Séminaire Lotharingien de Combinatoire 85 (2020), Article B85a

THE A·B·C·Ds OF SCHUBERT CALCULUS

COLLEEN ROBICHAUX, HARSHIT YADAV, AND ALEXANDER YONG

Abstract. We collect Atiyah–Bott Combinatorial Dreams (A·B·C·Ds) in Schubert calculus. One result relates equivariant structure coefficients for two isotropic flag manifolds, with consequences to the thesis of C. Monical. We contextualize using work of N. Bergeron and F. Sottile, of S. Billey and M. Haiman, of P. Pragacz, and of T. Ikeda, L. Mihalcea, and I. Naruse. The relation complements a theorem of A. Kresch and H. Tamvakis in quantum cohomology. Results of A. Buch and V. Ravikumar rule out a similar correspondence in K-theory.

1. Introduction

1.1. Conceptual framework. Each generalized flag variety G/B has finitely many orbits under the left action of the (opposite) Borel subgroup B_{−} of a complex reductive Lie group G. They are indexed by elements w of the Weyl group W ≅ N(T)/T, where T = B ∩ B_{−} is a maximal torus. The Schubert varieties are closures X_{w} of these orbits. The Poincaré duals of the Schubert varieties {σ_{w}}_{w∈W} form a ℤ-linear basis of the cohomology ring H^{⋆}(G/B). The Schubert structure coefficients are nonnegative integers, defined by

σ_{u} ⌣ σ_{v} = ∑_{w∈W} c_{u,v}^{w}σ_{w}.

Geometrically, c_{u,v}^{w} ∈ ℤ_{≥0} counts intersection points of generic translates of three Schubert varieties. The main problem of modern Schubert calculus is to combinatorially explain this positivity. For Grassmannians, this is achieved by the Littlewood–Richardson rule [13].

The title alludes to a principle, traceable to M. Atiyah and R. Bott [6], that equivariant cohomology is a lever on ordinary cohomology. In our case, each X_{w} is T-stable, so it admits a class ξ_{w} in H_{T}^{⋆}(G/B), the T-equivariant cohomology ring of G/B. These classes are a basis for H_{T}^{⋆}(G/B) as a module over the base ring H_{T}^{⋆}(pt). If Δ = {α_{1}, …, α_{r}} are the simple roots of the root system Φ = Φ^{+}∪Φ^{−} associated to our pinning of G, H_{T}^{⋆}(pt) ≅ ℤ[α_{1}, …, α_{r}].

Define the equivariant Schubert structure coefficient C_{u,v}^{w} ∈ H_{T}^{⋆}(pt) by

(1)  ξ_{u} · ξ_{v} = ∑_{w∈W} C_{u,v}^{w} ξ_{w}.

If ℓ(u) + ℓ(v) = ℓ(w), C_{u,v}^{w} = c_{u,v}^{w}. In fact, C_{u,v}^{w} ∈ ℤ_{≥0}[α_{1}, …, α_{r}]; see (III) below. Thus we have a harder positivity problem, with #Δ-many parameters. Does the added complication make the problem simpler? We initiate our systematic exploration of the question.

The inclusion (G/B)^{T} ↪ G/B induces an injective map

(2)  H_{T}^{⋆}(G/B) ↪ H_{T}^{⋆}((G/B)^{T}) ≅ ⨁_{w∈W} ℤ[α_{1}, …, α_{r}].

Thus, each ξ_{w} is identified with a #W-size list of polynomials (ξ_{w}|_{v})_{v∈W}. Multiplication in H_{T}^{⋆}(G/B) is thereby pointwise multiplication of these lists. Moreover, there is a positive combinatorial formula for the equivariant restriction ξ_{w}|_{v} due to H. Andersen. J. Jantzen, and W. Soergel [4], and rediscovered by S. Billey [8].^{1} Let I = s_{\underline{α}_{1}}s_{\underline{α}_{2}} ⋯ s_{\underline{α}_{ℓ(v)}} be a reduced word for v ∈ W, where s_{\underline{α}_{i}} is the reflection through the hyperplane perpendicular to \underline{α}_{i} := α_{j} ∈ Δ (for some j depending on i). Then,

(3)  ξ_{w}|_{v} = ∑_{J⊆I} ∏_{I}(\underline{α}_{i}^{⟨i∈J⟩}s_{\underline{α}_{i}}) · 1;

cf. [22, Theorem 1]. The sum is over subwords J that are reduced words for w. Also, \underline{α}_{i}^{⟨i∈J⟩} means \underline{α}_{i} appears only if i ∈ J. Let us illustrate this formula.

Example 1.1. Fix the reduced word I = s_{1}s_{2}s_{3}s_{2} for v = 2431 ∈ S_{4}. Then for w = 2314,

ξ_{w}|_{v} = α_{1}s_{1}α_{2}s_{2}s_{3}s_{2} · 1 + α_{1}s_{1}s_{2}s_{3}α_{2}s_{2} · 1 = α_{1}s_{1}α_{2} + α_{1}s_{1}s_{2}s_{3}α_{2} = α_{1}(α_{1}+α_{2}) + α_{1}(α_{3}). □

The combination of (3) and (2) provides the combinatorial definition of H_{T}^{⋆}(G/B) we use.

This description of H_{T}^{⋆}(G/B) permits a non-positive linear algebraic computation of C_{u,v}^{w}, see, e.g., [8, Section 6]. From this perspective, the solved combinatorics of equivariant restriction and the open problem of Schubert calculus seem far apart. However, we argue using an idealization that the concepts are closer than first supposed:

Atiyah–Bott Combinatorial Dream (A·B·C·D). A combinatorial (positivity) statement true for equivariant restrictions also holds for Schubert structure coefficients.

As with any maxim, this will not always hold, as illustrated in Section 2.3. However, the principle suggests new ideas. First we begin with retrospective examples to explore the connection between equivariant restrictions and Schubert structure coefficients:

(I) Sometimes combining (3) with basic Coxeter theory realizes an A·B·C·D. Bruhat order ≤ on W is geometrically defined by w ≤ v if X_{w} ⊇ X_{v}. Fix a reduced word I of v. The subword property of Bruhat order states that w ≤ v if and only if there exists a subword J of I that is a reduced word of w. Hence from (3),

(4)  ξ_{w}|_{v} = 0 unless w ≤ v.

Combining (4) and (2), we obtain

(5)  C_{u,v}^{w} = 0 unless u ≤ w and v ≤ w.

(II) The converse of the A·B·C·D, i.e., a combinatorial (positivity) statement true for Schubert structure coefficients also holds for equivariant restrictions, is true. Following [22, Lemma 1], by (4) and (5),

ξ_{v}|_{v} · ξ_{w}|_{v} = C_{v,w}^{v} ξ_{v}|_{v}.

By (3), ξ_{v}|_{v} ≠ 0. Hence

(6)  C_{v,w}^{v} = ξ_{w}|_{v}.

This is a tantalizing clue about an eventual combinatorial rule for C_{u,w}^{v}. More concretely, in [25], (6) implies a recurrence that, with additional combinatorics, proves an equivariant Littlewood–Richardson rule sans symmetric functions.

1Equivariant restriction is part of GKM-theory [15], a subject of extensive investigation; see, e.g., J. Tymoczko’s exposition [39], and the references therein, for an account germane to our discussion. However the case of Schubert varieties is found in work of B. Kostant and S. Kumar [26, 27].

(III) Here is a deep instance ([8], cf. [39, Section 2]). It is a textbook fact (see [18, Section 1.7]) that

(7)  Inv(v^{−1}) := \{α ∈ Φ^{+} : v^{−1}(α) ∈ Φ^{−}\} = \{s_{\underline{α}_{1}}s_{\underline{α}_{2}} ⋯ s_{\underline{α}_{k−1}} \underline{α}_{k} : 1 ≤ k ≤ ℓ(v)\}.

Since each positive root is a positive linear combination of simples, by (7) and (3),

ξ_{w}|_{v} ∈ ℤ_{≥0}[α_{1}, …, α_{r}].

Indeed, D. Peterson conjectured, and W. Graham [16] geometrically proved that

(8)  C_{u,v}^{w} ∈ ℤ_{≥0}[α_{1}, …, α_{r}].

(IV) This is closely related to (III), but is folklore. Since ℓ(v) = ℓ(v^{−1}) = #Inv(v^{−1}), by (7), s_{\underline{α}_{1}}s_{\underline{α}_{2}} ⋯ s_{\underline{α}_{k−1}} \underline{α}_{k} ∈ Φ^{+} are all distinct. Hence from (3), ξ_{w}|_{v} is square-free when expressed in the positive roots. As A. Knutson (private communication) points out, the proof in [16] shows this to be true of C_{u,v}^{w} as well.

(V) For any G/B, there is a recurrence, due to B. Kostant and S. Kumar to compute ξ_{w}|_{v}; it has an analogue to non-positively compute C_{u,v}^{w}, due to A. Knutson. See [23, Theorem 1] and [22, Section 1]. In turn, special cases of Knutson’s recurrence give “descent cycling” relations on the ordinary Schubert structure constants [21]. We also point to recent work of R. Goldin and A. Knutson [14] which generalizes (3) to give another non-positive formula for C_{u,v}^{w}.

1.2. Does the A·B·C·D suggest anything new? Our main instance is of different flavor than (I)–(V). Using an instance of the A·B·C·D, we relate all structure coefficients of one isotropic flag variety to those of another; this has consequences. The results are neither explicit in the literature nor seem well-known. The correspondence generalizes, with a new proof, non-equivariant results of P. Pragacz [34] and of N. Bergeron and F. Sottile [7] (who rely on S. Billey and M. Haiman’s work [9], which in turn generalizes [34]). We emphasize that the correspondence can also be derived from T. Ikeda, L. Mihalcea, and H. Naruse’s paper [19]; see the discussion of Section 3.

Consider the classical groups G = SO_{2n+1} and G = Sp_{2n} of non-simply laced type. These are automorphism groups preserving a non-degenerate bilinear form ⟨·, ·⟩. In the former case it is a symmetric form on W = ℂ^{2n+1} whereas in the latter case it is a skew symmetric form on W = ℂ^{2n}. A subspace V ⊆ W is isotropic if, for all v_{1}, v_{2} ∈ V, ⟨v_{1}, v_{2}⟩ = 0. The maximum dimension of an isotropic space is n. Any flag of isotropic subspaces ⟨0⟩ ⊂ F_{1} ⊂ F_{2} ⊂ ⋯ ⊂ F_{n} extends to a complete flag in W by ⟨0⟩ ⊂ F_{1} ⊂ F_{2} ⊂ ⋯ ⊂ F_{n} ⊆ F_{n}^{⊥} ⊂ F_{n−1}^{⊥} ⊂ ⋯ ⊂ F_{1}^{⊥} ⊂ W, where F_{k}^{⊥} is the orthogonal complement of F_{k}. Then the flag manifolds X = SO_{2n+1}/B and Y = Sp_{2n}/B consist of complete flags of this form.

The root systems for SO_{2n+1} (type B_{n}) and Sp_{2n} (type C_{n}) are rank r = n. Let {β_{1}, …, β_{n}} and {γ_{1}, …, γ_{n}} be a choice of simple roots labeled by their respective Dynkin diagrams

1∘ < 2∘ 3∘· · · · · · n−1 ∘ ∘ n and 1∘ > 2∘ 3∘· · · · · · n−1 ∘ ∘ n

The two root systems share the hyperoctahedral group B_{n} as their common Weyl group. We represent B_{n} as signed permutations of \{1, 2, …, n\}, e.g., \underline{2} 1 \underline{3}. Define

s(w) := #{1 ≤ i ≤ n : w(i) < 0}.

Let \overline{f} ∈ ℤ[β_{1}, β_{2}, …, β_{n}] be f ∈ ℤ[γ_{1}, γ_{2}, …, γ_{n}] with γ_{1} ↦ 2β_{1} and γ_{i} ↦ β_{i} for 1 < i ≤ n.

Theorem 1.2. C_{u,v}^{w}(X) = 2^{s(w)−s(u)−s(v)}\overline{C_{u,v}^{w}(Y)}.

Proof.This equivalence is from the definitions:

ξ_{w}(Y)|_{x} = ∑_{J⊆I} ∏_{I}(\underline{γ}_{i}^{⟨i∈J⟩}s_{\underline{α}_{i}}) · 1 ⟺ \overline{ξ_{w}(Y)|_{x}} = ∑_{J⊆I} 2^{#\{1∈J\}} ∏_{I}(\underline{β}_{i}^{⟨i∈J⟩}s_{\underline{α}_{i}}) · 1.

The Coxeter combinatorics needed is merely this: since J is a reduced word for w, it is true that #{1 ∈ J} = s(w). Therefore,

(9)  \overline{ξ_{w}(Y)|_{x}} = 2^{s(w)} ∑_{J⊆I} ∏_{I}(\underline{β}_{i}^{⟨i∈J⟩}s_{\underline{α}_{i}}) · 1 = 2^{s(w)}ξ_{w}(X)|_{x};

i.e., a “power of two relationship” between the restrictions. Applying (1), (2) and (3) to Y,

ξ_{u}(Y)|_{x} · ξ_{v}(Y)|_{x} = ∑_{w∈B_{n}} C_{u,v}^{w}(Y) ξ_{w}(Y)|_{x} ∀x ∈ B_{n}
⟺ \overline{ξ_{u}(Y)|_{x}} · \overline{ξ_{v}(Y)|_{x}} = ∑_{w∈B_{n}} \overline{C_{u,v}^{w}(Y)} \overline{ξ_{w}(Y)|_{x}} ∀x ∈ B_{n}
⟺ (2^{−s(u)}\overline{ξ_{u}(Y)|_{x}}) · (2^{−s(v)}\overline{ξ_{v}(Y)|_{x}}) = ∑_{w∈B_{n}} 2^{s(w)−s(u)−s(v)}\overline{C_{u,v}^{w}(Y)}(2^{−s(w)}\overline{ξ_{w}(Y)|_{x}}) ∀x ∈ B_{n}
⟺ ξ_{u}(X)|_{x} · ξ_{v}(X)|_{x} = ∑_{w∈B_{n}} 2^{s(w)−s(u)−s(v)}\overline{C_{u,v}^{w}(Y)} ξ_{w}(X)|_{x} ∀x ∈ B_{n} [by (9)].

We are now done by (1), (2) and (3) applied to X, i.e., uniqueness of the equivariant structure coefficients. □

Example 1.3. Consider u = 3 \underline{2} 1, v = \underline{3} \underline{2} 1 and w = \underline{2} \underline{3} 1 in B_{3}. Then s(w)−s(u)−s(v) = −1 and C_{u,v}^{w}(Y) = 2γ_{1}γ_{2}^{2} + 2γ_{1}γ_{2}γ_{3} + 4γ_{2}^{3} + 6γ_{2}^{2}γ_{3} + 2γ_{2}γ_{3}^{2}, so

\overline{C_{u,v}^{w}(Y)} = 4β_{1}β_{2}^{2} + 4β_{1}β_{2}β_{3} + 4β_{2}^{3} + 6β_{2}^{2}β_{3} + 2β_{2}β_{3}^{2}.

We also have

C_{u,v}^{w}(X) = 2β_{1}β_{2}^{2} + 2β_{1}β_{2}β_{3} + 2β_{2}^{3} + 3β_{2}^{2}β_{3} + β_{2}β_{3}^{2}.

Hence C_{u,v}^{w}(X) = 2^{−1}\overline{C_{u,v}^{w}(Y)}, in agreement with Theorem 1.2. □

Since the correspondence of Theorem 1.2 respects Graham-positivity, we obtain the following equivalence.

Corollary 1.4. We have

[β_{1}^{i_{1}} ⋯ β_{n}^{i_{n}}]C_{u,v}^{w}(X) = 0 ⟺ [γ_{1}^{i_{1}} ⋯ γ_{n}^{i_{n}}]C_{u,v}^{w}(Y) = 0.

In particular, C_{u,v}^{w}(X) = 0 ⟺ C_{u,v}^{w}(Y) = 0.

Let X′ = OG(n, 2n + 1) be the maximal orthogonal Grassmannian of n-dimensional subspaces of ℂ^{2n+1} that are isotropic with respect to a nondegenerate symmetric form. Also, let Y′ = LG(n, 2n) be the Lagrangian Grassmannian of n-dimensional subspaces of ℂ^{2n} that are isotropic with respect to a nondegenerate skew symmetric form. A strict partition is an integer partition λ = (λ_{1} > λ_{2} > ⋯ > λ_{ℓ}). The Schubert varieties and their (equivariant) cohomology classes are indexed by such λ with λ_{1} ≤ n and ℓ ≤ n. Let ℓ(λ) be the number of (nonzero) parts of a strict partition λ.

Corollary 1.5 (cf. Conjecture 5.1 of [32]). C_{λ,μ}^{ν}(X′) = 2^{ℓ(ν)−ℓ(λ)−ℓ(μ)}\overline{C_{λ,μ}^{ν}(Y′)}.

Proof.The map X ↠ X′ that forgets all subspaces of a complete flag in X except the nth induces H_{T}(X′) ↪ H_{T}(X) sending Schubert classes to Schubert classes. The image of ξ_{λ}(X′) is ξ_{w_{λ}}(X) where w_{λ} ∈ B_{n} is the unique ascending signed permutation beginning as −λ_{1}, −λ_{2}, …, −λ_{ℓ}, followed by positive integers in increasing order. Therefore, C_{λ,μ}^{ν}(X′) = C_{w_{λ},w_{μ}}^{w_{ν}}(X). Similarly, C_{λ,μ}^{ν}(Y′) = C_{w_{λ},w_{μ}}^{w_{ν}}(Y). Hence, the result follows from Theorem 1.2 since by definition of w_{λ}, ℓ(λ) = s(w_{λ}). □

Example 1.6. Let n = 3 and λ = (3, 2), μ = (2, 1), and ν = (3, 2, 1).^{2} Then ℓ(ν)−ℓ(λ)−ℓ(μ) = −1. Now, C_{λ,μ}^{ν}(Y′) = 3γ_{1}^{2} + 10γ_{1}γ_{2} + 8γ_{2}^{2} + 5γ_{1}γ_{3} + 8γ_{2}γ_{3} + 2γ_{3}^{2}, so

\overline{C_{λ,μ}^{ν}(Y)} = 12β_{1}^{2} + 20β_{1}β_{2} + 8β_{2}^{2} + 10β_{1}β_{3} + 8β_{2}β_{3} + 2β_{3}^{2}.

We also have

C_{λ,μ}^{ν}(X′) = 6β_{1}^{2} + 10β_{1}β_{2} + 4β_{2}^{2} + 5β_{1}β_{3} + 4β_{2}β_{3} + β_{3}^{2},

so C_{λ,μ}^{ν}(X′) = 2^{−1}\overline{C_{λ,μ}^{ν}(Y′)}, agreeing with Corollary 1.5. □

Corollary 1.5 says that the open problems of giving (Graham positive) combinatorial rules to compute C_{λ,μ}^{ν}(X′) and C_{λ,μ}^{ν}(Y′) are equivalent.

Moreover, Corollary 1.5 makes exact a conjecture stated in the thesis of C. Monical [32, Conjecture 5.1]. In that thesis, one also finds [32, Conjecture 5.3], a conjectural recursive list of inequalities characterizing nonzeroness of C_{λ,μ}^{ν}(X′) (and implicitly, C_{λ,μ}^{ν}(Y′)). That conjecture generalizes work of K. Purbhoo and F. Sottile [35]. It is an analogue of D. Anderson, E. Richmond, and A. Yong’s result [3] that extends work of A. Klyachko [20] and of A. Knutson and T. Tao [24] on the eigenvalue problem for sums of Hermitian matrices. Corollary 1.5 implies the following fact.

Corollary 1.7. (cf. [32, Conjecture 5.3]) C. Monical’s inequalities characterize C_{λ,μ}^{ν}(X′) ≠ 0 if and only if they characterize C_{λ,μ}^{ν}(Y′) ≠ 0.

Here is another consequence of Theorem 1.2. C. Li and V. Ravikumar [31] prove equivariant Pieri rules for (submaximal) isotropic Grassmannians of classical type B, C, D. Their type B and C rules are proved by separate geometric analyses. Theorem 1.2 immediately implies a Pieri rule for type C from the type B rule (or vice versa).

The “power of two” relationship between X′ and Y′ does not hold (in any obvious way) in the Grothendieck (K-theory) ring of algebraic vector bundles; see work of A. Buch and V. Ravikumar [11, Examples 4.9, 5.8]. On the other hand, Theorem 1.2 may be compared to the quantum cohomology result of A. Kresch and H. Tamvakis [29, Theorem 6].

2. More examples of realized A·B·C·Ds

2.1. Inclusion of Dynkin diagrams. Suppose we have an inclusion^{3} of (finite) Dynkin diagrams D ↪ E where the nodes 1, 2, …, r(D) of D are sent to the nodes 1^{∘}, 2^{∘}, …, r(D)^{∘} of E, respectively. Let

Δ(D) = {α_{1}, …, α_{r(D)}} and Δ(E) = {β_{1^{∘}}, …, β_{r(D)^{∘}}, β_{(r(D)+1)^{∘}}, … β_{r(E)^{∘}}}.

2Hence w_{λ} = \underline{3} \underline{2} 1 = s_{2}s_{1}s_{3}s_{2}s_{1}, w_{μ} = \underline{2} \underline{1} 3 = s_{1}s_{2}s_{1} and w_{ν} = \underline{3} \underline{2} \underline{1} = s_{1}s_{2}s_{1}s_{3}s_{2}s_{1}.

3a (multi)-graph theoretic injection that respects arrows

Given w ∈ W(D) we can unambiguously define w^{∘} ∈ W(E) by taking a reduced word I for w and replacing s_{α_{i}} with s_{β_{i^{∘}}} to obtain a reduced word I^{∘} for w^{∘}. Let

ψ_{D,E} : ℤ[α_{1}, …, α_{r(D)}] → ℤ[β_{1^{∘}}, …, β_{r(D)^{∘}}]

be defined by α_{i} ↦ β_{i^{∘}}.

Theorem 2.1. ψ_{D,E}(C_{u,v}^{w}(D)) = C_{u^{∘},v^{∘}}^{w^{∘}}(E).

Proof.We start with the restriction version of the statement, i.e.,

Claim 2.2. ψ_{D,E}(ξ_{w}(D)|_{v}) = ξ_{w^{∘}}(E)|_{v^{∘}}.

Proof of Claim 2.2: This is immediate from (3) using I and I^{∘} respectively in computing ξ_{w}(D)|_{v} and ξ_{w^{∘}}(E)|_{v^{∘}}. This is since the inclusion of Dynkin diagrams induces a canonical isomorphism of the root system of D with a subroot system of E that maps α_{i} to β_{i^{∘}}, and a canonical isomorphism of W(D) with the parabolic subgroup W(E)_{D} of W(E) generated by s_{β_{i}^{∘}} for 1 ≤ i ≤ r(D); see, e.g., [18, Section 5.5]. □

Claim 2.3. If w ∈ W(E) − W(E)_{D} and v ∈ W(D) then ξ_{w}(E)|_{v^{∘}} = 0.

Proof of Claim 2.3: Since w ∈ W(E) − W(E)_{D}, by definition any reduced word for w involves a s_{β_{t^{∘}}} for some t > r(D). Fix any reduced word I^{∘} of v^{∘}. Since v^{∘} ∈ W(E)_{D}, I^{∘} does not involve s_{β_{t^{∘}}}. Hence no subword of I^{∘} can be a reduced word for w. Now the claim follows from (3). □

Combining Claims 2.2 and 2.3, we see that, for any u, v, x ∈ W(D),

(10)  ξ(E)_{u^{∘}}|_{x^{∘}} · ξ(E)_{v^{∘}}|_{x^{∘}} = ∑_{w^{∘}∈W(E)_{D}} C_{u^{∘},v^{∘}}^{w^{∘}}(E) ξ(E)_{w^{∘}}|_{x^{∘}} ∀x ∈ W(D).

By Claim 2.2, for all y ∈ W(D),

ξ(E)_{y^{∘}}|_{x^{∘}} ∈ ℤ_{≥0}[β_{1^{∘}}, …, β_{r(D)^{∘}}].

Therefore by this nonnegativity and W. Graham’s theorem (8), it must be that

C_{u^{∘},v^{∘}}^{w^{∘}}(E) ∈ ℤ_{≥0}[β_{1^{∘}}, …, β_{r(D)^{∘}}].

Therefore, it makes sense to apply ψ_{D,E}^{−1} to both sides of (10) to obtain

(11)  ξ(D)_{u}|_{x} · ξ(D)_{v}|_{x} = ∑_{w∈W(D)} ψ_{D,E}^{−1}(C_{u^{∘},v^{∘}}^{w^{∘}}(E)) ξ(D)_{w}|_{x} ∀x ∈ W(D).

We can now conclude as in the proof of Theorem 1.2. By uniqueness of the structure constants, (11) asserts

ψ_{D,E}^{−1}(C_{u^{∘},v^{∘}}^{w^{∘}}(E)) = C_{u,v}^{w}(D).

Apply ψ_{D,E} to both sides to conclude the proof. □

Example 2.4. The Dynkin diagram for F_{4} is 1∘ 2∘ < 3∘ 4∘ . Now, there is an embedding of D = B_{3} into E = F_{4} given by 1 ↦ 1^{∘} = 2, 2 ↦ 2^{∘} = 3, 3 ↦ 3^{∘} = 4. One computes

C_{s_{1}s_{2}s_{1},s_{2}s_{3}s_{1}}^{s_{1}s_{2}s_{3}s_{1}}(B_{3}) = 2β_{1}^{2} + 3β_{1}β_{2} + β_{2}^{2} and C_{s_{2}s_{3}s_{2},s_{3}s_{4}s_{2}}^{s_{2}s_{3}s_{4}s_{2}}(F_{4}) = 2ζ_{2}^{2} + 3ζ_{2}ζ_{3} + ζ_{3}^{2}.

These are equal after β_{i} ↦ ζ_{i+1} for 1 ≤ i ≤ 3, in agreement with Theorem 2.1. □

Besides being computationally useful, Theorem 2.1 is a guiding property in the search for an eventual combinatorial rule for C_{u,v}^{w}. See [37, Section 5.2] for hints of this in the rootsystem uniform (non-equivariant) rule for the special case of minuscule flag varieties.

There are coincidences between types B_{n} and D_{n+1}, since the Dynkin diagram of the former is the “folding” of the Dynkin diagram for the latter:

[figure: 1; 2; ∘; ∘; 3; ∘; 4; ∘; ⋯; ⋯; n; ∘; ∘; n+1]

Example 2.5. C_{s_{1}s_{2}s_{1},s_{1}s_{2}s_{1}}^{s_{1}s_{2}s_{1}}(B_{2}) = β_{1}(2β_{1} + β_{2})(β_{1} + β_{2}). It is natural to compare s_{1}s_{2}s_{1} ∈ W(B_{2}) with s_{1}s_{3}s_{2} ∈ W(D_{3}). Indeed,

C_{s_{1}s_{3}s_{2},s_{1}s_{3}s_{2}}^{s_{1}s_{3}s_{2}}(D_{3}) = δ_{1}(δ_{1} + δ_{2} + δ_{3})(δ_{1} + δ_{3})

equals C_{s_{1}s_{2}s_{1},s_{1}s_{2}s_{1}}^{s_{1}s_{2}s_{1}}(B_{2}) under the “folding substitution” δ_{1}, δ_{2} ↦ β_{1} and δ_{3} ↦ β_{2}. □

Such a substitution gives a correspondence between OG(n, 2n + 1) restrictions and a subset of restrictions of OG(n + 1, 2n + 2) (the maximal isotropic Grassmannian of type D_{n+1}); see [17, Remark 5.7] and the references therein. By using a similar argument as in Theorem 1.2, one obtains a correspondence of structure coefficients. Unfortunately, this correspondence is not true in general, even for restrictions.

Example 2.6. One calculates that

ξ_{s_{2}s_{1}s_{2}s_{3}}|_{s_{2}s_{1}s_{2}s_{3}}(B_{3}) = 4β_{1}^{3}β_{2} + 10β_{1}^{2}β_{2}^{2} + 2β_{1}^{2}β_{2}β_{3} + 8β_{1}β_{2}^{3} + 3β_{1}β_{2}^{2}β_{3} + 2β_{2}^{4} + β_{2}^{3}β_{3}.

By direct search, there is no ξ_{v}|_{v}(D_{4}) which, after the folding substitution δ_{1}, δ_{2} ↦ β_{1}, δ_{3} ↦ β_{2}, δ_{4} ↦ β_{3}, has even the same monomial support as ξ_{s_{2}s_{1}s_{2}s_{3}}|_{s_{2}s_{1}s_{2}s_{3}}(B_{3}). □

2.2. Nonvanishing. The result is known, cf. [8, Corollary 4.5] which credits [26]. We include a proof to be self-contained.

Proposition 2.7. ξ_{w}|_{v} ≠ 0 for all w ≤ v ≤ w_{0}.

Proof.Suppose v ≤ v′ and fix a reduced word I′ for v′. By the subword property of Bruhat order, there is a subword I of I′ which is reduced for v. Any subword J of I that is a reduced word for w is also a subword of I′. Thus, by (3), any monomial appearing in ξ_{w}|_{v} associated to J corresponds to a maybe different monomial (in the positive roots) in ξ_{w}|_{v′}. Now use that (3) says ξ_{w}|_{w} is a nonzero monomial. □

Conjecture 2.8 (Applying the A·B·C·D to Proposition 2.7). Assume C_{u,v}^{w} ≠ 0.

(I) C_{u,s_{α}v}^{w} ≠ 0 when v < s_{α}v ≤ w and α ∈ Δ.

(II) If ℓ(w) < ℓ(u) + ℓ(v) then there exists s_{α} (α ∈ Δ) with s_{α}v < v such that C_{u,s_{α}v}^{w} ≠ 0.

Example 2.9. In Conjecture 2.8, the existential quantification in (II) is needed. In type B_{3},

C_{s_{2}s_{3},s_{1}s_{3}}^{s_{2}s_{1}s_{3}} = β_{2} + β_{3}, but C_{s_{2}s_{3},s_{1}(s_{1}s_{3})}^{s_{2}s_{1}s_{3}} = 0.

Now, C_{s_{2}s_{3},s_{3}(s_{1}s_{3})}^{s_{2}s_{1}s_{3}} = 1, as predicted. □

We exhaustively checked Conjecture 2.8 for A_{4}, B_{3} and G_{2} and for many examples in A_{5}, B_{4} and F_{4}. Conjecture 2.8 holds for Grassmannians, where it plays a key role in [3], which connects [12] to the equivariant structure coefficients. C. Monical’s extension, discussed in Section 1, motivates this conjecture.

Example 2.10. There is no “righthand version” of either part of Conjecture 2.8. For (I),

C_{s_{1}s_{2},s_{1}}^{s_{1}s_{2}s_{1}}(A_{2}) = 1 but C_{s_{1}s_{2},(s_{1})s_{2}}^{s_{1}s_{2}s_{1}}(A_{2}) = 0.

Whereas for (II), C_{s_{1}s_{2},s_{2}s_{1}}^{s_{1}s_{2}s_{1}}(A_{2}) = α_{1} + α_{2} yet C_{s_{1}s_{2},(s_{2}s_{1})s_{1}}^{s_{1}s_{2}s_{1}}(A_{2}) = 0. □

Proposition 2.7 implies that, for the classical types, the decision problem Restriction “ξ_{w}|_{v} ≠ 0?” is in the class P of polynomial time problems.^{4} This is since there is a polynomial time tableau criterion for deciding if w ≤ v for corresponding Weyl groups; see [10, Chapters 2, 8] (here the input size is bounded by a polynomial in r). Applying the A·B·C·D to this claim concerns the decision problem Nonvanishing: “C_{u,v}^{w} ≠ 0?” given input u, v, w ∈ W (in one line notation).

Conjecture 2.11. For each classical Lie type, Nonvanishing ∈ P.

Conjecture 2.11 is highly speculative. That said, it holds for Grassmannians [2]. In our opinion, this conjecture is related to the (testable) Conjecture 2.17 given below.

2.3. Unsuccessful A·B·C·Ds. It is interesting to study situations where the A·B·C·D is (seemingly) not realized. For instance, here is a true statement for restrictions.

Theorem 2.12 (Monotonicity). If w ≤ v ≤ v′ then ξ_{w}|_{v′} − ξ_{w}|_{v} ∈ ℤ_{≥0}[α_{1}, …, α_{r}].

Proof.It suffices to prove this when v′ covers v. Then

ξ_{w} · ξ_{v} = C_{w,v}^{v} ξ_{v} + C_{w,v}^{v′} ξ_{v′} + ∑_{ṽ≥v,ṽ≠v′} Cṽ ; w,v ξṽ.

Restricting the above equation at v′, using (5) and the fact (6) that C_{w,v}^{v} = ξ_{w}|_{v}, we get

(12)  ξ_{w}|_{v′} · ξ_{v}|_{v′} = ξ_{w}|_{v} · ξ_{v}|_{v′} + C_{w,v}^{v′}ξ_{v′}|_{v′}.

By Proposition 2.7, ξ_{v}|_{v′} ≠ 0, therefore,

(13)  ξ_{w}|_{v′} − ξ_{w}|_{v} = \frac{ξ_{v′}|_{v′}}{ξ_{v}|_{v′}} C_{w,v}^{v′}.

Fix a reduced word s_{i_{1}}s_{i_{2}} ⋯ s_{i_{m}} for v′. By the strong exchange property of Bruhat order [18, Section 5.8], there exists a unique 1 ≤ k ≤ m such that s_{i_{1}} ⋯ s_{i_{k−1}}s_{i_{k+1}} ⋯ s_{i_{m}} (s_{i_{k}} omitted) is a reduced word of v. Therefore by (3),

\frac{ξ_{v′}|_{v′}}{ξ_{v}|_{v′}} = s_{i_{1}} ⋯ s_{i_{k−1}} · α_{i_{k}} ∈ Φ^{+}.

Hence, by (8), ξ_{w}|_{v′} − ξ_{w}|_{v} = (s_{i_{1}} ⋯ s_{i_{k−1}} · α_{i_{k}})C_{w,v}^{v′} ∈ ℤ_{≥0}[α_{1}, …, α_{r}], as desired. □

Example 2.13 (Monotonicity counterexample). Thus, it is tempting to conjecture that if u, v, w ∈ W and s_{α} is a simple reflection such that u ≤ us_{α} := u′ and w ≤ ws_{α} := w′ then C_{u′,v}^{w′} − C_{u,v}^{w} ∈ ℤ_{≥0}[α_{1}, …, α_{r}]. In particular, this would imply c_{u′,v}^{w′} ≥ c_{u,v}^{w}. However, that is false in general. For instance in A_{5} if u = 351624, v = 214356, w = 631524 and s = s_{3} c_{u,v}^{w} = c_{351624,214356}^{631524} = 1 but c_{u′,v}^{w′} = c_{356124,214356}^{635124} = 0. □

4For complexity considerations, the exceptional types are ignored since they are finite in number.

A theorem of A. Arabia [5] states:

α divides ξ_{w}|_{s_{α}v} − ξ_{w}|_{v}.

(In general, this is the condition of [15] that describes the image of (2).)

Example 2.14 (Divisibility counterexample). Does α divide C_{s_{α}u,v}^{s_{α}w} − C_{u,v}^{w}? This is false in general. In type A_{3} let u = s_{3}, v = s_{2}s_{3}s_{1} and w = s_{2}s_{3}s_{1}. Then C_{u,v}^{w} = α_{2} + α_{3}. Let s_{α} = s_{1} and hence s_{α}u = s_{1}u = s_{1}s_{3}, s_{α}w = s_{1}w = s_{1}s_{2}s_{3}s_{1}. Now C_{s_{α}u,v}^{s_{α}w} = α_{1} + α_{2}, and thus C_{s_{α}u,v}^{s_{α}w} − C_{u,v}^{w} = α_{1} − α_{3} is neither α-positive nor divisible by α_{1}. □

A number of other simple variations on monotonicity and divisibility are false as well. Can the A·B·C·Ds for monotonicity/divisibility be realized, under a hypothesis?

2.4. Newton polytopes. The Newton polytope of

f = ∑_{(n_{1},...,n_{r})∈ℤ_{≥0}^{r}} c_{n_{1},...,n_{r}} ∏_{j=1}^{r} α_{j}^{n_{j}} ∈ ℝ[α_{1}, …, α_{r}]

is Newton(f) := conv{(n_{1}, …, n_{r}) : c_{n_{1},...,n_{r}} ≠ 0} ⊆ ℝ^{r}.

Proposition 2.15. Let w ∈ W and w ≤ v ≤ v′. Then Newton(ξ_{w}|_{v}) ⊆ Newton(ξ_{w}|_{v′}).

Proof.This is immediate from Theorem 2.12. □

f has saturated Newton polytope (SNP) [33] if c_{n_{1},...,n_{r}} ≠ 0 ⟺ (n_{1}, …, n_{r}) ∈ Newton(f).

Conjecture 2.16. Let v, w ∈ W, then ξ_{w}|_{v} has SNP.

Conjecture 2.17 (A·B·C·D applied to Conjecture 2.16). Let u, v, w ∈ W, then C_{u,v}^{w} has SNP.

We exhaustively checked these conjectures for A_{4}, B_{3}, D_{4}, G_{2} and many examples in A_{6} and B_{4}. A proof of either conjecture for Grassmannians would be interesting.

SNP is connected to computational complexity in [1, Section 1]. We suspect the concrete SNP claim of Conjecture 2.17 to be the combinatorial harbinger of the P assertion of Conjecture 2.11. Let Schubert be the decision problem “(n_{1}, …, n_{r}) ∈ Newton(C_{u,v}^{w})?”, given input u, v, w ∈ W and (n_{1}, …, n_{r}) ∈ ℤ_{≥0}^{r}. It is reasonable to conjecture existence of:

• a combinatorial rule for C_{u,v}^{w} that moreover implies counting C_{u,v}^{w} is a problem in the counting complexity class #P, and

• a halfspace description of Newton(C_{u,v}^{w}) where each individual inequality can be checked in polynomial time (even if there are exponentially many inequalities).

Conjecture 2.17 would then imply Schubert ∈ NP ∩ coNP. Often problems in NP ∩ coNP are in fact in P (see [1, Section 1.2] for a discussion). Schubert ∈ P implies the important case of Conjecture 2.11 for the non-equivariant c_{u,v}^{w} to be true.

3. Comparisons to Schubert polynomial theory

The theory of Schubert polynomials, introduced by A. Lascoux and M.-P. Schützenberger [30], is influential in the conversation of positivity in Schubert calculus.

These polynomials “lift” the Schur polynomials from the ring of symmetric polynomials to the ring of all polynomials. The study of Schur polynomials is backed by an extensive literature on Young tableaux, from which one obtains the Littlewood–Richardson rule. Thus one might hope for an analogous theory for Schubert polynomials; this remains unrealized. For the purposes of our discussion, let us call this the “lifting dream”.

Theorem 1.2 generalizes the identity

c_{u,v}^{w}(X) = 2^{s(w)−s(u)−s(v)}c_{u,v}^{w}(Y).

This seems to have been first stated in [7, (3.2)], who rely on the Schubert polynomials for classical groups of S. Billey and M. Haiman [9]. Similarly, Corollary 1.5 generalizes the equality

(14)  c_{λ,μ}^{ν}(X′) = 2^{ℓ(ν)−ℓ(λ)−ℓ(μ)}c_{λ,μ}^{ν}(Y′),

which is a consequence of P. Pragacz [34, Theorem 6.17] on the Schubert calculus interpretation of the Schur Q- and P -functions. Theorem 1.2 can also be deduced from T. Ikeda, L. Mihalcea, and H. Naruse’s paper [19] who give an equivariant generalization of the polynomials of [9]. Specifically one refers to Section 1.6, and the identity 𝔅_{w} = 2^{s(w)}ℭ_{w}, where 𝔅_{w} and ℭ_{w} are their double Schubert polynomials for types B_{n} and C_{n} respectively. Note however that this identity does not prove Theorem 1.2 without the appropriate change of variables that we describe. Specifically, it is not true that C_{u,v}^{w}(X) = 2^{s(w)−s(u)−s(v)}C_{u,v}^{w}(Y). This subtlety seems to not be explicitly addressed in [19].

Over the past three decades, within algebraic combinatorics, the emphasis has been on the Schubert polynomial rather than the list of many restrictions.^{5} Our proof replaces the effort of the polynomial constructions [34, 9, 19] with the general geometric result (2). This work suggests the A·B·C·D as an alternative to the “lifting dream” and one that opens up some new and testable possibilities.

Is there concrete evidence for preferring one approach to the other? For example, can one give a proof using the A·B·C·D of S. Robinson’s equivariant Pieri rule for GL_{n}/B [36]? Can one give a Schubert polynomial (in this case, factorial Schur polynomial) proof of one or more of the combinatorial rules [25, 28, 38] by giving an equivariant version of Schensted insertion? Based on earlier conversation of the third author with H. Thomas, this latter question seems quite nontrivial.

Acknowledgements

This paper was stimulated by the thesis of Cara Monical [32]; we made use of her related code. We thank the organizers of the Ohio State Schubert calculus conference (May 2018), which facilitated consultation with experts about the conjectures of [32]. Dylan Rupel provided helpful comments about an earlier draft. Sue Tolman explained to us the seminal role of [6] in equivariant symplectic geometry. We would also like to thank the anonymous referees for their helpful comments. AY was partially supported by an NSF grant, a U·I·U·C Campus Research Board grant, and a Simons Collaboration Grant. This material is based upon work of CR supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE – 1746047. This material was also supported by an NSF RTG 1937241 in Combinatorics.

5Not that the two viewpoints are unrelated: in type A for example, one can compute the restrictions as certain specializations of the double Schubert polynomials; see, e.g., [8].

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Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

  Email address: cer2@illinois.edu

Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Current address: Department of Mathematics, Rice University, Houston, TX 77005, USA

  Email address: hy39@rice.edu

Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

  Email address: ayong@illinois.edu