down this ribbon as many times as possible. Let a be the leftmost breaking box on the ribbon. Break the ribon at a as many times as possible. Then choose the next leftmost breaking box b and break the ribbon at b as many times as possible etc.

For some examples of maximal deformations of diagrams, see Example 4.7.

Proposition 3.11 Let D ⊂ D^{∘}_{μ} be such that r_{D} ∈ R(w_{λ}). If ∂_{μ}^{D}(E) ≠ 0 then

D is the maximal deformation of D^{∘}_{λ} ⊂ D^{∘}_{μ}.

Proof. The proof is by descending induction on the mark of a ribbon. Pick the ribbon with mark p. Assume that the ribbons with marks p + 1, …, l(λ) have been already maximally deformed. Suppose that we have either a possibility of pushing down of a ribbon or breaking a ribbon. In the case of the former operation we will refer to boxes of the three involved rows; in the case of the latter operation, we will refer to the two involved rows. We note that

• any box directly to the right of the end of the (deformed) ribbon is a ∼D-box;

• any box directly below a box of the (deformed) ribbon is a ∼D-box;

• ∼D-boxes in the n-th or n − 1-th column cannot be supplied by marks smaller that p.

This implies that if we not perform the operations in a maximal way, then we will either obtain a configuration

∘ ? ⋯ ? ∘ ∘

or, we will get one of the following two possibilities:

× × ×^{a} ^{b} ^{c} • • ⋯ •  × × ×^{a}  ^{b}^{c} • ⋯ • •

where a, b and c are ∼D-boxes. By Lemma 3.5 all these three configurations of ∼D-boxes give the vanishing. The obtained contradiction means that the maximal deformation of D^{∘}_{λ} ⊂ D^{∘}_{μ} is necessary to avoid the vanishing. □

As a corollary of this proposition, we get the following two results bounding the size of μ w.r.t. λ, if one wants to avoid the vanishing. We will need some additional notation. Given a strict partition μ of even length l, we denote by μ^{−} the strict partition (μ_{1} − 1, …, μ_{l} − 1). Note that (μ^{−})^{+} = μ. We also set l^{∘}(μ) := l(μ^{−}) . Setting for a strict partition α ⊂ ρ(n − 1), λ := α^{+},

we have l^{∘}(λ) = l(α) = #D_{α} = #D^{∘}_{λ}.

Proposition 3.12 If, for the maximal deformation D of D^{∘}_{λ} ⊂ D^{∘}_{μ}, one has

∂_{μ}^{D}(E) ≠ 0, then l^{∘}(μ) ≤ l^{∘}(λ) + 1 .

Proof. Pick the highest row from which some ribbon has been pushed down in the maximal deformation. After breaking some higher ribbons we get the following configuration of ∼D-boxes:

× ∘ ⋯ ∘ ⋯ ∘|  or:  ∘ × ∘ ⋯ ∘ ⋯ ∘|

If we suppose l^{∘}(μ) ≥ l^{∘}(λ) + 2, then there exists a ∼D box in some lower row than the rows of the displayed boxes. By Lemma 3.5, we get the vanishing – contradiction. □

Proposition 3.13 Assume that l^{∘}(μ) ≤ l^{∘}(λ) + 1. If for the maximal deformation D of D^{∘}_{λ} ⊂ D^{∘}_{μ} one has ∂_{μ}^{D}(E) ≠ 0, then D_{μ^{−}} \ D_{λ^{−}} is a horizontal strip.

Proof. Suppose that λ_{i} < μ_{i+1} for some i. We can assume that for some j < i,

μ_{i} = λ_{i−1}, μ_{i−1} = λ_{i−2}, …, μ_{j+2} = λ_{j+1} but λ_{j} > μ_{j+1} .

After the maximal deformation, we get in the consecutive rows with numbers i + 1, i, …, j + 1 the following configuration of ∼D-boxes:

∘ ⋯ ∘ ⋯ ∘ ⋯ ∘| ⋯ a  or:  ∘ × × • ∘ ⋯ ∘ ⋯ ∘ ⋯ ∘| ⋯ a

where a denotes a ∼D-box. We get the vanishing by Lemma 3.5 – contradiction. □

The maximal deformation is obtained by breaking each row such that λ_{i} = μ_{i+1}, at one breaking point.

Among the connected components of D^{∘}_{μ} \ D^{∘}_{λ} we have those which do not meet the n-th column: they are ordinary horizontal strips [Ma]. Those which meet the n-th component are of the form:

× ∘ × ∘ ⋯ ∘ ⋯ ∘ ⋯ ∘ ⋯ ∘ ⋯ ∘  (3)

or of the form:

∘ × × • ∘ ⋯ ∘ ⋯ ∘ ⋯ ∘ ⋯ ∘ ⋯ ∘  (4)

Note the following particular case of (4):

× ∘ × ∘ ⋯ ∘| × • • ⋯ • •  (5)

After the maximal deformation an ordinary horizontal strip becomes

∘ ⋯ ∘ ⋯ ∘|

In turn, configuration (4) which is not of the form (5), and configuration (3) are deformed to:

∘ × × • ∘ ⋯ ∘ ⋯ ∘|

Of course, configuration (5) does not change under the maximal deformation.

Proposition 3.14 Suppose that λ ⊂ μ ⊂ ρ(n) are strict partitions such that D_{μ^{−}} \ D_{λ^{−}} is a horizontal strip (in particular, l^{∘}(μ) ≤ l^{∘}(λ) + 1). Let D be the maximal deformation of D^{∘}_{μ} \ D^{∘}_{λ}. Then ∂_{μ}^{D}(E) = 2^{m}, where m is the number

of connected components of D^{∘}_{μ} \ D.

Proof. Different connected components of D^{∘}_{μ} \ D lie in separate rows and separate columns. Let us number these components from top to bottom. Pick a connected component of D^{∘}_{μ}\D. The part of ∂_{μ}^{D} associated with the boxes in the rows preceding the rows of the component, transform E into 2^{m′}E_{a}, where m′ is the number of components preceding the given one. If the first row of its appearance has odd (resp. even) number, then a = (1, 1, …, 1, ∗, …, ∗) (resp. a = (−1, 1, …, 1, ∗, …, ∗) ) and the cardinality of displayed ±1’s is the length of the first row of μ supporting the component, the count including the box ×. In turn, the operators of rows supporting the component transform E_{a} to 2E_{a′} for some a′. The multiplicity 2 comes from the highest leftmost box of the component; the operators of all remaining boxes give the multiplicity 1. If such highest leftmost box lies in the h-th column where h < n, then one gets the multiplicity 2 by applying ∂_{h} to E_{b} where b = (…, −1, 1, …), the displayed entries being in (h + 1)-th and h-th places. If the component is of the form (5), then we get the multiplicity 2 by applying ∂_{n} to E_{c} where c = (1, 1, …). This proves the proposition. □

We summarize the results of this section in the following theorem (observe that the connected components of D^{∘}_{μ} \ D are in one-to-one correspondence with the connected components of D_{μ^{−}} \ D_{λ^{−}}):

Theorem 3.15 Let λ ⊂ μ ⊂ ρ(n) be strict partitions. Then for D ⊂ D^{∘}_{μ}, one has ∂_{μ}^{D}(E) ≠ 0 iff D_{μ^{−}} \ D_{λ^{−}} is a horizontal strip (in particular l^{∘}(μ) <

l^{∘}(λ) + 1), and D is obtained by the maximal deformation of D^{∘}_{λ} ⊂ D^{∘}_{μ}. In this case, ∂_{μ}^{D}(E) = 2^{m}, where m is the number of connected components of D_{μ^{−}} \ D_{λ^{−}}.

4 A group-theoretic approach to Schubert calculus for CS_{n}.

We first introduce some notation. Recall that H = SO(2n, C) and B ⊂ H is a Borel subgroup of H. We denote by P - the maximal parabolic subgroup of H containing B and corresponding to the subset Σ of simple roots minus the right end root, by F – an “isotropic” orthogonal flag manifold H/B, and by G – the orthogonal Grassmannian H/P.

Moreover, the Schubert variety X_{w}, w ∈ W, is defined as the closure of the Schubert cell B^{−}wB/B in H/B (B^{−} is the opposite Borel subgroup to B). We record the following well-known result:

Lemma 4.1 X_{w} is a (closed) subvariety of H/B of (complex) codimension l(w).

Let X = {x_{1}, …, x_{n}} be a sequence of variables. For brevity, we denote also by the symbol X_{w} the class of the variety X_{w} in H^{2l(w)}(F) . Let α ⊂ ρ(n − 1) be a strict partition and put λ := α^{+}; one has X_{w_{λ}} ∈ H^{2|α|}(F). In fact, X_{w_{λ}} belongs already to H^{2|α|}(G) ⊂ H^{2|α|}(F). Let us denote this element in H^{2|α|}(G) (as well as the representing it Schubert variety in G), by σ_{α}.

There exists a surjective ring homomorphism c : Z[1/2][X] → H^{∗}(F) (called the Borel characteristic map) such that for a homogeneous f ∈ Z[X] one has

c(f) = ∑_{l(w)=deg f} ∂_{w}(f) X_{w} .  (6)

(The original Borel’s definition [Bo] of the characteristic map was different; this description comes from the work of Bernstein-Gelfand-Gelfand [BGG] and Demazure [De].)

Note (cf. e.g. [Bo]) that the ring H^{∗}(G) can be identified algebraically as

H^{∗}(G) = Z[X]^{S_{n}}/(e_{i}(X^{2}), i = 1, …, n − 1; x_{1}x_{2} ⋯ x_{n}) ,

where X^{2} = (x_{1}^{2}, …, x_{n}^{2}).

We have also another identification of H^{∗}(G) stemming from [Du1] and [P1]:

Lemma 4.2 Let S be the tautological rank n subbundle on G. The Chern classes c_{i}(S) are all divisible by 2, and one has the identification σ_{i} = \frac{1}{2}c_{i}(S^{∗}) for i = 1, …, n − 1. Moreover,

H^{∗}(G) = Z[σ_{1}, …, σ_{n−1}]/(R_{i}, 1 ≤ i ≤ n − 1) ,

where, with the convention σ_{i} = 0 for k > n − 1, the relations R_{i} are given by

R_{i} = σ_{i}^{2} − 2σ_{i−1}σ_{i+1} + 2σ_{i−2}σ_{i+2} − ⋯ + 2(−1)^{i−1}σ_{1}σ_{2i−1} + (−1)^{i}σ_{2i} .

It turns out that after restriction to Z[X]^{S_{n}} ⊗ Z[1/2], the map c goes onto H^{∗}(G), and we have the following fact. Let, from now on, e_{r} = e_{r}(X) denote the r-th elementary symmetric function in X = {x_{1}, …, x_{n}}.

Lemma 4.3 For every r = 1, …, n − 1, one has c(e_{r}) = 2σ_{r}.

Proof. We have

∂_{n−r} ⋯ ∂_{n−2}∂_{n}(e_{r}) = 2 .

Any other divided difference operator of degree r applied to e_{r} gives 0. This implies the assertion. □

For a strict partition α ⊂ ρ(n − 1), we choose a homogeneous f_{α} ∈ Z[1/2][X] such that c(f_{α}) = σ_{α}. Then, for w ∈ W^{∗} with l(w) = l(α), one has ∂_{w}(f_{α}) ≠ 0 iff w = w_{λ} and ∂_{w_{λ}}(f_{α}) = 1 for λ = α^{+}. We want to find the coefficients d_{β} in the expansion

c(f_{α} · e_{r}) = ∑ d_{β}σ_{β} .  (7)

Proposition 4.4 In the above notation, setting μ := β^{+}, one has

d_{β} = ∑ ∂_{μ}^{D}(e_{r}),

where the sum is over all D ⊂ D^{∘}_{μ} such that r_{D} ∈ R(w_{λ}) (here, λ = α^{+}).

Proof. We have d_{β} = ∂_{w_{μ}}(f_{α} ·e_{r}), and ∂_{w_{μ}} = ∂_{μ}^{∅}. The integer d_{β} = ∂_{μ}^{∅}(f_{α} ·e_{r}) is computed by a consecutive application of the Leibniz-type formula (1): we apply only the ∂_{i}’s (and the identity operators) to f_{α}, and both the s_{i}’s and ∂_{i}’s to the factor e_{r}. We get

d_{β} = ∑ ∂_{r_{D}}(f_{α}) · ∂_{μ}^{D}(e_{r}) ,

the sum over all D ⊂ D^{∘}_{μ}. The summand corresponding to a subset D ⊂ D^{∘}_{μ} is not zero only if #D = deg f_{α} and #(D_{μ} \ D) = r. By the choice of f_{α}, ∂_{r_{D}}(f_{α}) = 0 if r_{D} ∉ R(w_{λ}), and equals 1 if r_{D} ∈ R(w_{λ}), and thus we get the desired equality. □

Remark 4.5 This use of an iterated Leibniz-type formula to compute the multiplicities d_{β} stems from a series of papers of Ratajski and the second author (cf. [P2]). It was also known to Kostant and Kumar – see [KK].

Combining this proposition with Theorem 3.15, and taking into account Lemma 4.3, we get a group-theoretic proof of the following result (that is refered to as a “Pieri-type formula”):

Theorem 4.6 Let α ⊂ ρ(n − 1) be a strict partition. Then for any 1 ≤ r ≤ n − 1,

σ_{α} · σ_{r} = ∑_{β} 2^{m_{β}} σ_{β} ,

where the sum is over all strict partitions β ⊂ ρ(n − 1) such that D_{β} \ D_{α} is a horizontal strip of length r and m_{β} is the number of connected components of D_{β} \ D_{α} minus 1.

(Cf. also [P1, Theorem 6.17’].)

Example 4.7 Let n = 8. We examine the product σ_{5,3} · σ_{4}:

• • • • • • • •  times  ∘ ∘ ∘ ∘

On the LHS we depict the β’s; on the RHS we display the unique D ⊂ D^{∘}_{μ} such that ∂_{μ}^{D}(E) ≠ 0:

• • • • • • • • ∘ ∘ ∘ ∘  and  × • • • ∘ • • × • • • ∘ ∘ ∘

• • • • • • • • ∘ ∘ ∘ ∘  and  • × • • • • ∘ ∘ × • • • ∘ ∘ × ×

• • • • • • • • ∘ ∘ ∘ ∘  and  • × • • • • ∘ ∘ × • • • ∘ × ∘ ×

• • • • • • • • ∘ ∘ ∘ ∘  and  • × • • • ∘ ∘ × • • • ∘ • ∘ × ×

• • • • • • • • ∘ ∘ ∘ ∘  and  • × • • • • ∘ × • • • ∘ ∘ × ∘ ×

• • • • • • • • ∘ ∘ ∘ ∘  and  • × • • • • × • ∘ ∘ ∘ ∘ × • • ×

Thus we get:

σ_{5,3} · σ_{4} = σ_{7,5} + 4σ_{7,4,1} + 2σ_{7,3,2} + 2σ_{6,5,1} + 4σ_{6,4,2} + σ_{5,4,3} .

A fundamental invariant of a projective variety X ⊂ P_{C}^{N} is its degree, defined by

deg X = ∫_{X} ω_{X}^{n} ,

where n = dim_{C} X and ω_{X} is the restriction of the standard Kähler form on P^{N} to X. The importance of this invariant is seen from its various interpretations [GH, p.171]:

(i) The number deg X equals to the number of intersection points of X with a general linear subspace in P^{N} of complementary dimension.

(ii) The number n! deg X agrees with the volume of X . It is well known that σ_{1} is the generator of Pic(G) and also σ_{1} is the

Kähler class of G.

So by (i),

deg(σ_{α}) = σ_{α} · σ_{1}^{n(n−1)/2−|α|} .

We invoke Schur P -functions whose definition reads:

1) For a nonnegative integer i, P_{i} := ∑ s_{λ}, where the sum is over all hook partitions λ of i, and s_{λ} denotes the corresponding Schur S-function (cf., e.g., [Ma]).

2) For integers i > j > 0,

P_{(i,j)} := P_{i}P_{j} + 2 ∑_{1≤q≤i−1}(−1)^{q}P_{j+q}P_{i−q} + (−1)^{i+j}P_{i+j} .

3) For a strict partition λ = (λ_{1}, …, λ_{k}) written with an even k (by putting λ_{k} = 0 if necessary),

P_{(λ_{1},...,λ_{k})} := P f[P_{(λ_{p},λ_{q})}]_{1≤p<q≤k} .

(See [S], [Ma], [HH], and [P1] for more on Schur P -functions.)

Comparing the Pieri-type formula for P -functions, extracted in [P1] from [Mo], with Theorem 4.6, we get that deg σ_{α} is the coefficient of P_{ρ(n−1)} in

P_{α} · P_{1}^{n(n−1)/2−|α|} ,

or the coefficient of Pᾱ in

P_{1}^{n(n−1)/2−|α|} = P_{1}^{|ᾱ|} .

Here ᾱ is the partition whose part complement the parts of α in {1, …, n−1}. We define for a partition γ = (γ_{1}, γ_{2}, …),

g^{γ} = \frac{|γ|!}{γ!} ∏_{i<j} \frac{γ_{i} − γ_{j}}{γ_{i} + γ_{j}} ,  (8)

where γ! = γ_{1}!γ_{2}! · · ·.

Proposition 4.8 One has deg(σ_{α}) = g^{γ} for γ = ᾱ.

Remark 4.9 Certain special cases of this formula were obtained by Hiller [Hi]. Some related computations were performed by Tamvakis [Ta2] in the context of heights of homogeneous spaces in arithmetic intersection theory.

The proposition follows from the following lemma due essentially to Schur.

Lemma 4.10 One has

P_{1}^{k} = ∑ g^{γ}P_{γ} ,

the sum over strict partitions γ of k.

Proof. Here is a quick proof of the lemma invoking the following results from [Ma]: combine [Ma, Ex.III.8.1 p.259; Formula III.7.1 p.246; and Ex.III.8.12 p. 266]. □

Remark 4.11 As a matter of fact the key in the original Schur’s calculation [S] (see also [Ma, p.267] and in more detail [HH]) is the proof of the following equality:

g^{γ} = ∑ g^{γ^{(i)}} ,  (9)

where γ^{(i)} runs over strict partitions obtained from γ by subtracting 1 from a part of the partition γ. (The fact that this last equation implies the assertion of the lemma follows directly from the Pieri-type formula for P -functions.) The original argument rests on the expansion into partial fractions of the function

(2t − 1) ∏_{i} \frac{(t + γ_{i})(t − γ_{i} − 1)}{(t + γ_{i} − 1)(t − γ_{i})} .

For those who prefer Lagrange interpolation to the expansion into partial fractions, we sketch another proof of the lemma of Schur.

We use the following specialization. In the notation of [Ma], we put p_{1} = 1/2 and p_{i} = 0 for i ≥ 2 (in the notation of [L] this reads: ψ_{1} = 1/2 and ψ_{i} = 0 for i ≥ 2). To work with this specialization it is handy to define Schur Q-functions by Q_{λ} = 2^{l(λ)}P_{λ}, where λ is a strict partition. Under this specialization, Q_{i} = 1/i!. Denote the resulting under this specialization Q_{λ} by the same symbol Q_{λ} for brevity. The following equality was proved in [DP]:

Q_{λ} = \frac{1}{λ!} ∏_{i<j} \frac{λ_{i} − λ_{j}}{λ_{i} + λ_{j}} .  (10)

Now, we proceed as follows. By simple manipulations with Schur Q-functions, Equation (9) can be rewritten in the following form:

|γ|Q_{γ} = ∑_{i=1}^{l}(−1)^{i}Q_{γ_{i}−1}Q_{γ_{1},...,γ_{i−1},γ_{i+1},...,γ_{l}} ,  (11)

where γ = (γ_{1} > ⋯ > γ_{l} > 0). This is a formal identity in the γ’s, so we can assume that the γ’s are indeterminates and the last equation (to be proven!) holds in the field of rational functions Z(γ_{1}, …, γ_{l}). Equation (11) can be rewritten, with the help of (10), as the following equation:

|γ| ∏_{i<j}(γ_{i} − γ_{j}) = ∑_{p}(−1)^{p}γ_{p} ∏_{{i,j≠p; i<j}}(γ_{i} − γ_{j}) ∏_{i≠p}(γ_{p} + γ_{i}) .  (12)

But this is exactly (cf. [L]) the content of the Lagrange interpolation for |γ|. (We note that in this particular case, the formula can be obtained rather easily, by remarking that the RHS of (12) is an alternating function in the γ’s.)

Up to now, we have considered Schubert varieties as purely grouptheoretic objects. We end this section by recalling their interpretation in terms of Schubert-type conditions. This description is a recollection from [LSe] and [P1], and we will need it in the next section.

Let U be a 2n-dimensional vector space endowed with a nondegenerate orthogonal form ξ : U × U → C. Consider

Z = {L ⊂ U : L is maximal isotropic subspace in U} .

This subvariety is canonically embedded in the Grassmannian G_{n}(U). This last variety has Schubert (sub)varieties which are defined w.r.t. to flag

U_{1} ⊂ U_{2} ⊂ ⋯ ⊂ U_{2n} = U

(where dim(U_{i}) = i), in the following way: Given a sequence 1 ≤ i_{1} < ⋯ < i_{n} ≤ 2n, we set

Ω(i_{1}, …, i_{n}) = {L ∈ G_{n}(U) : dim(L ∩ U_{i_{p}}) ≥ p ∀p = 1, …, n} .

One has

dim Ω(i_{1}, …, i_{n}) = i_{1} + ⋯ + i_{n} − n(n + 1)/2 .

It is known that Z has two connected components which are isomorphic to G = H/P. Let v_{1}, …, v_{n}, w_{1}, …, w_{n} be a basis of U such that ξ(v_{i}, v_{j}) = ξ(w_{i}, w_{j}) = 0, ξ(v_{i}, w_{j}) = ξ(w_{j}, v_{i}) = δ_{i,j}. Let V_{i} be the vector space spanned by the first i vectors of the above basis. Then the Schubert varieties in G_{n}(U) (determined by the flag V_{1} ⊂ ⋯ ⊂ V_{2n} = V) which give rise to the Schubert varieties in G (in the sense of [LSe] and [P1]) are indexed by the sequences (i_{1}, …, i_{n}) where i_{p} ≠ 2n + 1 − i_{q} for p, q = 1, …, n, and if k denotes the largest number such that i_{k} ≤ n, then n − k is even. Let us denote by Ω[i_{1}, …, i_{k}] the Schubert variety in G determined (via restriction to G) by this Schubert variety in G_{n}(U), that is:

Ω[i_{1}, …, i_{k}] := {L ∈ G : dim(L ∩ V_{i_{p}}) ≥ p ∀ p = 1, …, k} .

(Instead refering to the flag V_{1} ⊂ ⋯ ⊂ V_{n}, we will also say that this Schubert variety is defined w.r.t. the ordered basis {v_{1}, …, v_{n}}.) One has

dim Ω[i_{1}, …, i_{k}] = i_{1} + ⋯ + i_{k} + n(n − k) − n(n + 1)/2 .

The Schubert classes in H^{∗}(G) determined by these Schubert varieties are related in following way to the Schubert classes σ_{α} considered earlier in this section. For a strict partition α = (α_{1}, …, α_{k}) ⊂ ρ(n − 1), one has σ_{α} = Ω[n − α_{1}, …, n − α_{k}] if n − k is even, and σ_{α} = Ω[n − α_{1}, …, n − α_{k}, n] if n − k is odd.

The corresponding Schubert variety σ_{α} in G can be defined in the following way w.r.t. the above flag V_{1} ⊂ ⋯ ⊂ V_{n}; it is:

{L ∈ G : dim(L ∩ V_{n−α_{p}}) ≥ i ∀ p = 1, …, k & codim_{V_{n}}(L ∩ V_{n}) is even}.

5 Schubert cycles of complex structures on R^{2n}.

We will adopt the following convention. Let R^{2n} be the real Euclidean 2nspace with the standard orthonormal basis {e_{1}, …, e_{2n}}.

The 2n-dimensional complex Euclidean space C^{2n} will be considered as the complexification of R^{2n}; C^{2n} = R^{2n} ⊗_{R} C. Note the following simple facts:

(a) The set {e_{1} ⊗ 1, …, e_{2n} ⊗ 1} is an orthonormal basis for C^{2n}.

(b) If L, K ⊂ R^{2n} are two linear subspaces satisfying dim_{R}(L ∩ K) ≥ i, then their complexifications L^{C}, K^{C} ⊂ C^{2n} satisfy dim_{C}(L^{C} ∩ K^{C}) ≥ i.

(c) Corresponding to an orthogonal decomposition L = L_{1} ⊕ L_{2} of a subspace L ⊂ R^{2n}, one has the orthogonal decomposition L^{C} = L_{1}^{C} ⊕ L_{2}^{C} of L^{C} ⊂ C^{2n}.

(d) An R-linear endomorphism of a subspace L ⊂ R^{2n} induces a C-linear endomorphism of the subspace L^{C} ⊂ C^{2n}.

Let V be an oriented even dimensional real Euclidean space and Iso(V), the group of orientation preserving isometries of V. Consider

CS(V) = {A ∈ Iso(V) : A^{2} = −Id_{V} } .

It is the space of complex structures on V.

If V = R^{2n}, one has the identification Iso(R^{2n}) = SO(2n, R), the special orthogonal group of order 2n, and

CS(R^{2n}) = {A ∈ SO(2n, R) | A^{2} = −I_{2n}} .

Note that if A ∈ CS(R^{n}) then A is a skew-symmetric matrix. Let us abbreviate CS(R^{2n}) by CS_{n} as is common. The space CS_{n} has two connected components which are distinguished by the Pfaffian function

P f : CS_{n} → {±1} .

We write CS_{n} = CS_{n}^{+} ⊔ CS_{n}^{−} with P f(CS_{n}^{±}) = ±1.( The symbol ⊔ denotes the disjoint union.) Both manifolds CS_{n}^{±} are isometric to SO(2n, C)/U(n).

Our goal in this section is to interpret Schubert varieties in SO(2n, C)/U(n) in terms of complex structures. We will give an interpretation, in terms of Schubert varieties, of the following Mahowald-Vassiljev-type formula

H_{p}(CS_{n}) = ⊕_{k=0}^{n}H_{p−k(k−1)}(G_{k}(C^{n})) ,

where G_{k}(C^{n}) is the Grassmannian of all complex k-planes through zero in C^{n} (see [DV] and also [V]).

We will now define Schubert varieties of complex structures. Let us fix a complex structure J_{0} ∈ CS_{n}. By convention, we will denote by CS_{n}^{+} the connected component of CS_{n} that contains J_{0}. We will work here with the component CS_{n}^{+}, leaving to the reader details concerning the other component CS_{n}^{−}. The results about the component CS_{n}^{−} will be summarized in Proposition 5.6.

Let

R^{2n} = L_{1} ⊕ L_{2} ⊕ ⋯ ⊕ L_{n} ,

where dim_{R} L_{i} = 2, be an invariant subspace decomposition of the orthogonal operator J_{0} : R^{2n} → R^{2n}. This yields a flag in R^{2n}

F_{1} ⊂ F_{2} ⊂ ⋯ ⊂ F_{n} = R^{2n} ,  (13)

where F_{i} = L_{1} ⊕ L_{2} ⊕ ⋯ ⊕ L_{i}. Furnishing R^{2n} with the complex structure J_{0}, we get an n-dimensional complex space C^{n} = (R^{2n}, J_{0}). Since each F_{i} is an invariant subspace w.r.t. J_{0}, the flag (13) gives rise to a complex flag

W_{1} ⊂ W_{2} ⊂ ⋯ ⊂ W_{n} = C^{n} ,

where dim_{C} W_{i} = i.

Consider the Grassmannian G_{l}(C^{n}) of all complex l-planes through zero in C^{n}. For a sequence 1 ≤ j_{1} < ⋯ < j_{l} ≤ n, one defines a Schubert variety

Ω(j_{1}, …, j_{l}) = {L ∈ G_{l}(C^{n}) : dim(L ∩ W_{i_{p}}) ≥ p ∀p = 1, …, l} .

One has

dim_{C} Ω(j_{1}, …, j_{l}) = j_{1} + ⋯ + j_{l} − l(l + 1)/2 .

For an even l, we define CS_{n}^{+}(j_{1}, …, j_{l}) as

{A ∈ CS_{n}^{+} : ∃L ∈ Ω(j_{1}, …, j_{l}) s.t. A(L_{R}) = L_{R} & A|L_{R}^{⊥} = J_{0}|L_{R}^{⊥}} , where L_{R}^{⊥} is the orthogonal complement of the real reduction L_{R} of L. This is a closed subvariety in CS_{n}^{+}. One has

dim_{C} CS_{n}^{+}(j_{1}, …, j_{l}) = j_{1} + ⋯ + j_{l} − l .

One verifies easily that the class of the variety CS_{n}^{+}(j_{1}, …, j_{l}) is independent of the choice of J_{0}. Indeed, a path in CS_{n}^{+} joining J_{0} to another J ∈ CS_{n}^{+} yields a one-parameter family of varieties from CS_{n}^{+}(j_{1}, …, j_{l}) attached to J_{0}, to that attached to J.

Define the complex structure J_{n} in the initial basis {e_{1}, …, e_{2n}} by

J_{n} = J_{1} ⊕ J_{1} ⊕ ⋯ ⊕ J_{1} n times ,

where

J_{1} = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) .

Example 5.1 Suppose that J_{0} = J_{n}. For a sequence 1 ≤ i_{1} < ⋯ < i_{k} ≤ n, we put

J(i_{1}, …, i_{k}) = ϵ_{1}J_{1} ⊕ ϵ_{2}J_{1} ⊕ ⋯ ⊕ ϵ_{n}J_{1}

where ϵ_{h} = −1 if h = i_{p} for 1 ≤ p ≤ k Then J(i_{1}, …, i_{k}) ∈ CS_{n}^{+} iff k is even.

We now want to identify the variety CS_{n}^{+}(j_{1}, …, j_{l}) with a suitable Schubert variety Ω[i_{1}, …, i_{k}] in G. To this end, we first describe an imbedding ι : CS_{n} → G_{n}(C^{2n}). Every A ∈ CS_{n} has two eigenvalues ±i (here i is the pure imaginary complex number) with equal multiplicities n. Thus, as an endomorphism of C^{2n}, (cf. (d)), A has the eigensubspace decomposition

C^{2n} = L(A, +) ⊕ L(A, −) where dim L(A, +) = dim L(A, −) = n ,

with

A(v) = iv for all v ∈ L(A, +) and A(v) = −iv for all v ∈ L(A, −) .

The embedding ι : CS_{n} → G_{n}(C^{2n}) defined by A → L(A, +) has as its image the Grassmannian of all isotropic subspaces of C^{2n} w.r.t. the orthogonal form induced by the scalar product.

For an Euclidean complex n-vector bundle p : E → X over a connected space X, let p_{R} : E_{R} → X be the real reduction of p, i.e. E_{x} = p_{R}^{−1}(x), x ∈ X, is the real reduction of the complex n-space p^{−1}(x). We regard a complex structure on p as a bundle isometry J : E_{R} → E_{R} with J^{2} = −id.

Consider the space CS(E) = {J_{x} ∈ CS(E_{x}) : x ∈ X}. The map π : CS(E) → X defined by π(J_{x}) = x, is a fiber bundle projection whose fiber over a point x ∈ X is CS(E_{x}). The set of all sections of π is in one-to-one correspondence with the set of all complex structures on the real 2n-vector bundle p_{R} : E_{R} → X. In particular, a complex structure K : E_{R} → E_{R} gives a section σ : X → CS(E).

Let C^{n} = (R^{2n}, J_{0}), so we can form G_{l}(C^{n}) – the Grassmannian of all complex l-planes through zero in C^{n}. Let R be the tautological complex l-bundle on G_{l}(C^{n}). Of course, J_{0} : R^{2n} → R^{2n} induces a complex structure on R, denoted by the same symbol. Let CS^{+}(R) denote the connected component of CS(R) containing the section σ corresponding to this complex structure J_{0} : R → R. So we have the fibrations

π : CS(R) → G_{l}(C^{n}) or π^{+} : CS^{+}(R) → G_{l}(C^{n}) .

A point in CS(R) consists of a pair (L, J) with L ∈ G_{l}(C^{n}) and J : L_{R} → L_{R} – a complex structure on the real reduction L_{R} of L.

Finally, by furnishing R^{2n} with the complex structure J_{0} = J_{n}, we have an identification C^{n} = (R^{2n}, J_{n}). Let c_{h} = e_{2h−1} for h = 1, …, n, and consider the complex flag

W_{1} ⊂ W_{2} ⊂ ⋯ ⊂ W_{n} = C^{n} ,

where W_{h} = span{c_{1}, …, c_{h}}. Let Ω(j_{1}, …, j_{l}) ⊂ G_{l}(C^{n}) be the Schubert variety defined w.r.t. to this flag; we will also say that this Schubert variety is defined w.r.t. the ordered basis {c_{1}, …, c_{n}} of C^{n}.

Assume that l is even. Let CS^{+}(R; j_{1}, …, j_{l}) ⊂ CS^{+}(R) be the inverse image of Ω(j_{1}, …, j_{l}) under π^{+}. An easy dimension count shows that

dim CS^{+}(R; j_{1}, …, j_{l}) = j_{1} + ⋯ + j_{l} − l .

Consider now the map φ : CS^{+}(R) → CS_{n}^{+} defined on a point (L, J) by

φ((L, J)) = J ⊕ (J_{n}|L_{R}^{⊥}) : R^{2n} = L_{R} ⊕ L_{R}^{⊥} → R^{2n} = L_{R} ⊕ L_{R}^{⊥} .

(Note that for each L ∈ G_{l}(C^{n}), the real reduction L_{R} ⊂ R^{2n}, and hence its orthogonal complement L_{R}^{⊥} ⊂ R^{2n} is an invariant subspace of J_{n}.)

The restriction of φ to the subvariety CS^{+}(R; j_{1}, …, j_{l}) will be denoted by φ_{j}. By construction, φ_{j} maps CS_{n}^{+}(R; j_{1}, …, j_{l}) onto CS_{n}^{+}(j_{1}, …, j_{l}).

On the other hand, consider now the complexification C^{2n} = R^{2n} ⊗_{R} C. Complete the basis {c_{1}, …, c_{n}} of C^{n} to the basis {c_{1}, …, c_{n}, d_{1}, …, d_{n}} of C^{2n} by setting d_{h} = e_{2h} ⊗ 1 for h = 1, …, n.

Pick (L, J) ∈ CS_{n}^{+}(R; j_{1}, …, j_{n}). By using a simple linear algebra, one shows that

ι ∘ φ_{j}((L, J)) ∈ Ω(n + 1 − t_{n−l}, …, n + 1 − t_{1}, n + j_{1}, …, n + j_{l}) ,

where the Schubert variety is defined w.r.t. ordered basis

{c_{n}, …, c_{1}, d_{1}, …, d_{n}} ,

and t_{1} < ⋯ < t_{n−l} is the complement of j_{1} < ⋯ < j_{l} in {1, …, n}.

We point out the following fact which is used in this argument (see, e.g., [HP, p.321]).

Lemma 5.2 The map G_{l}(C^{n}) → G_{n−l}(C^{n}) defined by L → L^{⊥}, sends the Schubert variety Ω(j_{1}, …, j_{l}) ⊂ G_{l}(C^{n}) defined w.r.t. the ordered basis {c_{1}, …, c_{n}} to the Schubert variety Ω(n+1−t_{n−l}, …, n+1−t_{1}) ⊂ G_{n−l}(C^{n}) defined w.r.t. the ordered basis {c_{n}, …, c_{1}}.

Since one has

dim Im(φ_{j}) = dim CS_{n}^{+}(j_{1}, …, j_{n}) = dim Ω[n + 1 − t_{n−l}, …, n + 1 − t_{1}] ,

the image of φ_{j} must be actually equal to Ω[n + 1 − t_{n−l}, …, n + 1 − t_{1}], defined w.r.t. the ordered basis {c_{n}, …, c_{1}}.

Summing up, we have proved the following identification:

Proposition 5.3 Let l be even. Then one has

CS_{n}^{+}(j_{1}, …, j_{l}) = Ω[n + 1 − t_{n−l}, …, n + 1 − t_{1}] ,

where t_{1} < ⋯ < t_{n−l} is the complement of j_{1} < ⋯ < j_{l} in {1, …, n}.

Remark 5.4 The equality in this proposition is a refinement of a result announced by Dynnikov and Veselov in [DV], where some variant of Morse theory was used.

As a consequence of Proposition 4.8, we get:

Corollary 5.5 The degree of CS_{n}^{+}(j_{1}, …, j_{l}) is equal to the number g^{γ}, where γ = (j_{l} − 1, …, j_{1} − 1).

Since, for even l, Ω[n + 1 − t_{n−l}, …, n + 1 − t_{1}] is equal to

{A : ∃K ∈ Ω(j_{1}, …, j_{l}) s.t. A(K_{R}) = K_{R} & A|K_{R}^{⊥} = J_{0}|K_{R}^{⊥}} ,

then, rewriting it for even n − k, Ω[i_{1}, …, i_{k}] is equal to

{A : ∃K ∈ Ω(n+1−s_{n−k}, …, n+1−s_{1}) s.t. A(K_{R}) = K_{R} & A|K_{R}^{⊥} = J_{0}|K_{R}^{⊥}} .

By taking L = K^{⊥}, we can present this Ω[i_{1}, …, i_{k}] as

{A : ∃L ∈ Ω(i_{1}, …, i_{k}) s.t. A(L_{R}) = L_{R} & A|L_{R} = J_{0}|L_{R}} . (14)

This last identification (14) seems to be the most handy for applications. In the following proposition, keeping the notation from this section, we

collect properties of Schubert varieties in CS_{n}^{−}. For an odd l, we define CS_{n}^{−}(j_{1}, …, j_{l}) as

{A ∈ CS_{n}^{−} : ∃L ∈ Ω(j_{1}, …, j_{l}) s.t. A(L_{R}) = L_{R} & A|L_{R}^{⊥} = J_{0}|L_{R}^{⊥}} ,

where L_{R}^{⊥} is the orthogonal complement of the real reduction L_{R} of L.

Proposition 5.6 (i) CS_{n}^{−}(j_{1}, …, j_{l}) is a closed subvariety in CS_{n}^{−} of dimension j_{1} + ⋯ + j_{l} − l. (ii) CS_{n}^{−}(j_{1}, …, j_{l}) can be identified with the restriction to CS_{n}^{−}, properly embedded in G_{n}(C^{2n}), of the Schubert variety

Ω(n + 1 − t_{n−l}, …, n + 1 − t_{1}, n + j_{1}, …, n + j_{l})

in this last Grassmannian. (iii) CS_{n}^{−}(j_{1}, …, j_{l}) is also identified with

{A ∈ CS_{n}^{−} : ∃L ∈ Ω(i_{1}, …, i_{k}) s.t. A(L_{R}) = L_{R} & A|L_{R} = J_{0}|L_{R}} .

(iv) The degree of CS_{n}^{−}(j_{1}, …, j_{l}) is equal to g^{γ}, where γ = (j_{l} −1, …, j_{1}−1).

Example 5.7 We describe the Schubert varieties in CS_{n}^{+} which are divisors. We have different description according to the parity of n. If n is odd, then the divisor σ_{1} = Ω[n − 1] is

{A : ∃L ⊂ W_{n−1} s.t. dim_{C} L = 1, A(L_{R}) = L_{R} & A|L_{R} = J_{0}|L_{R}} .

If n is even, then the divisor σ_{1} = Ω[n − 1, n] is

{A : ∃L ⊂ W_{n} s.t. dim_{C} L = 2, A(L_{R}) = L_{R} & A|L_{R} = J_{0}|L_{R}} .

We will end this paper with some applications. The identification made in (14) allows us to solve enumerative problems about the number of general complex structures satisfying some constrains. To this end, we need the following definition.

Definition 5.8 Let A and B be two orthogonal operators on R^{2n}. A linear subspace L ⊂ R^{2n} is said to be a common k-space of A and B iff

A(L) = B(L) = L , A|L = B|L & dim_{R} L = k .

We will work in CS_{n}^{+}. Let n and 2 ≤ k ≤ n be even integers. Let 1 ≤ i_{1} < ⋯ < i_{k} ≤ n be a sequence of integers. Set d = dim Ω[i_{1}, …, i_{k}]. Suppose that a list {B_{i}}, 0 ≤ i ≤ d, of general complex structures on R^{2n} is given. Then the number of complex structures A ∈ CS_{n}^{+} s.t. A has a common 2k-space from Ω(i_{1}, …, i_{k}) with B_{0}, and A has a common 4-space with any other B_{i} from the list, is equal to deg Ω[i_{1}, …, i_{k}].

As a particular case, we have:

Proposition 5.9 Let n and 2 ≤ k ≤ n be even integers. Suppose that a list {B_{i}} of general complex structures on R^{2n} is given, where

0 ≤ i ≤ 1 + (n − k)(n − k − 1)/2 .

Then the number of complex structures A ∈ CS_{n}^{+} having a common fixed 2k-space (W_{k})_{R} with B_{0} and a common 4-space with any other B_{i} is given by

g^{(n−k−1,n−k−2,...,2,1)} = [(n − k)(n − k − 1)/2]! ∏_{i=1}^{n−k−1} \frac{(i − 1)!}{(2i − 1)!} .

Indeed, this is a restatement of the formula about the degree of

Ω[1, 2, …, k − 1, k] = CS_{n}^{+}(1, 2, …, n − k) ,

which is simplified in this case of a triangular partition, cf. [DP].

Example 5.10 For n = 8 and k = 4, a list of 7 complex structures {B_{0}, B_{1}, B_{2}, B_{3}, B_{4}, B_{5}, B_{6}} is given. There exist exactly 2 complex structures A ∈ CS_{n}^{+} s.t. A has a common fixed 8-space with B_{0} and at least a common 4-space with every B_{i}, where 1 ≤ i ≤ 6.

If n = 10 and k = 4, a list of 16 structures {B_{0}, B_{1}, …, B_{15}} is given. There exist exactly 286 complex structures A ∈ CS_{n}^{+} s.t. A has a common fixed 8-space with B_{0} and at least a common 4-space with every B_{i}, where 1 ≤ i ≤ 15.

Let now n and 2 < k ≤ n be odd integers. Let 1 ≤ i_{1} < ⋯ < i_{k} ≤ n be a sequence of integers. Put d = dim Ω[i_{1}, …, i_{k}]. Suppose that a list {B_{i}}, 0 ≤ i ≤ d, of general complex structures on R^{2n} is given. Then the number of complex structures A ∈ CS_{n}^{+} s.t. A has a common 2k-space from Ω(i_{1}, …, i_{k}) with B_{0}, and A has a common 2-space from (W_{n−1})_{R} with any other B_{i} from the list, is equal to deg Ω[i_{1}, …, i_{k}].

We leave it to the reader to deduce from it a result analogous to the one in the last proposition.

We will give now another example of enumerating complex structures satisfying some constraints and state some conjecture.

For a complex structure B_{0} ∈ CS_{n}, we have an n-dimensional complex space C^{n} = (R^{2n}, B_{0}). Note that for all L ∈ G_{k}(C^{n}), both L_{R} and L_{R}^{⊥} are invariant subspaces of B_{0}. We have then an embedding α : G_{k}(C^{n}) → CS_{n} defined by

α(L) = (B_{0}|L_{R}) ⊕ (−B_{0}|L_{R}^{⊥}) .

Without loss of generality, we can assume that the image of α lies in CS_{n}^{+}.

Let R be the canonical complex k-bundle over G_{k}(C^{n}) and R^{⊥} its orthogonal complement in the trivial complex n-bundle. Let S be the canonical complex n-bundle over CS_{n}^{+}. From the definition of α we have

α^{∗}S = R ⊕ \overline{R^{⊥}} ,

where \overline{R^{⊥}} denotes the complex conjugation of R^{⊥}. We infer that the pullback of the total Chern class of S, α^{∗}(1 + c_{1}(S) + ⋯ + c_{n}(S)), is equal to

(1 + c_{1}(R) + ⋯ + c_{k}(R))(1 − c_{1}(R) + c_{2}(R) − ⋯ + (−1)^{k}c_{k}(R))^{−1} .

From this we get that the induced homomorphism

α^{∗} : H^{∗}(CS_{n}) → H^{∗}(G_{k}(C^{n}))

satisfies α^{∗}(\frac{1}{2}c_{1}(S)) = −α^{∗}(σ_{1}) = c_{1}(R). That is, the embedding α preserves the classes of hyperplane sections, or, the Kähler classes of both varieties.

As a consequence, we get results summarized in the following proposition.

Proposition 5.11 (i) Let n be an even integer. Suppose that 2 ≤ k ≤ n is another integer. Let {B_{i}}, 0 ≤ i ≤ k(n − k), be a list of general complex structures on R^{2n}. Then the number of complex structures A ∈ CS_{n}^{+} s.t. A and B_{0} have a common 2k-space, A and (−B_{0}) have a common 2(n − k)- space, and A and each B_{i}, i ≥ 1 have at least a common 4-space, is equal to the degree of G_{k}(C^{n}). (ii) Let now n be an odd integer. Suppose that 2 ≤ k ≤ n is another integer. Let {B_{i}}, 0 ≤ i ≤ k(n − k), be a list of general complex structures on R^{2n}. Then the number of complex structures A ∈ CS_{n}^{+} s.t. A and B_{0} have a common 2k-space, A and (−B_{0}) have a common 2(n − k)-space, and A and each B_{i}, i ≥ 1 have at least a common 2-space in (W_{n−1})_{R}, is equal to the degree of G_{k}(C^{n}).

(Recall that

deg G_{k}(C^{n}) = \frac{1!2! ⋯ (k − 1)![k(n − k)]!}{(n − k)!(n − k + 1)! ⋯ (n − 1)!} ,

a result which goes back to Schubert (1886).)

It is well known that the Grassmannian G_{k}(C^{n}) is an approximation space for the classifying space BU(k) of all complex k-bundles. On the other hand, the space CS_{n} serves as the classifying space for all complex n-bundles with a trivial real reduction [Du1]. Thus a homotopy classification of continuous maps G_{k}(C^{n}) → CS_{n} may suggest possible interesting operators between these two vector bundle theories.

Let β : G_{k}(C^{n}) → CS_{n} be a continuous map. Since σ_{1} = −\frac{1}{2}c_{1}(S) ∈ H^{∗}(CS_{n}^{±}) = Z and c_{1}(R) ∈ H^{∗}(G_{k}(C^{n})) = Z are the only generators in dimension 2, then the induced map β^{∗} : H^{∗}(CS_{n}^{±}) → H^{∗}(G_{k}(C^{n})) satisfies

β^{∗}(\frac{1}{2}c_{1}(S)) = −β^{∗}(σ_{1}) = m · c_{1}(R)

for some m ∈ Z.

We finish this paper by stating the following conjecture. Conjecture 5.12 If m ≠ 0, then the map β^{∗} : H^{∗}(CS_{n}^{±}) → H^{∗}(G_{k}(C^{n})) is given by

β^{∗}(x) = m^{p}α^{∗}(x) ,

for x ∈ H^{p}(CS_{n}^{±}).

We refer the reader to [Du2] and [Ho] for some background related to this conjecture. Note that this conjecture is purely algebraic.

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H.D.: Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, P.R. of China.

e-mail address: dhb@math.ac.cn

P.P. (on leave from Institute of Mathematics, Polish Academy of Sciences): Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany.

e-mail address: pragacz@impan.gov.pl