PIERI-TYPE FORMULAS FOR MAXIMAL ISOTROPIC GRASSMANNIANS VIA TRIPLE INTERSECTIONS

FRANK SOTTILE

Abstract. We give an elementary proof of the Pieri-type formula in the cohomology of a Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic vector space. This proof proceeds by explicitly computing a triple intersection of Schubert varieties. The decisive step is an exact description of the intersection of two Schubert varieties, from which the multiplicities (which are powers of 2) in the Pieri-type formula are immediately obvious.

Introduction

The goal of this paper is to give an elementary geometric proof of Pieri-type formulas in the cohomology of Grassmannians of maximal isotropic subspaces of odd orthogonal or symplectic vector spaces. For this, we explicitly compute a triple intersection of Schubert varieties, where one is a special Schubert variety. Previously, Sertöz [16] had studied such triple intersections in orthogonal Grassmannians, but was unable to determine the intersection multiplicities and obtain a formula.

These multiplicities are either 0 or powers of 2. Our proof explains them as the intersection multiplicity of a linear subspace (defining the special Schubert variety) with a collection of quadrics and linear subspaces (determined by the other two Schubert varieties). This is similar to triple intersection proofs of the classical Pieri formula (cf. [7][5, p. 203][4, §9.4]) where the multiplicities (0 or 1) count the number of points in the intersection of linear subspaces. A proof of the Pieri-type formula for classical flag varieties [17] was based upon those ideas. Similarly, the ideas here provide a basis for a proof of Pieri-type formulas in the cohomology of symplectic flag varieties [1].

These Pieri-type formulas are due to Hiller and Boe [6], whose proof used the Chevalley formula [2]. Another proof, using the Leibnitz formula for symplectic and orthogonal divided differences, was given by Pragacz and Ratajski [14]. These formulas also arise in the theory of projective representations of symmetric groups [15, 9] as product formulas for Schur P - and Q-functions, and were first proven in this context by Morris [11]. The connection of Schur P - and Q-functions to geometry was established by Pragacz [12] (see also [13]).

In Section 1, we give the basic definitions and state the Pieri-type formulas in both the orthogonal and symplectic cases, and conclude with an outline of the proof in the orthogonal case. Since there is little difference between the proofs in each case, we only do the orthogonal case in full. In Section 2, we describe the intersection of two Schubert varieties, which we use in Section 3 to complete the proof.

Date: June 20, 1997.

1991 Mathematics Subject Classification. 14M15.

Key words and phrases. isotropic Grassmannian, Pieri’s formula, Schubert varieties.

Research supported in part by NSF grant DMS-90-22140 and NSERC grant OPG0170279.

1. The Grassmannian of maximal isotropic subspaces

Let V be a (2n + 1)-dimensional complex vector space equipped with a nondegenerate symmetric bilinear form β and W a 2n-dimensional complex vector space equipped with a non-degenerate alternating bilinear form, also denoted β. A subspace H of V or of W is isotropic if the restriction of β to H is identically zero. Isotropic subspaces have dimension at most n. The Grassmannian of maximal isotropic subspaces B_{n} or B(V) (respectively C_{n} or C(W)) is the collection of all isotropic n-dimensional subspaces of V (respectively of W). The group So_{2n+1}ℂ = Aut(V, β) acts transitively on B_{n} with the stabilizer P_{0} of a point a maximal parabolic subgroup associated to the short root, hence B_{n} = So_{2n+1}ℂ/P_{0}. Similarly, C_{n} = Sp_{2n}ℂ/P_{0}, where P_{0} is a maximal parabolic associated to the long root.

Both B_{n} and C_{n} are smooth complex manifolds of dimension \binom{n+1}{2}. While they are not isomorphic if n > 1, they have identical Schubert decompositions and Bruhat orders. Another, deeper, connection is discussed in Remark 2.8. We describe the Schubert decomposition. For an integer j, let \overline{ȷ} denote −j. Choose bases \{e\overline{n}, …, e_{n}\} of V and \{f\overline{n}, …, f_{n}\} of W for which:

β(e_{i}, e_{j}) = \{ \begin{matrix}1 & if i = \overline{ȷ} \\ 0 & otherwise \end{matrix} and β(f_{i}, f_{j}) = \{ \begin{matrix}j/|j| & if i = \overline{ȷ} \\ 0 & otherwise \end{matrix} .

For example, β(e_{1}, e_{0}) = β(f\overline{2}, f_{1}) = 0 and β(e_{0}, e_{0}) = β(f\overline{1}, f_{1}) = −β(f_{1}, f\overline{1}) = 1. Schubert varieties are determined by sequences

λ : n ≥ λ_{1} > λ_{2} > ⋯ > λ_{n} ≥ \overline{n}

whose set of absolute values {|λ_{1}|, …, |λ_{n}|} equals {1, 2, …, n}. Let 𝕊𝕐_{n} denote this set of sequences. The Schubert variety X_{λ} of B_{n} is

{H ∈ B_{n} | dim H ∩ ⟨e_{λ_{j}}, …, e_{n}⟩ ≥ j for 1 ≤ j ≤ n}

and the Schubert variety Y_{λ} of C_{n}

{H ∈ C_{n} | dim H ∩ ⟨f_{λ_{j}}, …, f_{n}⟩ ≥ j for 1 ≤ j ≤ n}.

Both X_{λ} and Y_{λ} have codimension |λ| := λ_{1} + ⋯ + λ_{k}, where λ_{k} > 0 > λ_{k+1}. Given λ, μ ∈ 𝕊𝕐_{n}, we see that

X_{μ} ⊃ X_{λ} ⟺ Y_{μ} ⊃ Y_{λ} ⟺ μ_{j} ≤ λ_{j} for 1 ≤ j ≤ n.

Define the Bruhat order by μ ≤ λ if μ_{j} ≤ λ_{j} for 1 ≤ j ≤ n. Note that μ ≤ λ if and only if μ_{j} ≤ λ_{j} for those j with 0 < μ_{j}.

Example 1.1. Suppose n = 4. Then X_{3 2 \overline{1} \overline{4}} consists of those H ∈ B_{4} such that

dim H ∩ ⟨e_{3}, e_{4}⟩ ≥ 1, dim H ∩ ⟨e_{2}, e_{3}, e_{4}⟩ ≥ 2, and dim H ∩ ⟨e\overline{1}, …, e_{4}⟩ ≥ 3.

Define P_{λ} := [X_{λ}], the cohomology class Poincaré dual to the fundamental cycle of X_{λ} in the homology of B_{n}. Likewise set Q_{λ} := [Y_{λ}]. Since Schubert varieties are closures of cells from a decomposition into (real) even-dimensional cells, these Schubert classes {P_{λ}}, {Q_{λ}} form bases for integral cohomology:

H^{∗}B_{n} = ⨁_{λ} P_{λ} · ℤ and H^{∗}C_{n} = ⨁_{λ} Q_{λ} · ℤ.

Each λ ∈ 𝕊𝕐_{n} determines and is determined by its diagram, also denoted λ. The diagram of λ is a left-justified array of |λ| boxes with λ_{j} boxes in the jth row, for λ_{j} > 0. Thus

3 2 \overline{1} \overline{4} ⟷  and  4 2 1 \overline{3} ⟷  .

The Bruhat order corresponds to inclusion of diagrams. Given μ ≤ λ, let λ/μ be their set-theoretic difference. For instance,

4 2 1 \overline{3}/3 2 \overline{1} \overline{4} ⟷  and  3 2 \overline{1} \overline{4}/1 \overline{2} \overline{3} \overline{4} ⟷  .

Two boxes are connected if they share a vertex or an edge; this defines components of λ/μ. We say λ/μ is a skew row if λ_{1} ≥ μ_{1} ≥ λ_{2} ≥ ⋯ ≥ μ_{n} equivalently, if λ/μ has at most one box in each column. Thus 4 2 1 \overline{3}/3 2 \overline{1} \overline{4} is a skew row, but 3 2 \overline{1} \overline{4}/1 \overline{2} \overline{3} \overline{4} is not.

The special Schubert class p_{m} ∈ H^{∗}B_{n} (q_{m} ∈ H^{∗}C_{n}) is the class whose diagram consists of a single row of length m. Hence, p_{2} = P_{2 \overline{1} \overline{3} \overline{4}}. A special Schubert variety X_{K} (Y_{K}) is the collection of all maximal isotropic subspaces which meet a fixed isotropic subspace K nontrivially. If dim K = n + 1 − m, then [X_{K}] = p_{m} and [Y_{K}] = q_{m}.

Theorem 1.2 (Pieri-type Formula). For any μ ∈ 𝕊𝕐_{n} and 1 ≤ m ≤ n,

1. P_{μ} · p_{m} = ∑_{λ/μ skew row} 2^{δ(λ/μ)−1} P_{λ}  and

2. Q_{μ} · q_{m} = ∑_{λ/μ skew row} 2^{ε(λ/μ)} Q_{λ},

where δ(λ/μ) counts the components of the diagram λ/μ and ε(λ/μ) counts the components of λ/μ which do not contain a box in the first column.

Example 1.3. For example,

P_{3 2 \overline{1} \overline{4}} · p_{2} = 2 · P_{4 2 1 \overline{3}} + P_{4 3 \overline{2} \overline{1}}  and
Q_{3 2 \overline{1} \overline{4}} · q_{2} = 2 · Q_{4 2 1 \overline{3}} + 2 · Q_{4 3 \overline{2} \overline{1}}.

For λ, μ, ν ∈ 𝕊𝕐_{n}, there exist integral constants g_{μ,ν}^{λ} and h_{μ,ν}^{λ} defined by the identities

P_{μ} · P_{ν} = ∑_{λ} g_{μ,ν}^{λ} P_{λ} and Q_{μ} · Q_{ν} = ∑_{λ} h_{μ,ν}^{λ} Q_{λ}.

These constants were first given a combinatorial formula by Stembridge [18].

Define λ^{c} by λ_{j}^{c} := \overline{λ_{n+1−j}}. Let [pt] be the class dual to a point. The Schubert basis is self-dual with respect to the intersection pairing: If |λ| = |μ|, then

(1) P_{μ} · P_{λ^{c}} = Q_{μ} · Q_{λ^{c}} = \{ \begin{matrix}[pt] & if λ = μ \\ 0 & otherwise \end{matrix} .

Define the Schubert variety X′_{λ^{c}} to be

\{H ∈ B_{n} | dim H ∩ ⟨e\overline{n}, …, e_{λ_{j}}⟩ ≥ n + 1 − j for 1 ≤ j ≤ n\}.

This is a translate of X_{λ^{c}} by an element of So_{2n+1}ℂ. In a similar fashion, define Y′_{λ^{c}}, a translate of Y_{λ^{c}} by an element of Sp_{2n}ℂ. For any λ, μ, X_{μ} ⋂ X′_{λ^{c}} is generically transverse [10]. This is because if X_{μ} and X′_{λ^{c}} are any Schubert varieties in general position, then there is a basis for V such that these varieties and the quadratic form β are as given.

We see that to establish the Pieri-type formula, it suffices to compute the degree of

X_{μ} ⋂ X′_{λ^{c}} ⋂ X_{K} and Y_{μ} ⋂ Y′_{λ^{c}} ⋂ Y_{K}

where K is a general isotropic (n + 1 − m)-plane and |λ| = |μ| + m.

We only do the (more difficult) orthogonal case of Theorem 1.2 in full, and indicate the differences for the symplectic case. We first determine when X_{μ} ⋂ X′_{λ^{c}} ≠ ∅. Let μ, λ ∈ 𝕊𝕐_{n}. Then, by the definition of Schubert varieties, H ∈ X_{μ} ⋂ X′_{λ^{c}} implies dim H ⋂⟨e_{μ_{j}}, …, e_{λ_{j}}⟩ ≥ 1, for every 1 ≤ j ≤ n. Hence μ ≤ λ is necessary for X_{μ} ⋂ X′_{λ^{c}} to be nonempty. In fact,

X_{μ} ⋂ X′_{λ^{c}} = \{ \begin{matrix}⟨e_{λ_{1}}, …, e_{λ_{n}}⟩ & if λ = μ \\ ∅ & otherwise, \end{matrix}

which establishes (1).

Suppose μ ≤ λ in 𝕊𝕐_{n}. For each component d of λ/μ, let col(d) index the columns of d together with the column just to the left of d, which is 0 if d meets the first column, in that it has a box in the first column. For each component d of λ/μ, define a quadratic form β_{d}:

β_{d} := ∑_{j∈col(d)} x_{j}x\overline{ȷ},

where x\overline{n}, …, x_{n} are coordinates for V dual to the basis e\overline{n}, …, e_{n}. For each fixed point of λ/μ (j such that λ_{j} = μ_{j}), define the linear form α_{j} := x\overline{λ_{j}}. If there is no component meeting the first column, then we say that 0 is a fixed point of λ/μ and define α_{0} := x_{0}. Let Z_{λ/μ} be the common zero locus of these forms α_{j} and β_{d}.

Lemma 1.4. Let μ ≤ λ and H ∈ X_{μ} ⋂ X′_{λ^{c}}. Then H ⊂ Z_{λ/μ}.

Let Q be the isotropic points in V, the zero locus of β. For each 0 ≤ i ≤ n, there is a unique form among the α_{j}, β_{d} in which one (or both) of the coordinates x_{i}, x\overline{ı} appears. Thus β is in the ideal generated by these forms α_{j}, β_{d} and we see that they are dependent on Q. However, if δ = δ(λ/μ) counts the components of λ/μ and φ the number of fixed points, then the collection of φ forms α_{j} and δ − 1 of the forms β_{d} are independent on Q. Moreover, Lemma 2.2 shows that

n + 1 = φ + δ + #columns of λ/μ.

Thus, if m = |λ| − |μ|, then φ + δ − 1 ≤ n − m, with equality only when λ/μ is a skew row. Since Q has dimension 2n, it follows that a general isotropic (n + 1 − m)-plane K meets Z_{λ/μ} only if λ/μ is a skew row. We deduce

Theorem 1.5. Let μ, λ ∈ 𝕊𝕐_{n} and suppose K is a general isotropic (n+1−m)-plane with |μ| + m = |λ|. Then

X_{μ} ⋂ X′_{λ^{c}} ⋂ X_{K} ≠ ∅

only if μ ≤ λ and λ/μ is a skew row.

Under the hypotheses of Theorem 1.5, the forms α_{j} and β_{d} define 2^{δ(λ/μ)−1} isotropic lines in K. Theorem 3.1 asserts that a general isotropic line in Z_{λ/μ} determines a unique H ∈ X_{μ} ⋂ X′_{λ^{c}}, which completes the proof of Theorem 1.2.

Example 1.6. We show that if K ⊂ Q is a general 3-plane, then

# X_{3 2 \overline{1} \overline{4}} ⋂ X′_{(4 2 1 \overline{3})^{c}} ⋂ X_{K} = 2.

Note that 2 is the coefficient of P_{4 2 1 \overline{3}} in the product P_{3 2 \overline{1} \overline{4}} · p_{2} of Example 1.3.

First, the local coordinates for X_{3 2 \overline{1} \overline{4}} ⋂ X′_{(4 2 1 \overline{3})^{c}} described in Lemma 3.3 show that, for any x, z ∈ ℂ, the row span H of the matrix with rows g_{i} and columns e_{j}

e\overline{4} e\overline{3} e\overline{2} e\overline{1} e_{0} e_{1} e_{2} e_{3} e_{4}
g_{1} 0 0 0 0 0 0 0 −x 1
g_{2} 0 0 0 0 0 0 1 0 0
g_{3} 0 0 0 1 2z −2z^{2} 0 0 0
g_{4} x 1 0 0 0 0 0 0 0

is in X_{3 2 \overline{1} \overline{4}} ⋂ X′_{(4 2 1 \overline{3})^{c}}. Suppose K is the row span of the matrix with rows v_{i}

e\overline{4} e\overline{3} e\overline{2} e\overline{1} e_{0} e_{1} e_{2} e_{3} e_{4}
v_{1} 0 1 0 1 0 0 1 0 1
v_{2} 1 1 0 1 2 −2 1 1 −1
v_{3} 0 0 1 0 0 −1 0 0 0

Then K is an isotropic 3-plane, and the forms

β_{0} = 2x\overline{1}x_{1} + x_{0}^{2}
β_{d} = x\overline{4}x_{4} + x\overline{3}x_{3}
α_{2} = x\overline{2}

define the 2 isotropic lines ⟨v_{1}⟩ and ⟨v_{2}⟩ in K. Lastly, for i = 1, 2, there is a unique H_{i} ∈ X_{3 2 \overline{1} \overline{4}} ⋂ X′_{(4 2 1 \overline{3})^{c}} with v_{i} ∈ H_{i}. In these coordinates,

H_{1} : x = z = 0 and H_{2} : x = z = 1.

In the symplectic case, isotropic K are not contained in a quadric Q, the form α_{0} = x_{0} does not arise, only components which do not meet the first column give quadratic forms β_{d}, and the analysis of Lemma 3.2 (2) in Section 3 is (slightly) different.

2. The intersection of two Schubert varieties

We study the intersection X_{μ} ⋂ X′_{λ^{c}} of two Schubert varieties. Our main result, Theorem 2.4, expresses X_{μ} ⋂ X′_{λ^{c}} as a product whose factors correspond to components of λ/μ, and each factor is itself an intersection of two Schubert varieties. These factors are described in Lemmas 2.5 and 2.6, and in Corollary 2.7. These are crucial to the proof of the Pieri-type formula that we complete in Section 3. Also needed is Lemma 2.1, which identifies a particular subspace of H ∩ ⟨e_{1}, …, e_{n}⟩ for H ∈ X_{μ} ⋂ X′_{λ^{c}}.

For Lemma 2.1, we work in the (classical) Grassmannian G_{k}(V^{+}) of k-planes in V^{+} := ⟨e_{1}, …, e_{n}⟩. For basic definitions and results see any of [8, 5, 4]. Schubert subvarieties Ω_{σ}, Ω′_{σ^{c}} of G_{k}(V^{+}) are indexed by partitions σ ∈ 𝕐_{k}, where n−k ≥ σ_{1} ≥ ⋯ ≥ σ_{k} ≥ 0. For σ ∈ 𝕐_{k} define σ^{c} ∈ 𝕐_{k} by σ_{j}^{c} = n − k − σ_{k+1−j}. For σ, τ ∈ 𝕐_{k}, define

Ω_{τ} := {H ∈ G_{k}(V^{+}) | dim H ∩ ⟨e_{k+1−j+τ_{j}}, …, e_{n}⟩ ≥ j, 1 ≤ j ≤ k}
Ω′_{σ^{c}} := {H ∈ G_{k}(V^{+}) | dim H ∩ ⟨e_{1}, …, e_{j+σ_{k+1−j}}⟩ ≥ j, 1 ≤ j ≤ k}.

Let λ, μ ∈ 𝕊𝕐_{n} with μ ≤ λ, and suppose μ_{k} > 0 > μ_{k+1}. Define partitions σ and τ in 𝕐_{k} (which depend upon λ and μ) by

τ := μ_{1} − k ≥ ⋯ ≥ μ_{k} − 1 ≥ 0
σ := λ_{1} − k ≥ ⋯ ≥ λ_{k} − 1 > 0.

Lemma 2.1. Let μ ≤ λ ∈ 𝕊𝕐_{n}, and define σ, τ ∈ 𝕐_{k}, and k as above. If H ∈ X_{μ} ⋂ X′_{λ^{c}}, then H ∩ V^{+} contains a k-plane L ∈ Ω_{τ} ⋂ Ω′_{σ^{c}}.

Proof. Suppose first that H ∈ X_{μ} with dim H ∩ ⟨e_{μ_{j}}, …, e_{n}⟩ = j for j = k and k + 1. Since μ_{k} > 0 > μ_{k+1}, we see that L := H ∩ V^{+} has dimension k. If H ∈ X′_{λ^{c}} in addition, it is an exercise in the definitions to verify that L ∈ Ω_{τ} ⋂ Ω′_{σ^{c}}. The lemma follows as such H are dense in X_{μ}.

The first step towards Theorem 2.4 is the following combinatorial lemma.

Lemma 2.2. Let φ count the fixed points and δ the components of λ/μ. Then

n + 1 = φ + δ + #columns of λ/μ,

and μ_{j} > λ_{j+1} precisely when |μ_{j}| is an empty column of λ/μ.

Proof. Suppose k is a column not meeting λ/μ. Thus, there is no i for which μ_{i} < k ≤ λ_{i}. Let j be the index such that |μ_{j}| = k. If μ_{j} = k, then we must also have λ_{j+1} < k, as μ_{j+1} < k. Either μ_{j} = λ_{j} is a fixed point of λ/μ or else μ_{j} < λ_{j}, so that k is the column immediately to the left of a component d which does not meet the first column.

If μ_{j} = −k, then λ_{j} = −k, for otherwise k = λ_{i} for some i, and for this i we must necessarily have μ_{i} < k, contradicting k being an empty column. This proves the lemma, as 0 is either a fixed point of λ/μ or else λ/μ has a component meeting the first column, but not both.

Let d_{0} be the component of λ/μ meeting the first column (if any). Define mutually orthogonal subspaces V_{φ}, V_{0}, and V_{d}, for d a component of λ/μ not meeting the first column (0 ∉ col(d)) as follows:

V_{φ} := ⟨e_{μ_{j}}, e\overline{μ_{j}} | μ_{j} = λ_{j}⟩,
V_{0} := ⟨e_{0}, e_{k}, e\overline{k} | k ∈ col(d_{0})⟩,
V_{d}^{−} := ⟨e_{k} | k ∈ col(d)⟩,
V_{d}^{+} := ⟨e\overline{k} | k ∈ col(d)⟩,

and set V_{d} := V_{d}^{−} ⊕ V_{d}^{+}. Then

V = V_{φ} ⊕ V_{0} ⊕ ⨁_{0∉col(d)} V_{d}.

For each fixed point μ_{j} = λ_{j} of λ/μ, define the linear form α_{j} := x\overline{μ_{j}}. For each component d of λ/μ, let the quadratic form β_{d} be the restriction of the form β to V_{d}. Composing with the projection of V to V_{d} gives a quadratic form (also written β_{d}) on V. If there is no component meeting the first column, define α_{0} := x_{0} and call 0 a fixed point of λ/μ. If 0 ∉ col(d), then the form β_{d} identifies V_{d}^{+} and V_{d}^{−} as dual vector spaces.

Lemma 2.3. Let H ∈ X_{μ} ⋂ X′_{λ^{c}}. Then 1. H ⋂ V_{φ} = ⟨e_{μ_{j}} | μ_{j} = λ_{j}⟩.

2. dim H ⋂ V_{0} = #col(d_{0}).

3. For all components d of λ/μ which do not meet the first column,

dim H ⋂ V_{d}^{+} = #rows of d,
dim H ⋂ V_{d}^{−} = #col(d) − #rows of d,

and (H ⋂ V_{d}^{−})^{⊥} = H ⋂ V_{d}^{+}.

Proof of Lemma 2.3. Let H ∈ X_{μ} ⋂ X′_{λ^{c}}. Suppose μ_{j} > λ_{j+1} so that |μ_{j}| is an empty column of λ/μ. Then the definitions of Schubert varieties imply

H = H ∩ ⟨e\overline{n}, …, e_{λ_{j+1}}⟩ ⊕ H ∩ ⟨e_{μ_{j}}, …, e_{n}⟩.

Suppose d is a component not meeting the first column. If the rows of d are j, …, k, then

H ∩ V_{d}^{+} = H ∩ ⟨e_{μ_{k}}, …, e_{λ_{j}}⟩
= H ∩ ⟨e\overline{n}, …, e_{λ_{j}}⟩ ∩ ⟨e_{μ_{k}}, …, e_{n}⟩,

and so has dimension at least k − j + 1.

Similarly, if l, …, m are the indices i with \overline{λ_{j}} ≤ μ_{i}, λ_{i} ≤ \overline{μ_{k}}, then H ⋂ V_{d}^{−} has dimension at least l − m + 1. Hence dim V_{d}/2 = #col(d) = k + m − l − j + 2, as λ_{j}, …, λ_{k}, \overline{λ_{l}}, …, \overline{λ_{m}} are the columns of d.

Since H is isotropic, dim H_{d}^{+} + dim H_{d}^{−} ≤ #col(d), which proves the first part of (3). Moreover, H ⋂ V_{d}^{+} = (H ⋂ V_{d}^{−})^{⊥}: Since H is isotropic, we have ⊂, and equality follows by dimension counting.

Similar arguments prove the other statements.

For H ∈ X_{μ} ⋂ X′_{λ^{c}}, define H_{φ} := H ⋂ V_{φ}, H_{0} := H ⋂ V_{0}, H_{d}^{+} := H ⋂ V_{d}^{+}, and H_{d}^{−} := H ⋂ V_{d}^{−}. Then H_{φ} ⊂ V_{φ} is the zero locus of the linear forms α_{j}, H_{0} is isotropic in V_{0}, and, for each component d of λ/μ not meeting the first column, H_{d} := H_{d}^{+} ⊕ H_{d}^{−} is isotropic in V_{d}, which proves Lemma 1.4. Moreover, H is the orthogonal direct sum of H_{φ}, H_{0}, and the H_{d}.

Theorem 2.4. The map

{H_{0} | H ∈ X_{μ} ⋂ X′_{λ^{c}}} × ∏_{0∉col(d)}{H_{d} | H ∈ X_{μ} ⋂ X′_{λ^{c}}} ⟶ X_{μ} ⋂ X′_{λ^{c}}

defined by

(H_{0}, …, H_{d}, …) ⟼ ⟨H_{φ}, H_{0}, …, H_{d}, …⟩

is an isomorphism.

Proof. By the previous discussion, it is an injection. For surjectivity, note that both sides have the same dimension. Indeed, dim X_{μ} ⋂ X′_{λ^{c}} = |λ| − |μ|, the number of boxes in λ/μ. Lemmas 2.5 and 2.6 show that the factors of the domain each have dimension equal to the number of boxes in the corresponding components.

Suppose there is a component, d_{0}, meeting the first column. Let l be the largest column in d_{0}, and define λ(0), μ(0) ∈ 𝕊𝕐_{l} as follows: Let j be the first row of d_{0} so that l = λ_{j}. Then, since d_{0} is a component, for each j ≤ i < j + l − 1, we have λ_{i+1} ≥ μ_{i} and l = \overline{μ_{j+l−1}}. Set

μ(0) := μ_{j} > ⋯ > μ_{j+l−1}
λ(0) := λ_{j} > ⋯ > λ_{j+l−1}

Define λ(0)^{c} by λ(0)_{p}^{c} := \overline{λ(0)_{l+1−p}} = \overline{λ_{j+l−p}}. The following lemma is a straightforward consequence of these definitions.

Lemma 2.5. With the above definitions,

{H_{0} | H ∈ X_{μ} ⋂ X′_{λ^{c}}} = X_{μ(0)} ⋂ X′_{λ(0)^{c}},

as subvarieties of B_{k} ≃ B(V_{0}), and λ(0)/μ(0) has a unique component meeting the first column and no fixed points.

We similarly identify {H_{d} | H ∈ X_{μ} ⋂ X′_{λ^{c}}} as an intersection X_{μ(d)} ∩ X′_{λ(d)^{c}} of Schubert varieties in B_{#col(d)} ≃ B(⟨e_{0}, V_{d}⟩). Let j, …, k be the rows of d and l, …, m be the indices i with \overline{λ_{j}} ≤ μ_{i}, λ_{i} ≤ \overline{μ_{k}}, as in the proof of Lemma 2.3. Let p = #col(d) and define λ(d), μ(d) ∈ 𝕊𝕐_{p} as follows. Set a = μ_{k}, and define

μ(d) := μ_{j} − a + 1 > ⋯ >  1  > μ_{l} + a − 1 > ⋯ > μ_{m} + a − 1
λ(d) := λ_{j} − a + 1 > ⋯ > λ_{k} − a + 1 > λ_{l} + a − 1 > ⋯ > λ_{m} + a − 1

As with Lemma 2.5, the following lemma is straightforward.

Lemma 2.6. With these definitions,

{H_{d} | H ∈ X_{μ} ⋂ X′_{λ^{c}}} ≃ X_{μ(d)} ⋂ X′_{λ(d)^{c}}

as subvarieties of B_{p} ≃ B(⟨e_{0}, V_{d}⟩) and λ(d)/μ(d) has a unique component not meeting the first column and no fixed points.

Suppose now that μ, λ ∈ 𝕊𝕐_{n} where λ/μ has a unique component d not meeting the first column and no fixed points. Suppose λ has k rows. A consequence of Lemma 2.3 is that the map H_{d}^{+} ↦ ⟨H_{d}^{+}, (H_{d}^{+})^{⊥}⟩ gives an isomorphism

{H_{d}^{+} | H ∈ X_{μ} ⋂ X′_{λ^{c}}} ⟶^{∼} X_{μ} ⋂ X′_{λ^{c}}. (2)

The following corollary of Lemma 2.1 identifies the domain.

Corollary 2.7. With μ, λ as above and σ, τ, and k as defined in the paragraph preceding Lemma 2.1, we have:

{H_{d}^{+} | H ∈ X_{μ} ⋂ X′_{λ^{c}}} = Ω_{τ} ⋂ Ω_{σ^{c}},

as subvarieties of G_{k}(V^{+}).

Remark 2.8. The symplectic analogs of Lemma 2.6 and Corollary 2.7, which are identical save for the necessary replacement of Y for X and C_{p} for B_{p}, show an unexpected connection between the geometry of the symplectic and orthogonal Grassmannians. Namely, suppose λ/μ has no component meeting the diagonal. Then the projection map V ↠ W defined by

e_{i} ⟼ \{ \begin{matrix}0 & if i = 0 \\ f_{i} & otherwise \end{matrix}

and its right inverse W ↪ V defined by f_{j} ↦ e_{j} induce isomorphisms

X_{μ} ⋂ X′_{λ^{c}} ⟷^{∼} Y_{μ} ⋂ Y′_{λ^{c}}.

3. Pieri-type intersections of Schubert varieties

Fix λ/μ to be a skew row with |λ| − |μ| = m. Let Z_{λ/μ} ⊂ Q be the zero locus of the forms α_{j} and β_{d} of §2. If λ/μ has δ components, then as a subvariety of Q, Z_{λ/μ} is the generically transverse intersection of the zero loci of the forms α_{j} and any δ −1 of the forms β_{d}. It follows that a general (n + 1 − m)-plane K ⊂ Q meets Z_{λ/μ} in 2^{δ−1} lines. Thus if ⟨v⟩ ⊂ Z_{λ/μ} is a general line, then

#X_{μ} ⋂ X′_{λ^{c}} ⋂ X_{K} = 2^{δ} · #X_{μ} ⋂ X′_{λ^{c}} ⋂ X_{⟨v⟩}.

Theorem 1.2 is a consequence of this observation and the following:

Theorem 3.1. Let λ/μ be a skew row, Z_{λ/μ} be as above, and ⟨v⟩ a general line in Z_{λ/μ}. Then X_{μ} ⋂ X′_{λ^{c}} ⋂ X_{⟨v⟩} is a singleton.

Proof. Let Q_{0} be the cone of isotropic points in V_{0} and Q_{d} the cone of isotropic points in V_{d} for d ≠ d_{0}. Since

Z_{λ/μ} = H_{φ} ⊕ Q_{0} ⊕ ⨁_{0∉col(d)} Q_{d},

we see that a general non-zero vector v in Z_{λ/μ} has the form

v = ∑_{μ_{j}=λ_{j}} a_{j}e_{μ_{j}} + v_{0} + ∑_{0∉col(d)} v_{d},

where a_{j} ∈ ℂ^{×} and v_{0} ∈ Q_{0}, v_{d} ∈ Q_{d} are general vectors.

Thus, if H ∈ X_{μ} ⋂ X′_{λ^{c}} ⋂ X_{⟨v⟩}, we see that v_{0} ∈ H_{0} and v_{d} ∈ H_{d}. By Theorem 2.4, H is determined by H_{0} and the H_{d}, thus it suffices to prove that H_{0} and the H_{d} are uniquely determined. The identifications of Lemmas 2.5 and 2.6 show that this is just the case of the theorem when λ/μ has a single component, which is Lemma 3.2 below.

Lemma 3.2. Suppose λ, μ ∈ 𝕊𝕐_{n} where λ/μ is a skew row with a unique component and no fixed points.

1. If λ/μ does not meet the first column and v ∈ Q_{d} is a general vector, then X_{μ} ⋂ X′_{λ^{c}} ⋂ X_{⟨v⟩} is a singleton.

2. If λ/μ meets the first column and v ∈ Q is general, then X_{μ} ⋂ X′_{λ^{c}} ⋂ X_{⟨v⟩} is a singleton.

Proof of (1). Let v ∈ Q_{d} be a general vector. Since Q_{d} ⊂ V^{+} ⊕ V^{−}, v = v^{+} ⊕ v^{−} with v^{+} ∈ V^{+} and v^{−} ∈ V^{−}. Suppose μ_{k} > 0 > μ_{k+1}. Consider the set

{H^{+} ∈ G_{k}(V^{+}) | v ∈ H^{+} ⊕ (H^{+})^{⊥}} = {H^{+} | v^{+} ∈ H^{+} ⊂ (v^{−})^{⊥}}.

This is a Schubert variety Ω′′_{h(n−k,k)} of G_{k}V^{+}, where h(n − k, k) is the partition of hook shape with a single row of length n − k and a single column of length k.

Under the isomorphisms of (2) and Lemma 2.6, and with the identification of Corollary 2.7, we see that

X_{μ} ⋂ X′_{λ^{c}} ⋂ X_{⟨v⟩} ≃ Ω_{τ} ⋂ Ω′_{σ^{c}} ⋂ Ω′′_{h(n−k,k)},

where σ, τ are as defined in the paragraph preceding Corollary 2.7. For ρ ∈ 𝕐_{k}, let S_{ρ} := [Ω_{ρ}] be the cohomology class Poincaré dual to the fundamental cycle of Ω_{ρ} in H^{∗}G_{k}V^{+}. The multiplicity we wish to compute is

deg(S_{τ} · S_{σ^{c}} · S_{h(n−k,k)}). (3)

By a double application of the classical Pieri’s formula (as S_{h(n−k,k)} = S_{n−k} · S_{1^{k−1}}), we see that (3) is either 1 or 0, depending upon whether or not σ/τ has exactly one box in each diagonal. But this is the case, as the transformation μ, λ ⟼ τ, σ takes columns to diagonals.

Our proof of Lemma 3.2 (2) uses a system of local coordinates for X_{μ} ⋂ X′_{λ^{c}}. Let λ/μ be as in Lemma 3.2 (2), and suppose λ_{k+1} = 1. For y_{2}, …, y_{n}, x_{0}, …, x_{n−1} ∈ ℂ, define vectors g_{j} ∈ V as follows:

g_{j} := \{ \begin{matrix}e_{λ_{j}} + ∑_{i=μ_{j}}^{λ_{j}−1} x_{i} e_{i} & j ≤ k \\ −2x_{0}^{2}e_{1} + 2x_{0}e_{0} + e\overline{1} + ∑_{i=μ_{k+1}}^{\overline{2}} y_{i} e_{i} & j = k + 1 \\ e_{λ_{j}} + ∑_{i=μ_{j}}^{λ_{j}−1} y_{i} e_{i} & j > k + 1 \end{matrix} . (4)

Lemma 3.3. Let λ, μ ∈ 𝕊𝕐_{n} where λ/μ is a skew row meeting the first column with no fixed points, and define τ, σ ∈ 𝕐_{k}, and k as for Lemma 2.1. Then

1. For any x_{1}, …, x_{n−1} ∈ ℂ, we have ⟨g_{1}, …, g_{k}⟩ ∈ Ω_{τ} ⋂ Ω′_{σ^{c}}.

2. For and x_{0}, …, x_{n−1} ∈ ℂ with x\overline{μ_{k+1}}, …, x\overline{μ_{n−1}} ≠ 0, the condition that H := ⟨g_{1}, …, g_{n}⟩ is isotropic determines a unique H ∈ X_{μ} ⋂ X′_{λ^{c}}.

Moreover, these coordinates parameterize dense subsets of the intersections.

Proof. The first statement is immediate from the definitions.

For the second, note that each g_{j} ∈ Q. The conditions that ⟨g_{1}, …, g_{n}⟩ is isotropic are

β(g_{i}, g_{j}) = 0 for i ≤ k < j.

Only n − 1 of these are not identically zero. Indeed, for i ≤ k < j,

β(g_{i}, g_{j}) ≢ 0 ⟺ \{ \begin{matrix}either & \overline{λ_{j}} < μ_{i} < \overline{μ_{j}}, \\ or & μ_{i} < \overline{μ_{j}} < λ_{i}. \end{matrix}

Moreover, if we order the variables y_{2} < ⋯ < y_{n} < x_{0} < ⋯ < x_{n−1}, then, in the lexicographic term order, the leading term of β(g_{i}, g_{j}) for i ≤ k < j is

y_{λ_{i}}  if \overline{λ_{j}} < μ_{i} < \overline{μ_{j}},
y\overline{μ_{j}}x\overline{μ_{j}} if μ_{i} < \overline{μ_{j}} < λ_{i}, or
y_{n} = y\overline{μ_{n}} if i = 1, j = n.

Since \{2, …, n\} = \{λ_{2}, …, λ_{k−1}, \overline{μ_{k}}, …, \overline{μ_{n}}\}, each y_{l} appears as the leading term of a unique β(g_{i}, g_{j}) with i < k ≤ j, thus these n−1 non-trivial equations β(g_{i}, g_{j}) = 0 determine y_{2}, …, y_{n} uniquely.

These coordinates parameterize an n-dimensional subset of X_{μ} ⋂ X′_{λ^{c}}. Since dim(X_{μ} ⋂ X′_{λ^{c}}) = n and X_{μ} ⋂ X′_{λ^{c}} is irreducible [3], this subset is dense, which completes the proof.

Example 3.4. Let λ = 6 5 3 1 \overline{2} \overline{4} and μ = 5 3 1 \overline{2} \overline{4} \overline{6} so k = 3. We display the components of the vectors g_{i} in a matrix

e\overline{6} e\overline{5} e\overline{4} e\overline{3} e\overline{2} e\overline{1} e_{0} e_{1} e_{2} e_{3} e_{4} e_{5} e_{6}
g_{1} 0 0 0 0 0 0 0  0 0 0 0 x_{5} 1
g_{2} 0 0 0 0 0 0 0  0 0 x_{3} x_{4} 1 0
g_{3} 0 0 0 0 0 0 0 x_{1} x_{2} 1 0 0 0
g_{4} 0 0 0 0 y_{2} 1 2x_{0} −2x_{0}^{2} 0 0 0 0 0
g_{5} 0 0 y_{4} y_{3} 1 0 0  0 0 0 0 0 0
g_{6} y_{6} y_{5} 1 0 0 0 0  0 0 0 0 0 0

Then there are 5 non-zero equations β(g_{i}, g_{j}) = 0 with i ≤ 3 < j:

0 = β(g_{3}, g_{4}) = y_{2}x_{2} + x_{1}
0 = β(g_{3}, g_{5}) = y_{3} + x_{2}
0 = β(g_{2}, g_{5}) = y_{4}x_{4} + y_{3}x_{3}
0 = β(g_{2}, g_{6}) = y_{5} + x_{4}
0 = β(g_{1}, g_{6}) = y_{6} + x_{5}y_{5}

Solving, we obtain:

y_{2} = −x_{1}/x_{2}, y_{3} = −x_{2}, y_{4} = −y_{3}x_{3}/x_{4}, y_{5} = −x_{4}, and y_{6} = −x_{5}y_{5}.

Proof of Lemma 3.2 (2). Suppose λ, μ ∈ 𝕊𝕐_{n} where λ/μ is a skew row with a single component meeting the first column and no fixed points. Let v ∈ Q be a general vector and consider the condition that v ∈ H for H ∈ X_{μ} ⋂ X′_{λ^{c}}. Let σ, τ ∈ 𝕐_{k} be defined as in the paragraph preceding Lemma 2.1. We first show that there is a unique L ∈ Ω_{τ} ⋂ Ω_{σ^{c}} with L ⊂ H, and then argue that H is unique.

The conditions on μ and λ imply that μ_{n} = \overline{n} and μ_{j} = λ_{j+1} for j < n. We further suppose that λ_{k+1} = 1, so that the last row of λ/μ has length 1. This is no restriction, as the isomorphism of V defined by e_{j} ↦ e\overline{ȷ} sends X_{μ} ⋂ X′_{λ^{c}} to X_{λ^{c}} ⋂ X′_{(μ^{c})^{c}} and one of λ/μ or μ^{c}/λ^{c} has last row of length 1.

Let v ∈ Q be general. If necessary, scale v so that its e\overline{1}-component is 1. Let 2z be its e_{0}-component, then necessarily its e_{1}-component is −2z^{2}. Let v^{−} ∈ V^{−} be the projection of v to V^{−}. Similarly define v^{+} ∈ V^{+}. Set v′ := v^{+} + 2z^{2}e_{1}, so that β(v^{−}, v′) = 0 and

v = v^{−} + 2z(e_{0} − ze_{1}) + v′.

Let H ∈ X_{μ} ⋂ X′_{λ^{c}}, and suppose that v ∈ H. In the notation of Lemma 2.1, let L ∈ Ω_{τ} ⋂ Ω_{σ^{c}} be a k-plane in H ⋂ V^{+}. If H is general, in that

dim H ⋂⟨e\overline{n}, …, e_{λ_{k+2}}⟩ = dim H ⋂⟨e\overline{n}, …, e_{0}⟩ = n − k − 1,

then ⟨L, e_{1}⟩ is the projection of H to V^{+}. As v ∈ H, we have v^{+} ∈ ⟨L, e_{1}⟩. Since L ⊂ v^{⊥} ⋂ V^{+} = (v^{−})^{⊥}, we see that v′ ∈ L, and hence

v′ ∈ L ⊂ (v^{−})^{⊥}.

As in the proof of part (1), there is a (necessarily unique) such L ∈ Ω_{τ} ⋂ Ω_{σ^{c}} if and only if σ/τ has a unique box in each diagonal. But this is the case, as the transformation μ, λ ⟶ τ, σ takes columns (greater than 1) to diagonals.

To complete the proof, we use the local coordinates for X_{μ} ⋂ X′_{λ^{c}} and Ω_{τ} ⋂ Ω_{σ^{c}} of Lemma 3.3. Since v is general, we may assume that that the k-plane L ∈ Ω_{τ} ⋂ Ω_{σ^{c}} determined by v′ ∈ L ⊂ (v^{−})^{⊥} has non-vanishing coordinates x\overline{μ_{k+1}}, …, x\overline{μ_{n−1}}, so that there is an H ∈ X_{μ} ⋂ X′_{λ^{c}} in this system of coordinates with L = H ∩ V^{+}.

Such an H is determined up to a choice of coordinate x_{0}. The requirement that v ∈ H forces the projection ⟨e\overline{1} + 2x_{0}e_{0}⟩ of H to ⟨e\overline{1}, e_{0}⟩ to contain e\overline{1} + 2ze_{0}, the projection of v to ⟨e\overline{1}, e_{0}⟩. Hence x_{0} = z, and it follows that there is at most one H ∈ X_{μ} ⋂ X′_{λ^{c}} with v ∈ H. Let g_{1}, …, g_{n} be the vectors (4) determined by the coordinates x_{1}, …, x_{n−1} for L with x_{0} = z. We claim v ∈ H := ⟨g_{1}, …, g_{n}⟩.

Indeed, since v′ ∈ L and v^{−} ∈ L^{⊥} = ⟨g_{k+1} − 2z(e_{0} − ze_{1}), g_{k+2}, …, g_{n}⟩, we see there exists α_{1}, …, α_{n} ∈ ℂ with

v^{−} + v′ = α_{1}g_{1} + ⋯ + α_{k+1}(g_{k+1} − 2z(e_{0} − ze_{1})) + ⋯ + α_{n}g_{n}.

We must have α_{k+1} = 1, since the e\overline{1}-component of both v and g_{k+1} is 1. It follows that

v = ∑_{i=1}^{n} α_{i}g_{i} ∈ H.

Remarks.

1. Our desire to give elementary proofs led us to restrict ourselves to the complex numbers. With the appropriate modifications, these same arguments give the same results for Chow groups of these same varieties over any field of characteristic ≠ 2. For example, the appropriate intersection-theoretic constructions and the properness of a general translate provide a substitute for our use of transversality. Then one could argue for the multiplicity of 2^{δ−1} as follows:

If λ, μ ∈ 𝕊𝕐_{n} and K is a linear subspace in Q, then the scheme-theoretic intersection X_{μ} ⋂ X′_{λ^{c}} ⋂ X_{K} is pr_{∗}π^{∗}(K), where

[figure: K ↪ Q; C_{n}; Ξ = {(p, H) | p ∈ H ∈ X_{μ} ⋂ X′_{λ^{c}}}; π; pr]

Then intersection theory on the quadric Q (a homogeneous space) and Kleiman’s Theorem that the intersection with a general translate is proper [10] gives a factor of 2^{δ−1} from the intersection multiplicity of K and the subvariety Z_{λ/μ} of Q consisting of the image of π. The arguments of Section 3 show that π has degree 1 onto its image.

2. Conversely, similar to the proof of Lemma 3.3, we could give local coordinates for any intersection X_{μ} ⋂ X′_{λ^{c}}. Such a description would enable us to establish transversality directly, and to dispense with the intersection theory of the classical Grassmannian. This would work over any field whose characteristic is not 2, but would complicate the arguments we gave.

3. We have not investigated to what extent these methods would work in characteristic 2.

References

[1] N. Bergeron and F. Sottile, Pieri-type formulas for Lagrangian classes in the symplectic flag manifold. in preparation, 1997.

[2] C. Chevalley, Sur les décompositions cellulaires des espaces G/B, in Algebraic Groups and their Generalizations: Classical Methods, American Mathematical Society, 1994, pp. 1–23. Proceedings and Symposia in Pure Mathematics, vol. 56, Part 1.

[3] V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math., 79 (1985), pp. 499–511.

[4] W. Fulton, Young Tableaux, Cambridge University Press, 1996.

[5] P. Griffiths and J. Harris, Principles of Algebraic Geometry, J. Wiley and Sons, 1978.

[6] H. Hiller and B. Boe, Pieri formula for SO_{2n+1}/U_{n} and SP_{n}/U_{n}, Adv. in Math., 62 (1986), pp. 49–67.

[7] W. V. D. Hodge, The intersection formula for a Grassmannian variety, J. Lond. Math. Soc., 17 (1942), pp. 48–64.

[8] W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry, vol. II, Cambridge University Press, 1952.

[9] P. Hoffman and J. Humphreys, Projective representations of symmetric groups, Oxford, 1992.

[10] S. Kleiman, The transversality of a general translate, Comp. Math., 28 (1974), pp. 287–297.

[11] A. Morris, A note on the multiplication of Hall polynomials, J. London Math. Soc., 39 (1964), pp. 481–8.

[12] P. Pragacz, Enumerative geometry of degeneracy loci, Ann. Scient. Éc. Norm. Sup., 4e série, t. 21, (1988), pp. 413–454.

[13] P. Pragacz, Algebro-geometric applications of Schur S-and Q-polynomials, in Topics in Invariant Theory, Séminaire d’Algèbre Dubreil-Malliavin 1989-90, Springer-Verlag, 1991, pp. 130–191.

[14] P. Pragacz and J. Ratajski, Pieri type formula for isotropic Grassmannians; the operator approach, Manuscripta Math., 79 (1993), pp. 127–151.

[15] I. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math., 139 (1911), pp. 155–250.

[16] S. Sertöz, A triple intersection theorem for the varieties SO(n)/P_{d}, Fundamenta Mathematica, 142 (1993), pp. 201–220.

[17] F. Sottile, Pieri’s formula for flag manifolds and Schubert polynomials, Annales de l’Institut Fourier, 46 (1996), pp. 89–110.

[18] J. Stembridge, Shifted tableaux and the projective representations of the symmetric group, Adv. Math., 74 (1989), pp. 87–134.

Department of Mathematics, University of Toronto, 100 St. George Street, Toronto, Ontario M5S 3G3, Canada

E-mail address: sottile@math.toronto.edu