Mathematics, The Chinese University of Hong Kong, Hong Kong and

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International Mathematics Research Notices, Vol. 2012, No. 16, pp. 3706–3722 Advance Access Publication September 4, 2011

doi:10.1093/imrn/rnr157

Classical Aspects of Quantum Cohomology of Generalized Flag Varieties

Naichung Conan Leung

¹

and Changzheng Li

²

1

The Institute of Mathematical Sciences and Department of

Mathematics, The Chinese University of Hong Kong, Hong Kong and

2

School of Mathematics, Korea Institute for Advanced Study, 87 Hoegiro, Dongdaemun-gu, Seoul, Republic of Korea

Correspondence to be sent to: czli@kias.re.kr

We show that various genus zero Gromov–Witten invariants for flag varieties represent- ing different homology classes are indeed the same. In particular, many of them are classical intersection numbers of Schubert cycles.

1 Introduction

A generalized flag variety G/P is the quotient of a simply connected complex simple Lie group G by a parabolic subgroup P of G. The (small) quantum cohomology ring QH^∗(G/P)of G/P is a deformation of the ring structure on H^∗(G/P)by incorporating three-pointed, genus zero Gromov–Witten invariants of G/P. The presentation of the ring structure onQH^∗(G/P)in special cases have been studied by many mathematicians (see, e.g., [4,7,8,13,19,23,26,27] and references, and also an unpublished preprint of D.

Peterson). From the viewpoint of enumerative geometry, it is desirable to have (positive) combinatorial formulas for these Gromov–Witten invariants. Owing to the Peterson–

Woodward comparison formula [28], all these Gromov–Witten invariants for G/P can be recovered from the Gromov–Witten invariants for the special case of a complete flag varietyG/B, where Bis a Borel subgroup.

Received March 13, 2011; Accepted June 24, 2011 Communicated by Prof. Dragos Oprea

at The Chinese University of Hong Kong on August 11, 2013http://imrn.oxfordjournals.org/Downloaded from

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In [25], with the help of the Peterson–Woodward comparison formula, we established a natural filtered algebra structure on QH^∗(G/B). In this article, we use the structures of this filtration to obtain relationships among three-pointed genus zero Gromov–Witten invariants N_u^w,λ_,v for G/B. The Gromov–Witten invariants N_u^w,λ_,v are the structure coefficients of the quantum product

σ^u σ^v=

λ∈H2(G/B,Z),w

N_u^w,λ_,vq_λσ^w

of the Schubert cocyclesσ^uandσ^v in the quantum cohomology QH^∗(G/B). The evaluation ofqat the origin gives us the classical intersection product

σ^u∪σ^v=

w

N_u^w,0_,vσ^w.

LetΔ= {α1, . . . , αn}be a base of simple roots ofGand{α₁^∨, . . . , α^∨_n}be the simple coroots (see Section 2.1 and references therein for more details of the notation). The Weyl group W is a Coxeter group generated by simple reflections {s_α | α∈Δ}. For each α∈Δ, we introduce a map sgn_α:W→ {0,1}defined by sgn_α(w):=1 if(w)−(ws_α) >0, and 0 otherwise. Here :W→Z_≥0 denotes the length function. Let h be the dual of the vector spaceh^∗:= ⊕α∈ΔCαand·,·:h^∗×h→Cbe the natural pairing. Note thatH₂(G/B,Z)can be canonically identified with the coroot lattice Q^∨:=

α∈ΔZα^∨⊂h. We prove the following theorem.

Theorem 1.1. For anyu, v, w∈Wand for anyλ∈Q^∨, we have the following (1) N_u,v^w,λ=0 unless sgn_α(w)+ α, λ ≤sgn_α(u)+sgn_α(v)for allα∈Δ;

(2) suppose sgn_α(w)+ α, λ =sgn_α(u)+sgn_α(v)=2 for someα∈Δ, then

N_u^w,λ_,v =N_us^w,λ−α_α_,v_s_α^∨=

⎧⎨

⎩

N_u^w_,v^s^α_s^,λ−α_α ^∨, if sgn_α(w)=0, N_u,vs^w^s^α^,λ

α , if sgn_α(w)=1.

We obtain nice applications of the above theorem that demonstrate the so-called

“quantum to classical” principle.

The “quantum to classical” principle says that certain three-pointed genus zero Gromov–Witten invariants for a given homogeneous space are classical intersection numbers for a typically different homogeneous space. This phenomenon, probably for the first time, occurred in the proof of the quantum Pieri rule for partial flag varieties of

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type A by Ciocan-Fontanine [8], and later occurred in the elementary proof of the quantum Pieri rule for complex Grassmannians by Buch [2] as well as the works [21,22]

of Kresch and Tamvakis on Lagrangian and orthogonal Grassmannians. The phrase

“quantum to classical principle” was introduced by Chaput and Perrin [7] for the work [3] of Buch, Kresch, and Tamvakis on complex Grassmannians, Lagrangian Grassman- nians, and orthogonal Grassmannians, which says that any three-pointed genus zero Gromov–Witten invariant on a Grassmannian of aforementioned types is equal to a classical intersection number on a partial flag variety of the same Lie type. Recently, this principle has been developed by Buch et al. [4] for isotropic Grassmannians of classical types. For Grassmannians of certain exceptional types, this principle has also been studied by Chaput, Manivel, and Perrin [6,7]. For flag varieties of typeA, there are rele- vant studies by Coskun [10]. For the special case of computing the number of lines in a general complete varietyG/B, this principle has also been studied earlier by the second author and Mihalcea [25]. In addition, we note that this principle for certainK-theoretic Gromov–Witten invariants has been studied by Buch and Mihalcea [5].

Using Theorem 1.1, we not only recover most of the above results on the “quantum to classical” principle, but also get new and interesting results. For instance in a forthcoming paper we can see the applications of Theorem 1.1 in seeking quantum Pieri rules with respect to Chern classes of the dual of the tautological subbundles for Grass- mannians of classical types, which are not covered in [4] in general. In a joint work in progress with H. Duan and X. Zhao, we could also apply Theorem 1.1 to get a presentation of the quantum cohomology ring of a Grassmannian of typeF₄.

For the type A_n case, we note that the Weyl group W is canonically isomor- phic to the permutation group S_n+1 and G/B=Fn+1= {V₁≤ · · · ≤V_n≤Cⁿ⁺¹|dim_CV_j= j,j=1, . . . ,n}. An elementu∈W=S_n+1is called a Grassmannian permutationif there exists 1≤k≤nsuch thatσ^u∈H^∗(Fn+1)comes from the pull-backπ^∗:H^∗(Gr(k,n+1))→ H^∗(Fn+1)induced from the natural projection mapπ:Fn+1→ {Vk≤Cⁿ⁺¹|dimVk=k} = Gr(k,n+1). Equivalently, a Grassmannian permutation u∈W is an element such that all the reduced expressionsu=si1si2· · ·sim, where m=(u), end with the same simple reflectionsk. When written in “one-line” notation, Grassmannian permutations are precisely the permutations with a single descent (see Remark 2.16 for more details). As an application of Theorem 1.1, we have the following theorem.

Theorem 1.2. Letu, v, w∈Sn+1andλ∈Q^∨. Ifuis of Grassmannian type, then there exist v, w∈Sn+1such that

N_u,v^w,λ=N_u^w_,v^,0.

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It is also interesting to investigate the above theorem from the point of view of symmetries onQH^∗(Fn+1), analogous with the cyclic symmetries shown by Postnikov [26]. In addition, the proof of Theorem 1.2 will also describe how to findvandw(easily).

Special cases of Theorem 1.2 enable us to recover the quantum Pieri rule for partial flag varieties of type Aas in [8] and the “quantum to classical” principle for complex Grassmannians as in [3].

Geometrically, the Gromov–Witten invariants N_u^w,λ_,v count the number of stable holomorphic maps from the projective line P¹, or more generally a rational curve, to G/B. In particular, they are all nonnegative. There have been closed formulas/algorithms on the classical intersection product by Kostant and Kumar [20] and Duan [11] and on the quantum product by the authors [24]. Yet sign cancelations are involved in all these formulas/algorithms. For a complete flag variety (of general type), the problem of find- ing a positive formula on either side remains open. The quantum to classical principle, in many situations, helps us to reduce the quantum Schubert calculus to the classical Schubert calculus. WhenG=SL(n+1,C), we note that positive formulas on the classical intersection numbers have been given by Coskun [9].

The proof of Theorem 1.1 uses functorial relationships established by the authors in [25] and it is combinatorial in nature. However, for a special case (ofλ=α^∨) of Theorem 1.1, both a geometric proof of it and a combinatorial proof of its equivariant extension can be found in a joint work in progress of the second author and Mihalcea.

We also wish to see a geometric proof of this theorem in the future.

2 Proofs of Theorems

In this section, we first fix the notation in Section 2.1. Then we prove our first main theorem in Section 2.2. Finally, in Section 2.3 we obtain our second main theorem, as an application of the first main theorem.

2.1 Notations

More details on Lie theory can be found, for example, in [16,17].

Let G be a simply connected complex simple Lie group of ranknand B⊂G be a Borel subgroup. Let Δ= {α1, . . . , αn} ⊂h^∗ be the simple roots and {α₁^∨, . . . , α_n^∨} ⊂h be the simple coroots, wherehis the corresponding Cartan subalgebra of(G,B). LetQ^∨= n

i=1Zα^∨_i and ρ=n

i=1χi∈h^∗. Here χi’s are the fundamental weights, which, for any i,j, satisfyχi, α^∨_j =δi,j with respect to the natural pairing·,·:h^∗×h→C. The Weyl groupWis generated by{s1, . . . ,sn}, where eachsi=s_α_i is a simple reflection onh^∗defined

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bysi(β)=β− β, α_i^∨αi. The root system is given by R=W·Δ=R⁺(−R⁺), where R⁺= R∩_n

i=1Z_≥0αi is the set of positive roots. Each parabolic subgroup P⊃B is in one-to- one correspondence with a subsetΔP⊂Δ. In fact,ΔP is the set of simple roots of a Levi subgroup of P. Let :W→Z_≥0 be the length function, W_P denote the Weyl subgroup generated by{s_α|α∈ΔP}, andW^P denote the subset{w∈W|(w)≤(v),∀v∈wW_P}. Each coset inW/W_P has a unique minimal length representative inW^P.

The (co)homology of a (generalized) flag variety X=G/P has an additive basis of Schubert (co)homology classes indexed by W^P:H_∗(X,Z)=

v∈W^PZσv, H^∗(X,Z)=

u∈W^PZσûwithσû, σv =δu,vfor anyu, v∈W^P [1]. Note that eachσu(resp.σû) is a class in the 2(u)th-(co)homology. In particular, H2(X,Z)=

αi∈Δ\ΔPZσsi can be canonically identified withQ^∨/Q^∨_P, where Q^∨_P:=

αi∈ΔPZα_i^∨. For eachαj∈Δ\ΔP we introduce a formal variableq_α^∨_j₊Q^∨_P. For λP=

αj∈Δ\ΔPajα^∨_j +Q^∨_P∈H2(X,Z)we defineq_λ_P =

αj∈Δ\ΔPq_αâ∨^j j+Q^∨_P. The (small)quantum cohomologyQH^∗(X)=(H^∗(X)⊗Q[q], )ofXis a commutative ring and has aQ[q]-basis of Schubert classesσû=σû⊗1. Thequantum Schubert structure constants N_u,v^w,λ^P for the quantum product

σ^u σ^v=

w∈W^P,λP∈Q^∨/Q^∨_P

N_u^w,λ_,v^Pq_λ_Pσ^w

are three-pointed, genus zero Gromov–Witten invariants given by N_u,v^w,λ^P =

M¯0,3(X,λP)ev^∗₁(σ^u)∪ev^∗₂(σ^v)∪ev^∗₃((σ^w)). Here M¯0,3(X, λP) is the moduli space of stable maps of degree λP∈H₂(X,Z) of three-pointed genus zero curves into X, ev_i:M¯0,3(X, λP)→X is the ith canonical evaluation map, and {(σ^w) | w∈W^P}are the elements in H^∗(X) satisfying

X(σ^w)∪σ^w=δw,w for any w, w∈W^P [14]. Note that N_u^w,λ_,v^P =0 unlessq_λ_P ∈Q[q]. It is a well-known fact that these Gromov–Witten invariants N_u^w,λ_,v^P of the flag varietyXare enumerative, counting the number of certain holomorphic maps from P¹ to X. In particular, they are all nonnegative integers. (Thus, for the special case of a flag varietyX, we can also defineQH^∗(X)overZwhenever we wish.)

In analog with the classical cohomology, there is a natural Z-grading on the quantum cohomolgyQH^∗(X), making it aZ-graded ring:

QH^∗(X)=

n∈Z

⎛

⎝

deg(q_λPσ^w)=n

Qq_λPσ^w

⎞

⎠. (*)

Here the degree ofq_λ_Pσ^w, whereλP=

αj∈Δ\ΔPa_jα^∨_j +Q^∨_P∈H₂(X,Z), is given by

deg(q_λ_Pσ^w)=(w)+

αj∈Δ\ΔP

a_jσsj,c₁(X),

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in which·,·is the natural pairing between homology and cohomology classes, and an explicit description of the first Chern classc₁(X)can be found, for example, in [15]. When P=B, we haveΔP= ∅,Q^∨_P=0, W_P= {1}, andW^P=W. In this case, we simply defineλ= λP andq_j=q_α^∨_j. As a direct consequence of theZ-graded ring structure(∗)ofQH^∗(G/B), for anyu, v, w∈Wand for anyλ∈Q^∨, we have

N_u^w,λ_,v =0 unless(w)+ 2ρ, λ =(u)+(v).

2.2 Proof of Theorem 1.1

This subsection is devoted to the proof of Theorem 1.1. The main arguments are given in Section 2.2.3, based on the results in [25], which will be reviewed in Section 2.2.2.

We introduce the Peterson–Woodward comparison formula first in Section 2.2.1. This comparison formula not only plays an important role in obtaining the results in [25], but also shows us that it suffices to know all quantum Schubert structure constants N_u,v^w,λforG/Bin order to know all quantum Schubert structure constants for allG/P’s.

2.2.1 Peterson–Woodward comparison formula

We useP to distinguish the quantum products for different flag varietiesG/P’s (when needed). For anyu, v∈W^P we have σ^uP σ^v=

w∈W^P,λP∈Q^∨/Q^∨_P N_u^w,λ_,v^Pq_λ_Pσ^w. NoteW^P⊂W.

The classesσ^uandσ^v in QH^∗(G/P)can both be treated as classes in QH^∗(G/B)natu- rally. Whenever referring toN_u,v^w,λwhereλ∈Q^∨, we are considering the quantum product inQH^∗(G/B):σ^uB σ^v=

w∈W,λ∈Q^∨N_u,v^w,λq_λσ^w.

Proposition 2.1(Peterson–Woodward comparison formula [28]; see also [23]).

(1) Let λP∈Q^∨/Q^∨_P. Then there is a uniqueλB∈Q^∨ such thatλP=λB+Q^∨_P and γ, λB ∈ {0,−1}for allγ∈R⁺_P (=R⁺∩

β∈ΔPZβ). (2) For everyu, v, w∈W^P, we have

N_u^w,λ_,v^P=N_u^wω_,v^P^ω^P^,λ^B.

Here ωP (resp.ωP) is the longest element in the Weyl group W_P of P (resp.

WP ofPwhereΔP= {β∈ΔP | β, λB =0}).

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Owing to the above proposition, we obtain an injection of vector spaces

ψ_Δ,ΔP :QH^∗(G/P)−→QH^∗(G/B)defined byq_λ_Pσ^w→q_λ_Bσ^wω^P^ω^P.

For the special case of a singleton subset {α} ⊂Δ, we define P_α=P and simply define ψα=ψΔ,{α}. In this case, we note thatR⁺_P

α= {α},Q^∨_P

α=Zα^∨, and we have the natural fibra- tionP_α/B→G/B→G/P_αwith P_α/B∼=P¹.

Example 2.2. Let λP_α=β^∨+Q^∨_P

α where β∈Δ\ {α}. Then we have α, β^∨ ∈ {0,−1,−2,−3}. Furthermore, we have

ψα(q_λ_P_α)=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

q_β∨, ifα, β^∨ =0, s_αq_β∨, ifα, β^∨ = −1, q_α∨q_β∨, ifα, β^∨ = −2, s_αq_α∨q_β∨, ifα, β^∨ = −3.

More generally, we consider λP_α=λ+Q^∨_P

α∈Q^∨/Q^∨_P

α, where λ=

β∈Δ\{α}c_ββ^∨∈Q^∨. Settingm= α, λ, we have

ψα(q_λ_P_ασ^w)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ qλ−m

2^α^∨

σ^w, ifmis even,

q

λ−m+1 2 ^α^∨

σ^ws^α, ifmis odd.

2.2.2 Z²-filtrations on QH^∗(G/B)

As shown in [25], given any parabolic subgroup P ofGcontaining B, we can construct aZ^|Δ^P^|+¹-filtration on QH^∗(G/B). In this subsection, we review the main results in [25]

for the special case of a parabolic subgroup that corresponds to a singleton subset{α}. Using them, we prove Theorem 1.1 in the next subsection.

Recall that a natural basis of QH^∗(G/B)[q₁⁻¹, . . . ,q_n⁻¹] is given byq_λσ^w’s labeled by(w, λ)∈W×Q^∨. Note that q_λσ^w∈QH^∗(G/B)if and only ifq_λ∈Q[q] is a polynomial.

In order to obtain a filtration on QH^∗(G/B), we just need to define (nice) gradings for a given basis of it. Furthermore, as in the introduction, we have defined a map sgn_αwith

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respect to any given simple rootα∈Δas follows:

sgn_α:W→ {0,1}; sgn_α(w)=

⎧⎨

⎩

1, if(w)−(ws_α) >0, 0, if(w)−(ws_α)≤0.

Note that(w)−(ws_α)= ±1 and that(w)−(ws_α)=1 if and only ifw(α)∈ −R⁺, which holds if and only ifw=us_αfor a uniqueu∈W^P^α (see, e.g., [18]). We can define a grading map gr_α with respect to a given simple rootα∈Δas follows.

gr_α:W×Q^∨−→Z²;

gr_α(q_λσ^w)=(sgn_α(w)+ α, λ, (w)+ 2ρ, λ −sgn_α(w)− α, λ).

Here we are using lexicographical order on Z²; that is, a=(a₁,a₂) <b=(b₁,b₂)if and only if either (a₁=b₁ and a₂<b₂) or a₁<b₁ holds. The above Z²-grading of q_λσ^w can recover the degree grading of it as in Section 2.1. Precisely, if we write gr_α(q_λσ^w)=(i,j), then deg(q_λσ^w)=i+ j.

Remark 2.3. Following from Corollary 3.13 of [25], our grading map gr_αcoincides with the grading map in [25, Definition 2.8] by using the Peterson–Woodward lifting map

ψ_α=ψ_Δ,{α}.

As a consequence, we obtain a family F= {Fa}a∈Z² of vector subspaces of QH^∗(G/B), where Fa:=

gr_α(q_λσ^w)≤aQq_λσ^w⊂QH^∗(G/B), and the associated graded vector space Gr^F(QH^∗(G/B))=

a∈Z²Gr^F_a with respect toF, where Gr^F_a :=Fa/∪b<aFb.

Proposition 2.4([25, Theorem 1.2]). QH^∗(G/B)is a Z²-filtered algebra with respect to F; that is, we haveF_aF_b⊂F_a+bfor anya,b∈Z².

Define Gr^F_vert(QH^∗(G/B)):=

i∈ZGr^F₍_i_,₀₎ and Gr^F_hor(QH^∗(G/B)):=

j∈ZGr^F₍₀_,_j₎. We take the canonical isomorphism QH^∗(P¹)∼=^Q_x²^[x₋^,^t]_t for the fiber of the fibration P¹→ G/B→G/P_α.

Proposition 2.5([25, Theorem 1.4]). The following mapsΨvert^α andΨ_hor^α are well defined and they are algebra isomorphisms : (In terms of the notations in [25], Ψvert^α =Ψ1

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andΨ_hor^α =Ψ2.)

Ψ_vert^α :QH^∗(P¹)−→Gr^F_vert(QH^∗(G/B)); x→s_α, t→q_α∨,

Ψ_hor^α :QH^∗(G/P_α)−→Gr^F_hor(QH^∗(G/B)); q_λ_P_ασ^w→ψα(q_λ_P_ασ^w).

Here we note thats_α∈Gr^F₍₁_,₀₎⊂Gr^F_vert(QH^∗(G/B))denotes the graded component of σ^s^α+ ∪b<(1,0)Fb. Similar notations are taken whenever “( )” is used. In addition, we have the following proposition.

Proposition 2.6(Proposition 3.23 of [25]). For any u∈W^P^α we have σ^u σ^s^α=σ^us^α+

w,λb_w,λq_λσ^wwith gr_α(q_λσ^w) <gr_α(σ^us^α)wheneverb_w,λ=0.

The next lemma follows directly from the definition of the grading map gr_α.

Lemma 2.7. Letu, v, w∈Wandλ∈Q^∨. Then gr_α(σ^u)+gr_α(σ^v)=gr_α(q_λσ^w)if and only if both(w)+ 2ρ, λ =(u)+(v)and sgn_α(w)+ α, λ =sgn_α(u)+sgn_α(v)hold.

2.2.3 Proof of Theorem1.1

The first half of Theorem 1.1 is a direct consequence of Proposition 2.4. Indeed, if sgn_β(w)+ β, λ>sgn_β(u)+sgn_β(v) for some β∈Δ, then gr_β(q_λσ^w) >gr_β(σû)+gr_β(σ^v). Sinceσû σ^v∈Fgr_β(σû)Fgr_β(σ^v)⊂Fgr_β(σû)+gr_β(σ^v), we concludeN_u^w,λ_,v =0.

It remains to show the second half of Theorem 1.1. Note that sgn_αis a map from Wto{0,1}. Since sgn_α(u)+sgn_α(v)=2, we have sgn_α(u)=sgn_α(v)=1. Consequently,u:=

us_α andv:=vs_α are both elements in W^P^α. In the rest, we can assume (w)+ 2ρ, λ = (u)+(v). (Otherwise, both N_u^w,λ_,v andN_u^w,λ−α,v ^∨ would vanish, directly following from the standardZ-graded ring structure(∗)ofQH^∗(G/B).)

By Proposition 2.6 we have

σ^u σ^s^α=σ^u∈Gr^F_gr

α(σ^u) and σ^v σ^s^α=σ^v∈Gr^F_gr

α(σ^v).

Note thatQH^∗(G/B)is an associative and commutativeZ²-filtered algebra with respect toF. As a consequence, Gr^F(QH^∗(G/B))is an associative and commutativeZ²-graded algebra. Thus we have

LHS :=σ^u σ^s^α σ^v σ^s^α=σ^u σ^v=: RHS

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in Gr^F_gr

α(σ^u)+grα(σ^v). By Lemma 2.7 we have RHS=σ^u σ^v=

Nu^w,^˜,v^λ^˜q_λ_˜σ^w^˜ =

N_u^w,^˜_,v^λ^˜q_λ_˜σ^w^˜,

where the summation is over those(w,˜ λ)˜ ∈W×Q^∨ satisfying(w)˜ + 2ρ,λ =˜ (u)+(v) and sgn_α(w)˜ + α,λ =˜ 2. Letαdenote the quantum product for QH^∗(G/P_α). By Proposi- tion 2.5 we have

LHS=(σ^u σ^v) (σ^s^α σ^s^α)=Ψhor^α (σ^uασ^v) q_α∨=

N_u^w,v^,λ^P^αψα(σ^wq_λ_P_α)q_α∨,

the summation over those (w, λP_α)∈W^P ×Q^∨/Q^∨_P_α (with λP_α being effective). Then we conclude N_u^w,λ_,v =N_us^w,λ−α^∨

α,vsα by comparing coefficients of both sides. Indeed, for λP_α:=λ+ Q^∨_P

α, we have λB=λ−α^∨ via the Peterson–Woodward comparison formula (by noting α, λ−α^∨ = −sgn_α(w)∈ {0,−1}). Setw:=wif sgn_α(w)=0, orws_αif sgn_α(w)=1. Note that ψα(σ^wq_λ_P_α)q_α∨=σ^wq_λ. We conclude

N_u^w,λ_,v =N_u^w,v^,λ^P^α=N_u^w,λ−α,v ^∨. Note that, for anywˆ ∈W, we have

σ^w^ˆ σ^s^α=

⎧⎨

⎩

σ^w^ˆ^s^α, if sgn_α(w)ˆ =0, σ^ws^ˆ ^α σ^s^α σ^s^α=σ^ws^ˆ ^αq_α∨, if sgn_α(w)ˆ =1.

Hence, we have

σû σ^v=σû σ^v σ^s^α=σû σ^v σ^s^α

=

N_u^w,^ˆ_,v^ˆ^λq_ˆ_λσ^w^ˆ σ^s^α

=

N_u^w,^ˆ_,v^ˆ^λq_ˆ_λσ^ws^ˆ ^α+

N_u^w,^ˆ_,v^λ^ˆq_λ+α_ˆ ∨σ^ws^ˆ ^α,

the former (resp. latter) summation over those(w,ˆ λ)ˆ ∈W×Q^∨satisfying(w)ˆ + 2ρ,λ =ˆ (u)+(v), sgn_α(w)ˆ + α,λ =ˆ 1 and sgn_α(w)ˆ =0 (resp. 1); hence, if sgn_α(w)=0 (resp. 1), then we have N_u^w,λ_,vq_λσ^w=N_u,v^w,^ˆ ^λ^ˆq_λ+α_ˆ ∨σ^w^ˆ^s^α(resp.N_u^w,^ˆ_,v^ˆ^λq_ˆ_λσ^w^ˆ^s^α) for a unique(w,ˆ λ)ˆ in the latter (resp. former) summation. Thus we conclude thatN_u^w,λ_,v equals N_u^w_,v^s^α_s^,λ−α_α ^∨ if sgn_α(w)=0, or N_u^ws_,v^α_s^,λ_α if sgn_α(w)=1.

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2.3 Applications

In this subsection, we give applications of Theorem 1.1 forΔof type Acase. (See the introduction for possible further applications for other cases.) For convenience, we assume the Dynkin diagram of Δis given by ◦−−−◦ · · · ◦−−−◦

α1 α2 αn . The flag variety of type A_n, corresponding to a subsetΔ\ {αa1, . . . , αak}, parameterizes flags of linear subspaces {V_a₁≤ · · · ≤V_a_r≤Cⁿ⁺¹ | dim_CV_a_j =a_j,j=1, . . . ,r} where [a₁, . . . ,a_r] is a subsequence of [1, . . . ,n]. We fix an αk once and for all. Let P⊃B denote the parabolic subgroup that corresponds to ΔP=Δ\ {αk}. Note that the complete flag variety Fn+1=G/B and the complex Grassmannian Gr(k,n+1)=G/P correspond to the subsequences [1,2, . . . ,n] and [k], respectively, where G=SL(n+1,C). The natural projection π:G/B→G/P is just the forgetting map, sending a flagV1≤ · · · ≤Vn≤Cⁿ⁺¹inFn+1to the pointVk≤Cⁿ⁺¹ in Gr(k,n+1). Furthermore, the induced mapπ^∗:H^∗(G/P)→H^∗(G/B)sends a Schubert classσ_P^w∈H^∗(G/P)(wherew∈W^P) to the Schubert classπ^∗(σ_P^w)=σ_B^w∈H^∗(G/B). Such a classσ^winH^∗(G/B)(withw∈W^P) is called aGrassmannian class. By the abuse of notation, we skip the subscript “B” and “P”. Using Theorem 1.1, we can show Theorem 1.2 stated in the introduction, a reformulation of which is given as follows.

Theorem 1.2. For anyu∈W^P,v, w∈W, andλ∈Q^∨, there existv, w∈Wsuch that

N_u^w,λ_,v =N_u^w_,v^,⁰.

To prove the above theorem, we need the following two lemmas.

Lemma 2.8. Given any nonzeroλ=n

j=1a_jα^∨_j ∈Q^∨witha_j≥0 for all j, there existsm∈

{1, . . . ,n}such thatαm, λ>0 anda_m>0.

Proof. Assume αm, λ ≤0 for all m. Then λ is a nonpositive sum of fundamental coweights. As a consequence,λis a nonpositive sum of simple corootsα^∨_j’s (by Table 1 in Section 13.2 of [17]). Thusλ=0, which contradicts the assumptions.

Hence, there exists m such that αm, λ>0. Consequently, we have am>0, by noting thatαm, α^∨_jis positive if j=m, or nonpositive otherwise.

Remark 2.9. Lemma 2.8 works forΔof all types with the same proof.

Lemma 2.10. Letλ=n

j=1a_jα^∨_j ∈Q^∨ witha_j≥0 for all j. Ifαm, λ>0 for a uniquem,

then we haveαm, λ ≥2.

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Proof. Let Dyn(Δ)˜ denote the Dynkin diagram associated to a subbaseΔ˜⊂Δ.

Define Δ:= {αj |aj>0}. Clearly,αm∈Δ. We first conclude that Dyn(Δ)is connected. (Otherwise, we can writeΔ=Δ1Δ2 with Dyn(Δ1)being a connected component of Dyn(Δ). Thenλ=λ1+λ2withλ1(resp.λ2) belonging to the coroot sublattice of Δ1 (resp.Δ2). Note thatΔ1andΔ2are orthogonal to each other. For each j∈ {1,2}there existsαmj∈Δj such thatαmj, λj>0 by Lemma 2.8. This contradicts the uniqueness of αm.) ThusΔ= {αi, αi+1, . . . , αp}for some 1≤i≤m≤p≤n.

When i=p, the statements holds, by noting that αm, α_m^∨ =2 and λ=a_mα_m^∨ in this case. When i<p, we can assume i<m without loss of generality. Since 0≥ αi, λ =2a_i−a_i+1, we have a_i+1≥2a_i>a_i>0. Since 0≥ αi+1, λ = −a_i+2a_i+1−a_i+2, we havea_i+2≥a_i+1+(a_i+1−a_i) >a_i+1>0. By induction we concludea_m>a_m−1>0. Ifm=p, then we have αm, λ =2am−am−1≥2(am−1+1)−am−1>2. Ifm<p, then we can show am>am+1with the same arguments. As a consequence, we haveαm, λ = −am−1+2am− am+1≥ −am−1+(am−1+1+am+1+1)−am+1≥2.

Proof of Theorem1.2. Clearly, the statement holds ifN_u^w,λ_,v vanishes orλ=0.

Given nonzero λ=_n

j=1ajα^∨_j ∈Q^∨, we can assume aj≥0 for all j, that is, λ is effective, because otherwiseN_u,v^w,λvanishes. Sinceλ=0, there existsmsuch thatαm, λ>

0 by Lemma 2.8. We simply define sgn_m:=sgn_α_m, which is a map fromWto{0,1}defined in the introduction (see also Section 2.2.2).

If such anmis not unique, then we can take any one suchmthat is not equal to k. Sinceu∈W^P whereΔP=Δ\ {αk}, we have sgn_m(u)=0. If sgn_m(v) <sgn_m(w)+ αm, λ, then we have N_u^w,λ_,v =0 by Theorem 1.1(1); and hence we are done. Otherwise, we have sgn_m(v)=sgn_m(w)+ αm, λ =1 and sgn_m(w)=0. By Theorem 1.1(2) we have N_us^ws_m^m_,v^,λ= Nu^ws,v^msm^,λ−α^∨^m=N_u^w,λ_,v.

If such an m is unique, then we have αm, λ ≥2 by Lemma 2.10. Thus either sgn_m(u)+sgn_m(v) <sgn_m(w)+ αm, λ or sgn_m(u)+sgn_m(v)=sgn_m(w)+ αm, λ holds.

For the former case, N_u^w,λ_,v vanishes and then it is done. For the latter case, we con- cludem=k, sgn_k(v)=1, sgn_k(w)=0, and αk, λ =2, by noting that sgn_j(u)=1 if j=k, or 0 otherwise. Thus we have N_u,v^w,λ=N_us^w,λ−α_k_,vs_k^∨^k =N_u,vs^w^s^k^,λ−α_k ^k^∨, by using Theorem 1.1(2) again.

Hence, either of the following must hold: (i) N_u^w,λ_,v =0 (and then it is done); (ii) N_u^w,λ_,v =N_u^w_,v^s^m_s_m^,λ with λ=λ−αm^∨ =

ja_jα^∨_j, in which |λ| = |λ| −1 with a_j=a_j−1≥0 if j=m, or a_j otherwise. Here |λ|:=n

j=1a_j. Therefore, the statement holds, by using

induction on|λ|.